N Dupuis

L Canet

A Eichhorn

W Metzner

J M Pawlowski

M Tissier

N Wschebor

The nonperturbative functional renormalization group and its applications

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Linking descriptions of physics at various scales and relating the macroscopic physical properties of systems to the microscopic interactions and degrees of freedom is the primary goal of research in many areas of physics, from condensed matter and cold atoms to high-energy physics and quantum gravity. The aim of this review is to describe a theoretical approach, the nonperturbative functional renormalization group (FRG), which provides us with an efficient and versatile tool to bridge the gap between micro-and macroscopic scales and thus determine the physical properties of a wide variety of systems.

Strongly correlated systems, such as electrons in solids interacting via the Coulomb interaction or quarks in nucleons subjected to the strong interaction, although different in some respects share nevertheless a number of properties. When external parameters (temperature, density, etc.) are varied, they often exhibit rich phase diagrams due to competing collective phenomena. The theoretical study of their properties faces two major difficulties. First, there is often no small parameter that would allow for a systematic perturbative expansion. Second, whenever the degrees of freedom are correlated over distances much larger than the microscopic scales, collective effects become important at low energies. This implies that these systems may exhibit very distinct behavior on different energy scales and the relevant degrees of freedom which permit a simple formulation of the low-energy (macroscopic) properties may be different from the microscopic ones. This diversity of scales explains the difficulty of a straightforward numerical solution of microscopic models, since the interesting phenomena emerge only at low energies and in large-size systems. It is also responsible for the fact that perturbation theories are often plagued with infrared divergences and may be inapplicable even at weak coupling. Conversely, in fundamental physics one is often confronted with the opposite problem: While the low-energy description is known, one searches for a consistent underlying microphysics. Many of the technical challenges and conceptual insights -maybe surprisingly -resemble those of the previous examples.

Wilson's renormalization group

The renormalization group1 (RG) is a natural framework to study systems with many degrees of freedom correlated over long distances. In Wilson's modern formulation, fluctuations at short distances and high energies are progressively integrated out to obtain an effective (coarse-grained) description at long distances and low energies [START_REF] Kadanoff | Scaling laws for ising models near T c[END_REF][START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF][START_REF] Wilson | Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior[END_REF][START_REF] Wilson | The renormalization group and the expansion[END_REF][START_REF] Wegner | Renormalization Group Equation for Critical Phenomena[END_REF][START_REF] Polchinski | Renormalization and effective Lagrangians[END_REF][START_REF] Fisher | Renormalization group theory: Its basis and formulation in statistical physics[END_REF][START_REF] Bagnuls | Exact renormalization equations: an introductory review[END_REF]. The RG not only gives us an explanation of cooperative behavior and universality but also provides us with a practical tool to study systems where correlations and fluctuations play an important role, the prime example being systems in the vicinity of a second-order phase transition [START_REF] Wilson | The renormalization group and the expansion[END_REF][START_REF] Guida | Critical exponents of the N-vector model[END_REF][START_REF] Pelissetto | Critical phenomena and renormalization-group theory[END_REF]. In high-energy physics, the RG provides us with a powerful conceptual understanding of fundamental interactions, as it allows to distinguish effective theories (which break down in the UV limit, e.g., due to the triviality problem) from fundamental theories which hold over an infinite range of scales due to asymptotic freedom or safety.

In its standard formulation however, the RG usually relies on perturbation theory and is therefore restricted to weakly interacting systems where a small expansion parameter allows one to systematically compute the effects of fluctuations beyond the noninteracting limit or the mean-field theory. 2,3 For example in one of the most studied models, the O(N) model (ϕ 4 theory with O(N) symmetry), the critical exponents are computed either from a Ginzburg-Landau-Wilson functional in an = 4d expansion [START_REF] Wilson | The renormalization group and the expansion[END_REF][START_REF] Wilson | Critical Exponents in 3.99 Dimensions[END_REF] or from the nonlinear sigma model in an = d -2 expansion [START_REF]Phase transitions in gauge and spin-lattice systems[END_REF][START_REF] Polyakov | Interaction of goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills fields[END_REF][START_REF] Brézin | Renormalization of the Nonlinear σ Model in 2 + Dimensions-Application to the Heisenberg Ferromagnets[END_REF][START_REF] Brézin | Spontaneous breakdown of continuous symmetries near two dimensions[END_REF][START_REF] Nelson | Momentum-shell recursion relations, anisotropic spins, and liquid crystals in 2 + dimensions[END_REF] (with d being the space dimension). In the latter case the perturbative RG series is usually considered as useless due to the lack of Borel summability. Thus even high-order perturbative RG expansions cannot relate the two expansions, except in the large-N limit [START_REF] Moshe | Quantum field theory in the large N limit: a review[END_REF]. This is not crucial for the O(N) model where the critical behavior does not change qualitatively for 2 < d < 4 but in other models forbids a completely coherent picture of the physics between d = 2 and d = 4 (or, more generally, between the lower and upper critical dimensions).

Moreover, even when the field theoretical perturbative RG2 yields an accurate determination of universal quantities (e.g., the critical exponents or universal scaling functions), it is often not clear how to compute nonuniversal quantities such as a transition temperature, the phase diagram of equilibrium and out-of-equilibrium systems, the spectrum of bound states in strongly correlated theories, etc., since these properties depend on the underlying microscopic models.

Last, the perturbative RG is useless for genuinely nonperturbative problems such as the Berezinskii-Kosterliz-Thouless (BKT) transition [START_REF] Berezinskii | Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group i. classical systems[END_REF][START_REF] Berezinskii | Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. ii. quantum systems[END_REF][START_REF] Kosterlitz | Ordering, metastability and phase transitions in two-dimensional systems[END_REF][START_REF] Kosterlitz | The critical properties of the two-dimensional XY model[END_REF] in the two-dimensional O(2) model, 4 the growth of stochastically growing interfaces, turbulent flows, the confinement of quarks in quantum chromodynamics (QCD), gravity at the Planck scale, etc. In the latter case, a perturbative analysis indicates perturbative non-renormalizability, rendering non-perturbative tools necessary to describe quantum gravity beyond the Planck scale.

The functional renormalization group

Although various systems may be characterized by different microscopic energy scales (e.g., 10 -7 meV for a ultracold atomic gas, 1 eV for conduction electrons in a solid, 1 GeV for QCD, 125 GeV for the Higgs particle, 10 19 GeV for the Planck scale in quantum gravity), from a theoretical point of view there is no fundamental difference between relativistic quantum field theory (that describes elementary particles and their interactions) and statistical field theory (that describes the statistical properties of quantum or classical systems where the degrees of freedom are represented by fields). 5 In a modern language, both are formulated as functional integrals in d space dimensions or d + 1 spacetime dimensions. The primary goal of these functional approaches is to compute correlation functions as well as the free energy of the system.

The FRG combines the functional approach with the Wilson RG idea of integrating out fluctuations not all at once but progressively from high-to low-energy scales. The expression functional RG stems from the fact that one naturally deals with (possibly singular) functions of the field rather than a finite number of coupling constants. In the literature, the (nonperturbative) FRG is sometimes merely referred to as the nonperturbative RG. Both aspects (nonperturbative and functional) are actually crucial features of the method presented in this review. Note however that the RG approach can be functional and perturbative, or nonperturbative and nonfunctional. The FRG is also referred to as the Exact RG due to its one-loop exact (closed) functional form. This epithet does not imply that the flow equation can be solved exactly (except for models that can be solved more easily with other methods): Most nontrivial applications rely on an approximate solution.

Different versions of exact functional RG equations, such as the functional Callan-Symanzik, Wilson-Polchinski and Wegner-Houghton formulations, have already a long history [START_REF] Symanzik | Small distance behavior in field theory and power counting[END_REF][START_REF] Wilson | The renormalization group and the expansion[END_REF][START_REF] Wegner | Renormalization Group Equation for Critical Phenomena[END_REF][START_REF] Polchinski | Renormalization and effective Lagrangians[END_REF][START_REF] Hasenfratz | Renormalization group study of scalar field theories[END_REF][START_REF] Chang | Differential renormalization-group generators for static and dynamic critical phenomena[END_REF][START_REF] Parola | Liquid state theory and critical phenomena[END_REF][START_REF] Nicoll | Approximate Renormalization Group Based on the Wegner-Houghton Differential Generator[END_REF][START_REF] Nicoll | An exact one-particle-irreducible renormalization-group generator for critical phenomena[END_REF][START_REF] Parola | Liquid-State Theory for Critical Phenomena[END_REF]. However the application of these methods has been hindered for a long time by the complexity of functional differential equations and the difficulty to devise nonperturbative and reliable approximation schemes. Early works [START_REF] Wegner | Renormalization Group Equation for Critical Phenomena[END_REF][START_REF] Hasenfratz | Renormalization group study of scalar field theories[END_REF][START_REF] Nicoll | Approximate Renormalization Group Based on the Wegner-Houghton Differential Generator[END_REF][START_REF] Nicoll | Exact and approximate differential renormalization-group generators[END_REF][START_REF] Newman | Critical exponents by the scaling-field method: The isotropic n-vector model in three dimensions[END_REF][START_REF] Newman | q-state potts model by wilson's exact renormalization-group equation[END_REF][START_REF] Golner | Nonperturbative renormalization-group calculations for continuum spin systems[END_REF][START_REF] Hasenfratz | The cut-off dependence of the Higgs meson mass and the onset of new physics in the standard model[END_REF][START_REF] Zumbach | Almost second order phase transitions[END_REF][START_REF] Zumbach | Phase transitions with O(n) symmetry broken down to O(n-p)[END_REF][START_REF] Zumbach | The local potential approximation of the renormalization group and its applications[END_REF] were mainly based on the so-called local potential approximation (see Sec. 2 for a detailed discussion) and neglected the momentum dependence of the interaction vertices, with apparently no possibility of a systematic improvement. The FRG has nevertheless been useful in its perturbative formulation for the study of disordered systems [START_REF] Fisher | Random fields, random anisotropies, nonlinear σ models, and dimensional reduction[END_REF][START_REF] Narayan | Dynamics of sliding charge-density waves in 4-dimensions[END_REF][START_REF] Nattermann | Dynamics of interface depinning in a disordered medium[END_REF][START_REF] Chauve | Renormalization of Pinned Elastic Systems: How Does It Work Beyond One Loop?[END_REF][START_REF] Le Doussal | Functional renormalization group and the field theory of disordered elastic systems[END_REF][START_REF] Tarjus | Nonperturbative Functional Renormalization Group for Random-Field Models: The Way Out of Dimensional Reduction[END_REF]. The necessity of a functional approach in this context is due to an infinite number of operators being marginal at the upper or lower (whichever the case of interest) critical dimension. It is therefore not possible, in some disordered systems, to restrict oneself to a finite number of coupling constants and one must consider functions of the field (Sec. 3.3).

The formulation of the FRG based on a formally exact flow equation for a scale-dependent "effective action" Γ k [φ] (or Gibbs free energy in the language of statistical physics), the generating functional of one-particle irreducible vertices, has proven successful in devising nonperturbative approximation schemes [START_REF] Ringwald | Average action for the N-component ϕ 4 theory[END_REF][START_REF] Wetterich | Average action and the renormalization group equations[END_REF][START_REF] Wetterich | The average action for scalar fields near phase transitions[END_REF][START_REF] Wetterich | Improvement of the average action[END_REF][START_REF] Wetterich | Exact evolution equation for the effective potential[END_REF][START_REF] Berges | Non-perturbative renormalization flow in quantum field theory and statistical physics[END_REF][START_REF] Ellwanger | Flow equations for n point functions and bound states[END_REF][START_REF] Morris | The exact renormalization group and approximate solutions[END_REF][START_REF] Bonini | Perturbative renormalization and infrared finiteness in the Wilson renormalization group: the massless scalar case[END_REF]. The functional Γ k [φ] may be seen as a coarse-grained free energy that includes only fluctuations with momenta or energies larger than a scale k. 6 The field φ(r) or φ(r, t) represents the relevant microscopic degrees of freedom, e.g., the local magnetization in a solid. 7 If necessary, one may include additional fields corresponding to emerging low-energy (collective) degrees of freedom: a pairing field in a superconductor, a meson field in QCD, etc. The functional Γ k=0 [φ] includes fluctuations on all scales, and allows us to obtain the free energy and the one-particle irreducible vertices. In principle, all correlation functions can be deduced from Γ k=0 [φ]. Thus the FRG replaces the difficult determination of Γ k=0 [φ] from a direct calculation of the functional integral (that defines the partition function) by the solving of a functional differential equation ∂ k Γ k [φ] with an initial condition Γ Λ [φ], which often (but not always) corresponds to the bare action or the mean-field solution of the model, at some microscopic scale Λ. The flow equation ∂ k Γ k [φ] closely resembles a renormalization-group improved one-loop equation, but is exact. This close connection to perturbation theory, for which we have an intuitive understanding, is an important key for devising meaningful nonperturbative approximations.

Scope of the review

The aim of the review is to give an up-to-date nontechnical presentation of the nonperturbative FRG approach that emphasizes its applications in various fields of physics. 8 Section 2 is devoted to a general presentation of the FRG. We discuss the properties of the scale-dependent effective action Γ k [φ] and the main two approximations used for the solution of the exact flow equation: the derivative expansion and the vertex expansion. We also emphasize the applicability of the FRG to microscopic, classical or quantum, models. In the following sections, we show how the method can be used in practice in statistical mechanics, quantum many-body physics, high-energy physics and quantum gravity (Secs. [START_REF] Bogolyubov | QUANTUM FIELDS[END_REF][START_REF] Symanzik | Small distance behavior in field theory and power counting[END_REF][START_REF] Callan | Broken scale invariance in scalar field theory[END_REF][START_REF] Kadanoff | Scaling laws for ising models near T c[END_REF]. Although an exhaustive account of all applications is obviously impossible given the broad scope of the review, we have tried nevertheless to cover most subjects. A more technical presentation of the nonperturbative FRG approach, focusing on the derivative expansion, can be found in the Appendices.

We set = k B = c = 1 throughout the paper.

The FRG in a nutshell

While we wish to stress the general concepts of the FRG rather than its application to a particular model, we shall base our discussion in this section on the d-dimensional O(N) model (or ϕ 4 theory). Its partition function

Z[J] = ˆD[ϕ] e -S [ϕ]+ ´r J•ϕ (1)
can be written as a functional integral, with the action

S [ϕ] = ˆr 1 2 (∇ϕ) 2 + r 0 2 ϕ 2 + u 0 4! (ϕ 2 ) 2 , (2)
where ϕ = (ϕ 1 • • • ϕ N ) is an N-component real field, r a d-dimensional coordinate and ´r = ´dd r. We shall mostly use the language of classical statistical mechanics where S [ϕ] is simply the Hamiltonian H[ϕ] multiplied by the inverse temperature β = 1/T . J is an external "source" (e.g. a magnetic field for a magnetic system) which couples linearly to the field. The model is regularized by a UV momentum cutoff Λ which can be thought of as the inverse lattice spacing if the continuum model ( 2) is derived from a lattice model. We refer to the action (2) as the "microscopic" action, i.e., the action describing the physics at length scales ∼ Λ -1 . Assuming the value of u 0 fixed, the O(N) model exhibits a second-order phase transition between a disordered phase (r 0 > r 0c ) and an ordered phase (r 0 < r 0c ) where ϕ(r) 0 and the O(N) symmetry is spontaneously broken. r 0 is naturally related to the temperature by setting r 0 ≡ r0 (T -T 0 ); r 0c = r0 (T c -T 0 ) then defines the critical temperature T c while T 0 is the mean-field transition temperature.

We are typically interested in the Helmholtz free energy F[J] = -T ln Z[J] (usually for a vanishing source, J = 0) and the correlation functions such as the two-point one (or propagator)

G i j (r -r ) = ϕ i (r)ϕ j (r ) c = δ 2 ln Z[J] δJ i (r)δJ j (r ) J=0 , (3)
where ϕ i ϕ j c ≡ ϕ i ϕ j -ϕ i ϕ j .

The scale-dependent effective action

The main idea of Wilson's RG is to compute the partition function [START_REF] Stueckelberg De Breidenbach | Normalization of constants in the quanta theory[END_REF] by progressively integrating out shortdistance (or high-energy) degrees of freedom [START_REF] Kadanoff | Scaling laws for ising models near T c[END_REF][START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF][START_REF] Wilson | Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior[END_REF][START_REF] Wilson | The renormalization group and the expansion[END_REF][START_REF] Wegner | Renormalization Group Equation for Critical Phenomena[END_REF][START_REF] Polchinski | Renormalization and effective Lagrangians[END_REF][START_REF] Fisher | Renormalization group theory: Its basis and formulation in statistical physics[END_REF][START_REF] Bagnuls | Exact renormalization equations: an introductory review[END_REF]. In the FRG approach, one builds a family of models indexed by a momentum scale k such that fluctuations are smoothly taken into account as k is lowered from some initial scale k in ≥ Λ down to 0. In practice this is achieved by adding to the action S [ϕ] a quadratic term ∆S k [ϕ] defined by

∆S k [ϕ] = 1 2 ˆp N i=1 ϕ i (-p)R k (p)ϕ i (p) (4)
with ´p = ´dd p/(2π) d . The typical shape of the regulator function R k (p) is shown in Fig. 1; it is strongly suppressed for |p| k and of order k 2 for |p| k. In the language of high-energy physics, R k (p) can be interpreted as a momentum-dependent mass-like term that gives a mass of order k 2 to the low-energy modes and thus suppresses their fluctuations. The regulator function is discussed in more detail below.

Rather than considering the (now k-dependent) Helmholtz free energy F k [J] = -T ln Z k [J], one introduces the scale-dependent "effective action" (aka average effective action), or "Gibbs free energy" in the language of statistical mechanics,

Γ k [φ] = -ln Z k [J] + ˆr J • φ -∆S k [φ], (5)
defined as a (slightly modified) Legendre transform which includes the subtraction of ∆S k [φ] [START_REF] Ringwald | Average action for the N-component ϕ 4 theory[END_REF][START_REF] Wetterich | Average action and the renormalization group equations[END_REF][START_REF] Wetterich | The average action for scalar fields near phase transitions[END_REF][START_REF] Wetterich | Improvement of the average action[END_REF][START_REF] Wetterich | Exact evolution equation for the effective potential[END_REF][START_REF] Berges | Non-perturbative renormalization flow in quantum field theory and statistical physics[END_REF]. Here φ = ϕ ≡ φ k [J] is the order-parameter field and J ≡ J k [φ] in [START_REF] Callan | Broken scale invariance in scalar field theory[END_REF] should be understood as a functional of φ obtained by inverting φ k [J]. Thermodynamic properties can be obtained from the effective potential (V is the volume of the system) which is proportional to Γ k [φ] evaluated in a uniform field configuration φ(r) = φ.

U k (ρ) = 1 V Γ k [φ] φ unif. (6)
Because of the O(N) symmetry of the model, U k is a function of the O(N) invariant ρ = φ 2 /2. U k (ρ) may exhibit a minimum at ρ 0,k . Spontaneous breaking of the O(N) symmetry is characterized by a nonvanishing expectation value of the field ϕ J→0 + in the thermodynamic limit and occurs if lim k→0 ρ 0,k = ρ 0 > 0.

On the other hand correlation functions can be related to the one-particle irreducible vertices Γ (n) k [φ] defined as the nth-order functional derivatives of Γ k [φ] [START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF]. In particular the propagator defined by (3), G k [φ] = (Γ (2) k [φ] + R k ) -1 (written here in a matrix form), is simply related to the two-point vertex Γ (2) k .9

The regulator function R k

The regulator function R k is chosen such that Γ k smoothly interpolates between the microscopic action S for k = k in and the effective action of the original model [START_REF] Gell-Mann | Quantum electrodynamics at small distances[END_REF] for k = 0. It must therefore satisfy the following properties: i) At k = k in , R k in (p) = ∞. All fluctuations are then frozen and Γ k in [φ] = S [φ] [START_REF] Berges | Non-perturbative renormalization flow in quantum field theory and statistical physics[END_REF] as in Landau's mean-field theory of phase transitions. In practice, it is sufficient to choose k in Λ to ensure that R k in (p) is much larger than all microscopic mass scales in the problem.

ii) At k = 0, R k=0 (p) = 0 so that ∆S k=0 = 0. All fluctuations are taken into account and the effective action Γ k=0 [φ] ≡ Γ[φ] coincides with the effective action of the original model.

iii) For 0 < k < k in , R k (p) must suppress fluctuations with momenta below the scale k but leave unchanged those with momenta larger than k. In general one chooses a "soft" regulator (as opposed to the sharp regulator commonly used in the weak-coupling momentum-shell RG), see Fig. 1. Two popular choices are the exponential regulator R k (p) = αp 2 /(e p 2 /k 2 -1) and the theta regulator R k (p) = α(k 2p 2 )Θ(k 2p 2 ) [START_REF] Litim | Optimized renormalization group flows[END_REF] (with α a constant of order unity). The latter is a not a smooth function of p and cannot be used beyond the second order of the derivative expansion (see Sec. 2.3) [START_REF] Morris | Equivalence of local potential approximations[END_REF]. The generic form of the regulator is R k (p) = p 2 r(p 2 /k 2 ) with r(y) satisfying r(y → 0) ∼ 1/y and r(y 1) 1.

General properties of the effective action Γ k

i) The condition R k in = ∞, which ensures that Γ k in = S , can often be relaxed in particular when one is interested in universal properties of a model (e.g. the critical exponents or the universal scaling functions associated with a phase transition). In that case the microscopic physics can be directly parametrized by Γ Λ (with no need to specify the microscopic action) and one may simply choose k in = Λ. This is the most common situation and in the following, unless stated otherwise, we will assume k in = Λ and Γ Λ = S . However, in lattice models (and whenever one starts from a well-defined microscopic action) it is important to treat the initial condition at k = k in Λ carefully if one wants to compute nonuniversal quantities as a function of the microscopic parameters (see Sec. 2.5).

ii) Since k acts as an infrared regulator, somewhat similar to a box of finite size ∼ k -1 , the critical fluctuations are cut off by the R k term and the effective action Γ k is analytic for k > 0; there may be, however, some exceptions, e.g. in fermion systems (Sec. 4) or disordered systems (Sec. 3.3). The singularities associated with critical behavior therefore arise only for k = 0. This implies in particular that the vertices

Γ (n) k,i 1 •••i n [p 1 , • • • , p n ; φ] are smooth functions ∂ k Γ k = ∂ k Γ (1) k = ∂ k Γ (2) k = + Figure 2
: Diagrammatic representation of the RG equations satisfied by the effective action [Eq. [START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF]] and the vertices Γ (1) k and Γ (2) k . The solid line stands for the propagator G k , the cross for ∂ k R k and the dot with n legs for Γ (n) k . (Signs and symmetry factors are not shown explicitly.)

of the momenta and can be expanded in powers of p 2 i /k 2 or p 2 i /m 2 , whichever is the smallest, where m = ξ -1 is the smallest "mass" of the problem and ξ the correlation length. For the same reason, the effective action Γ k [φ] itself can be expanded in derivatives if one is interested only in the physics at length scales larger than either k -1 or ξ. This property of the effective action and the n-point vertices is crucial as it underlies both the derivative expansion (Sec. 2.3) and the Blaizot-Méndez-Galain-Wschebor approximation (Sec. 2.4).

iii) All linear symmetries of the model that are respected by the infrared regulator ∆S k are automatically symmetries of Γ k . As a consequence, Γ k can be expanded in terms of invariants of these symmetries.

iv) The effective action Γ k [φ] and the Wilsonian effective action S W k [ϕ] are related but carry different physical meanings [START_REF] Ellwanger | Flow equations for n point functions and bound states[END_REF][START_REF] Morris | The exact renormalization group and approximate solutions[END_REF][START_REF] Bonini | Perturbative renormalization and infrared finiteness in the Wilson renormalization group: the massless scalar case[END_REF][START_REF] Morris | Equivalence of local potential approximations[END_REF]. In the Wilson approach k plays the role of a UV cutoff for the low-energy modes that remain to be integrated out [START_REF] Wilson | The renormalization group and the expansion[END_REF][START_REF] Wegner | Renormalization Group Equation for Critical Phenomena[END_REF][START_REF] Polchinski | Renormalization and effective Lagrangians[END_REF]. S W k [ϕ] describes a set of different actions, parametrized by k, for the same model. The correlation functions are independent of k and have to be computed from S W k [ϕ] by functional integration. Information about correlation functions with momenta above k is lost. In contrast k acts as an infrared regulator in the effective action method. Moreover, Γ k is the effective action for a set of different models. The n-point correlation functions depend on k and can be obtained from the n-point vertices Γ (n) k . The latter are defined for any value of external momenta.

The exact flow equation

The FRG approach aims at relating the physics at different scales, e.g., in many cases of interest determining Γ[φ] ≡ Γ k=0 [φ] from Γ Λ [φ] using Wetterich's equation [START_REF] Wetterich | Exact evolution equation for the effective potential[END_REF][START_REF] Ellwanger | Flow equations for n point functions and bound states[END_REF][START_REF] Morris | The exact renormalization group and approximate solutions[END_REF][START_REF] Bonini | Perturbative renormalization and infrared finiteness in the Wilson renormalization group: the massless scalar case[END_REF]]

∂ k Γ k [φ] = 1 2 Tr ∂ k R k Γ (2) k [φ] + R k -1 , (7)
where Tr denotes a trace wrt space and the O(N) index of the field. 10 By taking successive functional derivatives of [START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF], one obtains an infinite hierarchy of equations for the 1PI vertices, the first two of which are shown in Fig. 2. It is sometimes convenient to introduce the (negative) RG "time" t = ln(k/Λ) and consider the equation

∂ t Γ k [φ] = k∂ k Γ[φ].
Let us point out important properties satisfied by the flow equation ( 7):

i) The standard perturbative expansion about the Gaussian model can be retrieved from Eq. ( 7) [START_REF] Papenbrock | Two-loop results from improved one loop computations[END_REF][START_REF] Bonini | Beta function and flowing couplings in the exact wilson renormalization group in yang-mills theory[END_REF][START_REF] Morris | Convergence of derivative expansions of the renormalization group[END_REF][START_REF] Kopietz | Two loop beta function from the exact renormalization group[END_REF].

ii) The flow equations for the Γ (n) k 's look very much like one-loop equations but where the vertices are the exact ones, Γ (n) k [φ] (see Fig. 2). Substitution of Γ (2) k [φ] by S (2) [φ] in [START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF] gives a flow equation which can be easily integrated out and yields the one-loop correction to the mean-field result Γ MF [φ] = S [φ]. The one-loop structure of the flow equation is important in practice as it implies that a single d-dimensional momentum integration has to be carried out in contrast to standard perturbation theory where l-loop diagrams require l-momentum integrals.

iii) Since the one-loop approximation is recovered by approximating Γ (2) k [φ] by S (2) [φ] in [START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF], any sensible approximation of the flow equation will be one-loop exact (in the sense that it encompasses the one-loop result when expanded in the coupling constants). This implies that all results obtained from a one-loop approximation must be iv) The presence of ∂ k R k (q) in the trace of Eq. [START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF] implies that only momenta q of order k or less contribute to the flow at scale k (provided that R k (q) decays sufficiently fast for |q| k), which implements Wilson's idea of momentum shell integration of fluctuations (with a soft separation between fast and slow modes). This, in particular, ensures that the momentum integration is UV finite. Furthermore, the R k term appearing in the propagator G k = (Γ (2) k + R k ) -1 acts as an infrared regulator and ensures that the momentum integration in ( 7) is free of infrared divergences. This makes the formulation well-suited to deal with theories that are plagued with infrared problems in perturbation theory, e.g. in the vicinity of a second-order phase transition. v) Different choices of the regulator function R k correspond to different trajectories in the space of effective actions. If no approximation were made on the flow equation, the final point Γ k=0 would be the same for all trajectories. However, once approximations are made, Γ k=0 acquires a dependence on the precise shape of R k (Fig. 3). 11,12 This dependence can be used to study the robustness of the approximations used to solve [START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF] [START_REF] Schnoerr | Error estimates and specification parameters for functional renormalization[END_REF].

vi) The flow equation [START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF] is a complicated functional integro-differential equation, which cannot (except in trivial cases) be solved exactly. Two main types of approximations have been designed: the derivative expansion, which is based on an ansatz for Γ k involving a finite number of derivatives of the field (Sec. 2.3), and the vertex expansion, which is based on a truncation of the infinite hierarchy of equations satisfied by the Γ (n) k 's (Sec. 2.4). vii) The flow equation -unlike the path-integral expression for Γ k -no longer depends on the microscopic action S . This enables a search for a consistent microscopic dynamics through a fixed-point search of Eq. [START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF]. This property, which will be discussed in more detail in Secs. 5 and 6, is key for the application in quantum gravity as well as in many beyond-Standard-Model settings.

The derivative expansion 13 (DE)

The DE is based on the regularity of the scale-dependent effective action at small momentum scales, |p| ≤ max(k, ξ -1 ) (Sec. 2.1.2). Since the ∂ k R k term in Eq. [START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF] implies that the integral over the internal loop momentum q is dominated by |q| ≤ k, an expansion in internal and external momenta of the vertices, corresponding to a DE of the effective action, makes sense and can be used to obtain the thermodynamics and the long-distance behavior of the system. As will be explained in detail in Sec. 2.3.3 and Appendix Appendix B, long-distance quantities computed within the DE converge very quickly to their exact physical values.

Local-potential approximation: LPA and LPA

To lowest order of the DE, the local potential approximation (LPA), the effective action

Γ LPA k [φ] = ˆr 1 2 (∇φ) 2 + U k (ρ) (8)
is entirely determined by the effective potential U k (ρ) whereas the derivative term keeps its bare (unrenormalized) form [START_REF] Reuter | The large-N limit and the high-temperature phase transition for the φ 4 theory[END_REF][START_REF] Morris | On truncations of the exact renormalization group[END_REF]. Despite its simplicity Γ LPA k is highly nontrivial from a perturbation theory point of view since it includes vertices to all orders: Γ LPA(n) k ∼ ∂ n φ U k for n ≥ 3. The effective potential satisfies the exact equation

∂ k U k (ρ) = 1 2 ˆq ∂ k R k (q)[G k,L (q, ρ) + (N -1)G k,T (q, ρ)] (9)
with initial condition U Λ (ρ) = r 0 ρ + (u 0 /6)ρ 2 , where G k,α (q, ρ) = [Γ (2) k,α (q, ρ) + R k (q)] -1 (α = L, T) are the longitudinal and transverse parts (wrt the order parameter φ) of the propagator G k (q, φ) evaluated in the uniform field configuration φ(r) = φ. The contribution of the transverse propagator appears with a factor N -1 corresponding to the number of transverse modes when ρ is nonzero. Within the LPA, one has

G k,L (q, ρ) = [q 2 + U k (ρ) + 2ρU k (ρ) + R k (q)] -1 , G k,T (q, ρ) = [q 2 + U k (ρ) + R k (q)] -1 . (10)
A negative r 0 corresponds to a system which is in the ordered phase at the mean-field level and the potential U Λ (ρ) then exhibits a minimum at ρ 0,Λ = -3r 0 /u 0 > 0. When r 0 is smaller than a critical value r 0c < 0, fluctuations are not sufficiently strong to fully suppress long-range order and the system is in the ordered phase, i.e., 0 < ρ 0 < ρ 0,Λ where ρ 0 = lim k→0 ρ 0,k . In that case the effective potential U k=0 (ρ) is flat for ρ < ρ 0 . The convexity of the effective potential U k=0 , which in the exact solution is a consequence of its definition as a Legendre transform, can be ensured in the LPA by a proper choice of the regulator. The approach to convexity of Γ k [φ] (which, being not a pure Legendre transform for k > 0, is not necessarily convex) is discussed in [START_REF] Berges | Non-perturbative renormalization flow in quantum field theory and statistical physics[END_REF][START_REF] Tetradis | Scale dependence of the average potential around the maximum in φ 4 theory[END_REF][START_REF] Tetradis | Analytical solutions of exact renormalization group equations[END_REF][START_REF] Peláez | Ordered phase of the O(N) model within the nonperturbative renormalization group[END_REF][START_REF] Litim | Convexity of the effective action from functional flows[END_REF]. For r 0 > r 0c the system is in the disordered phase, ρ 0 = 0, with a finite correlation length ξ.

At criticality (r 0 = r 0c ) the scale invariance due to the infinite correlation length can be made manifest by expressing all quantities in unit of the running momentum scale k. This amounts to defining dimensionless variables (coordinate, field and potential) as r = kr, ρ = k -(d-2) ρ, Ũk ( ρ) = k -d U k (ρ) [START_REF] Polchinski | Renormalization and effective Lagrangians[END_REF] (this is equivalent to the usual momentum and field rescaling in the standard formulation of the Wilsonian RG). The dimensionless effective potential Ũk ( ρ) of the critical system flows to a fixed point Ũ * ( ρ) of the flow equation ∂ k Ũk ( ρ). The fixed-point equation and its numerical solution are discussed in [START_REF] Bagnuls | Exact renormalization equations: an introductory review[END_REF][START_REF] Morris | On truncations of the exact renormalization group[END_REF][START_REF] Morris | Derivative expansion of the exact renormalization group[END_REF][START_REF] Ball | Scheme independence and the exact renormalization group[END_REF][START_REF] Comellas | Polchinski equation, reparameterization invariance and the derivative e xpansion[END_REF][START_REF] Morris | Derivative expansion of the renormalization group in O(N) scalar field theory[END_REF][START_REF] Zumbach | The renormalization group in the local potential approximation and its applications to the O (n) model[END_REF]. Linearizing the flow about the fixed-point value Ũ * gives the correlation-length exponent ν and the correction-to-scaling exponent ω. The LPA can also be used to study the ordered phase (r 0 < r 0c ) and the results [START_REF] Defenu | Truncation effects in the functional renormalization group study of spontaneous symmetry breaking[END_REF] agree with the Mermin-Wagner theorem forbidding spontaneous broken symmetry when N ≥ 2 and d ≤ 2 [START_REF] Mermin | Absence of Ferromagnetism or Antiferromagnetism in One-or Two-Dimensional Isotropic Heisenberg Models[END_REF][START_REF] Hohenberg | Existence of long-range order in one and two dimensions[END_REF][START_REF] Coleman | There are no goldstone bosons in two dimensions[END_REF]. The solution of the fixed-point equation for Ũk , which is independent of ŨΛ , is a paradigmatic example showcasing the power of the FRG in the search for scale-invariance that is central in the study of asymptotic safety in a high-energy context, see, e.g., [START_REF] Labus | Asymptotic safety in O(N) scalar models coupled to gravity[END_REF] for the generalization of Eq. ( 9) to the case with quantum gravity and the gravitationally dressed Wilson-Fisher fixed point.

An important limitation of the LPA is the absence of anomalous dimension, since G LPA k=0 (p, φ = 0) = 1/|p| 2 at criticality, whereas one expects ∼ 1/|p| 2-η with η > 0 when d < 4. This form can never be obtained in the DE (to all orders) since Γ (2) k (p, φ) is a regular function of p when |p| k (the domain of validity of the DE). The anomalous dimension η can nevertheless be obtained from a slight improvement of the LPA effective action,

Γ LPA k [φ] = ˆr Z k 2 (∇φ) 2 + U k (ρ) , (12)
which includes a field renormalization factor Z k . To allow for a scaling solution of the flow equations and therefore a fixed point at criticality, the regulator function must be defined as R k (p) = Z k p 2 r(p 2 /k 2 ). 14 One can then define a "running" anomalous dimension η k = -k∂ k ln Z k . At criticality lim k→0 η k ≡ η > 0 and Z k diverges as k -η . The two-point vertex Γ (2) k (p, φ) -Γ (2) k (0, φ) Z k p 2 is a regular function of p when |p| k as ensured by the infrared regulator R k . For |p| k, a momentum range which is outside the domain of validity of the DE, one expects the singular behavior Γ (2) k (p, φ) -Γ (2) k (0, φ) ∼ |p| 2-η for |p| smaller than the Ginzburg momentum scale p G ∼ u 1/(4-d) 0

. Since Z k ∼ k -η these two limiting forms (for |p| k and |p| k) match for |p| ∼ k. The fact that the anomalous dimension controls the divergence of Z k , and can therefore be obtained from the LPA , can be shown rigorously [START_REF] Blaizot | Nonperturbative renormalization group and momentum dependence of n-point functions. I[END_REF].

It is possible to simplify the LPA by expanding the effective potential to lowest (nontrivial) order about ρ 0,k ,

U k (ρ) = U 0,k + δ k (ρ -ρ 0,k ) + λ k 2 (ρ -ρ 0,k ) 2 , ( 13
)
where δ k = 0 if ρ 0,k > 0. This gives coupled ordinary differential equations for the coupling constants ρ 0,k , δ k , λ k and Z k (and a separate one for the free energy U 0,k ). The truncated LPA is still nonperturbative to the extent that

∂ k δ k , ∂ k λ k , etc.
, are nonpolynomial functions of the coupling constants. This simple truncation retains the main features of the LPA flow and turns out to be sufficient to recover the critical exponents to leading order in the large-N limit as well as to O( ) near four dimensions (d = 4 -) and two dimensions (d = 2 + ) as obtained from perturbative RG in the O(N) nonlinear sigma model. The LPA is not reliable for a precise estimate of the critical exponents but it shows that the FRG, even with a very simple truncation of the effective action, interpolates smoothly between two and four dimensions and suggests that with more involved truncations one can reliably explore the behavior of the system in any dimension and in particular d = 3 [START_REF] Codello | Scaling solutions in a continuous dimension[END_REF][START_REF] Codello | O(N)-Universality Classes and the Mermin-Wagner Theorem[END_REF][START_REF] Codello | Critical exponents of O(N) models in fractional dimensions[END_REF] (see Secs. 2.3.2, 2.4 and Appendix B).

Second order of the DE

To obtain reliable estimates of the critical exponents, it is necessary at least to consider the second order of the DE where the effective action,

Γ DE 2 k [φ] = ˆr 1 2 Z k (ρ)(∇φ) 2 + 1 4 Y k (ρ)(∇ρ) 2 + U k (ρ) , (14)
is defined by three functions of ρ [START_REF] Wetterich | The average action for scalar fields near phase transitions[END_REF][START_REF] Berges | Non-perturbative renormalization flow in quantum field theory and statistical physics[END_REF][START_REF] Morris | Convergence of derivative expansions of the renormalization group[END_REF][START_REF] Morris | Derivative expansion of the renormalization group in O(N) scalar field theory[END_REF][START_REF] Tetradis | Critical exponents from the effective average action[END_REF][START_REF] Aoki | Rapidly Converging Truncation Scheme of the Exact Renormalization Group[END_REF][START_REF] Morris | Three-dimensional massive scalar field theory and the derivative expansion of the renormalization group[END_REF]. In addition to the effective potential, there are two derivative terms, reflecting the fact that transverse and longitudinal fluctuations (wrt the local order parameter φ(r)) have different stiffness. For N = 1 the term Y k (ρ)(∇ρ) 2 should be omitted since it can be put in the form Z k (ρ)(∇φ) 2 .

As in the LPA it is convenient to use dimensionless variables,

Ũk ( ρ) = k -d U k (ρ), Zk ( ρ) = Z -1 k Z k (ρ), Ỹk ( ρ) = Z -2 k k d-2 Y k (ρ), (15)
where ρ = Z k k -(d-2) ρ. The field renormalization factor Z k is defined by imposing the condition Zk ( ρr ) = 1, where ρr is an arbitrary renormalization point (e.g. ρr = 0) and, as mentioned in the discussion of the LPA , must appear as a prefactor in the definition of the regulator function R k . We note that only if the factor Z k is introduced in the regulator does the fixed-point condition become identical to the Ward identity for scale invariance (in presence of the infrared regulator) [START_REF] Delamotte | Scale invariance implies conformal invariance for the three-dimensional Ising model[END_REF]. This implements Wilson's original idea identifying RG fixed point and scale invariance. By inserting the ansatz ( 14) into the flow equation ( 7) one obtains four coupled differential equations for the three functions Ũk , Zk and Ỹk and the running anomalous dimension

η k = -k∂ k ln Z k .
The choice of the regulator function R k is crucial when looking for accurate estimates of critical exponents. One usually considers a family of functions depending on one or more parameters {α i } (see, e.g., the exponential and theta regulators defined in Sec. 2.1.1 which depend on a single parameter α). We determine the optimal value of {α i } from the principle of minimal sensitivity, that is by demanding that locally critical exponents be independent of {α i }, e.g. dν/dα i = 0 for the correlation-length exponent. The renormalization point ρr is usually taken fixed (for numerical convenience) and, provided that the fixed point exists, a change in ρr is equivalent to a change in the amplitude of R k (which is usually one of the α i 's) so that the critical exponents are independent of ρr [START_REF] Balog | Convergence of nonperturbative approximations to the renormalization group[END_REF]. The optimization of the regulator choice is discussed in [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Litim | Optimized renormalization group flows[END_REF][START_REF] Morris | Equivalence of local potential approximations[END_REF][START_REF] Codello | Scheme dependence and universality in the functional renormalization group[END_REF][START_REF] Polsi | Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group[END_REF][START_REF] Balog | Convergence of nonperturbative approximations to the renormalization group[END_REF][START_REF] Litim | Optimisation of the exact renormalization group[END_REF][START_REF] Litim | Mind the gap[END_REF][START_REF] Litim | Critical exponents from optimised renormalisation group flows[END_REF][START_REF] Litim | Universality and the renormalisation group[END_REF][START_REF] Liao | Optimization of renormalization group flow[END_REF][START_REF] Canet | Optimization of the derivative expansion in the nonperturbative renormalization group[END_REF][START_REF] Canet | Nonperturbative renormalization group approach to the Ising model: A derivative expansion at order ∂ 4[END_REF][START_REF] Canet | Optimization of field-dependent nonperturbative renormalization group flows[END_REF][START_REF] Nandori | Spontaneous symmetry breaking and optimization of functional renormalization group[END_REF][START_REF] Pawlowski | Physics and the choice of regulators in functional renormalisation group flows[END_REF].

Results for the critical exponents of the three-dimensional O(N) universality class obtained from the LPA and the DE to second, fourth [START_REF] Polsi | Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group[END_REF][START_REF] Canet | Nonperturbative renormalization group approach to the Ising model: A derivative expansion at order ∂ 4[END_REF][START_REF] Litim | Ising exponents from the functional renormalization group[END_REF] and sixth [START_REF] Balog | Convergence of nonperturbative approximations to the renormalization group[END_REF] orders are shown in Table 1 for N = 0, 1, 2, 3, 4 and compared to Monte Carlo simulations, fixed-dimension perturbative RG, -expansion and conformal bootstrap (the two-Table 1: Critical exponents ν, η and ω for the three-dimensional O(N) universality class obtained in the FRG approach from DE to second [START_REF] Seide | Equation of state near the endpoint of the critical line[END_REF][START_REF] Gersdorff | Nonperturbative renormalization flow and essential scaling for the Kosterlitz-Thouless transition[END_REF], fourth [START_REF] Polsi | Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group[END_REF] and sixth [START_REF] Balog | Convergence of nonperturbative approximations to the renormalization group[END_REF] orders, LPA [START_REF] Hasselmann | Effective-average-action-based approach to correlation functions at finite momenta[END_REF][START_REF] Rose | Nonperturbative renormalization-group approach preserving the momentum dependence of correlation functions[END_REF] and BMW approximation [START_REF] Benitez | Solutions of renormalization group flow equations with full momentum dependence[END_REF][START_REF] Benitez | Nonperturbative renormalization group preserving full-momentum dependence: Implementation and quantitative evaluation[END_REF], compared to Monte Carlo (MC) simulations [START_REF] Hasenbusch | Finite size scaling study of lattice models in the three-dimensional Ising universality class[END_REF][START_REF] Campostrini | Theoretical estimates of the critical exponents of the superfluid transition in 4 He by lattice methods[END_REF][START_REF] Campostrini | Critical exponents and equation of state of the three-dimensional Heisenberg universality class[END_REF][START_REF] Hasenbusch | Monte Carlo study of an improved clock model in three dimensions[END_REF][START_REF] Clisby | High-precision estimate of the hydrodynamic radius for self-avoiding walks[END_REF][START_REF] Clisby | Scale-free monte carlo method for calculating the critical exponent γ of self-avoiding walks[END_REF], d = 3 perturbative RG (PT) [START_REF] Guida | Critical exponents of the N-vector model[END_REF], -expansion at order 6 ( -exp) [START_REF] Kompaniets | Minimally subtracted six loop renormalization of O(n)-symmetric φ 4 theory and critical exponents[END_REF] and conformal bootstrap (CB) [START_REF] Shimada | Fractal dimensions of self-avoiding walks and Ising high-temperature graphs in 3D conformal bootstrap[END_REF][START_REF] Kos | Precision islands in the Ising and O(N) models[END_REF][START_REF] Simmons-Duffin | The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT[END_REF][START_REF] Echeverri | The effective bootstrap[END_REF][START_REF] Chester | Carving out ope space and precise o(2) model critical exponents[END_REF] (when several estimates are available in the literature, we show the one with the smallest error bar). [START_REF] Fisher | Renormalization group theory: Its basis and formulation in statistical physics[END_REF] 0.72 0.765 0.774(20) 0.794(9) 0.817 [START_REF] Berezinskii | Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group i. classical systems[END_REF] dimensional O(1) model (Ising universality class) is discussed in [START_REF] Morris | The renormalization group and two-dimensional multicritical effective scalar field theory[END_REF][START_REF] Ballhausen | Critical phenomena in continuous dimension[END_REF][START_REF] Defenu | Scaling solutions in the derivative expansion[END_REF]). In the large-N limit the DE to second order becomes exact for the critical exponents and the functions U k (ρ) and Z k (ρ) [START_REF] Reuter | The large-N limit and the high-temperature phase transition for the φ 4 theory[END_REF][START_REF] Tetradis | Analytical solutions of exact renormalization group equations[END_REF][START_REF] Tetradis | Critical exponents from the effective average action[END_REF][START_REF] Rose | Nonperturbative renormalization-group approach preserving the momentum dependence of correlation functions[END_REF][START_REF] Attanasio | Large N and the renormalization group[END_REF]. 15Since the DE is a priori valid in all dimensions and for all N, it can be applied to the two-dimensional O(2) model where the transition, as predicted by BKT [START_REF] Berezinskii | Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group i. classical systems[END_REF][START_REF] Berezinskii | Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. ii. quantum systems[END_REF][START_REF] Kosterlitz | Ordering, metastability and phase transitions in two-dimensional systems[END_REF][START_REF] Kosterlitz | The critical properties of the two-dimensional XY model[END_REF], is driven by topological defects (vortices). The FRG approach requires a fine tuning of the regulator function R k in order to reproduce, stricto sensu, the line of fixed points in the low-temperature phase but otherwise recovers most universal features of the BKT transition [START_REF] Gersdorff | Nonperturbative renormalization flow and essential scaling for the Kosterlitz-Thouless transition[END_REF][START_REF] Gräter | Kosterlitz-Thouless Phase Transition in the Two Dimensional Linear σ Model[END_REF][START_REF] Jakubczyk | Reexamination of the nonperturbative renormalization-group approach to the Kosterlitz-Thouless transition[END_REF][START_REF] Jakubczyk | Thermodynamics of the two-dimensional XY model from functional renormalization[END_REF][START_REF] Ranc ¸on | Kosterlitz-Thouless signatures in the low-temperature phase of layered three-dimensional systems[END_REF][START_REF] Jakubczyk | Longitudinal fluctuations in the Berezinskii-Kosterlitz-Thouless phase[END_REF][START_REF] Jakubczyk | Renormalization theory for the Fulde-Ferrell-Larkin-Ovchinnikov states at T > 0[END_REF][START_REF] Defenu | Nonperturbative renormalization group treatment of amplitude fluctuations for ϕ 4 topological phase transitions[END_REF][START_REF] Krieg | Dual lattice functional renormalization group for the Berezinskii-Kosterlitz-Thouless transition: Irrelevance of amplitude and out-of-plane fluctuations[END_REF]. This significantly differs from more traditional studies, based on the Coulomb gas or Villain models [START_REF] Kosterlitz | The critical properties of the two-dimensional XY model[END_REF][START_REF] Villain | Theory of one-and two-dimensional magnets with an easy magnetization plane. ii. the planar, classical, two-dimensional magnet[END_REF], where vortices are introduced explicitly.

An important feature of the DE is that all linear symmetries can be implemented at the level of the effective action with the result that physical quantities satisfy these symmetries if the regulator does. The latter condition is trivially realized in the O(N) model but is more difficult to satisfy in other cases, e.g. in gauge theories (see Secs. 5 and 6).

The numerical solution of the flow equations is an important part of the FRG approach (for both the DE and the methods described in Sec. 2.4). Differential equations can be solved using the explicit Euler or Runge-Kutta methods (with a discretized RG time t = ln(k/Λ)) while momentum integrals can be computed using standard techniques. The variable ρ is often discretized but it is also possible to use pseudospectral methods (e.g. based on Chebyshev polynomials) for the field-dependent functions [START_REF] Fischer | Renormalization flow of Yang-Mills propagators[END_REF][START_REF] Borchardt | Solving functional flow equations with pseudospectral methods[END_REF][START_REF] Borchardt | Global solutions of functional fixed point equations via pseudospectral methods[END_REF][START_REF] Rose | Bound states of the φ 4 model via the nonperturbative renormalization group[END_REF], or discontinuous Galerkin methods, that combine the strength of finite-volume methods and pseudo-spectral methods, see [START_REF] Grossi | Resolving phase transitions with Discontinuous Galerkin methods[END_REF]. The DE can be simplified by truncating the effective action in powers of the field (as in Eq. ( 13)); the convergence of this expansion is discussed in [START_REF] Aoki | Rapidly Converging Truncation Scheme of the Exact Renormalization Group[END_REF][START_REF] Litim | Critical exponents from optimised renormalisation group flows[END_REF][START_REF] Liao | Optimization of renormalization group flow[END_REF][START_REF] Canet | Optimization of the derivative expansion in the nonperturbative renormalization group[END_REF][START_REF] Canet | Nonperturbative renormalization group approach to the Ising model: A derivative expansion at order ∂ 4[END_REF][START_REF] Litim | Derivative expansion and renormalisation group flows[END_REF].

Validity of the DE

The success of the DE can be partially understood by the functional form of the flow equations which makes the expansion nonperturbative in the coupling constants even to lowest order (LPA). But the convergence and the existence of a small parameter that would make the DE a fully controlled approximation is not a priori obvious.

In the previous sections the DE was (loosely) justified by the scale-dependent effective action Γ k [φ] being regular at small momentum scales |p| max(k, ξ -1 ) and the fact that its flow equation is insensitive to momenta larger than k due to the presence of ∂ k R k (p) in the momentum integrals. More precisely the convergence of the DE requires two properties: i) the expansion in p 2 /k 2 of the effective action Γ k [φ] must have a nonzero radius of convergence and ii) the momentum cutoff |p| p max due to ∂ k R k (p) must be sufficiently efficient for the parameter p 2 max /k 2 to be significantly smaller than the radius of convergence.

These two conditions are likely to be satisfied in all unitary theories (i.e., Euclidean theories whose analytic continuation in Minkowski space is unitary) for which the structure of nonanalyticities of correlation functions is known. The radius of convergence of the momentum expansion of a given correlation function is determined by the singularity closest to the origin p = 0. For the two-point correlation function, this singularity is located in the complex plane at p 2 = -m 2 where m is the mass (the inverse correlation length). The next singularity, which shows up in all correlation functions, is located at p 2 = -9m 2 and -4m 2 in the disordered and ordered phases, respectively, and corresponds to the threshold of the two-particle excitation continuum.

Consider now the regulated model defined by the action S +∆S k . Because of the regulator function R k , at criticality the two-point correlation function G k = (Γ (2) k + R k ) -1 exhibits a mass m k ≡ k for k → 0 due to the regulator R k (one can always redefine the running momentum scale such that m k = k). We therefore expect the Taylor expansion in p in that model to have the same radius of convergence as in the Ising model with a mass m ≡ k. The singularity at p 2 = -m 2 ≡ -k 2 determines the radius of convergence of G k but not that of Γ (2) k and higher-order vertices. The singularity closest to the origin in Γ (2) k corresponds to the threshold of the two-particle continuum and is expected to be in the range [-9k 2 , -4k 2 ]. This implies a radius of convergence for the p 2 /k 2 expansion of the effective action in the range [START_REF] Symanzik | Small distance behavior in field theory and power counting[END_REF][START_REF] Wilson | The renormalization group and the expansion[END_REF] and ensures that the condition (i) defined above is satisfied. The condition (ii) is then easily fulfilled by choosing a regulator function R k which cuts off momentum integrals (at least) exponentially for |p| k (which is the case of most regulators used in practice). 16 Finally one must also add that the corrections to second order and higher in the DE are suppressed by a factor η [START_REF] Polsi | Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group[END_REF][START_REF] Balog | Convergence of nonperturbative approximations to the renormalization group[END_REF], the anomalous dimension, which is a small number in the ϕ 4 theory. The convergence of the DE has been nicely illustrated by 4th-and 6th-order calculations in the three-dimensional O(N)-model universality class (see Sec. 2.3.2 and Table 1 as well as Appendix B).

This reasoning also explains why the DE exhibits poor convergence properties in the Wilson-Polchinski formulation, i.e., without performing the Legendre transform [START_REF] Morris | Convergence of derivative expansions of the renormalization group[END_REF][START_REF] Yu | Solutions of the polchinski erg equation in the O(n) scalar model[END_REF], even if the LPA gives satisfactory results [START_REF] Morris | Equivalence of local potential approximations[END_REF][START_REF] Litim | Universality and the renormalisation group[END_REF]. The leading singularity being located at p 2 = -k 2 for the two-point correlation function, the radius of convergence in p 2 /k 2 is of order one and thus of the same order as p 2 max /k 2 ∼ 1, 16 so that there is no small parameter.

2.3.4.

Further results obtained from the DE 17In Sec. 2.3.2 we have emphasized the computation of critical exponents but the DE also allows one to compute the scaling functions determining the universal equation of state in the vicinity of a second-order phase transition both in classical and quantum systems [START_REF] Berges | Critical Equation of State from the Average Action[END_REF][START_REF] Ranc ¸on | Thermodynamics in the vicinity of a relativistic quantum critical point in 2+1 dimensions[END_REF][START_REF] Ranc ¸on | Quantum XY criticality in a two-dimensional Bose gas near the Mott transition[END_REF][START_REF] Ranc ¸on | Critical Casimir forces from the equation of state of quantum critical systems[END_REF]. The DE can be used to study the high-temperature disordered phase [START_REF] Morris | Three-dimensional massive scalar field theory and the derivative expansion of the renormalization group[END_REF] as well as the ordered phase [60, 93-95, 173, 174] of the O(N) model where for 2 < d ≤ 4 and N ≥ 2 the longitudinal susceptibility diverges due its coupling to transverse fluctuations, a general phenomenon in systems with a continuous broken symmetry [START_REF] Patasinskij | Longitudinal susceptibility and correlations in degenerate systems[END_REF][START_REF] Zwerger | Anomalous Fluctuations in Phases with a Broken Continuous Symmetry[END_REF].

Many authors have considered multicritical points in dimensions d ≤ 4 [109-111, 146, 150-152, 177-180] and the more speculative existence of critical fixed points for 4 < d < 6 [START_REF] Eichhorn | Critical o(n) models above four dimensions: Small-n solutions and stability[END_REF][START_REF] Percacci | Are there scaling solutions in the O(N)-models for large N in d > 4 ?[END_REF][START_REF] Mati | Vanishing beta function curves from the functional renormalization group[END_REF]. Some multicritical fixed points of the O(N) model for d < 4 show singularities in the form of cusps at N = ∞ in their effective potential that become a boundary layer at finite N and are therefore overlooked in the standard 1/N expansion [START_REF] Yabunaka | Surprises in o(n) models: Nonperturbative fixed points, large n limits, and multicriticality[END_REF][START_REF] Yabunaka | Why might the standard large n analysis fail in the O(n) model: The role of cusps in fixed point potentials[END_REF][START_REF] Katsis | Multicritical points of the O(N) scalar theory in 2 < d < 4 for large N[END_REF].

Let us also mention the following studies: O(N) models with long-range interactions [START_REF] Defenu | Fixed-point structure and effective fractional dimensionality for O(N) models with long-range interactions[END_REF][START_REF] Defenu | Criticality and phase diagram of quantum long-range O( N ) models[END_REF][START_REF] Goll | Renormalization group for ϕ 4 -theory with long-range interaction and the critical exponent η of the Ising model[END_REF][START_REF] Defenu | Criticality of spin systems with weak long-range interactions[END_REF]; fixed points with imaginary couplings [START_REF] Litim | Critical O(N) models in the complex field plane[END_REF][START_REF] Jüttner | Global Wilson-Fisher fixed points[END_REF]; non-polynomial perturbations to fixed points [START_REF] Halpern | Fixed-Point Structure of Scalar Fields[END_REF][191][START_REF] Morris | On the fixed point structure of scalar fields[END_REF][START_REF] Gies | Flow equation for Halpern-Huang directions of scalar O(N) models[END_REF][START_REF] Hamzaan Bridle | Fate of nonpolynomial interactions in scalar field theory[END_REF]; O(N) models in finite geometries and critical Casimir forces [START_REF] Ranc ¸on | Critical Casimir forces from the equation of state of quantum critical systems[END_REF][START_REF] Jakubczyk | Critical Casimir forces for O(N) models from functional renormalization[END_REF]; nonlinear sigma models [START_REF] Codello | Fixed Points of Nonlinear Sigma Models in[END_REF][START_REF] Flore | Functional renormalization group of the non-linear sigma model and the O(N) universality class[END_REF][START_REF] Percacci | Functional renormalization of N scalars with O(N) invariance[END_REF]; the Potts model [START_REF] Zinati | Functional rg approach to the potts model[END_REF].

The DE has been used to study the sine-Gordon model with emphasis on the BKT transition [START_REF] Nagy | Functional Renormalization Group Approach to the Sine-Gordon Model[END_REF][START_REF] Pangon | Structure of the broken phase of the sine-Gordon model using functional renormalisation[END_REF][START_REF] Pangon | Generating the mass gap of the sine-Gordon model[END_REF] or the central charge and the c-function [START_REF] Bacsó | c-function and central charge of the sine-Gordon model from the non-perturbative renormalization group flow[END_REF][START_REF] Oak | Exact renormalization group and sine gordon theory[END_REF]. It yields an accurate estimate of the (exactly known) soliton and soliton-antisoliton bound state masses in the massive phase of this model [START_REF] Daviet | Nonperturbative functional renormalization-group approach to the sine-Gordon model and the Lukyanov-Zamolodchikov conjecture[END_REF] and strongly supports the Lukyanov-Zamolodchikov conjecture [START_REF] Lukyanov | Exact expectation values of local fields in the quantum sine-Gordon model[END_REF] regarding the amplitude of the field fluctuations.

Computing momentum-dependent correlation functions

To obtain the full momentum dependence of correlation functions it is necessary to go beyond the DE, since the latter is restricted to the momentum range |p| max(k, ξ -1 ). Momentum-dependent correlation functions can be computed by means of a vertex expansion.

The flow equation ( 7) yields an infinite hierarchy of equations satisfied by the vertices Γ (n) k . The vertex expansion, in its simplest formulation, amounts to truncating this hierarchy by retaining a finite number of low-order vertices. This leads to a closed system of equations that can be solved. Retaining the momentum dependence of the vertices allows one to obtain that of the correlation functions [START_REF] Kopietz | Introduction to the Functional Renormalization Group[END_REF][START_REF] Blaizot | Nonperturbative renormalization group and momentum dependence of n-point functions. I[END_REF][START_REF] Blaizot | Non perturbative renormalization group, momentum dependence of n-point functions and the transition temperature of the weakly interacting Bose gas[END_REF][START_REF] Blaizot | Nonperturbative renormalization group and momentum dependence of n-point functions. II[END_REF][START_REF] Blaizot | Non-perturbative renormalization group calculation of the scalar self-energy[END_REF][START_REF] Ledowski | Self-energy and critical temperature of weakly interacting bosons[END_REF][START_REF] Hasselmann | Two-parameter scaling of correlation functions near continuous phase transitions[END_REF][START_REF] Sinner | Functional renormalization group in the broken symmetry phase: momentum dependence and two-parameter scaling of the self-energy[END_REF]. Systematic vertex expansion schemes with full momentum dependence have been also used in condensed matter systems, QCD and gravity, for more details see Secs. 4, 5 and 6.

Keeping only a finite number of vertices is however not always sufficient. In some problems, it is necessary to keep both the momentum dependence of low-order vertices and the full set of vertices in the zero-momentum sector (which amounts to considering the full effective potential). A possible approximation [START_REF] Hasselmann | Effective-average-action-based approach to correlation functions at finite momenta[END_REF][START_REF] Rose | Nonperturbative renormalization-group approach preserving the momentum dependence of correlation functions[END_REF][START_REF] Guerra | Correlation functions in the non perturbative renormalization group and field expansion[END_REF], inspired by the LPA , is defined by the effective action

Γ LPA k [φ] = ˆr 1 2 (∂ µ φ) • Z k (-∇ 2 )(∂ µ φ) + 1 4 (∂ µ ρ)Y k (-∇ 2 )(∂ µ ρ) + U k (ρ) , (16)
with a sum over µ = 1 for |p| p G when the system is critical. The value of the critical exponents ν and η is shown in Table 1. This approximation scheme, which is sometimes referred to as the LPA , is not numerically more costly than the DE and usually allows for an easy implementation of the symmetries, and for these reasons has been used in various contexts, see e.g. [START_REF] Hasselmann | Nonlocal effective-average-action approach to crystalline phantom membranes[END_REF][START_REF] Mathey | Anomalous scaling at nonthermal fixed points of Burgers' and gross-pitaevskii turbulence[END_REF][START_REF] Canet | General framework of the non-perturbative renormalization group for non-equilibrium steady states[END_REF][START_REF] Canet | Fully developed isotropic turbulence: Nonperturbative renormalization group formalism and fixedpoint solution[END_REF][START_REF] Feldmann | Critical Wess-Zumino models with four supercharges in the functional renormalization group approach[END_REF].

A more elaborate approximation scheme, which also keeps all vertices in the zero-momentum sector, has been proposed by Blaizot, Méndez-Galain and Wschebor (BMW) [START_REF] Benitez | Solutions of renormalization group flow equations with full momentum dependence[END_REF][START_REF] Benitez | Nonperturbative renormalization group preserving full-momentum dependence: Implementation and quantitative evaluation[END_REF][START_REF] Blaizot | A new method to solve the non-perturbative renormalization group equations[END_REF][START_REF] Benitez | Calculations on the two-point function of the O(N) model[END_REF][START_REF] Rose | Higgs amplitude mode in the vicinity of a (2 + 1)-dimensional quantum critical point: A nonperturbative renormalization-group approach[END_REF] and, in the context of liquid theory, by Parola and Reatto [START_REF] Parola | Liquid state theory and critical phenomena[END_REF][START_REF] Parola | Liquid-State Theory for Critical Phenomena[END_REF]. The flow equation of the two-point vertex Γ (2) k (p, φ) in a uniform field φ involves Γ (4) k (p, -p, q, -q, φ) and Γ (3) k (p, -q, -pq, φ) as well as ∂ k R k (q) (see Fig. 2). Because of the latter term, which restricts the integral over the loop momentum to |q| k, to leading order one can set q = 0 in Γ (4) k and Γ (3) k . Since Γ (3) k,il j (p, 0, -p, φ) = ∂Γ (2) k,i j (p, φ)/∂φ l (and a similar relation for Γ (4) k,i jlm (p, -p, 0, 0, φ)), one obtains a closed equation for Γ (2) k (p, φ) which must be solved together with the exact flow equation ( 9) of the effective potential. The BMW approximation scheme is numerically more involved than the DE and the LPA since Γ (2) k (p, φ) is a two-variable (|p| and ρ) function. Symmetries may also sometimes be difficult to implement.

The critical exponents obtained from BMW compare favorably with those derived from the DE to second order or the LPA . ν and η are within 0.41% and 7.45%, respectively, of the conformal bootstrap results for the threedimensional O(N) universality class and N = 1, 2, 3 (Table 1). For the two-dimensional Ising model, BMW gives ν 1.00 and η 0.254 [START_REF] Benitez | Nonperturbative renormalization group preserving full-momentum dependence: Implementation and quantitative evaluation[END_REF], to be compared with the exact values ν = 1, η = 1/4 [START_REF] Pogorelov | Renormalization group functions for two-dimensional phase transitions: To the problem of singular contributions[END_REF]. The BMW approximation becomes exact in the large-N limit [START_REF] Blaizot | A new method to solve the non-perturbative renormalization group equations[END_REF][START_REF] Rose | Higgs amplitude mode in the vicinity of a (2 + 1)-dimensional quantum critical point: A nonperturbative renormalization-group approach[END_REF]. 15 The small parameter of the order of 1/4-1/9 that has been mentioned in the context of the DE (Sec. 2.3.3) applies since the BMW approximation also corresponds to an expansion in momenta (although the internal ones).

The BMW approximation (and simplified versions of it [START_REF] Blaizot | Non-perturbative renormalization group calculation of the scalar self-energy[END_REF][START_REF] Guerra | Correlation functions in the non perturbative renormalization group and field expansion[END_REF][START_REF] Benitez | Calculations on the two-point function of the O(N) model[END_REF]) has been applied to the quantum O(N) model and interacting bosons (Sec. 3), as well as the Kardar-Parisi-Zhang and Navier-Stokes equations [START_REF] Canet | General framework of the non-perturbative renormalization group for non-equilibrium steady states[END_REF][START_REF] Canet | Fully developed isotropic turbulence: Nonperturbative renormalization group formalism and fixedpoint solution[END_REF].

Lattice models and realistic microscopic actions

The FRG approach to continuum models described in the preceding sections can be straightforwardly extended to lattice models [START_REF] Dupuis | Non-perturbative renormalization-group approach to lattice models[END_REF]. 18 The lattice is taken into account by replacing the p 2 dispersion in the bare propagator by the actual lattice dispersion 0 (p) and restricting the momentum to the first Brillouin zone. Alternatively, one can start from an initial condition of the RG flow corresponding to the local limit of decoupled sites [START_REF] Machado | From local to critical fluctuations in lattice models: A nonperturbative renormalization-group approach[END_REF]. The flow equation then implements an expansion about the single-site limit and is reminiscent, to some extent, of Kadanoff's idea of block spins [START_REF] Kadanoff | Scaling laws for ising models near T c[END_REF]. This lattice FRG captures both local and critical fluctuations and therefore enables us to compute nonuniversal quantities such as transition temperatures. It has been applied to the O(N) model defined on a lattice [START_REF] Machado | From local to critical fluctuations in lattice models: A nonperturbative renormalization-group approach[END_REF][START_REF] Caillol | Critical line of the theory on a simple cubic lattice in the local potential approximation[END_REF][START_REF] Caillol | Critical line of the Φ 4 scalar field theory on a 4D cubic lattice in the local potential approximation[END_REF][START_REF] Banerjee | Graph rules for the linked cluster expansion of the Legendre effective action[END_REF][START_REF] Banerjee | Critical behavior of the hopping expansion from the Functional Renormalization Group[END_REF] and to classical [START_REF] Machado | From local to critical fluctuations in lattice models: A nonperturbative renormalization-group approach[END_REF] and quantum [START_REF] Ranc ¸on | Nonperturbative renormalization group approach to quantum XY spin models[END_REF][START_REF] Krieg | Exact renormalization group for quantum spin systems[END_REF] spin models as well as the superfluid-Mott transition in the Bose-Hubbard model [START_REF] Ranc ¸on | Quantum XY criticality in a two-dimensional Bose gas near the Mott transition[END_REF][START_REF] Ranc ¸on | Nonperturbative renormalization group approach to the Bose-Hubbard model[END_REF][START_REF] Ranc ¸on | Nonperturbative renormalization group approach to strongly correlated lattice bosons[END_REF][START_REF] Ranc ¸on | Quantum criticality of a Bose gas in an optical lattice near the Mott transition[END_REF][START_REF] Ranc ¸on | Thermodynamics of a Bose gas near the superfluid-Mott-insulator transition[END_REF].

The possibility to start from an initial condition that already includes short-range fluctuations has been used in the Hierarchical Reference Theory of fluids [START_REF] Parola | Liquid state theory and critical phenomena[END_REF], an approach which bears many similarities with the lattice FRG. More recently, similar ideas have appeared in the RG approach to interacting fermions [START_REF] Reuther | Cluster functional renormalization group[END_REF][START_REF] Wentzell | Correlated starting points for the functional renormalization group[END_REF][START_REF] Taranto | From Infinite to Two Dimensions through the Functional Renormalization Group[END_REF].

Quantum models

There is no difficulty to extend the FRG approach to quantum bosonic models. In the Euclidean (Matsubara) formalism, the latter map onto (d+1)-dimensional classical field theories with a finite extension β = 1/T in the (d+1)th (imaginary time) direction [START_REF] Sachdev | Quantum Phase Transitions[END_REF]; the effective action Γ k becomes a functional of a space-and time-dependent bosonic field φ(r, τ) where τ ∈ [0, β] [START_REF] Reuter | The large-N limit and the high-temperature phase transition for the φ 4 theory[END_REF][START_REF] Tetradis | The high temperature phase transition for φ 4 theories[END_REF][START_REF] Litim | Non-perturbative thermal flows and resummations[END_REF]. Galilean-invariant bosons and the quantum O(N) model, the simplest quantum generalization of Eq. ( 2) with space-time Lorentz invariance, are discussed in Sec. 4.1.

Fermionic models are more difficult to deal with since the field φ(r, τ) = ϕ(r, τ) in that case is an anticommuting Grassmann variable. Functionals of Grassmann variables make sense only via their Taylor expansions and, for example, the very concept of an effective potential with a well-defined minimum is lost. Thus the only a priori available method for fermionic models is a vertex expansion where one retains a finite number of (momentum-dependent) vertices evaluated at φ = 0. It is however possible to introduce, via Hubbard-Stratonovich transformations, collective bosonic fields which can be treated nonperturbatively using the methods discussed in the previous sections (see .

A well-known difficulty in the study of quantum systems is the computation of real-frequency correlation functions from numerical data obtained in the Euclidean formalism, in particular at finite temperatures. At zero temperatures, the resonances-via-Padé method [START_REF] Schlessinger | Use of analyticity in the calculation of nonrelativistic scattering amplitudes[END_REF][START_REF] Vidberg | Solving the Eliashberg equations by means of N-point Padé approximants[END_REF][START_REF] Tripolt | Numerical analytic continuation of Euclidean data[END_REF] and other, Bayesian, reconstruction methods have been used in several works [START_REF] Rose | Nonperturbative renormalization-group approach preserving the momentum dependence of correlation functions[END_REF][START_REF] Rose | Bound states of the φ 4 model via the nonperturbative renormalization group[END_REF][START_REF] Rose | Higgs amplitude mode in the vicinity of a (2 + 1)-dimensional quantum critical point: A nonperturbative renormalization-group approach[END_REF][START_REF] Dupuis | Infrared behavior and spectral function of a Bose superfluid at zero temperature[END_REF][START_REF] Sinner | Functional renormalization-group approach to interacting bosons at zero temperature[END_REF][START_REF] Schmidt | Excitation spectra and rf response near the polaron-to-molecule transition from the functional renormalization group[END_REF][START_REF] Rose | Superuniversal transport near a (2 + 1)-dimensional quantum critical point[END_REF][START_REF] Tripolt | Threshold energies and poles for hadron physical problems by a model-independent universal algorithm[END_REF]. Alternative methods, where the analytic continuation is performed at the level of the flow equations, have been proposed [START_REF] Rohe | Pseudogap at hot spots in the two-dimensional Hubbard model at weak coupling[END_REF][START_REF] Jakobs | Nonequilibrium functional renormalization group with frequency-dependent vertex function: A study of the single-impurity Anderson model[END_REF][START_REF] Floerchinger | Analytic Continuation of Functional Renormalization Group Equations[END_REF][START_REF] Tripolt | Spectral functions from the functional renormalization group[END_REF][START_REF] Tripolt | Spectral Functions for the Quark-Meson Model Phase Diagram from the Functional Renormalization Group[END_REF][START_REF] Kamikado | Real-Time Correlation Functions in the O(N) Model from the Functional Renormalization Group[END_REF][START_REF] Haas | Gluon spectral functions and transport coefficients in Yang-Mills theory[END_REF][START_REF] Christiansen | Transport Coefficients in Yang-Mills Theory and QCD[END_REF][START_REF] Wambach | Spectral functions from the functional renormalization group[END_REF][START_REF] Pawlowski | Real time correlation functions and the functional renormalization group[END_REF][START_REF] Pawlowski | Finite temperature spectral functions in the O(N) model[END_REF][START_REF] Kedri | Nonequilibrium thermoelectric transport through vibrating molecular quantum dots[END_REF][START_REF] Cyrol | Reconstructing the gluon[END_REF]

Statistical mechanics

Why and when is the FRG useful?

Statistical mechanics is a natural area of application of the FRG since it aims at calculating macroscopic properties of a system from a microscopical model. This is the reason why the general presentation of Sec. 2 was framed in the language of equilibrium statistical mechanics by using examples drawn from the Ising and O(N) models.

Many models (including the O(N) model) can be treated by traditional means such as perturbation theory (in the critical regime, improved with perturbative RG). When does the FRG become useful compared to perturbative RG? There is no general answer but some general remarks can be made. First, the FRG is well suited for approximations where the functional form of various terms of the scale-dependent effective action plays a major role. This functional aspect becomes unavoidable when the considered functions develop nonanalyticities, such as "cusps" (see, e.g., Sec. 3.3 below). Second, in some cases, the physics to be studied is beyond the reach of approximations based on perturbation theory. This, for example, occurs in the study of the Kardar-Parisi-Zhang equation (Sec. 3.4). The main advantage of the FRG in this respect is that its flow equations are dominated by a small shell of momenta, making it extremely robust and flexible when employing approximations going beyond perturbation theory (see Sec. 2). This is at odds with other nonperturbative formulations such as Schwinger-Dyson equations which involve integrals over a large region of momenta. This "locality in momentum" is at the origin of the success of the expansion schemes described in Sec. 2. In particular, the decoupling of different momenta explains a posteriori the apparent convergence of the DE and the accurate results obtained by this method, even though a small expansion parameter has only recently been identified (Sec. 2.3.3). There is a third reason why the FRG proves useful: it can lead to the determination of nonuniversal properties, such as a phase diagram (see, e.g., Sec. 3.4.2), something which is often challenging in other RG approaches. Finally, as pointed out in Sec. 2, in the FRG framework it is particularly simple to vary the dimension of the theory. Even within very simple approximations, the behavior near the upper and lower dimensions is reproduced and one can therefore obtain results at intermediate dimensions that interpolate between controlled limiting cases. This typically makes the results much more robust than when only an extrapolation from the upper critical dimension is done.

In this section, we present applications of the FRG to classical statistical physics. Section 3.2 is devoted to classical equilibrium statistical mechanics, Sec. 3.3 to disordered systems, and out-of-equilibrium systems are discussed in Sec. 3.4. The subject has reached a mature level and it is therefore not possible to present a full account of all the topics that have been addressed within the FRG method. Some paradigmatic examples are nevertheless presented in detail and a survey of other works is briefly given.

Equilibrium statistical mechanics

The implementation of the FRG procedure relies on a microscopic Hamiltonian appropriate to describe the system under consideration. When studying universal features, it is sufficient to consider a general enough low-energy effective Hamiltonian respecting the symmetries of the problem and including the main infrared degrees of freedom. 19The derivation of the low-energy effective theory may be nontrivial and in some cases not even known. To illustrate this procedure in a concrete example, in the following section we discuss a paradigmatic problem at equilibrium: the nature of the phase transition in Stacked Triangular Antiferromagnets (STA), an important class of frustrated magnets.

Frustrated magnets as an example

The model. The STA model has been proposed to describe several frustrated magnets (we refer to [START_REF] Delamotte | Nonperturbative renormalization-group approach to frustrated magnets[END_REF] for a review). It describes N-component classical spins with antiferromagnetic nearest-neighbor interactions. The spins are located at the lattice sites of a d-dimensional lattice consisting of stacked two-dimensional triangular lattices. The corresponding Hamiltonian is O(N)-invariant:

H = J i, j S i • S j with J > 0. ( 17
)
The triangular planar structure induces frustration (it is not possible to minimize simultaneously all nearest-neighbor interactions). This makes the standard O(N) Ginzburg-Landau model unsuited for the description of the long-distance properties of STA. The derivation of a field theory that correctly describes the critical physics of a given model relies on the knowledge of the ground state, which is often nontrivial in presence of frustration. In the case of STA, the configuration which minimizes the energy is known and takes the form shown in Fig. 4 where the three spins of each triangular plaquette point 120 • one from another. The degree of freedom corresponding to the magnetization of each triangular block is therefore frozen and exhibits gapped excitations (even at the transition). Accordingly, in the critical regime, the degree of freedom associated with the block magnetization can be integrated out and the orientation of the spins on a plaquette can then be described by two N-component vectors that are not colinear. Indeed it is sufficient to consider one of the two spins and the projection of the second one in the direction orthogonal to the first one. As such, the block variables can be chosen to be two orthogonal vectors ϕ 1 and ϕ 2 , of unit norm. In terms of these block variables, the Hamiltonian becomes ferromagnetic. As a consequence, the effective Hamiltonian takes the form symmetry, where now the block variables are not constrained to have unit norm or being orthogonal. However, in order to be equivalent to the spin system one requires that the Ginzburg-Landau potential has its minimum in a configuration where the two vectors ϕ 1 and ϕ 2 are orthogonal and of the same norm. For that purpose it is convenient to introduce the 2 × N matrix Φ = ϕ 1 , ϕ 2 , in terms of which the Ginzburg-Landau Hamiltonian reads

H = -J I,J ϕ I 1 • ϕ J 1 + ϕ I 2 • ϕ J 2 , (18)
H GL = ˆr 1 2 Tr(∇Φ t ∇Φ) + r 2 ρ + λ 16 ρ 2 + µ 4 τ , (19)
with ρ = Tr(Φ t Φ) and τ = 1 2 Tr[(Φ t Φ -ρ 2 1 2 ) 2 ]
. The coupling constants µ and λ are chosen positive in order to ensure, as stated above, that the minimum of the potential corresponds to a configuration where the vectors ϕ 1 and ϕ 2 are mutually orthogonal and of the same norm. One can choose the coordinates such that the vacuum state takes the form

Φ 0 ∝                  1 0 0 1 0 0 . . . . . .                  . ( 20
)
This corresponds to a breaking of the O(N) × O(2) symmetry to O(N -2) × O(2) diag in the ordered phase. The Hamiltonian [START_REF] Feldman | Construction and Borel Summability of Infrared φ 4 in Four-dimensions by a Phase Space Expansion[END_REF] also describes spin-one bosons [START_REF] Debelhoir | Simulating frustrated magnetism with spinor Bose gases[END_REF][START_REF] Debelhoir | First-order phase transitions in spinor Bose gases and frustrated magnets[END_REF].

Main open problems in STA. The STA model has been studied in perturbation theory (d = 4 -, d = 2 + , fixed d), large-N, Monte-Carlo simulations, and within the FRG. There are also many experimental studies of materials expected to be in the STA universality class. The main results from these theoretical and experimental studies appear to be conflicting.

Results from perturbation theory near the upper critical dimension d c = 4 are schematically shown in Fig. 5. One observes two different cases. For values of N above a certain critical value N c (d), the critical regime is controlled by a nontrivial fixed point (denoted by C + in Fig. 5). Below N c (d) there is no stable fixed point anymore in the would-be critical surface and the transition is first-order (the O(2N)-invariant fixed point denoted by V in Fig. 5 has an unstable direction). The main problem is then to properly determine the function N c (d). Near d = 4, N c (d) can be determined from the -expansion [START_REF] Antonenko | Chiral transitions in three-dimensional magnets and higher order expansion[END_REF][START_REF] Calabrese | Five-loop expansion for O(n)×O(m) spin models[END_REF]:

N c (d) = 21.80 -23.43 + 7.09 2 -0.03 3 + 4.26 4 + O( 5 ). ( 21
)
For d = 3, Eq. ( 21) gives results that strongly oscillate with the order of the expansion. Moreover, resummation techniques [START_REF] Pelissetto | The Critical behavior of frustrated spin models with noncollinear order[END_REF][START_REF] Calabrese | Chiral phase transitions: Focus driven critical behavior in systems with planar and vector ordering[END_REF] yield very unstable predictions (at odds with the three-dimensional O(N) model). The model has also been studied in the fixed d = 3 resumed perturbative expansion at six loops, giving N c (d = 3) = 6.4(4) [START_REF] Pelissetto | The Critical behavior of frustrated spin models with noncollinear order[END_REF]. However, the d = 3 perturbative expansion finds that the transition becomes again of second order for N < 5.7(3), which disagrees with the conclusion of the nonperturbative FRG analysis (see below). The STA have also been studied within the conformal bootstrap program [START_REF] Nakayama | Bootstrapping phase transitions in QCD and frustrated spin systems[END_REF][START_REF] Stergiou | Bootstrapping MN and Tetragonal CFTs in Three Dimensions[END_REF][START_REF] Henriksson | Analytic and Numerical Bootstrap of CFTs with O(m) × O(n) Global Symmetry in 3D[END_REF] but this approach, even if it has been very successful in the calculation of critical exponents in cases where the transition is second order [START_REF] Poland | The conformal bootstrap: Theory, numerical techniques, and applications[END_REF], relies on the assumption that the transition is continuous and is therefore of no use for determining whether the transition is actually first or second order. On the experimental and numerical side, scaling is mostly observed. However, the exponents seem to be nonuniversal (see [START_REF] Delamotte | Nonperturbative renormalization-group approach to frustrated magnets[END_REF] for details). Large-scale simulations [START_REF] Itakura | Monte Carlo renormalization group study of the Heisenberg and XY antiferromagnet on the stacked triangular lattice and the chiral phi**4 model[END_REF][START_REF] Ngo | Phase transition in heisenberg stacked triangular antiferromagnets: End of a controversy[END_REF] indicate a weakly first-order transition.

FRG approach to STA. The phase transition in the O(N)×O(2) model can be studied within the FRG approach. Given that the anomalous dimension is small, the DE to second order, DE 2 , is expected to be a good approximation. To this order, the most general effective action compatible with symmetries reads [START_REF] Delamotte | The Wilson renormalization group approach of the principal chiral model around twodimensions[END_REF][START_REF] Tissier | A Nonperturbative approach of the principal chiral model between two-dimensions and fourdimensions[END_REF][START_REF] Tissier | Heisenberg frustrated magnets: A Nonperturbative approach[END_REF][START_REF] Tissier | XY frustrated systems: Continuous exponents in discontinuous phase transitions[END_REF][START_REF] Delamotte | Functional renormalization group approach to noncollinear magnets[END_REF].

Γ k [φ 1 , φ 2 ] = ˆr U k (ρ, τ) + Z k (ρ, τ) 2 (∇φ 1 ) 2 + (∇φ 2 ) 2 + Y (1) k (ρ, τ) 4 (φ 1 • ∇φ 2 -φ 2 • ∇φ 1 ) 2 + Y (2) k (ρ, τ) 4 (φ 1 • ∇φ 1 + φ 2 • ∇φ 2 ) 2 + 1 4 Y (3) k (ρ, τ) (φ 1 • ∇φ 1 -φ 2 • ∇φ 2 ) 2 + (φ 1 • ∇φ 2 + φ 2 • ∇φ 1 ) 2 . ( 22
)
To reduce the numerical cost of solving flow equations for two-variable functions, it is possible to expand U k (ρ, τ), Z k (ρ, τ) and Y (i) k (ρ, τ) about τ = 0 and ρ = ρ 0,k , which corresponds to the minimum of the effective potential [START_REF] Delamotte | The Wilson renormalization group approach of the principal chiral model around twodimensions[END_REF][START_REF] Tissier | A Nonperturbative approach of the principal chiral model between two-dimensions and fourdimensions[END_REF][START_REF] Tissier | Heisenberg frustrated magnets: A Nonperturbative approach[END_REF][START_REF] Tissier | XY frustrated systems: Continuous exponents in discontinuous phase transitions[END_REF]. As in the case of the O(N) model, the DE 2 approximation with a field expansion around the minimum of the potential reproduces the leading behavior near d = 4 and d = 2 and in the large-N limit. More recently, this truncation has been improved by expanding in the invariant τ but treating the full ρ-dependence of the various functions [START_REF] Delamotte | Functional renormalization group approach to noncollinear magnets[END_REF].

Figure 6 shows N c (d) obtained from DE 2 and three-loop calculation in the = 4d expansion improved by the exactly known condition N c (d = 2) = 2 [START_REF] Pelissetto | Large n critical behavior of O(n) × O(m) spin models[END_REF]. Both curves are qualitatively similar. In both cases, N c (d = 3) is larger than 3. Thus the FRG predicts a first-order phase transition in the O(3)×O(2) and O(2)×O(2) models in agreement with Monte-Carlo simulations [START_REF] Itakura | Monte Carlo renormalization group study of the Heisenberg and XY antiferromagnet on the stacked triangular lattice and the chiral phi**4 model[END_REF][START_REF] Ngo | Phase transition in heisenberg stacked triangular antiferromagnets: End of a controversy[END_REF] but in disagreement with fixed-dimension RG studies [START_REF] Pelissetto | Large n critical behavior of O(n) × O(m) spin models[END_REF].

The FRG also explains the pseudo-scaling observed in numerical simulations and experiments. When N > N c (d), the phase transition is second order and scaling (with universal critical exponents) is associated with the critical fixed point C + . The two fixed points C + and C -disappear by merging when N = N c (d), and for N < N c (d) two fixed points with complex coordinates appear. The RG flow becomes very slow in the vicinity of these two (unphysical) fixed points; hence a large, although finite, correlation length that varies as ξ ∼ (T -T c ) ν over a large temperature range with a (nonuniversal) pseudocritical exponent ν.

Being nonuniversal, the value of the pseudo-exponent ν depends on the initial condition of the flow. However, as explained above, there is a finite region of couplings that shows a quasi-fixed point behavior. As a consequence, if the bare couplings take natural values (that is, are of order one in units of the lattice spacing) the exponents lie within a small range of values. The typical value of these pseudo-critical exponents compares well with the results obtained in the six-loop perturbative expansion (even if the interpretation is quite different because the d = 3 perturbative expansion predicts a genuine second order transition [START_REF] Pelissetto | The Critical behavior of frustrated spin models with noncollinear order[END_REF]), see Table 2.

An FRG study of a lattice O(2)×O( 2) model [START_REF] Debelhoir | Simulating frustrated magnetism with spinor Bose gases[END_REF][START_REF] Debelhoir | First-order phase transitions in spinor Bose gases and frustrated magnets[END_REF] has confirmed the existence of a weak first-order transition and shown that the correlation length is larger than the typical size of systems studied in Monte Carlo simulations, thus explaining why the true nature of the phase transition may have been missed in some numerical calculations [START_REF] Calabrese | Critical behavior of O(2) ⊗ O(N) symmetric models[END_REF].

FRG studies of frustrated magnets are reviewed in [START_REF] Delamotte | Nonperturbative renormalization-group approach to frustrated magnets[END_REF]. Related works on phase transitions with matrix order parameters can be found in Refs. [274-278, 281, 282].

Critical phenomena and universal long distance regime

We now review some universal long-distance properties obtained from the DE in various models (and that were not already discussed in Sec. 2).

The critical regime of models with cubic symmetry and multiple scalar fields has been considered in detail in Refs. [START_REF] Bornholdt | Coleman-Weinberg phase transition in two scalar models[END_REF][START_REF] Bornholdt | High temperature phase transition in two scalar theories[END_REF][START_REF] Bornholdt | Flow of the coarse grained free energy for crossover phenomena[END_REF][START_REF] Tissier | Randomly dilute Ising model: A nonperturbative approach[END_REF][START_REF] Chlebicki | Criticality of the O(2) model with cubic anisotropies from nonperturbative renormalization[END_REF]. Other studies of classical models include some particular Potts models [START_REF] Zinati | Functional rg approach to the potts model[END_REF], the Yang-Lee model [START_REF] An | Functional renormalization group approach to the Yang-Lee edge singularity[END_REF][START_REF] Zambelli | Lee-Yang model from the functional renormalization group[END_REF], and the spontaneous breaking of space rotational invariance [START_REF] Lauscher | Rotation symmetry breaking condensate in a scalar theory[END_REF]. In some clock models, the DE has given the striking result that the exponents can be different on the two sides of the transition [START_REF] Léonard | Critical Exponents Can Be Different on the Two Sides of a Transition: A Generic Mechanism[END_REF]. A discussion of the relation between the LPA and Dyson's Hierarchical Model can be found in [START_REF] Meurice | Nonlinear Aspects of the Renormalization Group Flows of Dyson's Hierarchical Model[END_REF][START_REF] Litim | Towards functional flows for hierarchical models[END_REF].

The analysis of the existence of nontrivial critical points has been extended to very general models with a rich field content or symmetries [START_REF] Eichhorn | Stability of fixed points and generalized critical behavior in multifield models[END_REF][START_REF] Eichhorn | Discovering and quantifying nontrivial fixed points in multi-field models[END_REF][START_REF] Boettcher | Scaling relations and multicritical phenomena from Functional Renormalization[END_REF][START_REF] Borchardt | Universal behavior of coupled order parameters below three dimensions[END_REF].

FRG equations have also been employed to study various aspects of polymerized membranes, including the crumpling phase transition [START_REF] Kownacki | Crumpling transition and flat phase of polymerized phantom membranes[END_REF][START_REF] Essafi | Crumpled-to-Tubule Transition in Anisotropic Polymerized Membranes: Beyond the γ Expansion[END_REF][START_REF] Essafi | First-order phase transitions in polymerized phantom membranes[END_REF][START_REF] Essafi | Nonperturbative Renormalization Group Approach to Polymerized Membranes[END_REF][START_REF] Coquand | Flat phase of quantum polymerized membranes[END_REF][START_REF] Coquand | Glassy phase in quenched disordered crystalline membranes[END_REF], as well as the wetting transition [START_REF] Jakubczyk | Capillary-wave models and the effective-average-action scheme of functional renormalization group[END_REF][START_REF] Jakubczyk | Quantum interface unbinding transitions[END_REF][START_REF] Jakubczyk | Quantum wetting transitions in two dimensions: An alternative path to non-universal interfacial singularities[END_REF].

Critical properties have been studied within the DE not only in scalar theories but also in fermionic models (as well as models presenting both fermions and scalars) [START_REF] Hofling | Phase transition and critical behavior of the D = 3 Gross-Neveu model[END_REF][START_REF] Jaeckel | Flow equations without mean field ambiguity[END_REF][START_REF] Wetterich | Bosonic effective action for interacting fermions[END_REF][START_REF] Gies | Phase transition and critical behavior of d=3 chiral fermion models with left/right asymmetry[END_REF][START_REF] Gies | UV fixed-point structure of the three-dimensional Thirring model[END_REF][START_REF] Braun | Asymptotic safety: a simple example[END_REF][START_REF] Scherer | Renormalization Group Study of Magnetic Catalysis in the 3d Gross-Neveu Model[END_REF][START_REF] Janssen | Critical behavior of the (2+1)-dimensional Thirring model[END_REF][START_REF] Scherer | Many-flavor Phase Diagram of the (2+1)d Gross-Neveu Model at Finite Temperature[END_REF][START_REF] Strack | Fluctuations of imbalanced fermionic superfluids in two dimensions induce continuous quantum phase transitions and non-Fermi liquid behavior[END_REF][START_REF] Janssen | Antiferromagnetic critical point on graphene's honeycomb lattice: A functional renormalization group approach[END_REF][START_REF] Bauer | Nonperturbative renormalization group calculation of quasiparticle velocity and dielectric function of graphene[END_REF][START_REF] Gehring | Fixed-point structure of low-dimensional relativistic fermion field theories: Universality classes and emergent symmetry[END_REF][START_REF] Knorr | Ising and Gross-Neveu model in next-to-leading order[END_REF][START_REF] Janssen | Phase diagram of electronic systems with quadratic Fermi nodes in 2 < d < 4: 2 + expansion, 4expansion, and functional renormalization group[END_REF][START_REF] Ihrig | Critical behavior of Dirac fermions from perturbative renormalization[END_REF]. This includes the analysis of some properties of graphene [START_REF] Janssen | Antiferromagnetic critical point on graphene's honeycomb lattice: A functional renormalization group approach[END_REF][START_REF] Bauer | Nonperturbative renormalization group calculation of quasiparticle velocity and dielectric function of graphene[END_REF].

Nonuniversal properties

The FRG approach allows one to compute nonuniversal properties, in particular in the case of first-order phase transitions (including the problem of bubble nucleation) [START_REF] Delamotte | Nonperturbative renormalization-group approach to frustrated magnets[END_REF][START_REF] Litim | Scheme independence at first order phase transitions and the renormalization group[END_REF][START_REF] Seide | Equation of state near the endpoint of the critical line[END_REF][START_REF] Debelhoir | Simulating frustrated magnetism with spinor Bose gases[END_REF][START_REF] Debelhoir | First-order phase transitions in spinor Bose gases and frustrated magnets[END_REF][START_REF] Tissier | A Nonperturbative approach of the principal chiral model between two-dimensions and fourdimensions[END_REF][START_REF] Delamotte | Functional renormalization group approach to noncollinear magnets[END_REF][START_REF] Berges | Equation of state and coarse grained free energy for matrix models[END_REF][START_REF] Berges | Coarse graining and first order phase transitions[END_REF][START_REF] Strumia | A consistent calculation of bubble-nucleation rates[END_REF][START_REF] Strumia | The region of validity of homogeneous nucleation theory[END_REF][START_REF] Strumia | Bubble-nucleation rates for radiatively induced first-order phase transitions[END_REF][START_REF] Strumia | Testing nucleation theory in two dimensions[END_REF][START_REF] Tetradis | Renormalization-group study of weakly first-order phase transitions[END_REF][START_REF] Munster | Comparison of two methods for calculating nucleation rates[END_REF][START_REF] Tissier | Frustrated Heisenberg Magnets: A Nonperturbative Approach[END_REF][START_REF] Qin | Nonperturbative renormalization group for the Landau-de Gennes model[END_REF] as discussed in detail for frustrated magnets in Sec. 3.2.1.

A related topic is the description of nontrivial field configurations such as instantons. Even if the DE can reproduce in some cases the main consequences of the existence of topological excitations (e.g. vortices in the two-dimensional XY model or solitons in the sine-Gordon model, see Sec. 2.3), it does not seem to yield in general a proper quantitative description of nontrivial classical field configurations. For instance, it fails to reproduce all the features of the quantum-mechanical tunneling [START_REF] Kapoyannis | Quantum mechanical tunneling and the renormalization group[END_REF][START_REF] Zappala | Improving the renormalization group approach to the quantum mechanical double well potential[END_REF]; this would likely require to retain the full momentum dependence of correlation functions. 20We pointed out in Sec. 2.5 that the FRG approach can predict phase diagrams and transition temperatures provided that the precise form of the microscopic action is used as the initial condition of the flow equations. 21 This idea was used to determine the equation of state (and, possibly, the transition temperature to the low-temperature phase) of various classical and quantum fluids [START_REF] Parola | Liquid state theory and critical phenomena[END_REF][START_REF] Seide | Equation of state near the endpoint of the critical line[END_REF][START_REF] Ranc ¸on | Quantum criticality of a Bose gas in an optical lattice near the Mott transition[END_REF][START_REF] Ranc ¸on | Thermodynamics of a Bose gas near the superfluid-Mott-insulator transition[END_REF][START_REF] Bergerhoff | Phase diagram of superconductors[END_REF][START_REF] Bergerhoff | Phase transition of N component superconductors[END_REF][START_REF] Caillol | Non-perturbative renormalization group for simple fluids[END_REF][START_REF] Tarjus | Hierarchical reference theory of critical fluids in disordered porous media[END_REF][START_REF] Boettcher | Critical temperature and superfluid gap of the Unitary Fermi Gas from Functional Renormalization[END_REF][START_REF] Ranc ¸on | Universal thermodynamics of a two-dimensional Bose gas[END_REF].

Conformal invariance and c-theorem

Most FRG studies of critical phenomena do not consider the possible role of the conformal invariance in the critical regime. In recent years, this issue has been the subject of many publications. The implementation of conformal symmetry in the framework of FRG equations, together with the modified Ward identities due to the regularization of the theory in the infrared [START_REF] Ellwanger | Flow equations and BRS invariance for Yang-Mills theories[END_REF], has been considered in Refs. [START_REF] Delamotte | Scale invariance implies conformal invariance for the three-dimensional Ising model[END_REF][START_REF] Codello | Functional perturbative RG and CFT data in the -expansion[END_REF][START_REF] Codello | The Renormalization Group and Weyl-invariance[END_REF][START_REF] Codello | Functional and Local Renormalization Groups[END_REF][START_REF] Rosten | A Conformal Fixed-Point Equation for the Effective Average Action[END_REF][START_REF] Pagani | Products of composite operators in the exact renormalization group formalism[END_REF][START_REF] Polsi | Exact critical exponents for vector operators in the 3d Ising model and conformal invariance[END_REF][START_REF] Morris | Trace anomaly and infrared cutoffs[END_REF][START_REF] Rosten | On Functional Representations of the Conformal Algebra[END_REF][START_REF] Rosten | A Wilsonian Energy-Momentum Tensor[END_REF][START_REF] Sonoda | Conformal invariance for Wilson actions[END_REF][START_REF] Sonoda | Construction of the Energy-Momentum Tensor for Wilson Actions[END_REF][START_REF] Polsi | Conformal invariance and vector operators in the O(N) model[END_REF]. In particular, this has led to a proof that the critical point in the Ising-and O(N)-model universality classes is conformal invariant [START_REF] Delamotte | Scale invariance implies conformal invariance for the three-dimensional Ising model[END_REF][START_REF] Polsi | Conformal invariance and vector operators in the O(N) model[END_REF]. Closely related to the conformal symmetry, the trace of the energy-momentum tensor has been analyzed in the context of FRG regularizations [START_REF] Morris | Trace anomaly and infrared cutoffs[END_REF]. The structure of the Operator Product Expansion in the FRG context has been studied in Ref. [START_REF] Pagani | Operator product expansion coefficients in the exact renormalization group formalism[END_REF].

The global structure of the flow has also been studied by constructing quantities that behave as C-functions [START_REF] Zomolodchikov | Irreversibility" of the flux of the renormalization group in a 2D field theory[END_REF] in various approximations of FRG equations [START_REF] Bacsó | c-function and central charge of the sine-Gordon model from the non-perturbative renormalization group flow[END_REF][START_REF] Generowicz | C function representation of the local potential approximation[END_REF][START_REF] Codello | A functional RG equation for the c-function[END_REF].

Disordered systems

Impurities, defects and other types of imperfections are ubiquitous in realistic many-body systems. Such a random disorder can prevent phases of matter from forming, change the critical properties of a phase transition, etc. However disorder is also responsible for a variety of novel phenomena that often have no counterparts in clean systems: metastability, non-ergodicity, pinning, avalanches, chaotic behavior, slow dynamics and aging, etc.

Replica formalism and FRG

When the impurities have a relaxation time much larger than the typical time of the experiment, the disorder is said to be quenched and can be modeled by a static random field h(x) coupling to the density of particles or the order parameter field, etc. 22 W h [J] = ln Z[J; h] becomes a functional of both the external source J and the random field h (Z[J; h] is the partition function for a given random field h). The disorder averaged value of many observables (e.g. the order parameter or the susceptibility) depends on the mean free energy W 1 [J] = W h [J] (an overline indicates disorder averaging). However to fully characterize the disorder effect, in particular in the absence of self-averaging [START_REF] Aharony | Absence of Self-Averaging and Universal Fluctuations in Random Systems near Critical Points[END_REF], one also needs to consider higher-order cumulants W i of the random functional W h [J] and in particular the second-order one,

W 2 [J a , J b ] = W h [J a ]W h [J b ] -W h [J a ] W h [J b ]. ( 23
)
Note that the ith-order cumulant depends on i distinct sources: J a 1 • • • J a i . This allows one to obtain all physical observables of interest by taking functional derivatives of the W i 's and then setting the sources equal to zero. The standard method to compute the cumulants W i is to consider n copies (or replicas) of the system with the same random field h but with distinct sources J a . The partition function then reads where the functional W[{J a }] can be expanded in free replica sums,

e W[{J a }] = n a=1 Z[J a ; h] = exp        n a=1 W h [J a ]        , (24)
W[{J a }] = n a=1 W 1 [J a ] + 1 2 n a,b=1 W 2 [J a , J b ] + • • • (25)
the W i 's being the cumulants introduced above.

In the FRG formalism, the quantity of interest is the effective action Γ[{φ a }] defined as the Legendre transform of W[{J a }] where φ a (x) = δW[J]/δJ a (x) [START_REF] Tissier | Nonperturbative functional renormalization group for random field models and related disordered systems. iii. superfield formalism and ground-state dominance[END_REF]. Γ[{φ a }] can be expanded in free replica sums,

Γ[{φ a }] = n a=1 Γ 1 [φ a ] - 1 2 n a,b=1 Γ 2 [φ a , φ b ] + • • • (26)
Γ 1 is the Legendre transform of W 1 and contains all information on the average order parameter, susceptibility, etc. Γ 2 is simply related to W 2 and encodes the effective disorder correlator h(x)h(x ).

An essential feature of disordered systems is that the disorder correlator, W 2 [J a , J b ] or Γ 2 [φ a , φ b ], may assume a nonanalytic (cuspy) functional form that encodes the existence of metastable states and the ensuing glassy properties of the system [START_REF] Balents | The Large Scale Energy Landscape of Randomly Pinned Objects[END_REF]. In order to take into account such nonanalyticities, it is necessary to implement a RG approach that retains the functional form of the "cumulants"

Γ i [φ a 1 • • • φ a i ].
Perturbative implementations of the FRG in disordered systems have a long history [START_REF] Fisher | Random fields, random anisotropies, nonlinear σ models, and dimensional reduction[END_REF][START_REF] Narayan | Dynamics of sliding charge-density waves in 4-dimensions[END_REF][START_REF] Nattermann | Dynamics of interface depinning in a disordered medium[END_REF][START_REF] Chauve | Renormalization of Pinned Elastic Systems: How Does It Work Beyond One Loop?[END_REF][START_REF] Le Doussal | Functional renormalization group and the field theory of disordered elastic systems[END_REF][START_REF] Tarjus | Nonperturbative Functional Renormalization Group for Random-Field Models: The Way Out of Dimensional Reduction[END_REF][START_REF] Balents | Localization of elastic layers by correlated disorder[END_REF][START_REF] Fisher | Interface fluctuations in disordered systems: 5expansion and failure of dimensional reduction[END_REF][START_REF] Balents | Large-n expansion of (4-)-dimensional oriented manifolds in random media[END_REF][START_REF] Narayan | Critical behavior of sliding charge-density waves in 4-dimensions[END_REF][START_REF] Chauve | Creep and depinning in disordered media[END_REF][START_REF] Feldman | Critical exponents of the random-field o(N) model[END_REF][START_REF] Doussal | Exact results and open questions in first principle functional RG[END_REF]. In the following section, we focus on the three-dimensional randomfield Ising model, a system that is far from the upper critical dimension d uc = 6 and where nonperturbative effects are crucial (see [START_REF] Tarjus | Random-field Ising and O(N) models: theoretical description through the functional ren ormalization group[END_REF] for a recent review).

The Random Field Ising Model

The random-field Ising model (RFIM) [START_REF] Imry | Random-field instability of the ordered state of continuous symmetry[END_REF] is defined by the Hamiltonian

H = -J i j S i S j - i S i h i (J > 0), (27)
where the quenched disorder appears through a random magnetic field h i which is Gaussian distributed, with zero mean and variance ∆. This model is used to describe, among others, diluted antiferromagnets in a homogeneous field [START_REF] Cardy | Random-field effects in site-disordered Ising antiferromagnets[END_REF] and the critical point of a fluid adsorbed in a porous matrix [START_REF] Brochard | Phase transitions of binary mixtures in random media[END_REF]. It is characterized by two control parameters: the temperature (or equivalently the exchange J) and the variance of the disorder. The phase diagram is shown in Fig. 7. The temperature is known to be a dangerously irrelevant term, which flows to zero like a power law at the fixed point describing the critical physics of the model. This indicates that sample-to-sample fluctuations dominate the thermal fluctuations. As a consequence, the transition can be studied at vanishing temperature, by changing the variance of the disorder.

The critical behavior of the phase transition in the RFIM can also be studied from the field theory defined by the action

S [ϕ] = ˆdd x 1 2 (∂ µ ϕ) 2 + r 0 2 ϕ 2 + u 0 4! ϕ 4 -hϕ , (28)
where h(x) is a Gaussian distributed random field. Perturbative calculations performed in the 70's on the RFIM unveiled the very surprising property of dimensional reduction: The critical exponents of the RFIM are identical, to all orders of perturbation theory, to those of the pure Ising model in 2 dimensions less [START_REF] Grinstein | Ferromagnetic phase transitions in random fields: The breakdown of scaling laws[END_REF][START_REF] Aharony | Comments on the critical behavior of random systems[END_REF][START_REF] Young | On the lowering of dimensionality in phase transitions with random fields[END_REF]. This result was first derived by comparing Feynman diagrams in the two theories. It was later understood that dimensional reduction is a consequence of a hidden (super)symmetry [START_REF] Parisi | Random magnetic fields, supersymmetry, and negative dimensions[END_REF]. Surprisingly for a result true to all orders in perturbation theory, dimensional reduction is invalid in three dimensions [START_REF] Imbrie | Lower critical dimension of the random-field ising model[END_REF][START_REF] Bricmont | Lower critical dimension for the random-field ising model[END_REF][START_REF] Bricmont | Phase transition in the 3d random field ising model[END_REF]. The failure of dimensional reduction was expected to be related to the existence of many metastable states, which were overlooked in the supersymmetric construction of [START_REF] Parisi | Random magnetic fields, supersymmetry, and negative dimensions[END_REF]. Moreover, in several other disordered systems, such as random manifolds pinned by impurities, it was shown that the existence of these many metastable states can lead to nonanalycities in the vertex functions [START_REF] Nattermann | Dynamics of interface depinning in a disordered medium[END_REF][START_REF] Balents | The Large Scale Energy Landscape of Randomly Pinned Objects[END_REF][START_REF] Fisher | Interface fluctuations in disordered systems: 5expansion and failure of dimensional reduction[END_REF][START_REF] Narayan | Critical behavior of sliding charge-density waves in 4-dimensions[END_REF][START_REF] Feldman | Critical exponents of the random-field o(N) model[END_REF]. This possibility was not considered in the diagrammatic derivation performed in [START_REF] Grinstein | Ferromagnetic phase transitions in random fields: The breakdown of scaling laws[END_REF][START_REF] Young | On the lowering of dimensionality in phase transitions with random fields[END_REF][START_REF] Aharony | Lowering of dimensionality in phase transitions with random fields[END_REF] and is probably at the origin of the invalid prediction of dimensional reduction.

The strategy followed to study the RFIM in the nonperturbative FRG approach is standard [START_REF] Tissier | Two-loop functional renormalization group of the random field and random anisotropy O(N) models[END_REF][START_REF] Tissier | Unified picture of ferromagnetism, quasi-long-range order, and criticality in random-field models[END_REF][START_REF] Tarjus | Nonperturbative functional renormalization group for random field models and related disordered systems. i. effective average action formalism[END_REF][START_REF] Tissier | Nonperturbative functional renormalization group for random field models and related disordered systems. II. Results for the random field O(N) model[END_REF][START_REF] Tissier | Supersymmetry and its spontaneous breaking in the random field ising model[END_REF][START_REF] Tissier | Nonperturbative functional renormalization group for random field models and related disordered systems. iv. supersymmetry and its spontaneous breaking[END_REF][START_REF] Tarjus | Critical scaling in random-field systems: 2 or 3 independent exponents?[END_REF]. We first introduce a regulator in the construction described in Sec. We then propose an ansatz for this quantity and derive the approximate flow equations. Before embarking in this discussion, we have to ensure that supersymmetry, which is responsible for dimensional reduction is not explicitly broken. In this respect, dimensional regularization which is often used in perturbative approaches is very powerful because it satisfies automatically many symmetries, may they be explicit or hidden. On the contrary, the regulator introduced in the FRG may break some symmetries and we have to be cautious about that. 23In Refs. [START_REF] Tissier | Nonperturbative functional renormalization group for random field models and related disordered systems. iii. superfield formalism and ground-state dominance[END_REF][START_REF] Tissier | Nonperturbative functional renormalization group for random field models and related disordered systems. iv. supersymmetry and its spontaneous breaking[END_REF], it was shown that the nonperturbative FRG can indeed reproduce supersymmetry and dimensional reduction, if appropriate truncations are performed. Supersymmetry shows up as relations between Γ i and Γ i+1 , which are preserved by the renormalization-group flow, under the assumptions that the vertex functions are sufficiently regular. The simplest of these relations reads

Γ (1,1) 2 [q 2 ; φ, φ] = ∆ B ∂ q 2 Γ (2) 1 [q 2 ; φ] (29)
where the superscripts indicate functional derivatives. A similar property must be fulfilled by the regulating term ∆S k [Eq. ( 4)]. This implies that we need to introduce a regulator R in the 1-replica part and a related regulator R in the 2-replica part.

The minimal truncation that preserves this symmetry consists in neglecting Γ i when i > 2 and assuming

Γ 1,k [φ] = ˆdd x U k (φ) + 1 2 Z k (φ)(∂ µ φ) 2 , (30)
Γ 2,k [φ 1 , φ 2 ] = ˆdd x V k (φ 1 , φ 2 ), (31)
which amounts to implementing the DE to second order in Γ 1,k and to LPA order in Γ 2,k . The flow equations for the three functions appearing in this truncation have been integrated numerically. For space dimensions larger than d dr ∼ 5.13, the flow attains a fixed point after fine tuning the initial condition. During the whole flow, the functions U k (φ) and Z k (φ) remain equal to the associated functions in the pure Ising model in 2 dimensions less, computed at the same level of truncation (up to unimportant multiplicative factors). The critical exponents fulfill the dimensional reduction property. For lower dimensions, a nonanalyticity appears at a finite RG scale, called the Larkin length. Below this scale, V (11) k (φ a , φ b ) develops a cuspy behavior ∝ |φ a -φ b |, the flow of U k and Z k depart from those of the pure Ising model in two dimension less and the fixed point does not present dimensional reduction. The appearance of such a cusp is common to many systems governed by a zero-temperature fixed point. It is associated with the presence of avalanches, or shocks, which correspond to reorganizations of the ground state on large scales when the external parameters are slightly modified.

This change of behavior in dimension is depicted in Fig. 8, where the two anomalous dimensions η and η, associated with the long-distance power-law decay of the correlation functions,

ϕ(x)ϕ(y) -ϕ(x) ϕ(y) ∼ 1 |x -y| d-2+η , (32) ϕ
(x) ϕ(y) -ϕ(x) ϕ(y) ∼ 1 |x -y| d-4+η , (33)
are shown as a function of the dimension. The agreement with lattice simulations is very good. The picture emerging from the nonperturbative FRG approach is that dimensional reduction is broken in low enough dimensions but is realized in high dimensions, in particular in the vicinity of the upper critical dimension 6. This may seem surprising at first sight because there is no qualitative difference between these two regimes, in what concerns the metastable states which are responsible for the breaking of supersymmetry and dimensional reduction. A criterion was derived in [START_REF] Tarjus | Avalanches and dimensional reduction breakdown in the critical behavior of disordered systems[END_REF] to quantify under which conditions metastable states destabilize the dimensional reduction property.

Other works in the same line aimed at studying the universality class of the RFIM in presence of long range interactions and/or correlations of the disorder [START_REF] Balog | Critical behaviour of the random-field ising model with long-range interactions in one dimension[END_REF][START_REF] Baczyk | Dimensional reduction and its breakdown in the three-dimensional long-range random-field ising model[END_REF]. The RFIM presents also interesting dynamical properties, which were studied both at equilibrium [START_REF] Balog | Activated dynamic scaling in the random-field ising model: A nonperturbative functional renormalization group approach[END_REF] and out-of-equilibrium [START_REF] Balog | Same universality class for the critical behavior in and out of equilibrium in a quenched random field[END_REF][START_REF] Balog | Criticality of the random field ising model in and out of equilibrium: A nonperturbative functional renormalization group description[END_REF].

Other disordered systems

The perturbative FRG has been used to study many disordered systems, in particular random manifolds pinned by impurities, at and out of equilibrium [START_REF] Nattermann | Dynamics of interface depinning in a disordered medium[END_REF][START_REF] Le Doussal | Functional renormalization group and the field theory of disordered elastic systems[END_REF][START_REF] Balents | The Large Scale Energy Landscape of Randomly Pinned Objects[END_REF][START_REF] Balents | Localization of elastic layers by correlated disorder[END_REF][START_REF] Fisher | Interface fluctuations in disordered systems: 5expansion and failure of dimensional reduction[END_REF][START_REF] Balents | Large-n expansion of (4-)-dimensional oriented manifolds in random media[END_REF][START_REF] Narayan | Critical behavior of sliding charge-density waves in 4-dimensions[END_REF][START_REF] Chauve | Creep and depinning in disordered media[END_REF][START_REF] Feldman | Critical exponents of the random-field o(N) model[END_REF].

Other works used the nonperturbative aspect of Wetterich's equation [START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF]. The random elastic manifold model in and out of equilibrium is studied in [START_REF] Balog | Benchmarking the nonperturbative functional renormalization group approach on the random elastic manifold model in and out of equilibrium[END_REF]. The authors of Ref. [START_REF] Tissier | Randomly dilute Ising model: A nonperturbative approach[END_REF] considered the random bond model where the disorder appears as an inhomogeneity of the exchange term (or, equivalently, of the ϕ 2 term in the continuum field theory [START_REF] Nelson | Momentum-shell recursion relations, anisotropic spins, and liquid crystals in 2 + dimensions[END_REF]). In Ref. [START_REF] Biroli | Role of fluctuations in the phase transitions of coupled plaquette spin models of glasses[END_REF], a first attempt to study the spin-glass to paramagnetic transition was performed. The authors of Ref. [START_REF] Coquand | Glassy phase in quenched disordered crystalline membranes[END_REF] studied the influence of disorder on graphene and crystalline membranes and a glass phase was identified. The disordered Bose fluid is discussed in [START_REF] Dupuis | Glassy properties of the Bose-glass phase of a one-dimensional disordered Bose fluid[END_REF][START_REF] Dupuis | Bose-glass phase of a one-dimensional disordered bose fluid: Metastable states, quantum tunneling, and droplets[END_REF][START_REF] Dupuis | Is there a mott-glass phase in a one-dimensional disordered quantum fluid with linearly confining interactions?[END_REF].

Classical non-equilibrium systems and FRG

In this section, we mostly consider classical systems out of equilibrium. The FRG formalism has also been developed for nonequilibrium quantum systems where it has led to fruitful applications (see e.g. [START_REF] Schoeller | Real-time renormalization group and charge fluctuations in quantum dots[END_REF][START_REF] Jakobs | Nonequilibrium Functional Renormalization Group for Interacting Quantum Systems[END_REF][START_REF] Gasenzer | Towards far-from-equilibrium quantum field dynamics: A functional renormalisation-group approach[END_REF][START_REF] Pietroni | Flowing with Time: a New Approach to Nonlinear Cosmological Perturbations[END_REF][START_REF] Berges | Nonthermal fixed points and the functional renormalization group[END_REF][START_REF] Berges | Introduction to the nonequilibrium functional Renormalization Group[END_REF][407]. We also refer the reader to Secs. 3.4.3 and 4.2.4 of this review.

In a classical system, thermodynamics and dynamics are decoupled, which means that the dynamics has to be specified through an equation of motion. In statistical mechanics, there are two standard ways of defining a (stochastic) dynamics: a microscopic description via a master equation or an effective approach based on a Langevin type of equation. Both descriptions can be recast into a path integral formulation, adapted to field-theoretical treatments. Systems of both classes have been studied within the FRG framework. In the following, we put emphasis on the four most studied and paradigmatic examples, which we present in details, while other works are only briefly reviewed.

Langevin stochastic dynamics

A Langevin equation (in a broad sense) describes the time evolution of a mesoscopic (coarse-grained) field φ(t, x), subjected to some microscopic noise. For simplicity, we consider a scalar field, the generalization to more complex situations is straightforward. The Langevin equation has the generic form

∂ t φ(t, x) = -F [φ(t, x)] + η(t, x) , (34)
where the functional F [φ] encompasses the deterministic part of the dynamics and η models the randomness. This noise is in general chosen with a Gaussian probability distribution, of zero mean η = 0 and variance

η(t, x)η(t , x ) = 2D[φ(t, x)]δ(t -t )δ d (x -x ) ( 35
)
where D is a (possibly field-dependent) diffusion coefficient 24 . This form can be generalized to include spatial or temporal correlations of the microscopic noise. The Langevin equation ( 34) can be cast into a field theory using the Martin-Siggia-Rose-Janssen-de Dominicis (MSRJD) formalism [START_REF] Martin | Statistical dynamics of classical systems[END_REF][START_REF] Janssen | On a lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties[END_REF][START_REF] De Dominicis | Techniques de renormalisation de la théorie des champs et dynamique des phénomènes critiques[END_REF]. This procedure involves the introduction of a response field φ, and yields an action with the general structure

S[φ, φ] = ˆt,x φ ∂ t φ + F [φ] -φD[φ] φ , (36)
where the deterministic part of the dynamics is linear in φ and the noise correlator is encoded in the quadratic term in φ.

a-FRG formalism for Langevin stochastic dynamics

The equilibrium FRG formalism can be simply extended to study classical nonequilibrium systems, by including the additional response fields, and taking into account Ito's prescription and causality issues [START_REF] Canet | General framework of the non-perturbative renormalization group for non-equilibrium steady states[END_REF][START_REF] Canet | Nonperturbative renormalization-group study of reaction-diffusion processes[END_REF][START_REF] Benitez | Branching and annihilating random walks: Exact results at low branching rate[END_REF]. The minimal FRG scheme consists in implementing only a spatial coarse-graining, which amounts to adding to the action a quadratic scale-dependent term which depends on momentum but not on frequency, of the form

∆S k = 1 2 ˆω,q Φ i [R k ] i j (q)Φ j (37)
where Φ has component Φ 1 = φ and Φ 2 = φ. In most cases, the integrals in the frequency domain are convergent, and this scheme suffices to achieve the progressive integration of the fluctuation modes and to regularize the theory.

A frequency regularization has been implemented up to now only in [START_REF] Duclut | Frequency regulators for the nonperturbative renormalization group: A general study and the model a as a benchmark[END_REF]. The effective average action Γ k is then defined as usual and its exact flow equation reads

∂ s Γ k = 1 2 Tr ˆω,q ∂ s R k • Γ (2) k + R k -1 . ( 38
)
with ∂ s ≡ k∂ k 25 , and where R k and Γ (2) k are 2 × 2 matrices. This formalism has been used to study relaxation spin models (Model A, reviewed below, and Model C [START_REF] Mesterházy | Dynamic universality class of model c from the functional renormalization group[END_REF] in the classification by Hohenberg and Halperin [START_REF] Hohenberg | Theory of dynamic critical phenomena[END_REF]), and famous non-linear partial differential equations such as the Kardar-Parisi-Zhang equation and the stochastic Navier-Stokes equation (see below), or Burgers equation [START_REF] Mathey | Anomalous scaling at nonthermal fixed points of Burgers' and gross-pitaevskii turbulence[END_REF].

b-Critical Dynamics: Model A 24 The Langevin equation is written here using the Ito discretization for time derivatives. 25 The notation s is used in this section for the RG "time" since t denotes the physical time.

Model A describes the purely dissipative relaxation towards equilibrium of a non-conserved scalar order parameter φ(t, x) with Ising symmetry. The model is defined by the Langevin equation [START_REF] Chaikin | Principles of Condensed Matter Physics[END_REF] with a simple additive noise D[φ] ≡ D. The functional F ≡ D δH δφ derives from the equilibrium Ising Hamiltonian H where the coefficient D is identical to the amplitude of the noise in order to fulfill Einstein's relation, and ensure that the system evolves towards the equilibrium state. According to Eq. [START_REF] Chang | Differential renormalization-group generators for static and dynamic critical phenomena[END_REF], the action of model A is then given by

S[φ, φ] = ˆt,x φ ∂ t φ -D∇ 2 φ + DV (φ) -D φ2 , ( 39
)
where V is the φ 4 potential. When approaching the continuous phase transition of the model, the relaxation time of the order parameter diverges. This behavior reflects the critical slowing down of the dynamics near the critical point, characterized by the dynamical exponent z which relates the divergence of the relaxation time τ to the divergence of the correlation length ξ in the vicinity of the critical point as τ ∼ ξ z ∼ |T -T c | -zν . The value of the exponent z depends on the precise relaxation mechanism and differs in the different models A, B, C . . . . For model A, its mean field value is z = 2. The classical action (39) possesses a time-reversal symmetry (in the long-time limit), that can be expressed as an invariance of the action (39) under the following field transformation [START_REF] Andreanov | Dynamical field theory for glass-forming liquids, self-consistent resummations and time-reversal symmetry[END_REF][START_REF] Canet | A non-perturbative approach to critical dynamics[END_REF]

         t → -t φ → φ φ → φ -1 D ∂ t φ. (40)
The time-reversal symmetry can also be expressed as a supersymmetry [START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF], and the corresponding supersymmetric FRG formalism for Model A is expounded in Ref. [START_REF] Canet | General framework of the non-perturbative renormalization group for non-equilibrium steady states[END_REF]. The consequence of this time-reversal symmetry (or supersymmetry) is the existence of a fluctuation-dissipation theorem, which yields the decoupling of the statics and of the dynamics. The static critical exponents ν and η for Model A are thus those of the equilibrium Ising model. This model has been studied within the second order of DE (DE 2 ). The minimal regularization, which is a straightforward extension of the static case, consists in introducing only off-diagonal and frequency-independent cutoff terms [R k ] 12 (q) = [R k ] 21 (q) in [START_REF] Parola | Liquid state theory and critical phenomena[END_REF]. If one also implements a frequency regularization, then a diagonal cutoff term, satisfying [R k ] 12 (t, x) = 2θ(-t)∂ t [R k ] 22 (t, x) must be included in order to preserve the time-reversal symmetry [START_REF] Parola | Liquid-State Theory for Critical Phenomena[END_REF] and causality [START_REF] Duclut | Frequency regulators for the nonperturbative renormalization group: A general study and the model a as a benchmark[END_REF]. The symmetry (40) also imposes constraints on the ansatz for Γ k , which must have the following structure [START_REF] Canet | A non-perturbative approach to critical dynamics[END_REF] Γ

k [ϕ, φ] = ˆx,t X k (ϕ) φ ∂ t ϕ -φ2 + φ U k (ϕ) -Z k (ϕ) ∇ 2 ϕ - 1 2 ∂ ϕ Z k (ϕ)(∇ϕ) 2 . ( 41
)
where the time derivative and quadratic term in φ are renormalised in the same way and the rest derives from the standard ansatz for the equilibrium Ising model at DE 2 . At the bare level, X Λ = 1/D. The flow equations for Model A have been integrated numerically at different orders for the dynamical parts, referred to as LPA, LPA and DE 2 , which all correspond to the complete DE 2 for the static part, and respectively X k = 1 (LPA), X k (ρ) = X k (LPA ), and X k (ρ) (DE 2 ). At DE 2 , the integration has been performed both with and without frequency regularization (using different frequency regulators) [START_REF] Duclut | Frequency regulators for the nonperturbative renormalization group: A general study and the model a as a benchmark[END_REF]. As an illustration, the results obtained for the critical exponents in d = 3 in the different schemes are reported in Table 3, and compared with the best estimates available in the literature. The agreement is satisfactory, and the frequency regularization turns out to improve the results. The critical exponents of Model A have also been obtained in d = 2 and for N = 2 and 3 [START_REF] Duclut | Frequency regulators for the nonperturbative renormalization group: A general study and the model a as a benchmark[END_REF]. The relaxational critical dynamics of a nonconserved order parameter coupled to a conserved scalar density, which corresponds to Model C, has been studied in [START_REF] Mesterházy | Dynamic universality class of model c from the functional renormalization group[END_REF]. The properties of the phase diagram for the dynamic critical behavior are determined. In this work, different scaling regimes corresponding to different critical exponents are identified and characterized.

c-Kardar-Parisi-Zhang equation

A prominent example of non-equilibrium and non-linear Langevin dynamics is the Kardar-Parisi-Zhang (KPZ) equation. It was originally derived to describe the critical roughening of stochastically growing interfaces, whose [START_REF] Canet | A non-perturbative approach to critical dynamics[END_REF] and DE 2 without a or with b frequency regularization [START_REF] Duclut | Frequency regulators for the nonperturbative renormalization group: A general study and the model a as a benchmark[END_REF]), compared with results from other field theoretical methods (FT) [START_REF] Guida | Critical exponents of the N-vector model[END_REF][START_REF] Krinitsyn | Calculations of the dynamical critical exponent using the asymptotic series summation method[END_REF] and Monte Carlo simulations (MC) [START_REF] Hasenbusch | Finite size scaling study of lattice models in the three-dimensional Ising universality class[END_REF][START_REF] Ito | Nonequilibrium relaxation of fluctuations of physical quantities[END_REF][START_REF] Grassberger | Damage spreading and critical exponents for "model a" ising dynamics[END_REF].

d = 3 ν η z LPA 0.
dynamics was modeled by [START_REF] Kardar | Dynamic scaling of growing interfaces[END_REF] ∂h(t, x)

∂t = ν ∇ 2 h(t, x) + λ 2 ∇h(t, x) 2 + η(t, x) (42)
where the diffusion term provides the interface with a smoothening mechanism, and the non-linear term encompasses an enhanced growth along the local normal to the surface, and leads to critical roughening. Note that the deterministic part of Eq. ( 42) is not the functional derivative of some Hamiltonian, and the amplitude of the noise D is no longer related to the diffusion part, such that this dynamics leads to intrinsically non-equilibrium steady-states. In fact, a remarkable feature of the KPZ equation is generic scale invariance, also termed self-organized criticality, in any dimension. For d ≤ 2, the interface always becomes rough (and critical) as it grows, without fine-tuning any control parameter, contrary to usual equilibrium phase transitions. The rough interface is characterized by two universal critical exponents, the roughening exponent χ and the dynamical exponent z. Due to its simplicity, KPZ universality class arises in connection with an extremely large class of nonequilibrium or disordered systems, such as randomly stirred fluids (Burgers equation) [START_REF] Forster | Large-distance and long-time properties of a randomly stirred fluid[END_REF], directed polymers in random media [START_REF] Kardar | Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities[END_REF], or even driven-dissipative Bose-Einstein condensation [407,[START_REF] Squizzato | Kardar-Parisi-Zhang universality in the phase distributions of one-dimensional exciton-polaritons[END_REF], to cite only a few. The KPZ equation has thereby emerged as one of the fundamental theoretical models to investigate nonequilibrium scaling phenomena and phase transitions [START_REF] Halpin-Healy | Kinetic roughening phenomena, stochastic growth, directed polymers and all that. aspects of multidisciplinary statistical mechanics[END_REF], and has witnessed an intense renewal of interests in recent years. Indeed, a considerable breakthrough has been achieved in the last decade regarding the characterization of the KPZ universality class in 1+1 dimensions, sustained by a wealth of exact results [START_REF] Corwin | The Kardar-Parisi-Zhang equation and universality classes[END_REF], and by high-precision measurements in turbulent convection of liquid crystals [START_REF] Takeuchi | Universal fluctuations of growing interfaces: Evidence in turbulent liquid crystals[END_REF][START_REF] Takeuchi | Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence[END_REF]. However, in higher dimensions, or in the presence of additional ingredients such as nondelta correlations of the microscopic noise, the integrability of the KPZ equation is broken, and controlled analytical methods to describe the rough phase are scarse. In this context, the FRG has been particularly useful since it is the only method which can access in a controlled way the strong-coupling KPZ fixed-point in any dimension [START_REF] Canet | Nonperturbative renormalization group for the Kardar-Parisi-Zhang equation[END_REF].

Following the MSRJD procedure, the field theory associated with the KPZ equation reads

S[h, h]= ˆt,x h ∂ t h -ν ∇ 2 h - λ 2 (∇h) 2 -D h2 . ( 43
)
This action is invariant under shifts of the height field h → h + c where c is an arbitrary constant. It is also invariant under a Galilean transformation for Burgers' velocity v(t) ∝ ∇h, which corresponds for the height field to an infinitesimal tilt of the interface. This symmetry enforces the exact identity between critical exponents z + χ = 2 in all dimensions [START_REF] Halpin-Healy | Kinetic roughening phenomena, stochastic growth, directed polymers and all that. aspects of multidisciplinary statistical mechanics[END_REF].

In fact, both the Galilean and shift symmetries admit a stronger form, gauged in time [START_REF] Lebedev | Hidden symmetry, exact relations, and a small parameter in the Kardar-Parisi-Zhang problem with strong coupling[END_REF][START_REF] Canet | Nonperturbative Renormalization Group for the Kardar-Parisi-Zhang equation: General framework and first applications[END_REF]. The corresponding transformations read

h (t, x) = x • ∂ t v(t) + h(t, x + λv(t)) h (t, x) = h(t, x + λv(t)) (44)
h (t, x) = h(t, x) + c(t). ( 45
)
where c(t) and v(t) are infinitesimal time-dependent quantities 26 . More precisely, these time-gauged forms are extended symmetries, in the sense that the KPZ action is not strictly invariant under these transformations, but its variation is linear in the fields, and this entails important non-renormalization theorems and general Ward identities [START_REF] Canet | Nonperturbative Renormalization Group for the Kardar-Parisi-Zhang equation: General framework and first applications[END_REF]. Extended symmetries proved as an extremely powerful tool in the context of FRG, as illustrated below.

Furthermore, for a one-dimensional interface, the KPZ equation satisfies an 'accidental' time-reversal symmetry which, as shown in [START_REF] Canet | Nonperturbative Renormalization Group for the Kardar-Parisi-Zhang equation: General framework and first applications[END_REF][START_REF] Canet | Strong-coupling fixed point of the Kardar-Parisi-Zhang equation[END_REF], can be encoded in the discrete transformation

h (t, x) = -h(-t, x) h (t, x) = h(-t, x) + ν D ∇ 2 h(-t, x). ( 46
)
and which fixes the exponents exactly to χ = 1/2 and z = 3/2 in d = 1.

The KPZ field theory was studied within FRG using different approximations [START_REF] Canet | Nonperturbative renormalization group for the Kardar-Parisi-Zhang equation[END_REF][START_REF] Canet | Nonperturbative Renormalization Group for the Kardar-Parisi-Zhang equation: General framework and first applications[END_REF][START_REF] Canet | Strong-coupling fixed point of the Kardar-Parisi-Zhang equation[END_REF]. One first needs to specify the cutoff matrix in [START_REF] Parola | Liquid state theory and critical phenomena[END_REF]. An appropriate choice compatible with the symmetries is [R k ] 12 (q) = [R k ] 21 (q) = ν k q 2 r(q/k) and [R k ] 22 (q) = -2D k r(q/k) where ν k and D k are two running coefficients defined below. Note that the Galilean invariance precludes for implementing a frequency regularization, at least in a simply manageable way. The KPZ equation is an interesting example for which the LPA already goes beyond all-order perturbative RG but where a simple DE scheme is not enough to yield accurate results. Indeed, a rescaling of the time and the height leads to a single KPZ coupling g = λ 2 D/ν 3 . Its β-function was computed to all orders in perturbation [START_REF] Wiese | Critical discussion of the two-loop calculations for the Kardar-Parisi-Zhang equation[END_REF] and it fails to capture the strong-coupling fixed point describing the rough phase in d ≥ 2, whereas the LPA already contains this fixed point in any dimension. However, the critical exponents in d > 1 are poorly determined, even at DE 4 [START_REF] Canet | Strong-coupling fixed point of the Kardar-Parisi-Zhang equation[END_REF]. This is very unusual in the FRG context, and probably originates in the derivative nature of the KPZ vertex. Hence, one needs to resort to an alternative approximation scheme, such as the vertex expansion, in order to better account for the momemtum dependence. However, the application of the BMW approximation scheme, as presented in Sec. 2.4, is hindered by the symmetries for the KPZ problem, and in particular the Galilean one. Indeed, the associated Ward identities relate in a nontrivial way the momentum and frequency dependencies of vertex functions of different orders, preventing the direct BMW expansions of the vertices.

This difficulty has been successfully circumvented [START_REF] Canet | Nonperturbative renormalization group for the Kardar-Parisi-Zhang equation[END_REF][START_REF] Canet | Nonperturbative Renormalization Group for the Kardar-Parisi-Zhang equation: General framework and first applications[END_REF]. The strategy is to build an ansatz which automatically encodes the symmetries while leaving an arbitrary momentum and frequency dependence of the two-point functions. This is achieved by combining together the basic Galilean scalars of the theory -which are h, ∇ i ∇ j h and 2 , and arbitrary powers of their gradients and covariant time derivatives Dt = ∂ t -λ∇h•∇. Truncated at Second Order in the response field, the corresponding SO ansatz reads

D t h = ∂ t h-λ 2 (∇h)
Γ k [ϕ, φ] = ˆt,x φ f λ k D t ϕ -φ f D k φ - ν D φ f ν k ∇ 2 ϕ , (47)
where the three functions f X k , X = ν, D, λ, are functions of Dt and

∇, i.e f X k ≡ f X k (-D2 t , -∇ 2 )
and with ϕ = h and φ = h . Note that arbitrary powers of the field itself are included via the functional dependence in ϕ of the covariant derivative Dt . The gauged-shift symmetry furthermore imposes that f λ k (ω 2 , p 2 = 0) ≡ 1. One defines two running coefficients through the normalization conditions

f ν k (0, 0) = ν k and f D k (0, 0) = D k , which entail the anomalous dimensions η ν k = -∂ s ln ν k and η D k = -∂ s ln D k whose fixed point values are related to the critical exponents as χ = (2 -d -η ν * + η D * )/2 and z = 2 -η ν * .
This ansatz yields extremely accurate results for a one-dimensional interface [START_REF] Canet | Nonperturbative Renormalization Group for the Kardar-Parisi-Zhang equation: General framework and first applications[END_REF]. The numerical integration of the flow equations leads to a fully attractive, non-Gaussian fixed point, which means that the interface always become rough, and critical, as physically observed. As a consequence, one can demonstrate analytically, from the fixed point equation and the decoupling of the non-linear terms at large momentum and frequency, the existence of generic scaling. In particular, one can show that the two-point correlation function endows a scaling form C(ω, p) = p d+2+χ F(ω/p z ), and the scaling function F can be compared to exact results that were obtained in [START_REF] Prähofer | Exact scaling functions for one-dimensional stationary KPZ growth[END_REF], for f (k) = ´∞ 0 dτ cos(τk 3/2 )F(τ) and for its Fourier transform f (y). The comparison, which does not involve any free parameter, is excellent, as shown in Fig. 9. The FRG even captures very fine details of the tail of f (k) featuring a stretched exponential with superimposed oscillations.

Moreover, the FRG approach can be applied in all non-integrable cases. It has been used in particular in a simplified form for interfaces of d > 1 dimensions, where the critical exponents and scaling functions were calculated, and provided their first analytical estimates. It also yielded predictions for dimensionless ratios in d = 2 and 3 [START_REF] Kloss | Nonperturbative Renormalization Group for the stationary Kardar-Parisi-Zhang equation: Scaling functions and amplitude ratios in 1+1, 2+1, and 3+1 dimensions[END_REF] which were later accurately confirmed by large-scale numerical simulations [START_REF] Halpin-Healy | Extremal paths, the stochastic heat equation, and the three-dimensional Kardar-Parisi-Zhang universality class[END_REF][START_REF] Halpin-Healy | Erratum: Extremal paths, the stochastic heat equation, and the three-dimensional Kardar-Parisi-Zhang universality class[END_REF]. This framework was extended [START_REF] Canet | Nonperturbative Renormalization Group for the Kardar-Parisi-Zhang equation: General framework and first applications[END_REF][START_REF] Canet | Erratum: Nonperturbative Renormalization Group for the Kardar-Parisi-Zhang equation: General framework and first applications[END_REF] with the exact ones (black curves with squares) [START_REF] Prähofer | Exact scaling functions for one-dimensional stationary KPZ growth[END_REF]. The inset on the right panel shows the stretched exponential behavior of the tail with the superimposed oscillations, developing on the same scale k 3/2 . Note the vertical scale: this behavior develops with amplitudes below typically 10 -6 .

to study the influence of anisotropy [START_REF] Kloss | Strong-coupling phases of the anisotropic Kardar-Parisi-Zhang equation[END_REF], and of spatial correlations in the microscopic noise, following a power-law [START_REF] Kloss | Kardar-Parisi-Zhang equation with spatially correlated noise: A unified picture from nonperturbative Renormalization Group[END_REF] or with a finite length-scale [START_REF] Mathey | Kardar-Parisi-Zhang equation with short-range correlated noise: Emergent symmetries and nonuniversal observables[END_REF], and also the influence of temporal correlations of the noise [START_REF] Strack | Dynamic criticality far from equilibrium: One-loop flow of Burgers-Kardar-Parisi-Zhang systems with broken galilean invariance[END_REF][START_REF] Squizzato | Kardar-parisi-zhang equation with temporally correlated noise: A nonperturbative renormalization group approach[END_REF].

Let us emphasize that temporal correlations breaks at the microscopic level the Galilean invariance (and the timereversal symmetry in 1D) which are fundamental symmetries of the KPZ equation. Thus, it is not clear a priori whether the KPZ universality should survive even an infinitesimal breaking. This raises the important issue of its realisability in experimental systems since delta-correlations are an idealization and any real physical source of noise does possess some finite time correlation. This issue has been the subject of an active debate and conflicting answers have been given (see [START_REF] Squizzato | Kardar-parisi-zhang equation with temporally correlated noise: A nonperturbative renormalization group approach[END_REF] and references therein). The FRG analysis brought convincing evidence to settle this debate by showing that in fact, as long as the temporal correlations of the noise are not too long-ranged, the symmetries are dynamically restored along the flow, and the KPZ universal properties emerge at long distances and long times [START_REF] Squizzato | Kardar-parisi-zhang equation with temporally correlated noise: A nonperturbative renormalization group approach[END_REF].

d-Stochastic Navier-Stokes equation

Turbulence remains one of the major unsolved problems of classical physics. Whereas the constitutive model for fluid dynamics is well-known, no one has succeeded yet in deriving from it the actual statistical properties of the turbulent state [START_REF] Frisch | Turbulence: the legacy of A. N. Kolmogorov[END_REF]. The dynamics of an incompressible fluid is given by the Navier-Stokes (NS) equation

∂ t v + v • ∇v = - 1 ρ ∇p + ν∇ 2 v + f , with ∇ • v = 0 ( 48
)
where ν is the kinematic viscosity, ρ the density of the fluid, p the pressure field and f the forcing necessary to maintain a turbulence state. We here focus on isotropic, homogeneous and stationary turbulence. To describe its universal properties, it is convenient to consider a stochastic forcing, with a Gaussian distribution and concentrated in Fourier space at a large scale L, called the integral scale, where energy is injected

f α (t, x) f β (t , x ) = 2δ αβ δ(t -t )N L -1 (|x -x |) . ( 49
)
The phenomenology of turbulence is well-known from many experiments and Direct Numerical Simulations (DNS) [START_REF] Frisch | Turbulence: the legacy of A. N. Kolmogorov[END_REF]. One striking feature is the emergence of very robust power-law behaviors, observed over a wide range of scales called the inertial range, and irrespective of the nature of the fluid (liquids, gas, or even quantum fluids). This universal behavior was first explained by the pioneering statistical theory of turbulence proposed by Kolmogorov in 1941 [START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF][START_REF] Kolmogorov | Dissipation of energy in the locally isotropic turbulence[END_REF]. However, the power-laws observed in turbulence do not fit in the framework of standard scale invariance, but rather involve multi-scaling or multi-fractality, and this anomalous behavior is generically called intermittency. Another peculiar signature of intermittency in turbulence is the enhancement of extreme events at small scales, generally associated with quasi-singularities of the vorticity field at small-scales [START_REF] Frisch | Turbulence: the legacy of A. N. Kolmogorov[END_REF]. The calculation of the multi-fractal spectrum of anomalous exponents from first principles is an open issue.

The idea of applying RG methods to turbulence dates back to the late seventies and has a long history which is not reviewed here, but it has essentially failed to bridge this gap [START_REF] Adzhemyan | The Field Theoretic Renormalization Group in Fully Developed Turbulence[END_REF][START_REF] Zhou | Renormalization Group theory for fluid and plasma turbulence[END_REF]. One difficulty is the absence of upper critical dimension, and the necessary introduction of an expansion parameter through a power-law forcing profile, whose extrapolation to physical large-scale forcing turns out to be extremely problematic. This has thus motivated several authors to revisit turbulence using FRG [START_REF] Canet | Fully developed isotropic turbulence: Nonperturbative renormalization group formalism and fixedpoint solution[END_REF][START_REF] Tomassini | An exact renormalization group analysis of 3D well developed turbulence[END_REF]450]. In these works, it was shown that the turbulent state corresponds to a fixed point, for a physical large-scale forcing (no need to artificially introduce a power-law profile). More interestingly, the FRG has proven to be a very powerful tool in this context, since it allows for the derivation of exact results for the time-dependence (in the stationary state, i.e. the dependence in time delays) of any generic n-point correlation function in the limit of large wave-numbers, both in 3D [START_REF] Canet | Spatiotemporal velocity-velocity correlation function in fully developed turbulence[END_REF][START_REF] Tarpin | Breaking of scale invariance in the time dependence of correlation functions in isotropic and homogeneous turbulence[END_REF] and 2D [START_REF] Tarpin | Stationary, isotropic and homogeneous two-dimensional turbulence: a first non-perturbative renormalization group approach[END_REF]. This is all the more remarkable that exact results are scarce in turbulence. From a FRG perspective, turbulence also stands as a unique example where the RG flow equation for an arbitrary n-point correlation function can be closed exactly in the limit of large wave-numbers, without any further approximations. This is a compelling illustration of the importance of extended symmetries, which are the key ingredients in this derivation, within the FRG framework.

Upon considering a stochastic forcing, the NS equation ( 48) formally stands as a Langevin equation, and the MSRJD procedure yields the NS action [START_REF] Canet | Fully developed isotropic turbulence: Nonperturbative renormalization group formalism and fixedpoint solution[END_REF]

S = ˆt,x vα ∂ t v α -ν∇ 2 v α + v β ∂ β v α + 1 ρ ∂ α p + p ∂ α v α - ˆt,x,x vα N L -1 (|x -x |)v α . (50)
Similarly to the KPZ equation, the NS equation admits an extended (time-gauged) form of Galilean invariance, corresponding to the infinitesimal transformation:

δv α (t, x) = -˙ α (t) + β (t)∂ β v α (t, x) , δϕ(t, x) = β (t)∂ β ϕ(t, x) , (51)
where ϕ denotes any of the other fields vα , p and p. From this symmetry one can derive Ward identities which fix the value of any vertex function Γ (n) with one vanishing momentum carried by a velocity field in terms of lower-order ones. Another extended symmetry, unveiled in [START_REF] Canet | Fully developed isotropic turbulence: Symmetries and exact identities[END_REF], corresponds to a time-gauged shift of the response fields as

δv α (x) = ¯ α (t) , δ p(x) = v β (x)¯ β (t) . (52)
Interestingly, this symmetry yields Ward identities which impose that any vertex function with one zero-momentum carried by a response velocity is vanishing. Due to the time dependence of the parameter of these symmetries, both set of Ward identities hold for arbitrary frequencies.

To implememt FRG, the forcing term in (50) can be promoted to a regulator by replacing L -1 by the running RG scale k, and is then interpreted as an effective forcing. The usual off-diagonal regulator term can also be added (and is required in d = 2), and plays the role of an effective friction. For the NS field theory, the BMW scheme (introduced in Sec. 2.4) allows one to exactly close the flow equation of any n-point correlation function in the limit of large momentum. Indeed, consider the flow equation of a vertex function. In the regime where all external wavenumbers are large compared to k (which ultimately means large compared to L -1 ), then the internal (loop) momentum is negligible in all the vertex functions, which can therefore be expanded around q 0. At leading order, i.e. keeping only the zeroth order term of this expansion, all the vertex functions with one (or two) zero momentum are given exactly in terms of lower order ones thanks to the Ward identities. Compared to the usual BWM scheme, it is not necessary to keep a dependence in a background field to express these vertices. Moreover, the peculiarity of the NS flow equations is that the non-linear part of the flow is not negligible compared to the linear one in the limit of large momentum. This implies that there is no decoupling of the scales, and it yields a violation of standard scale invariance pertaining to intermittency. The physical origin of this non-decoupling of scales roots in what is known phenomenologically as the random sweeping effect, which is the random advection of small-(length)scale velocities by the large-(length)scale vortices of the flow [START_REF] Frisch | Turbulence: the legacy of A. N. Kolmogorov[END_REF].

Furthermore, it turns out that due to the structure of the Ward identities, the resulting flow equations endow a very simple form when expressed in terms of the correlation functions G (n) rather than in terms of the vertex functions Γ (n) (even if in the calculation the q expansion is applied only to vertex functions). As a result, the flow equation for an arbitrary n-point generalized correlation function (correlation and response) can be solved analytically at the fixed point, in both regimes of small and large time delays in the stationary state. This solution provides the exact time dependence of correlation function in turbulence at leading order in wave-numbers [START_REF] Canet | Spatiotemporal velocity-velocity correlation function in fully developed turbulence[END_REF][START_REF] Tarpin | Breaking of scale invariance in the time dependence of correlation functions in isotropic and homogeneous turbulence[END_REF][START_REF] Tarpin | Stationary, isotropic and homogeneous two-dimensional turbulence: a first non-perturbative renormalization group approach[END_REF].

For the 2-point correlation function, the FRG result yields a Gaussian decay in the variable tk where k is the wave-number and t a small time delay in the stationary state. This form can be physically understood as the effect of sweeping [START_REF] Frisch | Turbulence: the legacy of A. N. Kolmogorov[END_REF]. However, the FRG result shows that the Gaussian form is only valid at small time delays, and it predicts that the temporal decay crosses over to a simple exponential at large time delays, which was not known. Similar explicit expressions were obtained for the temporal dependence of any multi-point correlation. The related predictions for the two-point function have been compared to DNS of the NS equation [START_REF] Canet | Spatiotemporal velocity-velocity correlation function in fully developed turbulence[END_REF] and experiments [START_REF] Debue | Experimental test of the crossover between the inertial and the dissipative range in a turbulent swirling flow[END_REF][START_REF] Gorbunova | Analysis of the dissipative range of the energy spectrum in grid turbulence and in direct numerical simulations[END_REF], and the agreement is very accurate.

Master equations: reaction-diffusion processes

Reaction-diffusion processes are simple models which describe one or several species of particles, A, B, • • • , which diffuse on a lattice and can undergo some reactions at a certain rate when they meet (e.g. A+B → B+B). Owing to their simplicity, these models are widely used in physics (percolation or growth processes, cellular automata,. . . ), chemistry (chemical reactions), biology (dynamics of population or disease), and also economics (stockmarket evolution). This dynamics can be described by a master equation, which gives the time evolution of the probability distribution of the micro-states of the system (number of particles of each species on each lattice site). When the transition rates do not satisfy the detailed balance condition, the system reaches genuinely nonthermal (nonequilibrium) steady-states, whose probability distribution is not known in general.

Reaction-diffusion systems exhibit phase transitions between these nonequilibrium steady-states. The most common ones are absorbing phase transitions, which separate an active fluctuating steady-state from one, or several, absorbing states, with no fluctuations [START_REF] Hinrichsen | Non-equilibrium critical phenomena and phase transitions into absorbing states[END_REF][START_REF] Ódor | Universality classes in nonequilibrium lattice systems[END_REF][START_REF] Henkel | I: Absorbing Phase Transitions[END_REF]. An important challenge is to obtain a full classification of the associated non-equilibrium universality classes. When there is a unique absorbing state, and when the phase transition is characterized by a scalar order parameter (the mean density for instance), it generically belongs to the Directed Percolation (DP) universality class. Besides this prominent class, other universality classes can emerge, for instance in the presence of additional symmetries (see below).

Let us focus on one-species reaction-diffusion processes. They are generically defined as Branching and Annihilating Random Walks (BARW) which consist of identical particles A, with particle creation [START_REF] Bramson | The survival of branching annihilating random walk[END_REF][START_REF] Cardy | Theory of branching and annihilating random walks[END_REF][START_REF] Cardy | Field theory of branching and annihilating random walks[END_REF][463]. The standard procedure to derive a field theory from such microscopic rules is the Doi-Peliti formalism [START_REF] Doi | Stochastic theory of diffusion-controlled reaction[END_REF][START_REF] Peliti | Path integral approach to birth-death processes on a lattice[END_REF]. It consists in introducing creation and annihilation operators, as in second quantification, to turn the associated master equation into an imaginary-time Shrödinger equation. One then resorts to the coherent state formalism to cast it into a field theory. Details of this formalism can be found for instance in [START_REF] Cardy | Field theory of branching and annihilating random walks[END_REF]463,[START_REF] Täuber | Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior[END_REF]. The resulting action is formulated in terms of two fields φ and φ, as in the Langevin case, which are originally complex conjugate, but often generalized to independent fields. It has the following structure

A σ m --→ (m + 1)A and destruction nA λ n -→ ∅ [
S[φ, φ] = ˆt,x φ ∂ t -D ∇ 2 φ + U[φ, φ] (53)
where the quadratic dynamical part encodes the diffusion and the potential U encompasses all the microscopic reactions.

a-FRG formalism for reaction-diffusion processes

The FRG formalism to study reaction-diffusion processes is very similar to the Langevin case Eqs. [START_REF] Parola | Liquid state theory and critical phenomena[END_REF][START_REF] Nicoll | Approximate Renormalization Group Based on the Wegner-Houghton Differential Generator[END_REF]. The simplest approximation to study these processes is the DE. The generic ansatz for Γ k at LPA can be written as

Γ k [ϕ, φ] = ˆt,x φ X k ∂ t -Z k ∇ 2 ϕ + U k [ϕ, φ] (54)
where ϕ i = φ i , and with the two running coefficients Z k and X k leading to the anomalous dimensions

η Z k = -∂ s ln Z k and η X k = -∂ s ln X k .
The effective running potential U k must encode the symmetries of the specific reactions under study. For generic processes, the anomalous dimension η of the fields and the dynamic exponent z are defined such that at criticality, ϕ φ ∼ k d+η and ω ∼ k z , respectively [START_REF] Van Wijland | Wilson renormalization of a reaction-diffusion process[END_REF], thus one has η = η X * and z

= 2 + η X * -η Z * . b-Directed Percolation
The directed percolation universality class can be represented by the set of reactions

A σ -→ 2A , 2A λ - → ∅ (55)
which corresponds to the effective potential

U k [φ, φ] = -σ φ φ + 2λ φφ 2 -σφ φ2 + λ φφ 2 . ( 56
)
The fundamental symmetry of the DP process is the so-called rapidity symmetry which corresponds to the transformation φ(t) → -σ 2λ φ(-t), φ(t) → -2λ σ φ(-t). Hence, the effective potential U k can be parametrized in terms of the two invariants ρ ≡ φϕ and ξ ≡ ϕσ 2λ φ [START_REF] Canet | Nonperturbative renormalization-group study of reaction-diffusion processes[END_REF]. The critical exponents of the DP class in dimensions 1, 2 and 3, have been computed within the FRG at both LPA and LPA , and they compare accurately with results from other methods [START_REF] Canet | Nonperturbative renormalization-group study of reaction-diffusion processes[END_REF][START_REF] Buchhold | Background field functional renormalization group for absorbing state phase transitions[END_REF]. The DP exponents have also been obtained using FRG in the context of the Reggeon Field Theory, whose critical behavior belongs to the DP class [START_REF] Bartels | Could reggeon field theory be an effective theory for qcd in the regge limit?[END_REF]. Besides the value of the critical exponents, the FRG has been particularly useful to study nonuniversal aspects. 55) in dimensions 1 to 6 from FRG calculations (solid lines) and numerical simulations (symbols) from [START_REF] Canet | Quantitative phase diagrams of branching and annihilating random walks[END_REF]. For each dimension, the active phase lies on the left of the transition line, and the absorbing phase on its right.

For the DP process [START_REF] Ringwald | Average action for the N-component ϕ 4 theory[END_REF], mean field theory predicts that the active phase is always stable, and thus that there is no transition in this model. Perturbative RG studies have shown that fluctuations could induce an absorbing transition in low dimensions d ≤ 2 [START_REF] Cardy | Theory of branching and annihilating random walks[END_REF][START_REF] Cardy | Field theory of branching and annihilating random walks[END_REF]. The model was re-examined using FRG in [START_REF] Canet | Nonperturbative renormalization-group study of reaction-diffusion processes[END_REF][START_REF] Canet | Quantitative phase diagrams of branching and annihilating random walks[END_REF]. The results are in agreement with the perturbative ones in their region of validity, i.e. near vanishing reaction rates. However, a transition was found in d = 3, and also in all higher dimensions investigated up to d = 10. This result was confirmed by Monte Carlo simulations [START_REF] Canet | Quantitative phase diagrams of branching and annihilating random walks[END_REF][START_REF] Ódor | Role of diffusion in branching and annihilation random walk models[END_REF], and a remarkable agreement was found for the phase diagrams, as illustrated in Fig. 10, which is noticeable given that the critical rates are nonuniversal quantities, similar to critical temperatures.

The failure of perturbative RG to predict the transition in d > 2 is due to the existence of a finite threshold value (λ/D) th for the absorbing state to emerge at σ/D = 0, which is not accessible at any order in perturbation theory (performed around vanishing rates). These threshold values were observed to grow linearly with the dimension in the FRG analysis. This result was supported analytically by a single-site approximation of the model, which becomes exact in the large d limit [START_REF] Canet | Single-site approximation for reaction-diffusion processes[END_REF]. The model was eventually exactly solved at small σ but arbitrary λ in Ref. [START_REF] Benitez | Branching and annihilating random walks: Exact results at low branching rate[END_REF][START_REF] Benitez | Branching-rate expansion around annihilating random walks[END_REF]. The authors established that for hypercubic lattices, the thresholds grow as (λ/D) th ∼ 2da d-2 , where a is the lattice spacing, when d → ∞, confirming the FRG result. The mean field prediction of absence of phase transition for this DP model is hence only valid in infinite dimension.

c-Other processes

In the presence of an additional symmetry, which is the conservation of the parity of the number of particles, Within a perturbative RG analysis [START_REF] Cardy | Theory of branching and annihilating random walks[END_REF][START_REF] Cardy | Field theory of branching and annihilating random walks[END_REF], the existence of the absorbing phase transition has been associated with a change of stability of the Pure Annihilation (PA) fixed point (same model with σ = 0) occuring in an emerging critical dimension d c = 4/3. For d > d c , the branching σ is a relevant perturbation and only an active phase exists, whereas for d < d c , σ is irrelevant and an absorbing phase, which long-distance properties are controlled by the PA fixed point, is expected for small σ.

This model was investigated using FRG within LPA in [START_REF] Canet | Nonperturbative fixed point in a nonequilibrium phase transition[END_REF]. A genuinely nonperturbative fixed point (not connected to the Gaussian fixed point in any dimension) is found in d = 1 for finite σ/D and λ/D, driving an active-toabsorbing phase transition, as illustrated on Fig. 11. The associated critical exponents differ from the DP ones, and are compatible (at least for ν at LPA) with numerical simulations [START_REF] Canet | Nonperturbative fixed point in a nonequilibrium phase transition[END_REF]. Within LPA, this fixed point annihilates with the PA fixed point in d 4/3, thereby changing its stability. The LPA analysis hence confirms the perturbative scenario. However, an exact calculation at small σ (that is to first order in σ around the exactly solved PA fixed point, rather than around the Gaussian one) showed that the stability of the PA fixed point with respect to a perturbation in σ does not change between one and two dimensions [START_REF] Benitez | Branching and annihilating random walks: Exact results at low branching rate[END_REF][START_REF] Benitez | Branching-rate expansion around annihilating random walks[END_REF]. An alternative scenario with the existence of two fixed points was proposed to reconcile existing results, but is yet to be elucidated.

The last important one-species reaction-diffusion process which has been studied using FRG is the Pair Contact Process with Diffusion. It differs from the previous BARW models in that branching requires two particles to meet 2A σ -→ 3A, supplemented with pair annihilation 2A λ -→ ∅. This process has particularly resisted perturbative analysis [START_REF] Henkel | The non-equilibrium phase transition of the pair-contact process with diffusion[END_REF]. In fact, FRG studies, within the LPA approximation and using a semifunctional treatment of the effective potential U k [ϕ, φ], revealed that U k develops a nonanalyticity at a finite RG scale k c : linear terms, perturbatively forbidden, are dynamically generated. This scale corresponds to the scale where perturbative RG flows blow up. This is reminiscent of the appearance of a cusp in Random Field Ising Model and other disordered models (see Sec. 3.3.2). Let us finally mention that a two-species reaction-diffusion process called Diffusive Epidemic Process was studied using FRG in [START_REF] Tarpin | Nonperturbative renormalization group for the diffusive epidemic process[END_REF].

Non-equilibrium quantum systems

The understanding of non-equilibrium quantum phenomena is amongst the most pressing questions of modern physics. Potential applications ranging from the physics of the early universe over the early evolution of the fireball in heavy-ion collisions, the dynamics of the pair-creation in strong electromagnetic fields to the far-from-equilibrium dynamics of ultracold atomic systems. Non-equilibrium versions of the FRG may also help to understand the propagation of information, the generation of (entanglement-) entropy and the equilibration process.

A generic non-equilibrium process is often initiated with an over-occupied initial state with large occupancies n(t 0 , x)

1 encoded in the initial density matrix ρ(t 0 ). The statistical propagator {ϕ(t 0 , x) , ϕ(t 0 , 0)} ∝ n(t 0 , x) + 1/2 is first dominated by the large occupancies and the respective physics is well described by classical-statistical approximations to the full system. In O(N)-theories this classical-statistical regime is captured well by next-to-leading order computations in 1/N-expansions (s-channel resummations) of 2PI-hierarchies, for reviews see e.g. [START_REF] Berges | Nonequilibrium Quantum Fields: From Cold Atoms to Cosmology[END_REF][START_REF] Schmied | Non-thermal fixed points: Universal dynamics far from equilibrium[END_REF].

For occupancies n(t 0 , x) ≈ 1 the quantum 1/2 in n(t 0 , x) + 1/2 gets sizable. The system enters a regime with non-equilibrium quantum dynamics which requires a non-equilibrium quantum field theoretical description. This can be done within the Schwinger-Keldysh formalism on a closed time path C with an initial density matrix ρ(t 0 ) at the initial time t 0 . Then, the real-time FRG follows straightforwardly with the standard cutoff term quadratic in the fields. In the scalar case, the generating functional for non-equilibrium correlation functions is given by

Z k [J; ρ] = Tr ρ(t 0 ) exp i ˆC ϕ(x)J(x) + i 2 ˆϕ(x)R k (x, y)ϕ(y) , (57)
where x, y live on the Keldysh contour C. For an introduction to the formulation with a standard spatial-momentum cutoff see [START_REF] Berges | Introduction to the nonequilibrium functional Renormalization Group[END_REF]. A manifestly causal cutoff is obtained if closing the Keldysh contour at a finite time t = τ. Then the cutoff parameter is simply the maximal time k = τ. The respective flow equation does not integrate-out momentumshells but simply propagates the system in time, see [START_REF] Gasenzer | Towards far-from-equilibrium quantum field dynamics: A functional renormalisation-group approach[END_REF][START_REF] Pietroni | Flowing with Time: a New Approach to Nonlinear Cosmological Perturbations[END_REF][START_REF] Gasenzer | Far-from-equilibrium quantum many-body dynamics[END_REF][START_REF] Corell | Flowing with the Temporal Renormalisation Group[END_REF]. The regularisation can also be introduced via deformations of the occupancies n(t, x) → n k (t, x), see e.g. [START_REF] Schoeller | Real-time renormalization group and charge fluctuations in quantum dots[END_REF][START_REF] Jakobs | Nonequilibrium Functional Renormalization Group for Interacting Quantum Systems[END_REF] for applications in low-dimensional systems.

The Schwinger-Keldysh formulation of the FRG has been applied to the time evolution and turbulent cascades at non-thermal fixed points in O(N)-models, [START_REF] Gasenzer | Towards far-from-equilibrium quantum field dynamics: A functional renormalisation-group approach[END_REF][START_REF] Corell | Flowing with the Temporal Renormalisation Group[END_REF][START_REF] Berges | Nonthermal fixed points and the functional Renormalization Group[END_REF], to the classification of dynamical critical phenomena in models with and without driving force, [407,[START_REF] Mesterházy | Dynamic universality class of model c from the functional renormalization group[END_REF]482,[START_REF] Mesterházy | From quantum to classical dynamics: The relativistic O(N) model in the framework of the realtime functional renormalization group[END_REF], and to the cosmological power spectrum, [START_REF] Pietroni | Flowing with Time: a New Approach to Nonlinear Cosmological Perturbations[END_REF][START_REF] Lesgourgues | Non-linear Power Spectrum including Massive Neutrinos: the Time-RG Flow Approach[END_REF][START_REF] Bartolo | Signatures of Primordial non-Gaussianities in the Matter Power-Spectrum and Bispectrum: the Time-RG Approach[END_REF][START_REF] Audren | Non-linear matter power spectrum from Time Renormalisation Group: efficient computation and comparison with one-loop[END_REF][START_REF] Jürgens | Perturbation Theory Trispectrum in the Time Renormalisation Approach[END_REF][START_REF] Vollmer | Efficient implementation of the Time Renormalization Group[END_REF], see also [START_REF] Floerchinger | Renormalization-group flow of the effective action of cosmological large-scale structures[END_REF]. For further applications in low-dimensional systems see also Sec. 4.2.4.

Quantum many-particle systems

Quantum many-particle theory was developed to describe interacting particles in condensed matter, and is now relevant for ultracold atomic gases, too. According to the two types of quantum statistics (leaving anyons aside), this broad field is naturally divided in the theory of interacting bosons and interacting fermions. Applications of the FRG to quantum many-particle problems have already been summarized in two extensive review articles published in 2012, one in relation to cold atoms [START_REF] Boettcher | Ultracold atoms and the Functional Renormalization Group[END_REF], the other with a focus on interacting fermions in condensed matter [START_REF] Metzner | Functional renormalization group approach to correlated fermion systems[END_REF].

We note in passing that the FRG can also be applied to few-body systems. A beautiful example is the Efimov effect related to the formation of bound states of three or four interacting bosons or fermions, which is associated with a limit cycle in the renormalization group flow [START_REF] Floerchinger | Efimov physics from the functional renormalization group[END_REF].

Bosons

The first realization of an interacting quantum Bose system was liquid 4 He. Nowadays, ultracold bosonic atoms provide a vast additional playground for exploring interaction effects in Bose systems. Moreover, interacting Bose systems also emerge in effective low-energy theories. In the following we review a selection of systems where the FRG has made a major impact.

Superfluidity in a dilute Bose gas

The Euclidean action of a dilute Bose gas reads

S = ˆβ 0 dτ ˆdd r ψ * ∂ τ -µ - ∇ 2 2m ψ + g 2 (ψ * ψ) 2 , ( 58
)
where the complex field ψ satisfies periodic boundary conditions, ψ(r, τ) = ψ(r, τ + β), and µ is the chemical potential.

Most physical low-energy properties depend only on the boson mass m and the s-wave scattering length a ≡ a(g, Λ), which is a function of the repulsive interaction g and the upper momentum cutoff Λ in the model defined by [START_REF] Wetterich | Improvement of the average action[END_REF]. For any finite density (µ > 0), the ground state is superfluid and characterized by spontaneously broken U(1) symmetry: ψ(r, τ) 0. In three dimensions (and at zero temperatures in two dimensions) most physical properties of the superfluid phase, including the equation of state and the sound mode with linear dispersion, can be understood from the Bogoliubov theory [START_REF] Bogoliubov | On the theory of superfluidity[END_REF], based on a Gaussian-fluctuation approximation about the mean-field solution. Nevertheless, perturbation theory beyond the Bogoliubov approximation is plagued with infrared divergences [START_REF] Beliaev | Application of the methods of quantum field theory to a system of bosons[END_REF][START_REF] Beliaev | Energy spectrum of a non-ideal Bose gas[END_REF][START_REF] Hugenholtz | Ground-State Energy and Excitation Spectrum of a System of Interacting Bosons[END_REF][START_REF] Gavoret | Structure of the perturbation expansion for the Bose liquid at zero temperature[END_REF]. Although these divergences cancel out in gauge-invariant quantities (pressure, sound velocity, etc.), they have a definite physical origin since they reflect the divergence of the longitudinal susceptibility, a phenomenon which also occurs in the lowtemperature phase of the O(N ≥ 2) model (Sec. 2.3.2) but is not accounted for in the Bogoliubov theory. Popov has proposed an approach to superfluidity, based on an amplitude-phase representation of the boson field, which is free of infrared divergences and yields the correct low-energy behavior of the correlation functions but is restricted to the (low-momentum) hydrodynamic regime [START_REF] Popov | Low-frequency asymptotic form of the self-energy parts of a superfluid Bose system at T = 0[END_REF][START_REF] Popov | Functional Integrals in Quantum Field Theory and Statistical Physics[END_REF].

Field-theoretical diagrammatic methods [START_REF] Nepomnyashchii | Contribution to the theory of the spectrum of a Bose system with condensate at small momenta[END_REF][START_REF] Nepomnyashchii | Infrared divergence in field theory of a Bose system with a condensate[END_REF][START_REF] Nepomnyashchii | Concerning the nature of the λ-transition order parameter[END_REF] or perturbative RG constrained by Ward identities [START_REF] Castellani | Infrared Behavior of Interacting Bosons at Zero Temperature[END_REF][START_REF] Pistolesi | Renormalization-group approach to the infrared behavior of a zero-temperature Bose system[END_REF] can be used to handle the infrared divergences of perturbative theory. The FRG provides us with a simple alternative method, free of any divergences, which encompasses both the Bogoliubov theory (valid, stricto sensu, only at momenta larger than a "Ginzburg" scale) and Popov's hydrodynamic approach [START_REF] Dupuis | Unified Picture of Superfluidity: From Bogoliubov's Approximation to Popov's Hydrodynamic Theory[END_REF]. It has been used to understand various properties of three-and two-dimensional superfluid Bose systems, from the equation of state to the excitation spectrum and the damping of quasi-particles [START_REF] Dupuis | Infrared behavior in systems with a broken continuous symmetry: Classical O(N) model versus interacting bosons[END_REF][START_REF] Dupuis | Infrared behavior and spectral function of a Bose superfluid at zero temperature[END_REF][START_REF] Sinner | Functional renormalization-group approach to interacting bosons at zero temperature[END_REF][START_REF] Ranc ¸on | Universal thermodynamics of a two-dimensional Bose gas[END_REF][START_REF] Wetterich | Functional renormalization for quantum phase transitions with nonrelativistic bosons[END_REF][START_REF] Dupuis | Non-perturbative renormalization group approach to zero-temperature Bose systems[END_REF][START_REF] Sinner | Spectral Function and Quasiparticle Damping of Interacting Bosons in Two Dimensions[END_REF][START_REF] Floerchinger | Functional renormalization for Bose-Einstein condensation[END_REF][START_REF] Floerchinger | Superfluid Bose gas in two dimensions[END_REF][START_REF] Floerchinger | Nonperturbative thermodynamics of an interacting Bose gas[END_REF][START_REF] Eichler | Condensate density of interacting bosons: A functional renormalization group approach[END_REF][START_REF] Krieg | Thermodynamics and renormalized quasiparticles in the vicinity of the dilute Bose gas quantum critical point in two dimensions[END_REF][START_REF] Isaule | Application of the functional renormalization group to bose gases: From linear to hydrodynamic fluctuations[END_REF][START_REF] Isaule | Thermodynamics of bose gases from functional renormalization with a hydrodynamic low-energy effective action[END_REF].

Another important issue regarding dilute Bose gases is the determination of the superfluid transition temperature. In the absence of interactions (a = 0), the bosons undergo a Bose-Einstein condensation at the temperature T 0 c = (2π/m)[n/ζ(3/2)] 2/3 in three dimensions where n is the mean density. For weak interactions, the shift ∆T c /T 0 c = (T c -T 0 c )/T 0 c in the transition temperature is a universal function of na 3 . The dependence of ∆T c /T 0 c on a has remained a controversial issue for a long time and even the sign of the effect has been debated [START_REF] Lee | Low-Temperature Behavior of a Dilute Bose System of Hard Spheres. I. Equilibrium Properties[END_REF][START_REF] Toyoda | A microscopic theory of the lambda transition[END_REF][START_REF] Huang | Transition Temperature of a Uniform Imperfect Bose Gas[END_REF]. It is now understood that ∆T c /T 0 c = c(an 1/3 ) increases linearly with a [519-523] and the proportionality coefficient c has been estimated from various approaches. The difficulty in getting a precise estimate of c comes from the fact that it requires the knowledge of the one-particle propagator at T c in a large momentum range including the crossover region between the critical and noncritical regimes. The determination of c is therefore a nonperturbative problem which can be dealt with the FRG. Since the full knowledge of the momentum dependence of the propagator is required, the DE is not sufficient. The BMW approximation (Sec. 2.4) gives c 1.37 [START_REF] Benitez | Solutions of renormalization group flow equations with full momentum dependence[END_REF][START_REF] Benitez | Nonperturbative renormalization group preserving full-momentum dependence: Implementation and quantitative evaluation[END_REF][START_REF] Blaizot | Non-Perturbative Renormalization Group calculation of the transition temperature of the weakly interacting Bose gas[END_REF] in reasonable agreement with lattice results (c = 1.32(2) [START_REF] Arnold | BEC Transition Temperature of a Dilute Homogeneous Imperfect Bose Gas[END_REF] and c = 1.29(5) [START_REF] Kashurnikov | Critical Temperature Shift in Weakly Interacting Bose Gas[END_REF]) and seven-loop resummed calculations (c = 1.27 [START_REF] Wegner | Renormalization Group Equation for Critical Phenomena[END_REF] [START_REF] Kastening | Bose-Einstein condensation temperature of a homogenous weakly interacting Bose gas in variational perturbation theory through seven loops[END_REF]). Another FRG calculation, based on a vertex expansion, gives c = 1.23 [START_REF] Ledowski | Self-energy and critical temperature of weakly interacting bosons[END_REF][START_REF] Hasselmann | Critical behavior of weakly interacting bosons: A functional renormalization-group approach[END_REF]. In the case of a model with N-component real fields (Eq. ( 58) corresponds to N = 2), the value of c obtained from FRG is remarkably accurate for N ≥ 3, being within the error bars of lattice simulations and seven-loop calculations [START_REF] Benitez | Nonperturbative renormalization group preserving full-momentum dependence: Implementation and quantitative evaluation[END_REF].

Superfluid to Mott-insulator transition

The simplest model describing bosons moving in a d-dimensional hypercubic lattice is the one-band Bose-Hubbard model defined by the action [START_REF] Fisher | Boson localization and the superfluid-insulator transition[END_REF] The Monte Carlo data [START_REF] Dantchev | Critical Casimir force and its fluctuations in lattice spin models: Exact and Monte Carlo results[END_REF] have been rescaled so as to satisfy the correct asymptotic value for x→-∞, the bare data are shown in the inset. Reprinted from Ref. [START_REF] Ranc ¸on | Critical Casimir forces from the equation of state of quantum critical systems[END_REF].

S = ˆβ 0 dτ r ψ * r (∂ τ -µ)ψ r + U 2 n r (n r -1) -t r,r ψ * r ψ r + h.c. , (59)
where n r = ψ * r ψ r , U is the on-site repulsion, t the intersite hopping amplitude, and r, r denotes nearest-neighbor sites. When the average number n of bosons per site is integer and U t, the ground state is a Mott insulator with n bosons localized at each lattice site, a vanishing compressibility κ = ∂n/∂µ and a gap in the excitation spectrum. A transition to a superfluid ground state can be induced by either decreasing the ratio U/t or varying n. In the first case the transition is in the universality class of the (d+1)-dimensional XY (or O(2)) model with an upper critical dimension d + c = 3. In the second one, the transition is in the universality class of the dilute Bose gas and is mean-field-like above two dimensions (d + c = 2). The three-and two-dimensional Bose-Hubbard models have been studied within the FRG approach using the lattice formulation (Sec. 2.5) where the initial condition corresponds to the limit of decoupled sites [START_REF] Ranc ¸on | Nonperturbative renormalization group approach to the Bose-Hubbard model[END_REF][START_REF] Ranc ¸on | Nonperturbative renormalization group approach to strongly correlated lattice bosons[END_REF]. The FRG yields a phase diagram in very good agreement with the (numerically exact) quantum Monte Carlo simulations [START_REF] Capogrosso-Sansone | Phase diagram and thermodynamics of the three-dimensional Bose-Hubbard model[END_REF][START_REF] Capogrosso-Sansone | Monte Carlo study of the two-dimensional Bose-Hubbard model[END_REF] and with an accuracy similar to that obtained from dynamical-mean-field theory [START_REF] Anders | Dynamical mean-field theory for bosons[END_REF][START_REF] Panas | Numerical calculation of spectral functions of the bose-hubbard model using bosonic dynamical mean-field theory[END_REF]; see Fig. 12. It also allows one to compute nonuniversal quantities such as the velocity (or the effective mass when the transition is induced by a density change) of the critical fluctuations. It reproduces the two universality classes of the superfluid-Mott-insulator transition and gives an accurate determination of the universal scaling functions associated with the equation of state near the quantum critical point [START_REF] Ranc ¸on | Quantum XY criticality in a two-dimensional Bose gas near the Mott transition[END_REF][START_REF] Ranc ¸on | Quantum criticality of a Bose gas in an optical lattice near the Mott transition[END_REF][START_REF] Ranc ¸on | Thermodynamics of a Bose gas near the superfluid-Mott-insulator transition[END_REF].

Relativistic bosons and quantum O(N) model

The simplest quantum generalization of the O(N) model discussed in Sec. 2, with space-time Lorentz invariance, is defined by the Euclidean (imaginary-time) action

S [ϕ] = ˆβ 0 dτ ˆdd r 1 2 (∇ϕ) 2 + 1 2c 2 (∂ τ ϕ) 2 + r 0 2 ϕ 2 + u 0 4! (ϕ 2 ) 2 , ( 60
)
where the N-component real field satisfies periodic boundary conditions: ϕ(r, τ) = ϕ(r, τ + β). r 0 and u 0 are temperature-independent coupling constants and c is the (bare) velocity of the excitations. The quantum O(2) model describes relativistic bosons and is relevant for the superfluid-Mott-insulator transition (Sec. 4.1.2); the N = 3 model applies to quantum antiferromagnets, the N = 4 to QCD, etc. At zero temperature the quantum model is equivalent to its (d + 1)-dimensional classical counterpart and therefore exhibits a quantum phase transition between a disordered phase and an ordered phase where the O(N) symmetry is spontaneously broken. At nonzero temperature the equivalent classical model has a finite size L τ = β in the (d + 1)th dimension. The finite-temperature thermodynamic has been studied within the FRG approach [START_REF] Ranc ¸on | Thermodynamics in the vicinity of a relativistic quantum critical point in 2+1 dimensions[END_REF][START_REF] Blaizot | Perturbation theory and non-perturbative renormalization flow in scalar field theory at finite temperature[END_REF][START_REF] Blaizot | Calculation of the pressure of a hot scalar theory within the Non-Perturbative Renormalization Group[END_REF]. In the vicinity of the quantum critical point, the scaling functions determining the universal equation of state [START_REF] Ranc ¸on | Critical Casimir forces from the equation of state of quantum critical systems[END_REF] are in striking agreement with Monte Carlo simulations of (d + 1)-dimensional classical N-component spin models in a finite geometry (these models are in the same universality class as the two-dimensional quantum O(N) model) [START_REF] Vasilyev | Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations[END_REF][START_REF] Hucht | Aspect-ratio dependence of thermodynamic Casimir forces[END_REF][START_REF] Cardozo | Finite size scaling and the critical Casimir force : Ising magnets and binary fluids[END_REF][START_REF] Dantchev | Critical Casimir force and its fluctuations in lattice spin models: Exact and Monte Carlo results[END_REF]; see Fig. 13.

Momentum-frequency dependent correlation functions and spectral functions of the quantum O(N) model have also been obtained from the FRG. In the context of QCD, where the O(4) model is frequently used as a chiral effective model, spectral functions of sigma mesons and pions have been computed [START_REF] Tripolt | Spectral Functions for the Quark-Meson Model Phase Diagram from the Functional Renormalization Group[END_REF][START_REF] Kamikado | Real-Time Correlation Functions in the O(N) Model from the Functional Renormalization Group[END_REF][START_REF] Wambach | Spectral functions from the functional renormalization group[END_REF][START_REF] Pawlowski | Finite temperature spectral functions in the O(N) model[END_REF]. In statistical physics, the FRG approach has lead to the confirmation of the existence of a bound state in the ordered phase of the twodimensional quantum O(1) model [START_REF] Rose | Bound states of the φ 4 model via the nonperturbative renormalization group[END_REF], with a mass within 1% of previous Monte Carlo simulations and numerical diagonalization values [START_REF] Agostini | The spectrum of the 2 + 1-dimensional gauge ising model[END_REF][START_REF] Caselle | Non-perturbative states in the 3D ϕ 4 theory[END_REF][START_REF] Nishiyama | Universal critical behavior of the two-magnon-bound-state mass gap for the (2+1)-dimensional Ising model[END_REF]. In the ordered phase of the two-dimensional O(2) and O(3) models, the "Higgs" amplitude mode, whose existence and visibility in the vicinity of the quantum critical point have been overlooked for a long time [START_REF] Podolsky | Visibility of the amplitude (Higgs) mode in condensed matter[END_REF], is well described by FRG [START_REF] Rose | Higgs amplitude mode in the vicinity of a (2 + 1)-dimensional quantum critical point: A nonperturbative renormalization-group approach[END_REF][START_REF] Ranc ¸on | Higgs amplitude mode in the vicinity of a (2 + 1)-dimensional quantum critical point[END_REF], in quantitative agreement with Monte Carlo simulations [START_REF] Gazit | Fate of the Higgs Mode Near Quantum Criticality[END_REF]. The estimate of the "Higgs" mass in the quantum O(3) model from the BMW approach has been confirmed by subsequent exact diagonalizations [START_REF] Nishiyama | Critical behavior of the Higgs-and Goldstone-mass gaps for the two-dimensional {XY} model[END_REF][START_REF] Nishiyama | Universal scaled Higgs-mass gap for the bilayer Heisenberg model in the ordered phase[END_REF] and quantum Monte Carlo simulations of quantum spin models [START_REF] Lohöfer | Dynamical structure factors and excitation modes of the bilayer Heisenberg model[END_REF]. At zero-temperatures, the frequency-dependent conductivity and its universal features have also been studied [START_REF] Rose | Nonperturbative renormalization-group approach preserving the momentum dependence of correlation functions[END_REF][START_REF] Rose | Superuniversal transport near a (2 + 1)-dimensional quantum critical point[END_REF][START_REF] Rose | Nonperturbative functional renormalization-group approach to transport in the vicinity of a (2 + 1)-dimensional O(N)symmetric quantum critical point[END_REF].

Fermions

Interacting fermion systems display a particularly rich variety of phases and phenomena. The most important representatives are electrons in solids, liquid 3 He, and ultracold fermionic atoms. In this section we focus on nonrelativistic systems with spin- 1 2 fermions. For fermions moving in a continuum, the bare Euclidean action has the form [START_REF] Negele | Quantum Many-particle Systems[END_REF]

S [ψ, ψ * ] = ˆβ 0 dτ ˆdd r ψ * (r, τ) ∂ τ -µ - 1 2m ∇ 2 r ψ(r, τ) + V[ψ, ψ * ] , (61)
where ψ = (ψ ↑ , ψ ↓ ) and ψ * = (ψ * ↑ , ψ * ↓ ) are anticommuting spinor fields with antiperiodic boundary conditions in τ, and V[ψ, ψ * ] is an arbitrary interaction term. For two-particle interactions, V[ψ, ψ * ] is quartic in the fields. Note that ψ and ψ * are independent variables. For lattice fermions, the space variable r is replaced by a discrete lattice site variable j, and the gradient term by hopping amplitudes between sites j and j .

The Fermi-Dirac statistics of the particles and the existence of a Fermi surface at nonzero density leads to a number of difficulties that are not present in the bosonic case discussed in the previous section. First, the fermionic effective action Γ k [ψ, ψ * ] defined below is a functional of Grassmann variables. As pointed out in Sec. 2.6, one must therefore truncate Γ k [ψ * , ψ] in an expansion about ψ = ψ * = 0, which amounts to retaining only a finite number of vertices and performing a loop expansion of the flow equations. This makes the fermionic FRG fundamentally perturbative. Second, since the low-energy fermion states live near the Fermi surface, it is not possible to neglect the momentum dependence of the vertices Γ (n) k or perform a derivative expansion, i.e., an expansion about momentum p = 0. Thus the vertices are necessarily functionals of the momenta. 27 Third, the Grassmannian field is not an order parameter; order parameters are defined by composite fields: ψ * ψ for a charge-or spin-density wave, ψψ for a superconductor. A phase transition is signaled by a divergence of the associated susceptibility and the two-particle vertex Γ (4) k . The growth of Γ (4) k implies a breakdown of the (perturbative) RG and prevents, at least in a straightforward way, to continue the flow into the ordered phase. Furthermore, the fluctuations of the bosonic fields ψ * ψ and ψψ being not properly taken care of, the flow becomes uncontrolled whenever strong collective fluctuations with a large correlation length set in. We shall discuss below how one can, at least partially, overcome these difficulties, e.g., by introducing bosonic fields to take into account collective fluctuations or starting the FRG flow from a nonperturbative initial condition.

The FRG for interacting fermion systems and various applications have already been reviewed in Ref. [START_REF] Metzner | Functional renormalization group approach to correlated fermion systems[END_REF]. That review also contains a short history of the renormalization group for fermion systems which shall not be repeated here. In the meantime, there have been substantial methodological developments, and the range of applications has broadened considerably. A concise review covering also rigorous mathematical work has been published recently by Salmhofer [START_REF] Salmhofer | Renormalization in condensed matter: Fermionic systems -from mathematics to materials[END_REF].

d d dk d Γ (4) dk Γ (6) = Γ S (4) + Γ Γ Γ (4) = (4) + + Γ (6) = Γ (4) (4)
Γ (4) S S Γ (8) Γ (4) S G

k Σ d (6) k k k k k k k k k k k k k k k Γ k k k G k G k G k S k

S k

Figure 14: Diagrammatic representation of the flow equations for the self-energy Σ k , the two-particle vertex Γ (4) k , and the three-particle vertex Γ (6) k in the one-particle irreducible version of the functional RG. Lines with a dash correspond to the single scale propagator S k (shown as a line with a cross in Fig. 2), the other lines correspond to the full propagator G k .

Fermion flow equation

The scale dependent effective action and its flow equation are obtained in close analogy to the classical O(N) model described in Sec. II. The flow is defined by adding a scale dependent regulator term

∆S k = -(ψ * , R k ψ) = -´dx ´dx ψ * (x )R k (x , x)ψ(x)
to the bare action, where x and x include space and time variables (or, alternatively, momentum and frequency variables), and k is the flow parameter. 28 Here and in the following (., .) is a shorthand notation for the summation (or integration) over all fermion field indices in an arbitrary basis. The regulator term modifies the bare propagator

G 0 to G 0,k = (G -1 0 + R k ) -1 .
The effective action is defined by the Legendre transform of the free energy in the presence of Grassmann source fields

Γ k [ψ, ψ * ] = -ln Z k [η k , η * k ] + (η * k , ψ) + (ψ * , η k ) -∆S k [ψ, ψ * ] , (62)
where η k and η * k are k-dependent functions of ψ and ψ * , as determined by the equations

ψ = ∂ η * ln Z k [η, η * ] and ψ * = -∂ η ln Z k [η, η * ].
The fermionic version of Wetterich's exact flow equation for the effective action reads [START_REF] Berges | Non-perturbative renormalization flow in quantum field theory and statistical physics[END_REF][START_REF] Metzner | Functional renormalization group approach to correlated fermion systems[END_REF]

∂ k Γ k [ψ, ψ * ] = - 1 2 Tr ∂ k R k Γ (2) k [ψ, ψ * ] + R k -1 , (63)
where

Γ (2) k [ψ, ψ * ](x , x) =           ∂ 2 Γ k ∂ψ * (x )∂ψ(x) ∂ 2 Γ k ∂ψ * (x )∂ψ * (x) ∂ 2 Γ k ∂ψ(x )∂ψ(x) ∂ 2 Γ k ∂ψ(x )∂ψ * (x)           , (64)
and

R k (x , x) = diag[R k (x , x), -R k (x, x )]. Note that Γ (2)
k is a functional of the fields and a function of the field indices. The matrix structure of the two-point function generated by the conjugated variables ψ and ψ * gives rise to particle-particle and particle-hole channels in fermion systems.

Expanding the functional flow equation [START_REF] Bonini | Perturbative renormalization and infrared finiteness in the Wilson renormalization group: the massless scalar case[END_REF] in powers of the fields, one obtains an exact hierarchy of flow equations for the vertex functions [START_REF] Berges | Non-perturbative renormalization flow in quantum field theory and statistical physics[END_REF][START_REF] Metzner | Functional renormalization group approach to correlated fermion systems[END_REF]. For fermions, there are only vertex functions of even order in the fields. In Fig. 14 we show the Feynman diagrams describing the first three equations in the hierarchy. One of the internal lines corresponds to the single-scale propagator

S k = -G k (∂ k R k )G k = ∂ k G k Σ k const , ( 65
)
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 where

111111111
G k = (G -1 0,k -Σ k ) -1
is the full scale-dependent propagator. The non-interacting propagator in a translation and spin-rotation invariant system has the spin-independent and momentum-diagonal form

G 0 (p, iω n ) = iω n -( p -µ) -1 , ( 66
)
where p is the energy of single-particle states with momentum p, and ω n = (2n + 1)πT with integer n is the fermionic Matsubara frequency. At T = 0 the propagator diverges for ω n = 0 and p = µ, that is, for momenta on the Fermi surface. Hence, in fermion systems with dimensionality d > 1 the propagator exhibits an extended singularity, not just a divergence at a point in momentum space.

There are various possibilities to regularize the Fermi surface singularity in the FRG flow. One is to choose a momentum dependent regulator that suppresses contributions with momenta close to the Fermi surface. An extreme choice is a sharp momentum cutoff, where contributions from momenta p with | p -µ| < k are completely suppressed (see Fig. 15). This corresponds to a regularized bare propagator of the form G 0,k = Θ(| p -µ|k)G 0 . Of course one may also choose a smooth momentum cutoff. In any case, for fermion systems momentum cutoffs have two major drawbacks. First, the Fermi surface can actually be deformed by interactions, so that the regulator cannot be fixed a priori, if these deformations are taken into account. Second, the limit of vanishing momentum transfer q → 0 in interaction vertices does not commute with the limit k → 0 for momentum cutoffs [START_REF] Metzner | Fermi systems with strong forward scattering[END_REF]. Both of these problems can be avoided by choosing a frequency cutoff [START_REF] Husemann | Efficient parametrization of the vertex function, ω scheme, and the t, t hubbard model at van hove filling[END_REF], which is therefore the most popular choice nowadays. A frequency cutoff can also be used in systems without translation invariance [START_REF] Andergassen | Functional renormalization group for Luttinger liquids with impurities[END_REF]. Alternatively one may use the interaction [START_REF] Honerkamp | Interaction flow method for many-fermion systems[END_REF] or temperature [START_REF] Honerkamp | Temperature-flow renormalization group and the competition between superconductivity and ferromagnetism[END_REF] as flow parameters. The interaction flow is computationally convenient, but it does not regularize the Fermi surface singularity.

Competing instabilities

At low temperatures, the normal metallic state of interacting fermion systems is usually unstable toward magnetism, superconductivity, or more exotic forms of order. For systems with weak or moderate interaction strengths, the FRG has turned out to be a valuable tool for the unbiased detection of instabilities, especially in cases where several instabilities of distinct nature compete with each other.

Most instabilities are signalled by a divergence of the two-particle vertex Γ (4) k . For weak interactions, the flow of Γ (4) k can be truncated at second order, that is, contributions from the self-energy and the three-particle vertex can be neglected such that only the first Feynman diagram in the second line of Fig. 14 needs to be computed. Writing out the various terms arising from the matrix structure of Γ (2) k [ψ, ψ * ] in Eq. ( 64), one obtains three types of contributions to the flow of Γ (4) k , known as particle-particle, direct particle-hole, and crossed particle-hole channel, respectively (see Fig. 16).

The two-particle vertex is a function of four momenta and four Matsubara frequencies. The latter are constrained by energy conservation to three independent frequencies, and for translation invariant systems momentum conservation also reduces the number of independent momenta to three. The flow needs to be computed numerically, and a suitable parametrization of the two-particle vertex is a major issue. In weak coupling power counting, the frequency

2 1 1' 2' = pp 1 2 2 2' 2' 1 1' 1' 1 2 2 1' 1' 2' 2' 1 ph ph' Γ d d k (4

) k

Figure 16: Contributions to the flow of the two-particle vertex with particle-particle (pp) and particle-hole channels (ph and ph') written explicitly, without the contribution from Γ (6) k .

dependence and the momentum dependence perpendicular to the Fermi surface appear to be irrelevant [START_REF] Shankar | Renormalization-group approach to interacting fermions[END_REF]. Hence, in most concrete evaluations of the vertex flow the frequency dependence of the vertex was neglected, and the momentum dependence was discretized by a partition of momentum space in patches, paying particular attention to an accurate resolution of the dependences along the Fermi surface. With N patches one thus obtains O(N 3 ) "running couplings", where discrete symmetries can be exploited to reduce the precise number to a certain fraction of N 3 .

The nature of the instabilities can be read off from the momentum and spin dependence of the diverging twoparticle vertex. For example, pairing instabilities are signalled by a divergence in the Cooper channel, that is, for a vanishing total ingoing and outgoing momentum. To identify the instabilities it can also be helpful to analyze the corresponding susceptibilities, for which flow equations can be derived by adding external symmetry breaking fields to the bare action [START_REF] Metzner | Functional renormalization group approach to correlated fermion systems[END_REF].

Two-dimensional Hubbard model: The FRG stability analysis was originally developed for the two-dimensional one-band Hubbard model, describing tight-binding fermions with a local interaction on a square lattice. The Hamiltonian is given by

H = j,j σ t jj c † j σ c jσ + U j n j↑ n j↓ (67)
in standard second quantization notation, where t jj are hopping amplitudes between sites j and j and U is a local interaction. The two-dimensional Hubbard model was proposed by Anderson [START_REF] Anderson | The Resonating Valence Bond State in La 2 CuO 4 and Superconductivity[END_REF] as an effective model for electrons moving in the copper-oxygen planes of cuprate high temperature superconductors. At half-filling, that is, for an average fermion density n = 1, the ground state of the 2D Hubbard model is an antiferromagnetic insulator for sufficiently large U. The critical U depends on the structure of the hopping matrix. For pure nearest neighbor hopping any positive U leads to an antiferromagnetic ground state. While the magnetic instability in the Hubbard model is revealed already by conventional mean-field theory, pairing is fluctuation-driven and hence more difficult to capture. Simple qualitative arguments suggesting d-wave pairing driven by magnetic fluctuations were corroborated by the fluctuation exchange approximation, that is, a resummation of a certain class of Feynman diagrams in perturbation theory [START_REF] Scalapino | The case for d x 2 -y 2 pairing in the cuprate superconductors[END_REF]. However, only the unbiased stability analysis made possible by the FRG provided conclusive evidence for the existence of d-wave superconductivity with a sizable energy gap at moderate interaction strength, since the FRG treats all fluctuation channels on equal footing. The first FRG results for the Hubbard model were actually obtained from three distinct FRG versions, namely Polchinski's flow equation [START_REF] Zanchi | Weakly correlated electrons on a square lattice: A renormalization group theory[END_REF][START_REF] Zanchi | Weakly correlated electrons on a square lattice: Renormalization-group theory[END_REF], a Wick ordered flow equation [START_REF] Halboth | Renormalization-group analysis of the two-dimensional Hubbard model[END_REF][START_REF] Halboth | d-Wave Superconductivity and Pomeranchuk Instability in the Two-Dimensional Hubbard Model[END_REF], and the Wetterich equation [START_REF] Honerkamp | Breakdown of the Landau-Fermi liquid in two dimensions due to umklapp scattering[END_REF]. In Fig. 17 (left panel) we show the ground state phase diagram of the 2D Hubbard model for a nearest neighbor hopping amplitude t and a next-to-nearest neighbor hopping amplitude t = -0.01t in the weak coupling regime, as obtained from the FRG flow with a momentum cutoff [START_REF] Halboth | d-Wave Superconductivity and Pomeranchuk Instability in the Two-Dimensional Hubbard Model[END_REF]. Each symbol in the plot corresponds to a choice of µ and U for which the dominant instability was identified from divergences of the two-particle vertex and susceptibilities at a critical cutoff k c . In Fig. 17 The symbols indicate whether the dominant instability is magnetic (squares) or superconducting (circles); the solid line separates the magnetic from the pairing regime (from Ref. [START_REF] Halboth | d-Wave Superconductivity and Pomeranchuk Instability in the Two-Dimensional Hubbard Model[END_REF]). Right: Pseudocritical temperature obtained from the temperature flow for the 2D Hubbard model at van Hove filling as a function of t /t for U = 3t (from Ref. [START_REF] Honerkamp | Magnetic and Superconducting Instabilities of the Hubbard Model at the Van Hove Filling[END_REF]).

vertex diverges in a temperature flow [START_REF] Honerkamp | Magnetic and Superconducting Instabilities of the Hubbard Model at the Van Hove Filling[END_REF]. Here a fixed moderate coupling U = 3t was chosen, and the chemical potential was adjusted such that the Fermi surface touches the van Hove saddle points at (π, 0) and (0, π) for each t /t. The leading instability is antiferromagnetic for small |t /t|, then d-wave pairing, and finally ferromagnetic for t < -t/3. The temperature T c in this plot is not the true critical temperature, but rather the temperature scale at which strong magnetic or pairing correlations are formed. Order parameter fluctuations suppressing the actual transition temperature, especially in two dimensions, are not captured by the second order truncation of the flow. Remarkable is the drastic suppression of the pseudocritical scale at the boundary between the pairing and the ferromagnetic regime, which indicates a quantum critical point separating two symmetry-broken phases with distinct order parameters. Within Landau theory, such a behavior could be obtained only by tuning an additional parameter. The self-energy at this quantum critical point obeys an unconventional powerlaw as a function of frequency with an exponent near 0.74, for momenta on one of the van Hove points [START_REF] Giering | Self-energy flows in the two-dimensional repulsive Hubbard model[END_REF]. This implies that Landau quasi-particles are destroyed and Fermi liquid theory breaks down at this point.

The static approximation of the two-particle vertex turned out to be insufficient beyond the weak coupling regime. Improved parametrizations are based on a channel decomposition, where the fluctuation contributions to the vertex are decomposed in charge, magnetic, and pairing channels [START_REF] Husemann | Efficient parametrization of the vertex function, ω scheme, and the t, t hubbard model at van hove filling[END_REF]567]. In the charge and magnetic channels dependences on the momentum and energy transfer variable are usually stronger than those on the remaining variables, while in the pairing channel the total momentum and energy dependences require the highest resolution. Using the channel decomposition it was shown that the frequency (= energy) dependence of the vertex is actually important already at moderate coupling [START_REF] Husemann | Frequency-dependent vertex functions of the (t, t ) Hubbard model at weak coupling[END_REF]. Neglecting it leads, in particular, to an overestimation of the energy scale for pairing in the 2D Hubbard model. Moreover, the frequency dependence is not even separable, that is, each term in the channel decomposition exhibits a substantial dependence on all three frequency variables already for moderate coupling strengths [START_REF] Vilardi | Nonseparable frequency dependence of the two-particle vertex in interacting fermion systems[END_REF].

In view of the numerical difficulties posed already by the leading (second order) truncation of the vertex flow described above, it is clear that a complete inclusion of the three-particle or even higher order vertices is not feasible. However, an approximate evaluation of contributions from higher order vertices to the flow of the two-particle vertex was devised, and it was shown that the resulting multi-loop expansion sums up all parquet diagrams [START_REF] Kugler | Multiloop Functional Renormalization Group That Sums Up All Parquet Diagrams[END_REF][START_REF] Kugler | Multiloop functional renormalization group for general models[END_REF]. Employing an accurate and economic parametrization of momentum and frequency dependences [START_REF] Tagliavini | Multiloop functional renormalization group for the twodimensional Hubbard model: Loop convergence of the response functions[END_REF] it could be shown that the multi-loop expansion really (also "in practice") converges to the parquet approximation for the twodimensional Hubbard model. The results agree very well with numerically exact Quantum Monte Carlo results for U = 2t, and still decently for a moderate interaction U = 3t [START_REF] Hille | Quantitative functional renormalisation-group description of the two-dimensional Hubbard model[END_REF]. To extend the application range of the FRG to the strong coupling regime, one needs to start the flow from a non-perturbative starting point, as described in Sec. 4.2.5.

Other models: Competing instabilities and fluctuation induced superconductivity have been analyzed via FRG flows of the two-particle vertex for numerous other models. For multi-orbital systems, such as pnictide superconductors, the unitary transformation from orbital to band operators leads to momentum dependences already in the bare vertex, even for local (Hubbard-type) interactions [START_REF] Graser | Near-degeneracy of several pairing channels in multiorbital models for the Fe pnictides[END_REF][START_REF] Maier | Origin of gap anisotropy in spin fluctuation models of the iron pnictides[END_REF]. FRG flows have provided valuable information on instabilities of various compounds, especially of those where electron-electron interactions are weak or moderate. It is beyond the scope of this review to list all these applications here. Some of the earlier applications have been described in Ref. [START_REF] Metzner | Functional renormalization group approach to correlated fermion systems[END_REF]. A more extensive review of work on multi-orbital systems performed before 2013, in particular on pnictides and compounds with hexagonal lattices (triangular, honeycomb, and Kagome), has been published by Platt et al. [START_REF] Platt | Functional renormalization group for multi-orbital Fermi surface instabilities[END_REF]. Since then the range of applications has significantly broadened, often following experimental discoveries, including systems with spin-orbit coupling [START_REF] Schober | Functional renormalization and mean-field approach to multiband systems with spin-orbit coupling: Application to the Rashba model with attractive interaction[END_REF], new Dirac nodal-line materials [START_REF] Scherer | Excitonic instability and unconventional pairing in the nodal-line materials ZrSiS and ZrSiSe[END_REF] and, most recently, moiré heterostructures [START_REF] Classen | Competing phases of interacting electrons on triangular lattices in moiré heterostructures[END_REF], to name just a few.

Σ = = G d dk d dk G . k k k k S Γ k Γ Γ k k Γ k (4) (4) (4) (4)

Spontaneous symmetry breaking

In most interacting fermion systems a symmetry of the Hamiltonian is spontaneously broken at sufficiently low temperatures, or at least in the ground state. In the FRG flow, common types of symmetry breaking such as magnetic order or superconductivity are associated with a divergence of the two-particle vertex at a finite scale k c > 0. One may encounter a divergence even in cases where ultimately only quasi-long range order or two-particle bound states are formed. To continue the flow beyond this critical scale, an order parameter needs to be implemented. There are two distinct ways of doing this.

Fermionic flows: In one approach the FRG flow is computed in the presence of a small (ideally infinitesimal) symmetry breaking field added to the bare action, which develops into a finite order parameter below the scale k c [START_REF] Salmhofer | Renormalization Group Flows into Phases with Broken Symmetry[END_REF]. The external field regularizes the divergence of the vertex at k c . This field can be endowed with a scale-dependence, so that it can be removed completely at the end of the flow [START_REF] Eberlein | Effective interactions and fluctuation effects in spin-singlet superfluids[END_REF]. One then needs to apply a manageable truncation of the exact flow equations which does not break down at the critical scale where the two-particle vertex becomes large even for a weak bare interaction. A minimal requirement is that a truncation should at least provide a decent solution for mean-field models, such as the reduced BCS model. The simplest approximation satisfying this requirement is the Katanin truncation [START_REF] Katanin | Fulfillment of Ward identities in the functional renormalization group approach[END_REF] shown in Fig. 18. Here the single-scale propagator S k in the leading contribution to the flow of the two-particle vertex (cf. Fig. 14) has been replaced by ∂ k G k . This takes self-energy corrections generated by the three-particle vertex into account, while other contributions from the three-particle vertex are neglected. Spontaneous symmetry breaking leads to anomalous contributions to both the self-energy and the two-particle vertex. The flow obtained from Katanin's truncation solves the reduced BCS model and similar mean-field models for other types of order exactly [START_REF] Salmhofer | Renormalization Group Flows into Phases with Broken Symmetry[END_REF][START_REF] Gersch | Fermionic renormalization group flow into phases with broken discrete symmetry: charge-density wave mean-field model[END_REF]. In spite of the diverging two-particle interaction the neglected contributions do not contribute because of a zero-measure momentum integration region.

The Katanin truncation has been used as an approximation to compute ground state order parameters for the twodimensional Hubbard model with attractive and repulsive interactions. For the attractive Hubbard model, the s-wave pairing gap obtained from the FRG flow [START_REF] Eberlein | Effective interactions and fluctuation effects in spin-singlet superfluids[END_REF][START_REF] Gersch | Superconductivity in the attractive Hubbard model: functional renormalization group analysis[END_REF] is substantially suppressed compared to mean-field theory even at weak coupling, in agreement with results from other methods. For the repulsive Hubbard model, the d-wave pairing gap was computed for weak and moderate interactions, and it was shown that a sizable next-to-nearest neighbor hopping amplitude is beneficial for d-wave superconductivity [START_REF] Eberlein | Superconductivity in the two-dimensional t-t -Hubbard model[END_REF]. The Ward identity protecting gapless Goldstone modes in phases with a broken continuous symmetry is not automatically satisfied by the Katanin truncation. However, one may enforce it by a suitable projection procedure [START_REF] Eberlein | Effective interactions and fluctuation effects in spin-singlet superfluids[END_REF].

Flows with order-parameter fields: Spontaneous symmetry breaking in interacting fermion systems can also be treated by introducing collective bosonic order parameter fields via a Hubbard-Stratonovich decoupling of the twofermion interaction [START_REF] Popov | Functional integrals and collective excitations[END_REF], and applying the FRG to the resulting coupled boson-fermion action. This route to symme-try breaking was first used to treat the formation of an antiferromagnetic state in the repulsive 2D Hubbard model at half-filling [START_REF] Baier | Temperature dependence of antiferromagnetic order in the Hubbard model[END_REF]. The correct low-temperature behavior (usually described by a nonlinear sigma model) was recovered from from a simple truncation of the exact flow equation. Spontaneous symmetry breaking sets in at a critical scale k c , where the boson mass vanishes, but at finite temperatures the symmetry is gradually restored by bosonic fluctuations at lower scales, in agreement with the Mermin-Wagner theorem.

Subsequently, the coupled flow of fermions and bosonic order parameter fluctuations was applied to the superfluid phase of fermion systems with attractive interactions, using U(1)-symmetric truncations of the effective fermion-boson action. A simple truncation, where the bosonic part of the effective action has the form of a Bose gas with a local φ 4 interaction, yields already sensible results for the pairing gap and the transition temperature in three dimensions in the entire range from BCS superfluidity at weak coupling to Bose-Einstein condensation of tightly bound pairs at strong coupling [START_REF] Birse | Pairing in many-fermion systems: an exact renormalization group treatment[END_REF][START_REF] Diehl | Flow equations for the BCS-BEC crossover[END_REF][START_REF] Diehl | Renormalisation flow and universality for ultracold fermionic atoms[END_REF][START_REF] Krippa | Superfluidity in many fermion systems: Exact renormalisation group treatment[END_REF]. To include effects from particle-hole fluctuations, which tend to reduce the pairing gap, one has to take effective two-fermion interactions into account. Decoupling these at each step in the flow by a scale-dependent "dynamical bosonisation" [START_REF] Gies | Renormalization flow of bound states[END_REF][START_REF] Gies | Universality of spontaneous chiral symmetry breaking in gauge theories[END_REF], one obtains a substantial quantitative improvement [START_REF] Floerchinger | Particle-hole fluctuations in the BCS-BEC Crossover[END_REF]. Instead of using a U(1)-symmetric ansatz for the effective action, it is also possible to start from the hierarchy of flow equations for the vertex functions and implement the U(1) symmetry via Ward identities [START_REF] Bartosch | Renormalization of the BCS-BEC crossover by order-parameter fluctuations[END_REF].

To distinguish the Goldstone mode from longitudinal order parameter fluctuations in a U(1)-symmetric ansatz, one needs to include quartic gradient terms in the bosonic part of the effective action [START_REF] Tetradis | Critical exponents from the effective average action[END_REF]. A relatively simple truncation of the complete effective action (for fermions and bosons) yields the correct low-energy behavior of the superfluid ground state in dimensions d ≥ 2 [START_REF] Obert | Low-energy singularities in the ground state of fermionic superfluids[END_REF].

The interplay of antiferromagnetism and superconductivity in the two-dimensional repulsive Hubbard model was also analyzed by FRG flows with order-parameter fields. It was clarified how d-wave pairing mediated by antiferromagnetic fluctuations emerges from the coupled fermion-boson flow [START_REF] Krahl | Generation of d-wave coupling in the two-dimensional Hubbard model from functional renormalization[END_REF], and a phase diagram with magnetic and superconducting order was computed at weak coupling [START_REF] Friederich | Four-point vertex in the Hubbard model and partial bosonization[END_REF][START_REF] Friederich | Functional renormalization for spontaneous symmetry breaking in the Hubbard model[END_REF].

Renormalized mean-field theory: In case of competing instabilities and possible coexistence of two or more order parameters, both routes to symmetry breaking described above become quite involved. For a computation of order parameters in the ground state one may neglect low-energy fluctuations and combine flow equations at high scales with a mean-field approximation at low scales [START_REF] Reiss | Renormalized mean-field analysis of antiferromagnetism and d-wave superconductivity in the twodimensional Hubbard model[END_REF][START_REF] Wang | Competing order in correlated electron systems made simple: Consistent fusion of functional renormalization and mean-field theory[END_REF]. In this approach the flow of the two-particle vertex Γ (4) k is stopped at a scale k mf slightly above the critical scale k c , that is, before entering the symmetry-broken regime. The remaining low-energy degrees of freedom are treated in mean-field approximation, with a reduced effective interaction extracted from Γ (4) k mf . An application of this renormalized mean-field theory revealed broad doping regions with coexistence of d-wave superconductivity and Néel or incommensurate antiferromagnetic order in the ground state of the 2D Hubbard model [START_REF] Reiss | Renormalized mean-field analysis of antiferromagnetism and d-wave superconductivity in the twodimensional Hubbard model[END_REF][START_REF] Wang | Competing order in correlated electron systems made simple: Consistent fusion of functional renormalization and mean-field theory[END_REF][START_REF] Yamase | Coexistence of Incommensurate Magnetism and Superconductivity in the Two-Dimensional Hubbard Model[END_REF].

FRG in the two-particle irreducible (2PI) formalism: Although the 2PI formalism in the framework of the FRG has been considered in various contexts [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Nagy | Functional renormalization group for quantized anharmonic oscillator[END_REF][START_REF] Polonyi | Renormalization group in internal space[END_REF][START_REF] Blaizot | Exact renormalization group and Φ-derivable approximations[END_REF], few works have focused on interacting fermions. In the 2PI approach [START_REF] Luttinger | Ground-State Energy of a Many-Fermion System[END_REF][START_REF] Baym | Conservation Laws and Correlation Functions[END_REF][START_REF] Baym | Self-Consistent Approximations in Many-Body Systems[END_REF][START_REF] Dominicis | Stationary Entropy Principle and Renormalization in Normal and Superfluid Systems. I. Algebraic Formulation[END_REF][START_REF] Dominicis | Stationary Entropy Principle and Renormalization in Normal and Superfluid Systems. II. Diagrammatic Formulation[END_REF][START_REF] Cornwall | Effective action for composite operators[END_REF], the scale-dependent effective action Γ k [G] is a functional of the one-particle propagator and the Wetterich equation becomes a flow equation for the scale-dependent Luttinger-Ward functional Φ k [G] [START_REF] Wetterich | Bosonic effective action for interacting fermions[END_REF][START_REF] Dupuis | Renormalization group approach to interacting fermion systems in the two-particle-irreducible formalism[END_REF][START_REF] Dupuis | Nonperturbative renormalization-group approach to fermion systems in the two-particle-irreducible effective action formalism[END_REF][START_REF] Rentrop | Two-particle irreducible functional renormalization group schemes-a comparative study[END_REF]. The "classical" variable is thus bosonic in nature and is itself an order parameter. There are no conceptual difficulties to describe phases with spontaneously broken symmetries and one can even start the RG flow in a brokensymmetry phase (with the Hartree-Fock-RPA theory as initial condition). The 2PI formalism has been proposed as a possible approach to the Hubbard model [START_REF] Dupuis | Nonperturbative renormalization-group approach to fermion systems in the two-particle-irreducible effective action formalism[END_REF], in particular in combination with dynamical mean-field theory [START_REF] Katanin | Extended dynamical mean field theory combined with the two-particle irreducible functional renormalization-group approach as a tool to study strongly correlated systems[END_REF] (see Sec. 4.2.5). It has also been applied to quantum impurity models [START_REF] Rentrop | Renormalization group flow of the Luttinger-Ward functional: Conserving approximations and application to the Anderson impurity model[END_REF]. Finally, the 2PI FRG has been used in connection with density functional theory to study many-body systems in condensed or nuclear matter [START_REF] Polonyi | Effective action and density-functional theory[END_REF][START_REF] Kemler | Towards a renormalization group approach to density functional theory-general formalism and case studies[END_REF][START_REF] Yokota | Functional renormalization-group calculation of the equation of state of one-dimensional uniform matter inspired by the Hohenberg-Kohn theorem[END_REF][START_REF] Yokota | Ab initio description of excited states of 1D uniform matter with the Hohenberg-Kohn-theoreminspired functional-renormalization-group method[END_REF][START_REF] Yokota | Functional-renormalization-group aided density functional analysis for the correlation energy of the two-dimensional homogeneous electron gas[END_REF].

Quantum transport

Electronic transport through nanostructured devices such as quantum wires and quantum dots is an important area in modern condensed matter physics. The typical setup is a mesoscopic region coupled to two or more leads. While the electrons in the leads move more or less independently, interesting correlation effects due to the restricted geometry can occur in the mesoscopic region [START_REF] Hanson | Spins in few-electron quantum dots[END_REF].

In general, the calculation of transport properties in interacting systems requires the calculation of two-particle quantities such as current-current correlation functions [START_REF] Mahan | Many-Particle Physics[END_REF]. Under certain assumptions, the electronic transport through a mesoscopic region can be viewed as a scattering problem, and the electric conductance can be written in the Landauer-Büttiker form [START_REF] Landauer | Spatial Variation of Currents and Fields Due to Localized Scatterers in Metallic Conduction[END_REF][START_REF] Büttiker | Four-Terminal Phase-Coherent Conductance[END_REF]

C = - e 2 2π ζ ˆd T ζ ( ) f ( ) , (68)
where ζ labels the scattering channels, T ζ ( ) is the transmission probability for each channel, and f ( ) is the Fermi function. Note that we use natural units where = 1 such that the conductance quantum is e 2 /(2π). For a frequencyindependent self-energy, Eq. ( 68) is actually exact, and the transmission probability can be expressed by the oneparticle propagator at the contacts connecting the interacting region to the leads [START_REF] Oguri | Transmission Probability for Interacting Electrons Connected to Reservoirs[END_REF].

An important quantum transport problem involving correlated electrons is posed by electrons moving through a one-dimensional metallic wire with one or a few impurities [START_REF] Kane | Transmission through barriers and resonant tunneling in an interacting one-dimensional electron gas[END_REF][START_REF] Matveev | Tunneling in one-dimensional non-Luttinger electron liquid[END_REF]. Interacting electrons in one-dimensional metals form a Luttinger liquid [START_REF] Giamarchi | Quantum physics in one dimension[END_REF], which, unlike Fermi liquids, has no resemblance to a Fermi gas. While perturbation theory diverges at any interaction strength for these systems, a strikingly simple truncation of the FRG flow equations captures the most important effects related to non-magnetic impurities in one-dimensional metals [START_REF] Andergassen | Functional renormalization group for Luttinger liquids with impurities[END_REF][START_REF] Meden | Scaling behavior of impurities in mesoscopic Luttinger liquids[END_REF]. Indeed, already the first equation in the flow equation hierarchy (see Fig. 14), with the bare instead of the effective two-particle vertex, yields several power-laws describing the low-energy behavior, where the exponents are correct to leading order in the interaction strength. Since the self-energy is frequency independent in this first order truncation, the conductance can be computed without further approximations from the one-particle propagator [START_REF] Enss | Impurity and correlation effects on transport in one-dimensional quantum wires[END_REF]. The conductance through a wire with a single impurity vanishes with a power-law as a function of temperature as expected [START_REF] Kane | Transmission through barriers and resonant tunneling in an interacting one-dimensional electron gas[END_REF]. In case of two impurities a rich behavior involving resonant tunneling and crossovers between distinct power-laws could be obtained [START_REF] Enss | Impurity and correlation effects on transport in one-dimensional quantum wires[END_REF][START_REF] Meden | Correlation effects on resonant tunneling in one-dimensional quantum wires[END_REF].

The same FRG approach turned out to be equally fruitful in a computation of persistent currents in a mesoscopic Luttinger liquid ring [START_REF] Meden | Persistent currents in mesoscopic rings: A numerical and renormalization group study[END_REF][START_REF] Meden | Conductance of interacting nanowires[END_REF], in a study of quantum transport through a Y-junction of three Luttinger liquid wires [START_REF] Barnabé-Thériault | Junction of Three Quantum Wires: Restoring Time-Reversal Symmetry by Interaction[END_REF], and to describe transport through a correlated quantum dot setup dominated by charge fluctuations (absence of the Kondo effect) [START_REF] Karrasch | Functional renormalization group study of the interacting resonant level model in and out of equilibrium[END_REF].

The FRG was also extended to steady-state nonequilibrium transport through quantum wires and quantum dots in the presence of a finite voltage [START_REF] Jakobs | Nonequilibrium Functional Renormalization Group for Interacting Quantum Systems[END_REF][START_REF] Karrasch | Functional renormalization group study of the interacting resonant level model in and out of equilibrium[END_REF][START_REF] Gezzi | Functional renormalization group for nonequilibrium quantum many-body problems[END_REF]. This extension makes use of real-time or real-frequency Green functions on the Keldysh contour [START_REF] Rammer | Quantum field-theoretical methods in transport theory of metals[END_REF]. Compared to the equilibrium imaginary frequency FRG, in the steady state the only technical complication is an additional Keldysh index ±. An important issue is to define a flow that does not violate causality and Kubo-Martin-Schwinger (KMS) relations after truncation of the flow equation hierarchy [START_REF] Jakobs | Properties of multi-particle Green's and vertex functions within Keldysh formalism[END_REF]. Suitable choices are a judiciously designed imaginary frequency cutoff [START_REF] Jakobs | Nonequilibrium Functional Renormalization Group for Interacting Quantum Systems[END_REF][START_REF] Jakobs | Properties of multi-particle Green's and vertex functions within Keldysh formalism[END_REF] or scale dependent local auxiliary terms, which are removed at the end of the flow [START_REF] Jakobs | Nonequilibrium functional renormalization group with frequency-dependent vertex function: A study of the single-impurity Anderson model[END_REF][START_REF] Jakobs | Properties of multi-particle Green's and vertex functions within Keldysh formalism[END_REF]. The non-equilibrium FRG was applied to simple models of quantum dots [START_REF] Karrasch | Functional renormalization group study of the interacting resonant level model in and out of equilibrium[END_REF] and quantum wires [START_REF] Jakobs | Nonequilibrium Functional Renormalization Group for Interacting Quantum Systems[END_REF]. It is even possible to compute time-dependent non-equilibrium transport of setups with correlated mesoscopic regions using Keldysh-FRG. This approach was used to study the quench dynamics of quantum dots [START_REF] Kennes | Renormalization group approach to time-dependent transport through correlated quantum dots[END_REF] and quantum wires [START_REF] Kennes | Luttinger liquid properties of the steady state after a quantum quench[END_REF], as well as the dynamics of periodically driven systems [START_REF] Eissing | Renormalization in Periodically Driven Quantum Dots[END_REF]. The FRG on the Keldysh contour can also be used to compute real frequency quantities in equilibrium, avoiding thus the numerically unstable analytic continuation from imaginary to real frequencies [START_REF] Jakobs | Nonequilibrium functional renormalization group with frequency-dependent vertex function: A study of the single-impurity Anderson model[END_REF][START_REF] Kedri | Nonequilibrium thermoelectric transport through vibrating molecular quantum dots[END_REF].

Leap to strong coupling

A truncation of the FRG hierarchy of flow equations at some finite order can be justified only for weak interactions [START_REF] Salmhofer | Fermionic Renormalization Group Flows[END_REF], with the exception of mean-field models where phase space restrictions suppress higher order contributions [START_REF] Salmhofer | Renormalization Group Flows into Phases with Broken Symmetry[END_REF]. Although bare interactions are usually two-particle interactions, effective m-particle interactions with m > 2 are generated by the flow and affect the effective two-particle interaction and the self-energy, see Fig. 14. For strong bare interactions, these contributions become important already at high energy scales far above the critical scales for instabilities. One of the most important strong coupling phenomena in correlated lattice electron systems like the Hubbard model is the Mott metal-insulator transition. Cuprate high-temperature superconductors are doped Mott insulators [START_REF] Anderson | The Resonating Valence Bond State in La 2 CuO 4 and Superconductivity[END_REF]. Truncated FRG flow equations starting from the bare action do not capture the Mott transition.

For interacting fermion systems there are no non-perturbative approximations such as the derivative expansion of the exact flow equation for bosons. However, one may choose a suitable non-perturbative approximation as a starting point for the flow. The Mott transition is essentially a consequence of strong local correlations. As such, it is well described by the dynamical mean-field theory (DMFT), which treats local correlations non-perturbatively [START_REF] Metzner | Correlated Lattice Fermions in d = ∞ Dimensions[END_REF][START_REF] Georges | Hubbard model in infinite dimensions[END_REF][START_REF] Georges | Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions[END_REF]. The DMFT and the FRG can be consistently merged in the socalled DMF 2 RG [START_REF] Wentzell | Correlated starting points for the functional renormalization group[END_REF][START_REF] Taranto | From Infinite to Two Dimensions through the Functional Renormalization Group[END_REF]. In this approach the FRG flow does not start from the bare action of the system, but rather from the DMFT solution. Strong local correlations and the Mott physics are thus captured via the DMFT starting point, while the nonlocal correlations are generated by the FRG flow. While an obvious small expansion parameter is still lacking, the weaker nonlocal correlations may be captured by a manageable truncation of the exact FRG hierarchy. The flow can be defined such that it is generated exclusively by non-local correlations [START_REF] Vilardi | Antiferromagnetic and d-wave pairing correlations in the strongly interacting two-dimensional Hubbard model from the functional renormalization group[END_REF].

Concrete DMF 2 RG calculations have so far been performed with a truncation of the flow equations at the twoparticle level [START_REF] Taranto | From Infinite to Two Dimensions through the Functional Renormalization Group[END_REF][START_REF] Vilardi | Antiferromagnetic and d-wave pairing correlations in the strongly interacting two-dimensional Hubbard model from the functional renormalization group[END_REF], involving thus the two-particle vertex and the self-energy. For strong interactions, the twoparticle vertex exhibits strong frequency dependencies which cannot be reduced to one frequency per interaction channel. Nevertheless, numerical solutions of the DMF 2 RG flow equations have been obtained for the two-dimensional Hubbard model in the strong coupling regime, and evidence for d-wave pairing at the expected (for cuprates) temperature scale has been found [START_REF] Vilardi | Antiferromagnetic and d-wave pairing correlations in the strongly interacting two-dimensional Hubbard model from the functional renormalization group[END_REF].

Recently, a combination of the extended DMFT for non-local interactions with the 2PI version of the FRG has been proposed [START_REF] Katanin | Extended dynamical mean field theory combined with the two-particle irreducible functional renormalization-group approach as a tool to study strongly correlated systems[END_REF]. Encouraging first benchmarks were obtained for the conventional and the extended Hubbard model at half-filling.

Omissions

The range of applications of the FRG has become too broad to be covered in a single review. Here we briefly mention two important devopments related to fermion systems which we could not review in detail.

Spin systems: Using an auxiliary fermion representation of spin degrees of freedom, the FRG can also be applied to quantum spin systems [START_REF] Reuther | J 1 -J 2 frustrated two-dimensional Heisenberg model: Random phase approximation and functional renormalization group[END_REF][START_REF] Reuther | Functional renormalization group for the anisotropic triangular antiferromagnet[END_REF]. The pseudofermions carry no kinetic energy, and the interactions depend only on one space or momentum variable. Using the Katanin truncation [START_REF] Katanin | Fulfillment of Ward identities in the functional renormalization group approach[END_REF] of the flow equations and keeping the full frequency dependence of the self-energy and interaction vertex leads to an approximation scheme that works remarkably well for a variety of spin systems. This success of a perturbative truncatation for a strongly coupled system can be partially understood since leading orders in 1/N and 1/S expansions are captured by the approximation. The accuracy can be systematically improved by using decoupled clusters instead of single pseudo-fermions as starting point for an expansion [START_REF] Reuther | Cluster functional renormalization group[END_REF]. Alternative ways of treating quantum spin systems with the FRG have been developed by mapping spins on hard core bosons [START_REF] Ranc ¸on | Higgs amplitude mode in the vicinity of a (2 + 1)-dimensional quantum critical point[END_REF] or working directly with spin operators [START_REF] Krieg | Exact renormalization group for quantum spin systems[END_REF].

Quantum criticality: Numerous quantum many-body systems undergo a continuous transition between ground states with distinct symmetry or topology upon tuning a non-thermal control parameter. In the vicinity of the quantum critical point separating the two phases, quantum fluctuations are particularly important and lead to unconventional properties not only in the ground states, but also at low finite temperatures [START_REF] Sachdev | Universal relaxational dynamics near two-dimensional quantum critical points[END_REF]. In metallic systems quantum critical fluctuations prevent the existence of Landau quasi-particles. Fermi liquid theory thus breaks down and unconventional, so-called non-Fermi liquid behavior is observed [START_REF] Löhneysen | Fermi-liquid instabilities at magnetic quantum phase transitions[END_REF]. A purely bosonic order parameter theory for quantum critical metals as developed by Hertz [START_REF] Hertz | Quantum critical phenomena[END_REF] and Millis [START_REF] Millis | Effect of a nonzero temperature on quantum critical points in itinerant fermion systems[END_REF] is problematic since gapless fermionic excitations lead to singular interactions between the order parameter fluctuations. This problem is particularly serious in low-dimensional systems. Hence, for a complete and controlled treatment a coupled theory treating fermions (with a Fermi surface) and critical order parameter fluctuations on equal footing is required.

The FRG is an ideal tool to study quantum critical systems, and quantum critical fermion systems in particular. Some of the FRG work on quantum critical metals and semi-metals done before 2012 has been reviewed in Ref. [START_REF] Metzner | Functional renormalization group approach to correlated fermion systems[END_REF]. Hertz-Millis theory can be formulated as a simple truncation of the bosonic FRG with few running couplings [START_REF] Metzner | Functional renormalization group approach to correlated fermion systems[END_REF]. Some studies of metallic quantum critical points have been performed by using perturbative truncations for the coupled flow of fermions and order parameter fluctuations. This procedure helped clarifying, in particular, the intriguing case of quantum criticality at the onset of antiferromagnetic order in two dimensions [START_REF] Maier | Universality of antiferromagnetic strange metals[END_REF]. Since order parameter fluctuations are described by bosonic fields, their action is amenable to non-perturbative truncations. Non-perturbative truncations are required, in particular, to establish fluctuation-induced continuous quantum phase transitions in certain systems which exhibit only first order transitions on the mean-field level. Illustrative examples are the nematic phase transition in two-dimensional metals [START_REF] Jakubczyk | Turning a First Order Quantum Phase Transition Continuous by Fluctuations: General Flow Equations and Application to d-Wave Pomeranchuk Instability[END_REF], and gapless Dirac fermions coupled to a Z 3 symmetric order parameter within a two-dimensional Gross-Neveu model [START_REF] Classen | Fluctuation-induced continuous transition and quantum criticality in Dirac semimetals[END_REF]. More references, especially on relativistic critical fermion systems, are provided at the end of Sec. 3.2.2.

High-energy physics

Introduction: High-energy physics

High energy physics is very successfully described by the Standard Model (SM) of particle physics. Its different parts, QCD and the electroweak gauge theory with the Higgs sector give rise to many fascinating phenomena that either quantitatively or even qualitatively can only be described with nonperturbative methods. Most notably, together with quantum gravity discussed in Sec. 6, high energy physics is described with Abelian and non-Abelian gauge theories whose properties are responsible for quite some of its nonperturbative physics.

A paradigmatic example is QCD, and most of the conceptual investigations within the FRG as well as the physics results have been achieved here. Accordingly, we mostly concentrate on QCD in the following. At large momentum scales, QCD is asymptotically free, and the running strong coupling α s (p) = g 2 s (p)/(4π) decays logarithmically with the momentum scale p 2 . The converse of this behaviour is an increasing strong coupling in the low energy (infrared) regime of QCD. In this regime, QCD also exhibits spontaneous chiral symmetry breaking and confinement as well as the formation of a rich bound-state spectrum. While an increasing coupling does not necessarily entail the breakdown of perturbation theory and is in particular not tantamount to confinement, a full understanding of the latter as well as of spontaneous chiral symmetry breaking certainly requires nonperturbative techniques. Moreover, while highenergy QCD is well described with perturbation theory for weakly-interacting quarks and gluons due to asymptotic freedom, low-energy QCD is well-described by low-energy effective theories with weakly-interacting hadrons. This dynamical change of the relevant degrees of freedoms as well as the access to the emerging hadron spectrum are further challenges to be met.

For the physics of the early universe as well as that of heavy-ion collisions ('little big bangs') we also require an understanding of the rich phase structure of QCD at finite temperature T and density n or quark chemical potential µ q : At large temperatures the theory is deconfined and chirally symmetric (except for the current quark masses induced by the Higgs mechanism). Most of its properties are well-described by thermal perturbation theory of quarks and gluons. 29 At low temperatures we enter the hadronic low-energy regime with chiral symmetry breaking and confinement. At vanishing density and physical quark masses the chiral transition is a crossover, while at larger density the chiral crossover may turn into a 1st-order phase transition at a 2nd-order critical end point (CEP). Whether or not this CEP exists, as well as its location and further QCD properties at large densities such as possible inhomogeneous phases and so-called condensed matter phases with color superconductivity are pressing open questions. Their resolution gives e.g. access to the QCD equation of state at large densities which is required for the physics of neutron stars.

Finally, the physics of a heavy ion collision in its early stages as well as the QCD phase transition in the early universe are genuinely non-equilibrium processes. Their understanding necessitates the computation of real-time (in and out-of equilibrium) correlation functions. These correlation functions are relevant, e.g., for transport and hydrodynamic phases in the above-mentioned non-equilibrium evolutions. Moreover, time-like correlation functions are already required for unravelling the rich hadronic structure of QCD. More details can be found in Sec. 5.3.

The treatment of gauge theories within the FRG as required for high energy physics and quantum gravity is discussed in Sec. 5.2. Readers, who are either already familiar with the setup or are more interested in the physics of QCD and high energy physics in general may skip this part for a first reading and jump to Sec. 5.3 and beyond.

The functional renormalization group for gauge theories

In this section we briefly discuss some important features of the FRG-formulation of gauge theories at the example of an SU(N) gauge theory with the classical Euclidean Yang-Mills action,

S YM [A] = 1 2 ˆx tr F µν F µν = 1 4 ˆx F a µν F a µν , with F µν = F a µν t a , (69)
with ´x = ´d4 x. The fieldstrength tensor in [START_REF] Kopietz | Introduction to the Functional Renormalization Group[END_REF] is the curvature tensor

F µν = i/g s [D µ , D ν ],
where the covariant derivative and the generators t a of the group are given by

D µ = ∂ µ -ig s A µ , with A µ = A a µ t a , and [t a , t b ] = i f abc t c , tr t a t b = 1 2 δ ab . ( 70
)
The traces in ( 69), [START_REF] Braun | Fermion Interactions and Universal Behavior in Strongly Interacting Theories[END_REF] are taken in the fundamental representation. The non-Abelian gauge field in (70) has component fields A a µ with a = 1, ..., N 2 -1, and lives in the Lie algebra su(N) of the gauge group SU(N). The generators t a are normalized according to [START_REF] Braun | Fermion Interactions and Universal Behavior in Strongly Interacting Theories[END_REF] and satisfy the su(N)-Lie algebra in [START_REF] Braun | Fermion Interactions and Universal Behavior in Strongly Interacting Theories[END_REF]. The components of the fieldstrength tensor F µν used in [START_REF] Kopietz | Introduction to the Functional Renormalization Group[END_REF] are given by

F a µν = ∂ µ A a ν -∂ ν A a µ + g s f abc A b µ A c ν . (71)
The gauge theories introduced above encompass the Standard-Model with the gauge group U(1) × S U(2) × S U(3).

There, the coupling to matter is given by Dirac actions for the fermionic matter fields, leptons and quarks, in the fundamental representation. For the present example we restrict ourselves to QCD with the coupling of the gluon A µ ∈ su(3) to the quarks (q i ) A ξ . Here A = 1, 2, 3 is the color index in the fundamental representation and i = 1, ..., N f counts the quark flavors. The quarks are Dirac fermions with the Dirac index ξ = 1, ..., 4. The Dirac term in Euclidean spacetime is given by

S Dirac [q, q, A] = ˆx q i / D + m q + γ 0 µ q q , with / D = γ µ D µ , and {γ µ , γ ν } = 2δ µν , (72)
In [START_REF] Gies | Introduction to the Functional RG and Applications to Gauge Theories[END_REF] we have suppressed gauge group, Dirac and flavor indices, and introduced a mass matrix m q in flavor space as well as a quark chemical potential. The Yang-Mills action (69) and the matter term in [START_REF] Gies | Introduction to the Functional RG and Applications to Gauge Theories[END_REF] are invariant under gauge transformations with

S YM [A U ] + S Dirac [q, q, A U ] = S YM [A] + S Dirac [q, q, A].
The gauge group element is given by U(x) = e i ω(x) with ω ∈ su(N), and the gauge transformation reads

A µ → A U µ = i g U D µ U † = UA µ U † + i g U(∂ µ U † ) , q → Uq, q → qU † . (73)
Naively, we would simply adopt the general formulation of the FRG discussed in Sec. 2 for bosons and in Sec. 4.2 for fermions, and substitute the scalar field with the gauge field. However, the standard formulation of the functional renormalization group rests on a momentum cutoff with a cutoff term that is quadratic in the field, see [START_REF] Symanzik | Small distance behavior in field theory and power counting[END_REF]. Both properties are tightly related to the practical accessibility of the flow equation for the scale dependent effective action, [START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF], and giving them up comes at a high price. For this reason the commonly-used FRG-approach to gauge theories utilises a quadratic momentum cutoff term in the gauge field, which breaks gauge invariance for finite cutoff scales k 0. Moreover, the right hand side of (7) depends on the full propagator of the theory, and in particular its gauge field component

G AA (x 1 , x 2 ) = 1 Γ (2) + R k AA (x 1 , x 2 ) . (74)
Accordingly, the flow equation requires the existence of the inverse of the full kinetic operator of the theory in the presence of the infrared regulator, Γ (2) + R k .

Gauge-fixed flows and modified Slavnov-Taylor identities

The inverse in ( 74) is only defined with a gauge fixing due to gauge redundancies carried by the gauge field. Hence the standard FRG-formulation of gauge theories requires gauge-fixing. The most common choice in nonperturbative functional approaches (FRG, Dyson-Schwinger equations, n-particle irreducible approaches) is the Landau gauge, in which the covariant gauge is strictly implemented: The respective gauge-fixing term in the presence of a background field reads

S gf [ Ā, a] = 1 2ξ ˆx( Dµ a µ ) 2 , Dµ = ∂ µ -i g s Āµ ,
where

A µ = Āµ + a µ . (75)
In [START_REF] Morris | Equivalence of local potential approximations[END_REF] we have introduced the split of the full gauge field into the sum of an auxiliary background field Āµ and a dynamical fluctuation field a µ . The strict implementation of the gauge ( 75) is achieved for ξ = 0. Then, only configurations with Dµ a µ = 0 survive in the path integral. With a non-vanishing background, [START_REF] Morris | Equivalence of local potential approximations[END_REF] with ξ = 0 is also called the Landau-DeWitt gauge.30 In non-Abelian gauge theories or non-linear gauges in Abelian theories the transformation from an unrestricted integration over the gauge field to one over gauge-fixed configurations has a gauge-field dependent Jacobi-determinant ∆ F [ Ā, a] (Faddeev-Popov determinant) for a given gauge-fixing condition F [A] = 0. For Lorentz gauges we have

∂ t Γ k [Φ] = 1 2 - - + 1 2
F [A] = ∂ µ A µ .
Typically, it is taken into account as a Grassmann integral over auxiliary fermionic fields, the Faddeev-Popov ghosts. For the gauge fixing (75) the ghost action reads

S gh [ Ā, a, c, c] = -ˆx ca ( Dµ D µ ) ab c b , (76)
with a negative quadratic dispersion for the ghost field. The appeal of the background-field approach is the existence of a gauge-invariant effective action, the background field effective action Γ[A] = Γ[ Ā = A, a = 0] that can be defined at vanishing fluctuation gauge field a = 0. Here, for the sake of simplicity we have suppressed the dependence of Γ on all the other fields c, c, q, q, .... Note that the background field effective action Γ[A] inherits gauge invariance from gauge transformations of the auxiliary background field Āµ . The equivalence of this auxiliary symmetry with physical gauge invariance of the theory follows from (onshell) background-independence of the approach and the Slavnov-Taylor identities (STIs). The latter encode the gauge-transformation properties or BRST-symmetries of the dynamical fluctuation field. Background independence, typically encoded in Nielsen identities, and the STIs can also be used to show, that the background field correlation functions are related to S-matrix elements similarly to the fluctuation correlation functions, see e.g. [START_REF] Abbott | The Background Field Method and the S Matrix[END_REF].

The background effective action Γ[A] can be expanded in gauge-invariant operators leading to easily accessible approximation schemes. Consequently, many of the early applications have been performed in the background field approach, ranging from computations of the (cutoff-) running of the QCD and Yang-Mills couplings towards the infrared to computations of the infrared effective potential of tr F 2 µν . For applications to Yang-Mills theories see [START_REF] Reuter | Effective average action for gauge theories and exact evolution equations[END_REF][START_REF] Reuter | Indications for gluon condensation for nonperturbative flow equations[END_REF][START_REF] Wetterich | Integrating out gluons in flow equations[END_REF][START_REF] Reuter | Gluon condensation in nonperturbative flow equations[END_REF][START_REF] Gies | Running coupling in Yang-Mills theory: A flow equation study[END_REF][START_REF] Gies | Renormalizability of gauge theories in extra dimensions[END_REF][START_REF] Codello | Renormalization group flow equations for the proper vertices of the background effective average action[END_REF], for QED and scalar QED (Abelian Higgs model) see [START_REF] Gies | Renormalization flow of QED[END_REF] and [START_REF] Bergerhoff | Phase diagram of superconductors[END_REF][START_REF] Bergerhoff | Phase transition of N component superconductors[END_REF][START_REF] Reuter | Average action for the Higgs model with Abelian gauge symmetry[END_REF][START_REF] Reuter | Exact evolution equation for scalar electrodynamics[END_REF][START_REF] Freire | Charge crossover at the U(1) Higgs phase transition[END_REF] respectively. The electroweak sector of the Standard Model was discussed in [START_REF] Reuter | Running gauge coupling in three-dimensions and the electroweak phase transition[END_REF].

In the gauge-fixed approach with gauge fixing (75) and ghost action [START_REF] Papenbrock | Two-loop results from improved one loop computations[END_REF], the flow equation for the effective action of gauge theories with and without background field is based on a quadratic cutoff term for both the dynamical gauge field A µ and the ghost field. These cutoff terms read

∆S A,k [A] = ˆp A a µ (-p) R ab k,µν (p) A b ν (p) , ∆S gh,k [c, c] = ˆp ca (-p) R ab k (p) c b (p) , (77)
and the corresponding flows are given in the first two terms in Fig. 19. There we depict the flow equation for QCD with dynamical quarks and mesonic low-energy degrees of freedom as an illustrative example and for later use. The implementation of the latter low-energy degrees of freedom in first principles QCD is discussed in Sec. 5.3.3. Both cutoff terms in (77) explicitly break gauge invariance, as do the cutoff terms for the colored matter fields. This entails that the STIs are modified. Moreover, while background gauge invariance can be maintained with the substitution p 2 → -D2 in the cutoff terms, background independence is lost, also manifested in modified Nielsen identities. Finally, the propagator of the gauge field in the flow equation for the background field effective action Γ[A] is that of the fluctuation field a µ , since the flow

∂ t Γ k [A] is simply ∂ t Γ k [A, a = 0].
Here we have used the (inverse) RG-time t = ln k/Λ with some reference scale Λ. In summary, for a finite cutoff scale the underlying gauge invariance is broken by the cutoff terms and the gauge invariance of the background field effective action ceases to be physical.

In the limit k → 0 background independence as well as the standard STIs are recovered in a controlled way: First we note that for a non-linear gauge symmetry the breaking of gauge invariance is necessarily (at least) quadratic in the gauge field. This leads to loop terms in symmetry identities such as the Slavnov-Taylor identities. A derivation and discussion is beyond the scope of the present review, here we simply quote it in a very convenient form, the modified Quantum Master Equation (mQME) or modified Slavnov Taylor identity (mSTI) ˆx δΓ k δQ i (x)

δΓ k δΦ i (x) = Tr R i j G jl δ 2 Γ k [Φ, Q] δΦ l δQ i , Φ = (A µ , c, c, ....) . (78)
In [START_REF] Morris | Convergence of derivative expansions of the renormalization group[END_REF], the superfield Φ contains all fields of the given theory including auxiliary fields. The indices i, j, l comprise all internal (group, flavor) and Lorentz indices as well as the species of fields and are summed over. The trace on the right hand side sums over spacetime or momentum space. This condensed notation is similar to that used in [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF] (a variation of DeWitt's condensed notation). The left hand side of ( 78) is nothing but the Zinn-Justin equation in a concise form: In order to arrive at this form one has to augment the fields with an additional auxiliary field, the Nakanishi-Lautrup field B. In QCD this leads to Φ = (A µ , c, c, B, q, q). Moreover, the source term ´x J i Φ i in the path integral is amended by sources for BRST-transformations ´x Q i sΦ i . Here Q i are the BRST-sources of the fields Φ i , and(δΓ/δQ i )δ/δΦ i generates (quantum) BRST-transformations.

The right hand side is the modification of the STI in the presence of the cutoff term: it is proportional to the regulator function R k and the full propagator G jl of the fields Φ i , Φ l . This term resembles the flow equation itself. There, the loop carries the violation of the scaling relations of the underlying theory by the cutoff term, here it is the violation of BRST-symmetry.

This allows us to discuss the physical limit k → 0: Naturally, the right hand side of (78) vanishes for R k ≡ 0. Note however that this is not enough to guarantee a well-defined limit in the sense that we arrive at the original massless gauge theory. Another important property is the fact that the breaking of BRST-invariance in ( 78) is local in momentum space: the operator in the trace decays for large momenta due to the regulator. It is this important property that signifies the breaking of gauge invariance as a local perturbation. In summary this leads to a well-defined and smooth limit of the gauge theory within a gauge symmetry-breaking regularization to the gauge theory at k = 0. The existence of this limit can be studied in perturbation theory, where an iterative solution of the flow equation in terms of loop orders is nothing but the regularization and renormalization of the gauge theory with a momentum cutoff and the respective gauge-variant counter terms: a generalized BPHZ-scheme (Bogoliubov-Parasiuk-Hepp-Zimmermannscheme).

Eq. ( 78) guarantees the standard Master equation for the effective action at k = 0. This implies BRST-invariance of the effective action as well as the gauge invariance of observables. It is left to guarantee the gauge-independence of the approach. This property is encoded in Nielsen identities, which take a convenient form in the background gauge approach with the gauge fixing (75): the background effective action Γ[ Ā, a] carries an explicit dependence on the background field Ā, which vanishes onshell on the equations of motion. There the source term in the path integral vanishes and the Fadeev-Popov trick can be undone. The Nielsen identity (NI) monitors the difference of fluctuation field derivatives and background field derivatives. In the presence of the cutoff term the Nielsen identity gets modified similarly to the mSTI. The modified Nilsen identity (mNI) reads schematically

δΓ k δa - δΓ k δ Ā - δ(S gf + S gh ) δa - δ(S gf + S gh ) δ Ā = 1 2 Tr δR i j k δ Ā G ji , (79)
within the same condensed notation used for the mSTI, [START_REF] Morris | Convergence of derivative expansions of the renormalization group[END_REF]. At vanishing cutoff the right hand side of ( 79) vanishes and we are left with the standard NI. The fluctuation field derivative of the effective action vanishes onshell (vanishing currents). This entails that the Fadeev-Popov trick can be undon, and the expectation value on the left hand side vanishes. Consequently also the background field derivative vanishes. This is how the NI encodes the onshell background-independence of the background-field approach. In summary, the modified symmetry relations ( 78) and (79) monitor the violation of gauge invariance and gauge independence for k 0 and guarantee their reinstatement at k = 0. Background independence is also of also of crucial importance in quantum gravity, where the Nielsen identity is also called split Ward identity, see [START_REF] Hohenberg | Existence of long-range order in one and two dimensions[END_REF], the respective discussion in Sec. 6, and the recent review [START_REF] Pawlowski | Quantum gravity: a fluctuating point of view[END_REF].

Monitoring global and local symmetries, and in particular BRST-symmetry and gauge independence, and hence the physical gauge symmetry of the theory, is essential for the FRG-approach to gauge theories. Hence this topic has received much interest. Modified STIs and Master equations for Abelian, non-Abelian gauge theories and gravity (a) FRG Yang-Mills Gluon propagator G A (p 2 ), [START_REF] Cyrol | Landau gauge Yang-Mills correlation functions[END_REF], in comparison to lattice data, [START_REF] Sternbeck | Lattice study of the infrared behavior of QCD Green's functions in Landau gauge[END_REF]. The shaded area for small momenta potentially signals different infrared solutions of Landau gauge Yang-Mills theory, for more details see [START_REF] Cyrol | Landau gauge Yang-Mills correlation functions[END_REF]. have been discussed and applied in [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Ellwanger | Flow equations and BRS invariance for Yang-Mills theories[END_REF][START_REF] Reuter | Gluon condensation in nonperturbative flow equations[END_REF]. For gravity applications see also Sec. 6. The related fate of anomalies and topological terms in gauge theories have been discussed in [START_REF] Bonini | Axial anomalies in gauge theory by exact renormalization group method[END_REF][START_REF] Reuter | Renormalization of the topological charge in Yang-Mills theory[END_REF][START_REF] Pawlowski | Exact flow equations and the U(1) problem[END_REF].

We close this section with two remarks:

In the absence of the regulator term on the right hand side in [START_REF] Morris | Convergence of derivative expansions of the renormalization group[END_REF] one can use powerful cohomological methods for an iterative construction of the perturbative effective action. Beyond perturbation theory, the mQME or mSTI [START_REF] Morris | Convergence of derivative expansions of the renormalization group[END_REF] at R k = 0 at least provides algebraic relations between longitudinal and transverse correlation functions which are discussed in more detail in the next section Sec. (5.2.2). This has triggered some investigations of reformulations of gauge theories and the mSTI without the modification of (78) on the right hand side, see e.g. [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Igarashi | Exact symmetries realized on the renormalization group flow[END_REF][START_REF] Lavrov | On the Functional Renormalization Group approach for Yang-Mills fields[END_REF][START_REF] Igarashi | BRST in the exact renormalization group[END_REF]. Loosely speaking these attempts either work with non-linear gauges, non-linear field transformations or further auxiliary degrees of freedom. In the formulation with the mSTI [START_REF] Morris | Convergence of derivative expansions of the renormalization group[END_REF] this amounts to absorbing its right hand side in the symmetry transformations themselves. Typically then the generator of the symmetry looses its ultralocality, a well-studied problem for chiral symmetry transformations as well as supersymmetry in lattice formulation, for a point of view close to the FRG see [START_REF] Bergner | Blocking-inspired supersymmetric actions: a status report[END_REF]. Note that symmetries, and in particular the definition of (Noether) charges require locality (exponential decay in spacetime) of the generators of this symmetry. A well-studied example is chiral symmetry on the lattice: while chiral symmetry for Wilson fermions with explicit mass terms for the fermionic doublers can be formulated, the respective generator of chiral symmetry only has an algebraic decay in spacetime, see [START_REF] Bergner | Blocking-inspired supersymmetric actions: a status report[END_REF]. This leads to a slow convergence towards the continuum limit of observables with chiral properties. In turn, the generator of chiral symmetry for Ginsparg-Wilson fermions is local, though not ultralocal, and the continuum limit of observables with chiral properties is more rapid in comparison. In summary, in most of the FRG-formulations for gauge theories without the symmetry-breaking term on the right hand side of (78) it can be shown that this locality is lost. For the remaining formulations this issue has not been studied yet.

Most of the applications of the FRG to gauge theories have been performed in the Landau gauge or the Landau-DeWitt gauge (75) given its technical and numerical advantages. In turn, so called physical gauges as well as dual formulations are advantageous for approaching the underlying conceptual problems in strongly-correlated gauge theories. This has led to progress within Abelian gauges, [START_REF] Ellwanger | Confinement, monopoles and Wilsonian effective action[END_REF], Axial gauges, [START_REF] Litim | Flow equations for Yang-Mills theories in general axial gauges[END_REF][START_REF] Simionato | Gauge consistent Wilson renormalization group.2. NonAbelian case[END_REF][START_REF] Simionato | Gauge consistent Wilson renormalization group: Abelian case[END_REF][START_REF] Litim | Renormalization group flows for gauge theories in axial gauges[END_REF], the Polyakov gauge, [START_REF] Marhauser | Confinement in Polyakov Gauge[END_REF][START_REF] Kondo | Toward a first-principle derivation of confinement and chiral-symmetry-breaking crossover transitions in QCD[END_REF], within the Cho-Faddeev-Niemi decomposition, [START_REF] Gies | Wilsonian effective action for SU(2) Yang-Mills theory with Cho-Faddeev-Niemi-Shabanov decomposition[END_REF], the use of the Landau gauge with fieldstrength formulation for the correlator of the fieldstrength in [START_REF] Ellwanger | Field strength correlator and an infrared fixed point of the Wilsonian exact renormalization group equations[END_REF], the dual fieldstrength in [START_REF] Ellwanger | Monopole condensation and antisymmetric tensor fields: compact QED and the Wilsonian RG flow in Yang-Mills theories[END_REF], as well as the formulation of massive Yang-Mills theory in Abelian gauges in [START_REF] Ellwanger | Massive Yang-Mills theory in Abelian gauges[END_REF] and in the Abelian theory in [START_REF] Ellwanger | Confinement and mass gap in Abelian gauge[END_REF]. Hamiltonian FRG flows have been derived and applied to the infrared sector of Coulomb gauge Yang-Mills theory in [START_REF] Leder | Hamiltonian Flow in Coulomb Gauge Yang-Mills Theory[END_REF][START_REF] Leder | Color Coulomb Potential in Yang-Mills Theory from Hamiltonian Flows[END_REF].

Gauge invariance, locality & confinement

The functional setup developed in the last sections in terms of gauge-fixed correlation functions allows us to discuss some important aspects of functional approaches to gauge theories related to gauge invariance, locality as well as the regularity of correlation functions. These aspects are also relevant for the access to confinement in Yang-Mills theory and QCD in these approaches, mostly done in the Landau(-DeWitt) gauge, ξ = 0 in [START_REF] Morris | Equivalence of local potential approximations[END_REF].

The current argument follows the analysis for Yang-Mills theory in [START_REF] Cyrol | Landau gauge Yang-Mills correlation functions[END_REF] within the FRG approach. This work, together with corresponding ones for quenched QCD [START_REF] Mitter | Chiral symmetry breaking in continuum QCD[END_REF], unquenched QCD, [START_REF] Cyrol | Nonperturbative quark, gluon, and meson correlators of unquenched QCD[END_REF] as well finite temperature Yang-Mills theory, [START_REF] Cyrol | Nonperturbative finite-temperature Yang-Mills theory[END_REF][START_REF] Hajizadeh | Exploring the Tan contact term in Yang-Mills theory[END_REF] and three-dimensional Yang-Mills theory [START_REF] Corell | Correlation functions of three-dimensional Yang-Mills theory from the FRG[END_REF] constitute the technically most advanced FRG-QCD computations including the nonperturbative infrared regime of QCD. This line of works has been initiated in [START_REF] Ellwanger | The Running gauge coupling in the exact renormalization group approach[END_REF][START_REF] Ellwanger | Flow equations and BRS invariance for Yang-Mills theories[END_REF][START_REF] Ellwanger | Flow equations for the relevant part of the pure Yang-Mills action[END_REF][START_REF] Ellwanger | The Heavy quark potential from Wilson's exact renormalization group[END_REF][START_REF] Bergerhoff | Effective quark interactions and QCD propagators[END_REF]. While the results in [START_REF] Cyrol | Landau gauge Yang-Mills correlation functions[END_REF][START_REF] Cyrol | Nonperturbative quark, gluon, and meson correlators of unquenched QCD[END_REF][START_REF] Mitter | Chiral symmetry breaking in continuum QCD[END_REF][START_REF] Cyrol | Nonperturbative finite-temperature Yang-Mills theory[END_REF]] also include vertex dressings, we refer to the original works for the details. In Fig. 20 we only depict a selection of results for the gluon propagator as well as the running couplings in vacuum Yang-Mills theory and QCD. For momenta p 2 Λ 2 QCD the FRG-results are in quantitative agreement with the corresponding lattice results. For these results as well as results for Landau-gauge QCD with other functional methods we refer the reader to some reviews and works, see e.g. [657, 722, 725-727, 730, 735-750] and references therein.

The differences in the deep infrared are related to the Gribov ambiguity. The Gribov problem in gauge-fixed approaches is yet not fully resolved, for reviews and recent work see e.g. [START_REF] Maas | Describing gauge bosons at zero and finite temperature[END_REF][START_REF] Vandersickel | The Gribov problem and QCD dynamics[END_REF][START_REF] Capri | Renormalizability of the refined Gribov-Zwanziger action in linear covariant gauges[END_REF]. In the works [START_REF] Cyrol | Landau gauge Yang-Mills correlation functions[END_REF][START_REF] Cyrol | Nonperturbative quark, gluon, and meson correlators of unquenched QCD[END_REF][START_REF] Mitter | Chiral symmetry breaking in continuum QCD[END_REF][START_REF] Cyrol | Nonperturbative finite-temperature Yang-Mills theory[END_REF] a bootstrap approach for its resolution has been used: the BRST-symmetry underlying the mSTI and mNI in ( 78) and ( 79) is standard Landau gauge BRST-symmetry without infrared modifications as in the Gribov-Zwanziger approach. In this setup the Kugo-Ojima confinement scenario, [START_REF] Kugo | Local Covariant Operator Formalism of Nonabelian Gauge Theories and Quark Confinement Problem[END_REF][START_REF] Kugo | The universal renormalization factors Z(1) / Z(3) and color confinement condition in non-Abelian gauge theory[END_REF], applies if the respective BRST-charge is well-defined. Note that the STI only covers the invariance under infinitesimal BRST-transformations and not the existence of the BRST-charge, see [START_REF] Fischer | On the infrared behavior of Landau gauge Yang-Mills theory[END_REF]. Under the assumption that the BRST-charge exists, the ghost and gluon propagators are required to have a non-trivial infrared scaling. Moreover, this scaling is unique, [START_REF] Fischer | Uniqueness of infrared asymptotics in Landau gauge Yang-Mills theory[END_REF][START_REF] Fischer | Uniqueness of infrared asymptotics in Landau gauge Yang-Mills theory II[END_REF][START_REF] Alkofer | Infrared singularities in Landau gauge Yang-Mills theory[END_REF]. The initial conditions for the (classical) vertices and propagators in [START_REF] Cyrol | Landau gauge Yang-Mills correlation functions[END_REF][START_REF] Cyrol | Nonperturbative quark, gluon, and meson correlators of unquenched QCD[END_REF][START_REF] Mitter | Chiral symmetry breaking in continuum QCD[END_REF][START_REF] Cyrol | Nonperturbative finite-temperature Yang-Mills theory[END_REF] at the ultraviolet initial scale k = Λ UV have been fine-tuned such that this infrared scaling behaviour is obtained. A detailed discussion can be found in [START_REF] Cyrol | Nonperturbative finite-temperature Yang-Mills theory[END_REF]. This construction has passed several consistency checks: first the existence of such a scaling solution with the correct perturbative UV behaviour is by not trival. Second it has been shown that the STI consistence gets worse, if moving away from the scaling solution. Note however, that this is a bootstrap resolution based on the assumption of the existence of BRST-charges in the Landau gauge. We emphasise that a satisfactory resolution of the non-perturbative Gribov problem in gauge-fixed approaches has yet to be obtained not only for the FRG-approach, but for functional approaches in general as well as on the lattice.

Evidently, the results for the correlation functions depicted in Fig. 20 are gauge-dependent. We still have to compute observables from these correlation functions. Moreover, we have to check the underlying gauge-consistency of the numerical results obtained within approximations to the full hierarchy of flow equations. In perturbation theory the STIs simply provide checks for the computations and facilitate the relation of the computed gauge-dependent correlation functions to physical observables such as S-matrix elements. A well-known textbook example is the identification of different renormalized strong couplings α s extracted from different renormalization functions of primitively divergent (classical) vertices and propagators. Its momentum-dependent version follows from the STIs. Strictly speaking the respective running couplings are those of longitudinal projections of classical vertices since the STIs relate the dressings of longitudinal vertices to a combination of dressings of longitudinal and transverse correlation functions. Accordingly, an underlying implicit assumption within the STI construction of the running coupling is the identification of transverse and longitudinal dressings of the respective vertices. In the following we will call this the regularity assumption. A relevant example for a vertex that does violate the regularity assumption is the ghost-gluon vertex, [Γ (3) Acc ] abc µ (|p, |q|, cos θ). Here p is the gluon momentum, q is the anti-ghost momentum q, and θ is the angle between them. If the vertex still depends on θ in the limit |p| → 0, it is irregular. Such a vertex does not allow for a simple identification of different running couplings as described above. While in perturbation theory the vertices can shown to be regular, irregularities are required for confinement in the Landau gauge, for a detailed discussion see [START_REF] Cyrol | Landau gauge Yang-Mills correlation functions[END_REF].

As has been discussed in the last section, in nonperturbative functional approaches such as the FRG or Dyson-Schwinger equations, the STIs also provide non-trivial relations between correlation functions that can be used in order to maintain or monitor gauge invariance during the flow. This has been discussed for non-Abelian gauge theories in [START_REF] Fischer | On the infrared behavior of Landau gauge Yang-Mills theory[END_REF], for gravity see [START_REF] Denz | Towards apparent convergence in asymptotically safe quantum gravity[END_REF][START_REF] Christiansen | Curvature dependence of quantum gravity[END_REF]. In particular it should be noted, that gauge invariance alone is not necessarily the hallmark of a good approximation. This statement will now be discussed within the example of Landau gauge flows, and will lead to a refined criterion:

First we note that in the Landau gauge the propagator of the gauge field is transverse. Note also that this property is present in any approximation within the Landau gauge. Hence, internal gauge field legs of vertices are transversally projected as they are contracted with the transverse propagator of the gauge field. If additionally applying transverse projection operators to all external gauge field legs of the flow equations for a given correlation function Γ (n) , we are led to flow equations with completely transverse vertices Γ (n) ⊥ : all gauge field legs, internal and external ones are transversally projected. In summary, this leads us to a closed set of flow equations for the completely transverse correlation functions Γ (n) ⊥ . Schematically this reads

∂ t Γ (n) ⊥ = Flow (n) ⊥ [{Γ (m) ⊥ }] , (80)
where m ≤ n + 2 follows from the one-loop structure of the Wetterich equation. Accordingly, we can solve the flows (80) without the knowledge of the correlation functions with at least one longitudinal leg, in the following denoted with Γ (n) L . Note that some of the legs in Γ (n) L can still be transverse. This analysis entails that the full dynamics of the theory is carried by the transverse correlation functions and hence by [START_REF] Latorre | Exact scheme independence[END_REF].

We are led to the peculiar situation that we can always arrange for a BRST-invariant solution from a given set of transverse correlation functions computed from (80): we simply use the mSTIs for the correlation functions that follow from the functional mSTI [START_REF] Morris | Convergence of derivative expansions of the renormalization group[END_REF] to compute the corresponding longitudinal correlation functions. This construction of BRST-invariant solutions works even for transverse correlation functions, that even qualitatively do not have the correct behaviour. This already entails that BRST-invariance alone is no hallmark of a good approximation.

For collecting further information on the quality of the approximation we evaluate both, the flows of correlation functions Γ (n) L with at least one longitudinal gauge field leg as well as the corresponding mSTIs or mQMEs. Both sets of relations are not closed but require the transverse ones as input. Schematically this reads

∂ t Γ (n) L = Flow (n) L [{Γ (m) L } , {Γ (m) ⊥ }] , (81a)
Γ (n) L = mSTI (n) [{Γ (m) L } , {Γ (m) ⊥ }] . (81b)
The above equations relate (flows of) longitudinal correlation functions or vertices Γ (n) L on the left hand side to loops with sets of both, longitudinal correlation functions, {Γ (m) L } and transverse ones, {Γ (m) ⊥ }. Hence, in contradistinction to the closed relations for the completely transverse correlation functions, the relations for the longitudinal correlation functions can only be solved for a given input of transverse correlation functions.

As mentioned above, BRST-invariance within a given approximation is easily achieved by solving the mSTIs (81b) with the given input of a set of transverse correlation functions Γ (n) ⊥ computed from [START_REF] Latorre | Exact scheme independence[END_REF]. This leads to a gauge-invariant closure of the effective action. One may be tempted to argue that one should only consider approximations where the longitudinal correlation functions satisfy both relations in [START_REF] Latorre | Scheme Independence as an Inherent Redundancy in Quantum Field Theory[END_REF]. However, the comparison between the correlation functions from the mSTIs (81b) and the longitudinal flows (81a) only indirectly carries some information about physical gauge invariance. Instead, differences are to be expected in nonperturbative approximations. For example, if comparing the approximate solution of a set of flow equations with, e.g. the DSE or nPI solution for the same approximation, they will only agree if the given approximation leads to the same resummation of -nonperturbative -diagrams. This rarely happens, the only known examples are perturbation theory and self-consistent nPI schemes, [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Wetterich | Bosonic effective action for interacting fermions[END_REF][START_REF] Blaizot | Exact renormalization group and Φ-derivable approximations[END_REF][START_REF] Dupuis | Renormalization group approach to interacting fermion systems in the two-particle-irreducible formalism[END_REF][START_REF] Carrington | Renormalization group flow equations connected to the n-particle-irreducible effective action[END_REF][START_REF] Carrington | Renormalization group methods and the 2PI effective action[END_REF][START_REF] Carrington | The 2PI effective action at four loop order in ϕ 4 theory[END_REF][START_REF] Carrington | 2PI effective theory at next-to-leading order using the functional renormalization group[END_REF][START_REF] Carrington | Renormalization of the 4PI effective action using the functional renormalization group[END_REF][START_REF] Alexander | Alternative flow equation for the functional renormalization group[END_REF][START_REF] Alexander | A new functional RG flow: regulator-sourced 2PI versus average 1PI[END_REF]. Moreover, even in the known examples, perturbation theory and nPI schemes, a comparison is only possible if taking into account the generically different RG scheme in the functional RG, for related work see [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Latorre | Exact scheme independence[END_REF][START_REF] Arnone | Exact scheme independence at one loop[END_REF][START_REF] Arnone | Exact scheme independence at two loops[END_REF][START_REF] Litim | Scheme independence at first order phase transitions and the renormalization group[END_REF][START_REF] Pernici | Hard -soft renormalization and the exact renormalization group[END_REF][START_REF] Ellwanger | The Running gauge coupling in the exact renormalization group approach[END_REF][START_REF] Pernici | Wilsonian flow and mass independent renormalization[END_REF][START_REF] Rosten | Universality From Very General Nonperturbative Flow Equations in QCD[END_REF][START_REF] Codello | Scheme dependence and universality in the functional renormalization group[END_REF].

This leaves us with the yet only partially solved important question of how to evaluate the preservation of gauge invariance and the reliability within a given approximate solution for transverse correlation functions. Nonetheless, while the analysis is not fully conclusive, it provides us with the refined criterion promised above: Keeping in mind, that also for the definition of observables gauge invariance is not enough but also locality is required 31 , this is a sensible but only implicit criterium for judging the quality of given results: the results for the expectation values of local gauge invariant operator computed from the gauge-fixed correlation functions have to be local.

We close this section with a remark on irregularity of vertices and confinement. We first note that regularity implies, that the dressing of the longitudinal projection of a correlation function has to agree with that of the transverse projection for vanishing projection momentum. In particular, the regularity assumption together with the STI implies the vanishing of the inverse gluon propagator at vanishing cutoff, k = 0, and momentum: Γ (2) ⊥ (p = 0) = 0. However, confinement requires a gluon mass gap in covariant gauges, see [START_REF] Braun | Quark Confinement from Color Confinement[END_REF][START_REF] Fister | Confinement from Correlation Functions[END_REF] and the discussion in Sec. 5.3.1. Consequently, this mass gap can only be generated with irregularities in the vertices, for a discussion within the FRG approach see [START_REF] Cyrol | Landau gauge Yang-Mills correlation functions[END_REF]. We conclude that confinement also necessitates the irregularity of vertices, which has been already pointed out in [START_REF] Cornwall | Dynamical Mass Generation in Continuum QCD[END_REF]. In summary, the seemingly very formal (ir)regularity property has direct consequences for the confinement phenomenon. This property is also tightly related to the above discussion of locality and gauge invariance, as irregularities introduce non-localities. While the gauge-fixed correlation functions are not required to be local, they have to cancel out in local observables that are computed from these correlation functions.

Gauge-invariant flows & the quest for simplicity

The above intricacies have triggered many attempts to derive gauge-invariant flow equations. Such a framework with a gauge-invariant effective action has not only the appeal of a direct physics interpretation of the respective correlation functions but also that of technical simplicity: A gauge-invariant effective action can be expanded in gauge-invariant operators, e.g. powers of the fieldstrength F µν leading to ´x V(F). This potential is the analogue of the effective potential in a scalar theory and the respective expansion in gauge theories is the covariant analogue of the derivative expansion in scalar theories. Parts of it, ´x W(θ) with θ = tr F 2 , have been studied in the literature, see e.g. [START_REF] Reuter | Indications for gluon condensation for nonperturbative flow equations[END_REF][START_REF] Reuter | Gluon condensation in nonperturbative flow equations[END_REF][START_REF] Gies | Running coupling in Yang-Mills theory: A flow equation study[END_REF][START_REF] Eichhorn | Gluon condensation and scaling exponents for the propagators in Yang-Mills theory[END_REF]. Further terms are generalised kinetic terms such as ´x trF µν f µνρσ (-D)F ρσ , see [START_REF] Ellwanger | Flow equations for the relevant part of the pure Yang-Mills action[END_REF][START_REF] Ellwanger | The Heavy quark potential from Wilson's exact renormalization group[END_REF][START_REF] Bergerhoff | Effective quark interactions and QCD propagators[END_REF], or gauge-invariant matter terms such as ´x qD / n q, see [START_REF] Cyrol | Nonperturbative quark, gluon, and meson correlators of unquenched QCD[END_REF][START_REF] Mitter | Chiral symmetry breaking in continuum QCD[END_REF].

The above examples have all been employed in a gauge-fixed settings (either Landau gauge or Landau-DeWitt gauge) with the implicit assumption that the full effective action

Γ[ Ā, a] ≈ Γ Dyn [A] + Γ gf [ Ā, a] , (82a)
can be split into a dynamical gauge-invariant part Γ Dyn [A] and a nearly classical gauge fixing part

Γ gf ≈ S gf [ Ā, a] + S ghost [ Ā, a, c, c] . (82b)
While this underlying assumption has been checked to hold quantitatively in QCD for scales k 1 GeV for the above mentioned matter terms in [START_REF] Cyrol | Nonperturbative quark, gluon, and meson correlators of unquenched QCD[END_REF][START_REF] Mitter | Chiral symmetry breaking in continuum QCD[END_REF], respective checks for the pure gauge field dynamics are still lacking, and are highly warranted. These complications could be avoided within fully gauge-invariant formulations of the FRG-approach to gauge theories. Most of the respective constructions can be linked to the background field approach. A direct cousin is the geometrical or Vilkovisky-DeWitt effective action. This uses the geometry of the configuration space for defining gauge-invariant fields as tangent vectors of geodesics with respect to the Vilkovisky connection Γ V ( Ā) from a given background Ā to a general configuration A. The Vilkovisky connection is constructed with the demand of a maximal disentanglement of gauge degrees of freedom a gauge (gauge fibre) and the dynamical gauge-invariant ones a dyn (base space), with D • a dyn = 0. It can be shown that the effective action only depends on the gauge-invariant field a dyn and not on a gauge .

Naturally, the construction is highly non-local and Γ V ∝ (1/ D2 ) D. Its linear approximation is simply the standard linear split in the background field approach, A = Ā + a. Without such an approximation it involves an infinite series, schematically A = Ā + a -1/2Γ V • a 2 + O(a 3 ). In perturbation theory it is of limited use, and on the technical level it boils down to the standard background field computations in the Landau-DeWitt gauge. In the FRG-approach it allows for a gauge-invariant regularisation, as the dynamical field a dyn itself is gauge-invariant. Seemingly the geometrical approach depends on two fields, Ā and a, but it can be shown with Nielsen or split Ward identities (NIs, sWI) that the respective field derivatives are linearly related, in contradistinction to the standard background field approach. This property of the geometrical approach originates in its reparameterisation invariance. As already mentioned before, these attractive features come at the price, that the construction is highly non-local, both in the configuration space of the gauge field as well as in momentum space. We have already discussed and emphasised the importance of locality (and symmetry identities) at the end of the last Sec. 5.2.2. In the present framework it is carried by the Nielsen identities and the locality of the gauge-invariant effective action. As in the gauge-fixed approaches, the assessment of these properties is highly non-trivial.

The FRG-approach to the geometrical effective action has been studied in non-Abelian theories and gravity in [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Pawlowski | Geometrical effective action and Wilsonian flows[END_REF][START_REF] Donkin | The phase diagram of quantum gravity from diffeomorphism-invariant RG-flows[END_REF][START_REF] Safari | Splitting Ward identity[END_REF][START_REF] Branchina | The Price of an exact, gauge invariant RG flow equation[END_REF][START_REF] Demmel | RG flows of Quantum Einstein Gravity in the linear-geometric approximation[END_REF]. In the presence of the cutoff term, the Nielsen identities acquire a non-linear term similar to the right hand side of [START_REF] Morris | Convergence of derivative expansions of the renormalization group[END_REF], see [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Pawlowski | Geometrical effective action and Wilsonian flows[END_REF][START_REF] Safari | Splitting Ward identity[END_REF][START_REF] Safari | Covariant and single-field effective action with the background-field formalism[END_REF][START_REF] Safari | Covariant and background independent functional RG flow for the effective average action[END_REF]. This additional term mirrors a similar one in the Nielsen or split Ward identity in the standard background field approach, see [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Litim | Renormalization group flows for gauge theories in axial gauges[END_REF][START_REF] Litim | Wilsonian flows and background fields[END_REF][START_REF] Litim | Completeness and consistency of renormalisation group flows[END_REF][START_REF] Folkerts | Asymptotic freedom of Yang-Mills theory with gravity[END_REF][START_REF] Bridle | The local potential approximation in the background field formalism[END_REF][START_REF] Dietz | Background independent exact renormalization group for conformally reduced gravity[END_REF][START_REF] Labus | Background independence in a background dependent renormalization group[END_REF]. There, however, it adds to a non-linear term already present due to the gauge fixing. Importantly, in the geometrical approach gauge invariance is kept trivially at all stages of the construction thanks to the gauge invariance of the fields. The latter is the core ingredient of the construction: a parametrisation of the theory, that allows the distinction between gauge degrees of freedom and the dynamical ones without mixing terms. So far, the only nonperturbative application has been a diffeomorphism-invariant computation of UV-IR phase structure of asymptotically safe quantum gravity with the ultraviolet Reuter fixed point in [START_REF] Donkin | The phase diagram of quantum gravity from diffeomorphism-invariant RG-flows[END_REF].

Related approaches based on such a disentanglement have also been put forward in [START_REF] Wetterich | Gauge invariant flow equation[END_REF][START_REF] Wetterich | Gauge-invariant fields and flow equations for Yang-Mills theories[END_REF][START_REF] Pawlowski | Higgs scalar potential in asymptotically safe quantum gravity[END_REF][START_REF] Wetterich | Variable Planck mass from the gauge invariant flow equation[END_REF], based on a so called physical gauge fixing that allows to use the dynamical (mean) field in the regulator. This splitting is also at the root of a gauge-fixed approach with standard STIs put forward in [START_REF] Asnafi | BRST invariant RG flows[END_REF]. While these formulation can be embedded or related to the geometrical approach discussed above, they have the advantage of (relative) technical simplicity. In the case of the gauge-invariant flow [START_REF] Wetterich | Gauge invariant flow equation[END_REF] this has already led to first results in quantum gravity, see also Sec. 6.

Another promising approach has been put forward in [START_REF] Rosten | Universality From Very General Nonperturbative Flow Equations in QCD[END_REF][START_REF] Morris | A Gauge invariant exact renormalization group. 1[END_REF][START_REF] Morris | A Gauge invariant exact renormalization group. 2[END_REF][START_REF] Arnone | Gauge invariant regularization via SU(N-N)[END_REF][START_REF] Arnone | A Proposal for a manifestly gauge invariant and universal calculus in Yang-Mills theory[END_REF][START_REF] Arnone | A Generalised Manifestly Gauge Invariant Exact Renormalisation Group for SU(N) Yang-Mills[END_REF][START_REF] Arnone | Manifestly gauge invariant QED[END_REF][START_REF] Morris | A Manifestly gauge invariant, continuum calculation of the SU(N) Yang-Mills two-loop beta function[END_REF][START_REF] Morris | Manifestly gauge invariant QCD[END_REF][START_REF] Rosten | General Computations Without Fixing the Gauge[END_REF][START_REF] Rosten | A Manifestly gauge invariant and universal calculus for SU(N) Yang-Mills[END_REF][START_REF] Rosten | A Resummable beta-Function for Massless QED[END_REF], the manifestly gauge-invariant FRG. This approach utilizes a specific property of the Polchinski-RG for the Wilson effective action (or Schwinger functional), the generating functional for amputated connected correlation functions: the flow admits field-dependent regulators. It has been mostly used for gauge-invariant perturbative computations of β-functions, in particular the two-loop βfunction of Yang-Mills theory, [START_REF] Morris | A Manifestly gauge invariant, continuum calculation of the SU(N) Yang-Mills two-loop beta function[END_REF]. It also has been shown, how to set-up the approach for a computation of Wilson loop expectation values, see [START_REF] Rosten | General Computations Without Fixing the Gauge[END_REF]. Variants of such a flow have also been discussed more recently in [START_REF] De Alwis | Exact RG Flow Equations and Quantum Gravity[END_REF][START_REF] Bonanno | On Exact Proper Time Wilsonian RG Flows[END_REF]. In summary, this approach may give direct access to the flow of observables such as the confinement-deconfinement order parameter.

Very recently, a gauge-invariant, background-and scheme-independent FRG-approach for the Wilson effective action has suggested in [START_REF] Falls | Background independent exact renormalisation[END_REF]. It combines the geometrical approach to the FRG with the gauge-invariant FRG discussed in the previous two paragraphs. This promising approach may circumvent all the above mentioned obstacles.

A final possibility is to take more general regularisations than the standard momentum cutoff. This idea has been pursued at finite temperature in [START_REF] D'attanasio | Gauge invariant renormalization group at finite temperature[END_REF][START_REF] Comelli | Screening masses in SU(N) from Wilson renormalization group[END_REF]. In these works the temporal boundary conditions at finite temperature have been modified via the thermal distribution function. This does not lead to modifications of the Slavnov-Taylor identities. More generally, regularisations can be implemented in position space. This has been implemented within real-time approaches in [START_REF] Gasenzer | Towards far-from-equilibrium quantum field dynamics: A functional renormalisation-group approach[END_REF][START_REF] Gasenzer | Far-from-equilibrium quantum many-body dynamics[END_REF][START_REF] Corell | Flowing with the Temporal Renormalisation Group[END_REF] and [START_REF] Pietroni | Flowing with Time: a New Approach to Nonlinear Cosmological Perturbations[END_REF][START_REF] Lesgourgues | Non-linear Power Spectrum including Massive Neutrinos: the Time-RG Flow Approach[END_REF][START_REF] Bartolo | Signatures of Primordial non-Gaussianities in the Matter Power-Spectrum and Bispectrum: the Time-RG Approach[END_REF]. If applied to gauge theories, the standard STIs would be maintained in these approaches.

In summary, there are various vary promising options that allow to either keep the standard STIs or to obtain gauge-invariant flows. Still, we close this Section with a word of caution: the geometry of the configuration space in theories with non-linear symmetries structurally does not allow for a global linear disentanglement of the gauge degrees of freedom. In turn, any such disentanglement is inherently non-linear, if not non-local. This remark reiterated the importance of locality which cannot be emphasized enough. Hence it remains to be seen how far these approaches can be pushed in the necessary quest for quantitative precision.

QCD

We have already mentioned in the introduction Sec. 5.1, that QCD is a paradigmatic example for a gauge theory with many nonperturbative phenomena. As discussed there, QCD physics questions range from the confinement mechanism and that of chiral symmetry breaking, the determination of the bound state and resonance structure of QCD, to the phase structure of QCD including the condensed matter phases at high densities, as well as the real-time dynamics in and out of equilibrium. The resolution of these questions is highly relevant for an understanding of the QCD phase transition in the early universe, for nuclear astrophysics with intriguing links to gravitational-wave physics, the formation of matter around us as well as that of heavy-ion collisions. Therefore, QCD has been studied extensively with the FRG since the beginning of the 90ties. Apart from a plethora of research works this also has led to a number of QCD-related reviews that cover some or many of the relevant physics aspects, see [START_REF] Berges | Non-perturbative renormalization flow in quantum field theory and statistical physics[END_REF][START_REF] Polonyi | Lectures on the functional renormalization group method[END_REF][START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Rosten | Fundamentals of the Exact Renormalization Group[END_REF][START_REF] Braun | Fermion Interactions and Universal Behavior in Strongly Interacting Theories[END_REF][START_REF] Jungnickel | Nonperturbative flow equations, low-energy QCD and the chiral phase transition[END_REF][START_REF] Litim | On gauge invariant Wilsonian flows[END_REF][START_REF] Schaefer | Renormalization group approach towards the QCD phase diagram[END_REF][START_REF] Gies | Introduction to the functional RG and applications to gauge theories[END_REF][START_REF] Sonoda | The Exact Renormalization Group: Renormalization theory revisited[END_REF][START_REF] Schaefer | Fluctuations and the QCD Phase Diagram[END_REF][START_REF] Pawlowski | Equation of state and phase diagram of strongly interacting matter[END_REF][START_REF] Strodthoff | Phase Structure and Dynamics of QCD-A Functional Perspective[END_REF][START_REF] Klein | Modeling Finite-Volume Effects and Chiral Symmetry Breaking in Two-Flavor QCD Thermodynamics[END_REF]. Due to its paradigmatic nature we have already shown some results for Yang-Mills and QCD correlation functions in Fig. 20. Here we give a brief overview of the progress in describing QCD physics with the functional RG while also reporting on further technical advances within the FRG.

Confinement and the flow of composite operators

We start our investigation with a discussion of confinement, which is the peculiar property in QCD that the potential V static q q (r) between a static quark-anti-quark pair at a large spatial distance r grows linearly with that distance,

V static q q (r → ∞) σr , V static q q (r → 0)

α s (r) r . ( 83
)
For small differences, it resembles a Coulomb potential with a logarithmically decreasing coupling α s (r → 0) → 0 owing to asymptotic freedom. The proportionality factor σ is called the string tension and has the dimension of an inverse area. The linear rise in [START_REF] Arnone | Exact scheme independence at two loops[END_REF] entails that the energy between the quark and the anti-quark is stored in a flux tube between them. In QCD with dynamical quarks this asymptotic linear potential is not to be seen, since for sufficiently large distances r r sb , larger than the string-breaking scale r sb , the flux-tube energy is large enough to allow for the generation of a quark-anti-quark pair that shields the original q q-pair. Accordingly, for distances r > r sb , the quark-anti-quark potential is that of a color-dipole pair and levels off.

String breaking cannot happen in quenched QCD and pure Yang-Mills theory, and the potential or free energy of a static q q-pair indeed shows an asymptotic linear growth. The operator, that describes the generation of a static quark-anti-quark pair at a time t = 0 and distance r = x-y and its annihilation at time t, is the traced Wilson loop. The Wilson loop is the path-ordered exponential of the gauge field with the path being the rectangular wordline C t,r of the quark-antiquark pair, see [START_REF] Litim | Scheme independence at first order phase transitions and the renormalization group[END_REF]. The free energy or potential in Euclidean spacetime is proportional to the logarithm of the traced Wilson loop expectation value, see (84), V q q(r) ∝ log W(t, r) .

Accordingly, confinement in Euclidean spacetime is signalled by an area law of 'log W(t, r) ', and also implies a linear rise of this observable in both r and t. For the sake of simplicity we have restricted ourselves to rectangular paths of the static situation described above. Let us now also assume for the time being, that our Euclidean spacetime is a box T 4 = 4 i=0 [0, L i ] with periodic boundary conditions in all directions (four-dimensional torus). This is the spacetime manifold one considers on the lattice. Then, for large temporal distances t → L 0 the Wilson loop is proportional to the two-point correlation function of two Polyakov loops, the latter winding around the full temporal extent, to wit,

lim t→L 0 W(t, r) ∝ L( x)L( y) , with W(t, r) = 1 N c tr Pe -ig s ´Ct,r (x,y)A µ (z)dz µ , (84)
where P stands for path ordering. The traced Polyakov loop L( x) is the traced Wilson loop, that winds around the full temporal extent of the torus T 4 ,

L( x) = 1 N c tr P( x) with P( x) = Pe -ig s ´L0 0 A 0 (τ, x) dτ , (85)
The proportionality constant in the relation ( 84) tends to zero in the limit L 0 → ∞. In ( 84), ( 85) the traces are taken in the fundamental representation, since the quarks carry this representation. The relation ( 84) maps the Wilson loop to the two-point correlation function of a gauge-invariant operator that is local in position space. The area law for the logarithm of the Wilson loop expectation value is in one-to-one correspondence to a linear dependence on the distance r for the logarithm of the correlator of Polyakov loops. However, as the Polyakov loops span the whole temporal extent of the manifold, in the vacuum one typically uses the area law for the Wilson loop as a signature for confinement. Still, with a proper normalization or limit the Polyakov loop correlator signals confinement in the vacuum, and it is the confinement signature that has been studied in the FRG-literature. While it is not commonly used in the vacuum, the Polyakov loop correlation is typically used as an order parameter for the confinement-deconfinement phase transition at finite temperature T , with the identification T = 1/L 0 . There, for large distances we have declustering,

lim r→∞ L( x)L( y) → L( x) L( y) , (86)
and confinement entails the vanishing of the Polyakov loop expectation value itself, L( x) = 0. This happens for low temperatures and for T = 0 it is tantamount to confinement in the vacuum. In turn, in the deconfined phase for T → ∞ with small fluctuations of the temporal gauge field A 0 , the traced Polyakov loop is finite and tends to unity. The Polyakov loop also allows to easily identify the symmetry behind the confinement-deconfinement symmetry in Yang-Mills theory: the center symmetry of the gauge group. While the action is invariant under center transformations, the traced Polyakov loop is not. In SU(N c ) we have P( x) → z P( x) with z ∈ Z N c . The elements of the center group Z N c are the N c th roots of unity, z N c = 1. Accordingly, the symmetry group of the phase transition is the center Z N c , and we can deduce the universality class of the phase transitions for different N c . For SU(2) the center is Z 2 and the transition is a second order phase transition and lies in the Ising universality class. For SU(3) and all higher N c the phase transition is of first order. 32A challenge specific for the FRG-approach to QCD is the fact, that the above observables are infinite-order correlation function in the gauge field and the FRG-approach is based on the computation of finite-order correlation function. Indeed, the pivotal ones are the propagators, the two-point correlation functions of the theory. There are several ways out, all of which have been undertaken in the FRG:

A first option is the reformulation of QCD in terms of Wilson lines along the lines of [START_REF] Morris | A Gauge invariant exact renormalization group. 1[END_REF]. These reparameterisations of theories are very naturally implemented in the FRG as it is typically formulated such that it is quadratic in the degree of freedom that is regularised, see [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF] for a detailed discussion. Another possibility is the reparameterisation in terms of a suitable gauge fixing that simplifies the relation between the Wilson and Polyakov loops and the gauge field. In finite temperature gauge theories a natural choice is the Polyakov gauge, that has been studied in [START_REF] Marhauser | Confinement in Polyakov Gauge[END_REF][START_REF] Kondo | Toward a first-principle derivation of confinement and chiral-symmetry-breaking crossover transitions in QCD[END_REF] for S U(2). For this choice the flow equation for the effective potential V eff [A 0 ] resembles that of the effective potential of a real scalar field 33 , and the second order phase transition with Ising universality class in S U(2) Yang-Mills theory has been found.

The second option is to define variants of the standard order parameters, the Wilson loop W and the Polyakov loop L , that can be defined in particular from correlation functions within standard covariant gauges [START_REF] Morris | Equivalence of local potential approximations[END_REF], and in particular in the Landau(DeWitt) gauge. The latter is the best studied and numerically most advanced and successful formulation of QCD with functional methods, as discussed before.

In the Landau-DeWitt gauge we can compute the gauge-invariant background-effective action Γ[A] from the flow equation of the effective action with fluctuation fields. At finite temperature, Γ[A] includes a gauge-invariant effective potential V eff [A 0 ] of the temporal gauge field A 0 . This potential can be understood as an order-parameter potential: it inherits center symmetry from the effective action, but its minima A 0,EoM may or may not break center symmetry. Indeed, the Polyakov loop of the solution of the EoM, A 0,EoM , vanishes at the center-symmetric points of A 0,EoM . Accordingly, it vanishes for temperatures below the confinement temperature T c and is non-vanishing above T c ,

L[A 0,EoM ]        = 0 T < T c 0 T > T c . (87)
This functional approach to confinement is based on corresponding studies of the high temperature regime in perturbation theory founded in [START_REF] Gross | QCD and Instantons at Finite Temperature[END_REF][START_REF] Weiss | The Effective Potential for the Order Parameter of Gauge Theories at Finite Temperature[END_REF]. It has been put forward in [START_REF] Braun | Quark Confinement from Color Confinement[END_REF] within the FRG and has been extended later also to general functional approaches, [START_REF] Fister | Confinement from Correlation Functions[END_REF]. By now results have been obtained for generic S U(N c ) gauge groups in Yang-Mills theory, [START_REF] Braun | On the Nature of the Phase Transition in SU(N), Sp(2) and E(7) Yang-Mills theory[END_REF], and for full dynamical two-flavor QCD in the chiral limit, [START_REF] Braun | Phase Structure of Two-Flavor QCD at Finite Chemical Potential[END_REF]. For results within this approach obtained with other functional methods see [START_REF] Fister | Confinement from Correlation Functions[END_REF][START_REF] Reinosa | Deconfinement transition in SU(N) theories from perturbation theory[END_REF][START_REF] Fischer | Polyakov loop potential at finite density[END_REF][START_REF] Fischer | Phase structure of QCD for heavy quarks[END_REF][START_REF] Fischer | Phase structure of three and four flavor QCD[END_REF][START_REF] Reinosa | Deconfinement transition in SU(2) Yang-Mills theory: A two-loop study[END_REF][START_REF] Reinosa | Perturbative study of the QCD phase diagram for heavy quarks at nonzero chemical potential[END_REF][START_REF] Reinosa | Two-loop study of the deconfinement transition in Yang-Mills theories: SU(3) and beyond[END_REF][START_REF] Reinosa | Yang-Mills correlators across the deconfinement phase transition[END_REF][START_REF] Maelger | Perturbative study of the QCD phase diagram for heavy quarks at nonzero chemical potential: Two-loop corrections[END_REF][START_REF] Reinhardt | The effective potential of the confinement order parameter in the Hamilton approach[END_REF][START_REF] Reinhardt | The effective potential of the confinement order parameter in the Hamiltonian approach[END_REF][START_REF] Heffner | Finite-temperature Yang-Mills theory in the Hamiltonian approach in Coulomb gauge from a compactified spatial dimension[END_REF][START_REF] Quandt | Covariant variational approach to Yang-Mills Theory: effective potential of the Polyakov loop[END_REF]. Moreover, it has been shown in [START_REF] Braun | Quark Confinement from Color Confinement[END_REF], that confinement in covariant gauges necessitates a relative gapping of the gluon propagator to that of the ghost: confinement implies the vanishing of ( 87) and the part of the Polyakov loop potential stemming from the dressed gluon loop in the flow equation is deconfining. In turn, the contribution from the ghost loop is confining, which originates from the relative fermionic minus sign of the ghost loop. For large temperatures (or in perturbation theory) the gluon loop dominates. At small temperatures the ghost loop has to take over in order to guarantee confinement. However, it only dominates if the gluons are gapped relative to the ghost. The details of this mechanism have been corroborated and extended in [START_REF] Fister | Confinement from Correlation Functions[END_REF][START_REF] Braun | On the Nature of the Phase Transition in SU(N), Sp(2) and E(7) Yang-Mills theory[END_REF]. The gluon and ghost propagators in covariant gauges have this property, for a detailed discussion see Sec. 5.2.2. Respective FRG-results and applications for the gapped gluon propagator in Yang-Mills theory and QCD in the Landau gauge are given in [START_REF] Fischer | Renormalization flow of Yang-Mills propagators[END_REF][START_REF] Pawlowski | Geometrical effective action and Wilsonian flows[END_REF][START_REF] Cyrol | Landau gauge Yang-Mills correlation functions[END_REF][START_REF] Cyrol | Nonperturbative quark, gluon, and meson correlators of unquenched QCD[END_REF][START_REF] Corell | Correlation functions of three-dimensional Yang-Mills theory from the FRG[END_REF][START_REF] Fischer | On the infrared behavior of Landau gauge Yang-Mills theory[END_REF][START_REF] Fischer | Uniqueness of infrared asymptotics in Landau gauge Yang-Mills theory[END_REF][START_REF] Fischer | Uniqueness of infrared asymptotics in Landau gauge Yang-Mills theory II[END_REF], for related work see [732-734, 768, 826].

The results for [START_REF] Pernici | Wilsonian flow and mass independent renormalization[END_REF] show a second order phase transition for S U(2) Yang-Mills theory, while the phase transition is of first order for higher S U(N c > 2), and a crossover 34 for two-flavor QCD. This agrees with the expected orders of the phase transitions, and the critical temperatures are in quantitative agreement with the lattice results.

Finally, the results for L[A 0,EoM ] can be used within the flow of Wilson-and Polyakov-loop observables. The latter can be studied with the flow equation for general composite operators derived in [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Igarashi | Realization of Symmetry in the ERG Approach to Quantum Field Theory[END_REF][START_REF] Pagani | Note on scaling arguments in the effective average action formalism[END_REF],

∂ t O k [Φ] = - 1 2 Tr G k [Φ] ∂ t R k G k [Φ] O (2) k [Φ] . (88)
In ( 88) the operator O (2) k is the second derivative of O k with respect to the fields. This operator is contracted with

G k ∂ t R k G k in the trace. The set of composite operators O k [Φ]
with the flow (88) includes general correlation functions with their connected and disconnected parts as well as functions of the source J[Φ]. For the latter case further terms enter [START_REF] Rosten | Universality From Very General Nonperturbative Flow Equations in QCD[END_REF], see [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF]. An educative example for the former case of general correlation functions and the necessity of including the disconnected terms is the full two-point function G Φ 1 Φ 2 + Φ 1 Φ 2 . Eq. ( 88) has been put to work in Yang-Mills theories [START_REF] Herbst | Confinement order parameters and fluctuations[END_REF] for the traced Polyakov loop observables. In gravity it has been used in [START_REF] Pagani | Composite Operators in Asymptotic Safety[END_REF][START_REF] Becker | Fractal geometry of higher derivative gravity[END_REF][START_REF] Houthoff | On the scaling of composite operators in Asymptotic Safety[END_REF][START_REF] Kurov | On characterizing the Quantum Geometry underlying Asymptotic Safety[END_REF] for the study of the renormalization and scaling of composite operators, see Sec. 6.

Evidently, the expectation value of the traced Polyakov loop falls into the class of general correlation functions O[Φ] with the flow [START_REF] Rosten | Universality From Very General Nonperturbative Flow Equations in QCD[END_REF], and its flow is given by

∂ t L [Φ] = - 1 2 Tr G k [Φ] ∂ t R k G k [Φ] L (2) [Φ] . (89)
The flow [START_REF] Codello | Scheme dependence and universality in the functional renormalization group[END_REF] has been solved in a first approximation in an expansion of L [Φ] about L[A 0,EoM ] in [START_REF] Herbst | Confinement order parameters and fluctuations[END_REF] for S U(3) Yang-Mills theory. The result is already in quantitative agreement with the lattice results for the Polyakov loop expectation value. In summary by now the FRG-approach offers a well-developed and quantitative access to quite some aspects of the exciting phenomenon of confinement and in particular to the confinement-deconfinement phase transition at finite temperature. Remaining challenges concern in particular the thermodynamics of gauge theories, see e.g. [START_REF] Fister | Confinement from Correlation Functions[END_REF][START_REF] Fister | Yang-Mills correlation functions at finite temperature[END_REF][START_REF] Fister | On the Phase Diagram of QCD with Dynamical Quarks[END_REF]. Its resolution may also require a quantitative access to the subleading momentum and frequency dependence of correlation functions, for first work in this direction as well as non-relativistic analogues see [START_REF] Hajizadeh | Exploring the Tan contact term in Yang-Mills theory[END_REF]834].

Chiral symmetry breaking

The ultraviolet behaviour of QCD is governed by asymptotic freedom, the theory approaches the ultraviolet Gaußian fixed point with vanishing strong coupling α s (p → ∞) = 0 with α s = g 2 s /(4π). In turn, in the infrared the coupling grows strong and triggers chiral symmetry breaking. This is easily investigated from the flow of the dimensionless four-quark interaction λq ∝ λ q k 2 in the scalar-pseudoscalar channel: -λ q /4 ´x ( q q) 2 -( qiγ 5 τ q) 2 . Its flow reads schematically

∂ t λq = 2 λq -A k λ2 q -B k λq α s -C k α 2 s + tadpole-terms , (90)
with positive constants A k , B k , C k that depend on other parameters of the theory, and in particular on the mass scales. It is depicted in Fig. 21. In ( 90), 2 λq is the canonical scaling term and the other terms are quantum corrections computed from the respective diagrams in the flow. For α s = 0 the flow ( 90) is that of the four-quark coupling in an NJL-type model. Then, for large enough initial coupling λ2 q k=Λ > 2/A Λ at the UV cutoff scale Λ, the β-function ∂ t λq is negative and the coupling grows towards the infrared. It finally diverges at a pole that signals chiral symmetry breaking. In turn, for λq,Λ < 2/A Λ the coupling weakens towards the infrared and runs into the Gaußian fixed point without chiral symmetry breaking.

λ ∂ t λ α > α cr α = α cr α = 0 α = 0, T > 0 @ t ¯ q < l a t e x i t s h a 1 _ b a s e 6 4 = " Q J g w c 4 E 8 T a E D I G v I Y u h P x j I f 6 B o = " > A A A C A X i c b V D L S s N A F J 3 4 r P U V d S O 4 G S y C q 5 J I w c e q 4 M Z l B f u A J o S b y a Q d O n k 4 M x F K q B t / x Y 0 L R d z 6 F + 7 8 G y d t F t p 6 Y O B w z j 0 z c 4 + f c i a V Z X 0 b S 8 s r q 2 v r l Y 3 q 5 t b 2 z q 6 5 t 9 + R S S Y I b Z O E J 6 L n g 6 S c x b S t m O K 0 l w o K k c 9 p 1 x 9 d F 3 7 3 g Q r J k v h O j V P q R j C I W c g I K C 1 5 5 q G T g l A M u K e w 4 4 N w u M 4 G 4 N 1 7 Z s 2 q W 1 P g R W K X p I Z K t D z z y w k S k k U 0 V o S D l H 3 b S p W b F 7 c T T i d V J 5 M 0 B T K C A e 1 r G k N E p Z t P N 5 j g E 6 0 E O E y E P r H C U / V 3 I o d I y n H k 6 8 k I 1 F D O e 4 X 4 n 9 f P V H j h 5 i x O M 0 V j M n s o z D h W C S 7 q w A E T l C g + 1 g S I Y P q v m A x B A F G 6 t K o u w Z 5 f e Z F 0 z u p 2 o 3 5 5 2 6 g 1 r 8 o 6 K u g I H a N T Z K N z 1 E Q 3 q I X a i K B H 9 I x e 0 Z v x Z L w Y 7 8 b H b H T J K D M H 6 A + M z x + W Z J b 5 < / l a t e x i t > ↵ s = 0, T > 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " Y A j p s v 3 n Q V s Q W 7 P S z b c Y C i G Y S r M = " > A A A B + X i c b V D L S g N B E O z 1 G e N r 1 a O X w S B 4 k L A r A f W g B L x 4 j J A X J M v S O 5 k k Q 2 Y f z M w G w p I / 8 e J B E a / + i T f / x k m y B 0 0 s a C i q u u n u C h L B l X a c b 2 t t f W N z a 7 u w U 9 z d 2 z 8 4 t I + O m y p O J W U N G o t Y t g N U T P C I N T T X g r U T y T A M B G s F o 4 e Z 3 x o z q X g c 1 f U k Y V 6 I g 4 j 3 O U V t J N + 2 u y i S I f r q z r k k 9 X v i + H b J K T t z k F X i 5 q Q E O W q + / d X t x T Q N W a S p Q K U 6 r p N o L 0 O p O R V s W u y m i i V I R z h g H U M j D J n y s v n l U 3 J u l B 7 p x 9 J U p M l c / T 2 R Y a j U J A x M Z 4 h 6 q J a 9 m f i f 1 0 l 1 / 8 b L e J S k m k V 0 s a i f C q J j M o u B 9 L h k V I u J I U g l N 7 c S O k S J V J u w i i Y E d / n l V d K 8 K r u V 8 u 1 T p V S t 5 H E U 4 B T O 4 A J c u I Y q P E I N G k B h D M / w C m 9 W Z r 1 Y 7 9 b H o n X N y m d O 4 A + s z x 9 G t Z I Y < / l a t e x i t > ↵ s = ↵ s,cr
O i K 8 = " > A A A C C H i c d V D L S s N A F J 3 U V 6 2 v q k s X D h b B h Z R E C r Y L o e D G Z Q X 7 g C a E y X T S D p 0 8 m L k R S 8 j S j b / i x o U i b v 0 E d / 6 N 0 z Z C F T 1 w 4 X D O v d x 7 j x c L r s A 0 P 4 3 C 0 v L K 6 l p x v b S x u b W 9 U 9 7 d 6 6 g o k Z S 1 a S Q i 2 f O I Y o K H r A 0 c B O v F k p H A E 6 z r j S + n f v e W S c W j 8 A Y m M X M C M g y 5 z y k B L b n l Q 5 u I e E R c d Z G T V J 3 a w O 5 A B i m V W e a W K 2 b V n A E v k E a j b t U b 2 M q V C s r R c s s f 9 i C i S c B C o I I o 1 b f M G J y U S O B U s K x k J 4 r F h I 7 J k P U 1 D U n A l J P O H s n w s V Y G 2 I + k r h D w T F 2 c S E m g 1 C T w d G d A Y K R + e 1 P x L 6 + f g F 9 3 U h 7 G C b C Q z h f 5 i c A Q 4 W k q e M A l o y A m m h A q u b 4 V 0 x G R h I L O r q R D + P 4 U / 0 8 6 Z 1 W r V m 1 c 1 y r N W h 5 H E R 2 g I 3 S C L H S O m u g K t V A b U X S P H t E z e j E e j C f j 1 X i b t x a M f G Y f / Y D x / g U J C 5 q h < / l a t e x i t > ↵ s > ↵ s,cr < l a t e x i t s h a 1 _ b a s e 6 4 = " u 1 l H n F 4 u q C c u L l Z E K K s Y u 7 f d P d s = " > A A A C C H i c d V D L S g N B E J z 1 G e M r 6 t G D g 0 H w I M t u S D S 5 S M C L x w j m A U l Y Z i e z y Z D Z B z O 9 Y l j 2 6 M V f 8 e J B E a 9 + g j f / x k m y g o o W N B R V 3 X R 3 u Z H g C i z r w 1 h Y X F p e W c 2 t 5 d c 3 N r e 2 C z u 7 L R X G k r I m D U U o O y 5 R T P C A N Y G D Y J 1 I M u K 7 g r X d 8 c X U b 9 8 w q X g Y X M M k Y n 2 f D A P u c U p A S 0 7 h o E d E N C K O O s 9 I o k 5 6 w G 5 B + g m V a e o U i p Z Z q 1 Z q 5 S q 2 T G u G K S m d 1 i o 2 t j O l i D I 0 n M J 7 b x D S 2 G c B U E G U 6 t p W B P 2 E S O B U s D T f i x W L C B 2 T I e t q G h C f q X 4 y e y T F R 1 o Z Y C + U u g L A M / X 7
R E J 8 p S a + q z t 9 A i P 1 2 5 u K f 3 n d G L x q P + F B F A M L 6 H y R F w s M I Z 6 m g g d c M g p i o g m h k u t b M R 0 R S S j o 7 P I 6 h K 9 P 8 f + k V T L t s l m 7 K h f r 5 S y O H N p H h + g Y 2 e g M 1 d E l a q A m o u g O P a A n 9 G z c G 4 / G i / E 6 b 1 0 w s p k 9 9 A P G 2 y c s t Z q 4 < / l a t e x i t > ¯ q < l a t e x i t s h a 1 _ b a s e 6 4 = " 6 9 a u B r 7

O g Z 2 2 N t V p w Y C F h 4 c T S 1 M = " > A A A B 9 H i c b V D J S g N B F H w T t x i 3 q E c v j U H w F G Y k 4 H I K e P E Y w S y Q G c K b n p 6 k S c + S 7 p 5 A C P k O L x 4 U 8 e r H e P N v 7 C R z 0 M S C h q K q H u 9 1 + a n g S t v 2 t 1 X Y 2 N z a 3 i n u l v b 2 D w 6 P y s c n L Z V k k r I m T U Q i O z 4 q J n j M m p p r w T q p Z B j 5 g r X 9 4 f 3 c b 4 + Z V D y J n / Q k Z V 6 E / Z i H n K I 2 k u f 6 K F 1 h 4 g H 2 R r 1 y x a 7 a C 5 B 1 4 u S k A j k a v f K X G y Q 0 i 1 i s q U C l u o 6 d a m + K U n M q 2 K z k Z o q l S I f Y Z 1 1 D Y 4 y Y 8 q a L o 2 f k w i g B C R N p X q z J Q v 0 9 M c V I q U n k m 2 S E e q B W v b n 4 n 9 f N d H j j T X m c Z p r F d L k o z A T R C Z k 3 Q A I u G d V i Y g h S y c 2 t h A 5 Q I t W m p 5 I p w V n 9 8 j p p X V W d W v X 2 s V a p 3 + V 1 F O E M z u E S H L i G O j x A A 5 p A Y Q T P 8 A p v 1 t h 6 s d 6 t j 2 W 0 Y O U z p / A H 1 u c P z W a S H g = = < / l a t e x i t > ↵ s = 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " Z v N F 9 Q a s X + o k W S R H 2 7 F 5 + y A i X w s = " > A A A B 8 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k o B 6 E g h e P F e w H t q F M t p t 2 6 W Y T d j d C C f 0 X X j w o 4 t V / 4 8 1 / 4 7 b N Q V s f D D z e m 2 F m X p A I r o 3 r f j u F t f W N z a 3 i d m l n d 2 / / o H x 4 1 N J x q i h r 0 l j E q h O g Z o J L 1 j T c C N Z J F M M o E K w d j G 9 n f v u J K c 1 j + W A m C f M j H E o e c o
u E x S w y R d L A p T Q U x M Z u + T A V e M G j G x B K n i 9 l Z C R 6 i Q G h t S y Y b g L b + 8 S l o X V a 9 W v b 6 v V e q 1 P I 4 i n M A p n I M H l 1 C H O 2 h A E y h I e I Z X e H O 0 8 + K 8 O x + L 1 o K T z x z D H z i f P w 5 g k H 0 = < / l a t e x i t >
Figure 21: Flow ∂ t λq of the dimensionless four-quark coupling λq = λ q k 2 in the scalar-pseudoscalar channel as a function of the four-quark coupling for different values of the strong couplings α s . If the strong coupling is large enough, α s > α s,cr with the critical coupling α s,cr , the flow is negative for all λq . Then, chiral symmetry breaking happens for all initial values of λq , for more details see the discussion below [START_REF] Schnoerr | Error estimates and specification parameters for functional renormalization[END_REF].

When switching on the strong coupling, the β-function of the four-quark coupling in ( 90) is deformed: Firstly, the canonical running gets an anomalous part with 2 λq → (2 -B k α s ) λq . More importantly the whole β-function is shifted down globally by -C k α 2 s , see Fig. 21. This term originates from quark-gluon box diagrams and is negative. For large enough α s the β-function ∂ t λq is negative for all λq ,

∂ t λq < 0 ∀ λ if α s > α s,cr , with α s,cr = 2 B k + 2 √ A k C k , (91)
with the critical coupling α s,cr , see Fig. 21. Accordingly, if the growth of the strong coupling towards the infrared is unlimited, chiral symmetry breaking in QCD is always present, and is basically the converse of asymptotic freedom. More precisely it is the one-gluon exchange coupling in the quark-gluon box diagrams that has to satisfy (91) in the infrared. In this context it is important to mention that it is the gapping of the gluon related to confinement that stops the growth of the exchange couplings in the infrared and even leads to their decay for small momenta, see Fig. 20(b). The simple relation between the size of the strong coupling and chiral symmetry breaking also provides a simple explanation for the restoration at finite temperature: the strong coupling melts down and finally we have α s < α s,cr for all frequencies and momenta, and spontaneous chiral symmetry breaking cannot happen any more.

We emphasise that while the above argument has been discussed within a relatively simple approximation, its structure carries over straightforwardly to the full system of flow equations in QCD. For FRG-literature in gauge theories on the many facets of chiral symmetry breaking in the vacuum, [START_REF] Gies | Universality of spontaneous chiral symmetry breaking in gauge theories[END_REF][START_REF] Pawlowski | Exact flow equations and the U(1) problem[END_REF][START_REF] Cyrol | Nonperturbative quark, gluon, and meson correlators of unquenched QCD[END_REF][START_REF] Fu | QCD phase structure at finite temperature and density[END_REF][START_REF] Mitter | Chiral symmetry breaking in continuum QCD[END_REF][START_REF] Aoki | Nonperturbative renormalization group analysis of the chiral critical behaviors in QED[END_REF][START_REF] Aoki | Wilson renormalization group equations for the critical dynamics of chiral symmetry[END_REF][START_REF] Aoki | Analysis of the Wilsonian effective potentials in dynamical chiral symmetry breaking[END_REF][START_REF] Aoki | Nonladder extended renormalization group analysis of the dynamical chiral symmetry breaking[END_REF][START_REF] Meggiolaro | Evolution equations for the effective four quark interactions in QCD[END_REF][START_REF] Aoki | Evaluation of the spontaneous chiral symmetry breaking scale in general gauge theories with non-perturbative renormalization group[END_REF][START_REF] Aoki | Evaluation of the spontaneous chiral symmetry breaking scale in general gauge theories with non-perturbative renormalization group[END_REF][START_REF] Aoki | Solving the QCD non-perturbative flow equation as a partial differential equation and its application to the dynamical chiral symmetry breaking[END_REF][START_REF] Aoki | Weak solution of the non-perturbative renormalization group equation to describe dynamical chiral symmetry breaking[END_REF][START_REF] Braun | From Quarks and Gluons to Hadrons: Chiral Symmetry Breaking in Dynamical QCD[END_REF][START_REF] Rennecke | Vacuum structure of vector mesons in QCD[END_REF].

In combination the results and mechanisms discussed in the last two sections lead to the interesting interplay between chiral symmetry breaking and confinement. While being short of a proof, the following picture unfolds: While a growing one-gluon exchange coupling leads to chiral symmetry breaking, confinement requires a gapped gluon which stops this growth and even leads to exchange couplings that decay in the infrared. In turn, massless quarks lead to a massless dispersion in the gluon (for N f large enough) and hence no confinement. These properties relate to the 't Hooft anomaly matching argument of 'No confinement without chiral symmetry breaking' (for N f ≥ 3).

Vacuum QCD and hadronic bound states

The mechanisms of chiral symmetry breaking and confinement as described in such a gauge-fixed functional approach lead to a very robust access to both phenomena. In particular the above mechanism of chiral symmetry breaking yields very robust results for strong couplings α s that far exceed the infrared bound α s > α * in [START_REF] Reuter | The large-N limit and the high-temperature phase transition for the φ 4 theory[END_REF] for low cutoff scales. The strength of chiral symmetry breaking is measured in the chiral condensate ∆ q ∝ ´x q(x)q(x) , which agrees well with that in full QCD as measured on the lattice or within quantitative computations with functional methods already in crude approximations.

These approximations have to be improved for a reliable quantitative, and even for a qualitative, access to the hadron structure as well as the phase structure of QCD. In the ongoing and crucial quest for quantitative precision it has turned out that the chiral and confinement dynamics of QCD leads to gluon exchange couplings close to the bound [START_REF] Reuter | The large-N limit and the high-temperature phase transition for the φ 4 theory[END_REF]. In this regime the size of the chiral symmetry breaking, i.e. the size of the chiral condensate ∆ q strongly depends on the size of the running coupling: variations of α s by a few percent lead to variations of ∆ q by factors 1/2 to 2 or more, see [START_REF] Cyrol | Nonperturbative quark, gluon, and meson correlators of unquenched QCD[END_REF][START_REF] Mitter | Chiral symmetry breaking in continuum QCD[END_REF]. In short, QCD is living on the edge. One may be tempted to classify this behaviour as a merely technical challenge, but it has far-reaching physics consequences: Even small changes of external parameters such as temperatures, density or (chromo-) electric and (chromo-) magnetic background fields potentially have a large impact on observables.

Accordingly, the qualitative and even more so quantitative access to strongly correlated infrared QCD from first principles (the only input being the fundamental parameters of QCD, the current quark masses) requires advanced approximation schemes. Advanced numerical and computer algebraic tools for the derivation of large systems of flow equations, Dyson-Schwinger equations and Slavnov-Taylor identities, as well as their numerical solution have been developed in [START_REF] Huber | Algorithmic derivation of functional renormalization group equations and Dyson-Schwinger equations[END_REF][START_REF] Huber | DoFun 3.0: Functional equations in Mathematica[END_REF][START_REF] Cyrol | FormTracer -A Mathematica Tracing Package Using FORM[END_REF][START_REF] Cyrol | FormTracer GitHub Repository[END_REF]. Related first principles work can be found in [START_REF] Cyrol | Landau gauge Yang-Mills correlation functions[END_REF][START_REF] Cyrol | Nonperturbative quark, gluon, and meson correlators of unquenched QCD[END_REF][START_REF] Mitter | Chiral symmetry breaking in continuum QCD[END_REF][START_REF] Hajizadeh | Exploring the Tan contact term in Yang-Mills theory[END_REF][START_REF] Corell | Correlation functions of three-dimensional Yang-Mills theory from the FRG[END_REF]. While being developed in the context of QCD, they also can be readily applied to generic relativistic and non-relativistic systems.

One of the chiefly interesting properties of QCD is the transformation of a weakly-interacting almost chiral theory of quarks and gluons for large momentum scales into a weakly-interacting theory of hadrons with large masses and the light pions at small momentum scales, governed by chiral perturbation theory. The formation of the hadronic bound states and the related dynamical change of the relevant degrees of freedom is very well captured with dynamical hadronization [START_REF] Gies | Renormalization flow of bound states[END_REF][START_REF] Gies | Universality of spontaneous chiral symmetry breaking in gauge theories[END_REF], see also [START_REF] Ellwanger | FLow equations for N point functions and bound states[END_REF][START_REF] Ellwanger | Evolution equations for the quark -meson transition[END_REF]. Further formal developments can be found in [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Fu | QCD phase structure at finite temperature and density[END_REF][START_REF] Floerchinger | Exact flow equation for composite operators[END_REF], applications to QCD are found in [START_REF] Cyrol | Nonperturbative quark, gluon, and meson correlators of unquenched QCD[END_REF][START_REF] Fu | QCD phase structure at finite temperature and density[END_REF][START_REF] Mitter | Chiral symmetry breaking in continuum QCD[END_REF][START_REF] Braun | From Quarks and Gluons to Hadrons: Chiral Symmetry Breaking in Dynamical QCD[END_REF][START_REF] Alkofer | Bound state properties from the functional renormalization group[END_REF]. It is reminiscent of a Hubbard-Stratonovich transformation, but does not share the potential double-counting problems of the latter that are well-known from applications in low-energy effective field theories.

Here we explain this problem at the example of the scalar channel of the four-quark interaction already discussed in the last section Sec. 5.3.2 on chiral symmetry breaking. The Hubbard-Stratonovich transformation of the four-quark term is a reformulation in terms of effective exchange fields (here the scalar σ-field) and a Yukawa interaction,

- λ q 4 ( q q) 2 = 1 2 m 2 φ σ 2 + 1 2 h σ σ qq σ EoM , with h 2 σ 2m 2 φ = λ q and σ EoM (q, q) = h σ 2m 2 φ qq , (92)
where σ EoM (q, q) is the solution of the σequation of motion (EoM). Note that the Yukawa coupling and meson mass function m φ can also be chosen such that only a part of the four-quark interaction is captured by the right hand side. Double counting may arise due to the fact that the original four-quark interaction can now be represented both in a purely fermionic or a mixed fermion-scalar form. While this is easily disentangled on the classical level, it is hard to resolve on the quantum level for the quantum effective action. Within the FRG-approach this problem is avoided with dynamical hadronization. There, the dynamical redistribution of the full scalar four-quark interaction in the effective action is described by the flow of the four-quark coupling,

∂ t λq -2 1 + η q λq -hσ Ȧ = Flow (4) ( qq) 2 . ( 93
)
In [START_REF] Tetradis | Scale dependence of the average potential around the maximum in φ 4 theory[END_REF] the right hand side stands for the diagrams of the scalar four-quark flow. Now this flow also includes diagrams with the effective meson field, and the subscript ( qq) 2 indicates the projection on the scalar part of the interaction. Moreover, for the time being we only consider vanishing momenta. The term proportional to Ȧ is the analogue of the choice for h 2 σ /(2m 2 φ ) in the Hubbard-Stratonovich transformation. The typical choice for the dynamical hadronization parameter Ȧ is such that the flow of the four-quark coupling vanishes identically, ∂ t λq ≡ 0. This leads us to Ȧ = -Flow (4) ( qq) 2 /h σ . Then, with the initial four-quark interaction Γ (4) ( q q) 2 ,Λ ≡ 0 at a large initial cutoff scale Λ the full scalar channel of the four-quark interaction is parametrised in terms of the exchange fields σ and π. For a diagrammatic depiction see Fig. 22(a), where we have dropped higher order loop terms.

In QCD the above simple example of a scalar four-quark coupling has to be embedded in the more complicated four-quark interaction structure of the theory. For example, in two-flavor QCD we have 10 momentum-independent four-quark tensor structures that respect the symmetries of the theory, and in 2+1-flavor QCD this number increases to 27 momentum-independent tensor structures. Each tensor structure comes with a scalar dressing function depending on all momenta. The size of the full basis of tensor structures including momentum-dependent ones is even larger by an order of magnitude, see e.g. [START_REF] Eichmann | Four-point functions and the permutation group S4[END_REF]. The latter extension is important for quantitative studies of the hadron-resonance structure. A fully reliable treatment of even the momentum-independent tensor structures necessitates to disentangle all of them in order to avoid the Fierz ambiguity. This will be discussed further in Sec. 5.4.3, for a respective review see [START_REF] Braun | Fermion Interactions and Universal Behavior in Strongly Interacting Theories[END_REF]. where = 0 (φ)

(p1 + p3) 2 = 0 (p2 + p4) 2 = 0 p 1 p 2 p 3 p 4 + = s
(a) Full four-quark interaction in the scalar-pseudoscalar channel as an effective scalar-pseudoscalar meson exchange and a residual fourquark interaction (square). The meson exchange diagram provides the scalar-pseudoscalar four-quark coupling at (p 1 + p 3 ) 2 = (p 2 + p 4 ) 2 = 0 (s-channel). Higher order contributions in the flow (93) are dropped. In QCD, dynamical hadronization has been used for the scalar-pseudoscalar (σ -π) channel of the four-quark interaction. On the level of the effective action this leads to an extension of the field content by the mesonic O(4)-field φ = (σ, π). We emphasize, that the effective action Γ DynHad with these additional mesonic fields is still an effective action of first principles QCD: The standard effective action of QCD, Γ QCD [A µ , c, c, q, q], can be obtained from Γ DynHad with

Γ QCD [A, c, c, q, q] = Γ DynHad [A, c, c, q, q, φ EoM ] , (94)
where the mesonic solution of the EoM depends on the fundamental fields, φ EoM = φ EoM [A, c, c, q, q]. This is reminiscent of going from the 2PI-effective action to the 1PI-effective action with the EoM for the propagator. Indeed, technically Γ DynHad is the 2PPI (two-point particle) effective action. For more details see in particular [START_REF] Fu | QCD phase structure at finite temperature and density[END_REF]. Dynamical hadronization is nothing but a dynamical change of the field basis in the theory. Within an optimal use it allows to expand the dynamics in terms of weakly interacting degrees of freedom at all cutoff scales and hence speedsup the convergence of a given expansion scheme. In particular, the low energy limit of higher order scatterings of resonant interaction channels is now described very efficiently in terms of an effective potential V eff (ρ) of the composite mesonic field φ with ρ = φ 2 /2. In the original formulation in terms of quarks the higher order terms ρ 2 , ρ 3 , ρ 4 , ... are related to channels of multi-quark scattering vertices, ( qq) 3 , ( qq) 4 , ( qq) 5 , ... which are technically less accessible. Note however, that dynamical hadronization does not describe the complete four-quark interaction, but only a given momentum channel of a specific tensor structure. The remnant vertex has to be kept, in particular if discussing competing order effects, see for example [START_REF] Fukushima | Emergent Hadrons and Diquarks, and Dense QCD Matter[END_REF][START_REF] Denz | Partial bosonisation for the two-dimensional Hubbard model: How well does it work?[END_REF]. This is depicted in Fig. 22(a). A complete bosonization in terms of a series expansion has been discussed in [START_REF] Jakovac | Bound states in Functional Renormalization Group[END_REF][START_REF] Jakovac | Interacting two-particle states in the symmetric phase of the chiral Nambu-Jona-Lasinio model[END_REF].

In Fig. 22(b) one also sees the sequential decoupling of the different degrees of freedom within the flow equation for first principles QCD with dynamical hadronization of the (σ -π) channel depicted in Fig. 19: The full fielddependent propagators of gluons, quarks and mesons and hence the respective exchange couplings are gapped with different mass scales. In the ultraviolet regime for k 1 GeV, the quark-gluon dynamics is dominant as the gluonexchange couplings dominate over the other couplings, see Fig. 22(b). At about 600 MeV, the quark-meson dynamics takes over and the gluon dynamics decouples due to the gluon mass gap m gap . The latter is the confinement scale in disguise, see also Sec. 5.3.1. Finally only the dynamics of the pseudo-Goldstone bosons, the pions, survives. Below the pion mass also the latter dynamics dies out. Note that the auxiliary ghost field is massless but only interacts with the matter fields via the gluon. Accordingly the ghost decouples together with the gluon from the matter sector.

Further hadronic resonances (and heavier quarks) are far too heavy to give sizeable effects to the Euclidean vacuum dynamics of QCD. We rush to add that this decoupling only concerns the offshell dynamics in the loops in the flow equation, see Fig. 19. It does not entails the irrelevance of onshell hadrons for observables. However, the respective correlation functions of hadrons or more generally asymptotic states can be built from the offshell dynamics of ghosts, gluons, quarks and light mesons. This sequential decoupling already explains the success and the natural emergence of chiral perturbation theory from QCD. It also allows for the computation of its low energy parameters within the FRG, for recent works see [START_REF] Eser | Low-energy limit of the O(4) quark-meson model from the functional renormalization group approach[END_REF][START_REF] Divotgey | Dynamical generation of low-energy couplings from quark-meson fluctuations[END_REF]. The concept of dynamical hadronization straightforwardly carries over to gluonic excitations such as glueballs or to effective degrees of freedom inside hadrons such as the pomeron. For first FRG-steps in the latter direction see [START_REF] Bartels | A functional RG approach for the BFKL Pomeron[END_REF][START_REF] Bartels | The Odderon in QCD with running coupling[END_REF], for the description of baryonic degrees of freedom see [START_REF] Weyrich | Chiral mirror-baryon-meson model and nuclear matter beyond mean-field approximation[END_REF]. Hence, dynamical hadronization is a very natural way to describe the quark-hadron duality in QCD.

Another very interesting link is that to bound-state equations of QCD, Bethe-Salpeter equations (BSEs), Faddeev equations (FEs), and higher ones for e.g. tetraquarks, for a recent review see [START_REF] Eichmann | Baryons as relativistic three-quark bound states[END_REF]. To see this link more clearly, we evaluate the pseudoscalar channel of the four-quark interaction close to the pion pole with p 2 = -m 2 π,pole . There, the pion propagator grows large and the higher order terms in Fig. 22(a) are suppressed. This leaves us with the exchange term depicted in Fig. 22(a), which is given by the pion propagator close to the pole, 1/(p 2 + m 2 π,pole ) → ∞, sandwiched by two Yukawa vertices. Note that these vertices are not simply h σ but carry all allowed tensor structures with momentum-dependent dressings. A comparison with the respective BSE expressions reveals that the Yukawa vertices Γ (3) q πq can be identified with the Bethe-Salpeter wave function of the pion. The latter carries low energy constants of QCD such as the pion decay constant f π . These relations carry over to generic BSE, FE and higher hadronic bound state equations if the respective channels are hadronized. In summary, dynamical hadronization allows for a natural implementation of bound-state equations of QCD within an effective action approach. Together with the recent developments for real-time flow equations, see Sec. 5.4.2, this opens a yet unexplored door to a uniform access to one of the major challenges in QCD, the description of the spectrum of hadron resonances. In particular this would encompass the higher states, also including decays and scattering processes. The setup is applicable to general theories and generic composite fields. It does not only allow for the hadronization of mesons, but also, e.g., baryons and tetraquarks.

In the context of condensed matter and statistical physics systems the present setup is called dynamical condensation/pairing. In particular ultracold atomic systems with their very exciting strongly correlated physics and pairing phenomena are ideally-suited for the application of the techniques described above. With the plethora of precise experimental results they also provide an ideal testbed for the techniques described above. For FRG-works on two-, three-, and four-body resonances with and without dynamical pairing, and related work on limit cycles in non-relativistic systems see e.g. [588-590, 594, 834, 864-875], for introductory reviews see [START_REF] Boettcher | Ultracold atoms and the Functional Renormalization Group[END_REF][START_REF] Floerchinger | Efimov physics from the functional renormalization group[END_REF].

Phase structure and dynamics of QCD

A major challenge in QCD concerns its phase structure and dynamics. This is important for the explanation of heavy-ion collisions (HICs), the QCD phase transition in the early universe, and for neutron star physics.

The related physics offers several intricate and yet unsolved questions ranging from the equilibrium phase structure of QCD at finite density to the non-equilibrium dynamics at early times of a heavy ion collision. Specific questions concern in particular the existence and location of a critical end point (CEP) for the chiral and confinementdeconfinement crossovers at large density, the possible existence of mixed or inhomogeneous phases as well as the phase structure beyond nuclear densities. In the latter case most likely competing order dynamics has to be resolved, see also Sec. 4.2.2. At these large densities we are also interested in the equation of state (EoS), which is chiefly important for a resolution of the mass range of neutron stars. Finally, for the description of HICs we need access to the dynamics of strongly correlated QCD from the first far from equilibrium phase over kinetic and hydrodynamical phases to transport close to equilibrium. It is in particular the non-equilibrium (real-time) physics as well as the high density regime that are at present not accessible with lattice simulations due to sign-problems. In turn, in the vacuum at T, µ = 0 and at finite temperature lattice simulations provide quantitative results that can be used for benchmarking computations and approximations within the FRG-approach to QCD.

The unreasonable effectiveness of low-energy effective theories

The highly exciting phenomena mentioned above in the introduction of Sec. 5.4 require an access to QCD at momentum and cutoff scales p 2 , k 2 1 (GeV) 2 . In this regime QCD is strongly correlated, and nonperturbative 60 [START_REF] Resch | Mass sensitivity of the three-flavor chiral phase transition[END_REF]. At µ = 0 there is a small first-order region around the chiral limit. The anomalous U A (1)-breaking is chiefly important for this results, without the anomaly the phase structure is significantly changed, see [START_REF] Resch | Mass sensitivity of the three-flavor chiral phase transition[END_REF]. methods are required. In Sec. 5.3.3 we have already discussed how dynamical low energy degrees of freedom of QCD emerge naturally in the FRG-approach to QCD with dynamical hadronization, including the sequential decoupling of the gluonic and quark degrees of freedom, see in particular Fig. 22.

Below the gluon-decoupling scale of about 600 MeV the flow of the QCD effective action is only driven by diagrams with quarks and mesons. Note however, that this dynamics takes place in a gluonic background, the solution of the gluonic equations of motion. As discussed in Sec. 5.3.1, at finite temperature and density such a background, A 0,EoM 0, gives rise to a non-trivial Polyakov-loop expectation value. This property of QCD can be included in low-energy effective theories (LEFTs) of QCD in terms of a Polyakov-loop background L, or that of a temporal gauge field A 0 . In these Polyakov-loop enhanced models this background couples to the quark fields and the effective action of the LEFT is augmented with the effective potential in QCD,

V Pol [L] or V eff [A 0 ].
Clearly the Polyakov-loop enhanced models are not confining themselves but the Polyakov-loop background carries over confinement information from QCD. This is usually called Statistical Confinement. Note however, that the QCD-embedding of the models discussed above entails that, while it is not the only confining property to be taken into account, it carries a good deal of it. For a recent review on Polyakov loop-enhanced models see [START_REF] Fukushima | Polyakov loop modeling for hot QCD[END_REF].

Note also that one of the consequences of the QCD-embedding with and without dynamical hadronization is the formal equivalence of fermionic NJL-type models and models with quark and hadronic degrees of freedom such as the quark-meson model (QM). The renormalizability of the latter in comparison to the former models is a red herring: Firstly, both models have a physical ultraviolet cutoff: one due to non-renormalizability (NJL), the other due to a Landau pole (QM). Moreover, they cease to describe QCD even for scales smaller than the ultraviolet cutoff, see e.g. [START_REF] Alkofer | Bound state properties from the functional renormalization group[END_REF]. Still one should keep in mind that models with hadronic degrees of freedom are better suited to describe the infrared dynamics of QCD, in particular that of pions.

In summary, the flow of full QCD naturally leads to an emergent low-energy effective theory of quarks and hadrons as well as a background temporal gluon or Polyakov loop in the infrared. These QCD-assisted models can be improved within systematic expansion schemes towards QCD, for a more detailed description see [START_REF] Fu | QCD phase structure at finite temperature and density[END_REF][START_REF] Alkofer | Bound state properties from the functional renormalization group[END_REF][START_REF] Springer | QCD-inspired determination of NJL model parameters[END_REF][START_REF] Braun | Renormalization group consistency and low-energy effective theories[END_REF] and references therein. This allows to study many phenomena in low energy QCD within these LEFTs. The latter allow for an easier access to the dynamics both in Euclidean spacetime and also with genuinely real-time FRG-methods. The LEFTs are also an ideal testbed for conceptual advances in the FRG to be used later also in QCD.

The phase structure of QCD from low-energy effective models

This has led to a plethora of FRG-works in low-energy effective models (LEFT) on the phase structure of QCD with NJL-type and quark-meson models (QM-model) with and without Polyakov-loop background. Below we provide a brief recollection of FRG-works on the phase structure of QCD, and single out only a few of the many highlights.

The chiral phase structure has been studied in many works, starting with works at finite temperature and vanishing density, see [START_REF] Jungnickel | Effective action for the chiral quark-meson model[END_REF][START_REF] Berges | Two flavor chiral phase transition from nonperturbative flow equations[END_REF][START_REF] Berges | The Chiral phase transition at high baryon density from nonperturbative flow equations[END_REF][START_REF] Berges | Quark and nuclear matter in the linear chiral meson model[END_REF][START_REF] Papp | On the convergence of the expansion of renormalization group flow equation[END_REF][START_REF] Bergerhoff | The Thermal renormalization group for fermions, universality, and the chiral phase transition[END_REF][START_REF] Bohr | Renormalization group flow equations and the phase transition in O(N) models[END_REF][START_REF] Schaefer | Finite temperature gluon condensate with renormalization group flow equations[END_REF][START_REF] Braun | Thermodynamics of QCD low-energy models and the derivative expansion of the effective action[END_REF][START_REF] Fukushima | Second-order and Fluctuation-induced First-order Phase Transitions with Functional Renormalization Group Equations[END_REF]. The chiral phase structure at finite density or baryon-chemical potential including the volume- (b) Quark spectral functions from [START_REF] Tripolt | Fermionic excitations at finite temperature and density[END_REF] with the aFRG. The temperature is T = 180 MeV, close but above the crossover temperature, see also [START_REF] Tripolt | Fermionic spectral functions with the Functional Renormalization Group[END_REF][START_REF] Wang | Fermion spectral function in hot strongly interacting matter from the functional renormalization group[END_REF]. The spectral function shows quasi-particle and plasmino peaks as well as the ultrasoft mode. dependence of the chiral phase structure has been considered in [START_REF] Schaefer | Fluctuations and the QCD Phase Diagram[END_REF][START_REF] Resch | Mass sensitivity of the three-flavor chiral phase transition[END_REF][START_REF] Berges | The Chiral phase transition at high baryon density from nonperturbative flow equations[END_REF][START_REF] Berges | Quark and nuclear matter in the linear chiral meson model[END_REF], superfluid phases with gapless fermionic excitations at finite (isospin) density and their counterparts in non-relativistic systems have been studied in [START_REF] Boettcher | Phase structure of spin-imbalanced unitary Fermi gases[END_REF][START_REF] Boettcher | Sarma phase in relativistic and non-relativistic systems[END_REF]. Here we want to single out one of the many important and interesting results, which concerns the chiral phase structure of QCD as a function of the masses of the pseudo-Goldstone bosons of chiral symmetry breaking, the pions and kaons, [START_REF] Resch | Mass sensitivity of the three-flavor chiral phase transition[END_REF]. Their masses can be varied by varying the current quark masses that originate in the Higgs mechanism. This analysis is specifically important as it also constrains the QCD phase structure finite density, for FRG-studies of the relevant Yang-Lee edge singularities and related work see [START_REF] Litim | Critical O(N) models in the complex field plane[END_REF][START_REF] An | Functional renormalization group approach to the Yang-Lee edge singularity[END_REF][START_REF] Zambelli | Lee-Yang model from the functional renormalization group[END_REF][START_REF] Connelly | Universality driven analytic structure of QCD crossover: radius of convergence and QCD critical point[END_REF][START_REF] Connelly | Universal location of the Yang-Lee edge singularity in O(N) theories[END_REF]. Moreover, the axial anomaly plays an important role for this investigation. It is a genuine nonperturbative problem, where a good grip on the scaling behaviour of light or massless degrees of freedom in the chiral limit is crucial. The chiral limit of QCD is still a major challenge for lattice simulations, and hence this problem is specifically well-suited for a renormalization group study. For more details see the 2 + 1-flavor investigation within the quark-meson model in [START_REF] Resch | Mass sensitivity of the three-flavor chiral phase transition[END_REF]. The respective phase structure is depicted in Fig. 23.

Further investigations with the FRG include not only the chiral dynamics within the phase structure, but also statistical confinement in terms of a background Polyakov loop or a temporal background gluon. The phase structure of QCD and the interplay between confinement and chiral symmetry at finite temperature and density have been studied in [START_REF] Skokov | Meson fluctuations and thermodynamics of the Polyakov loop extended quark-meson model[END_REF][START_REF] Herbst | The phase structure of the Polyakov-quark-meson model beyond mean field[END_REF][START_REF] Braun | Dynamical Locking of the Chiral and the Deconfinement Phase Transition in QCD[END_REF][START_REF] Morita | Role of mesonic fluctuations in the Polyakov loop extended quark-meson model at imaginary chemical potential[END_REF][START_REF] Braun | On the Relation of the Deconfinement and the Chiral Phase Transition in Gauge Theories with Fundamental and Adjoint Matter[END_REF][START_REF] Herbst | Phase structure and thermodynamics of QCD[END_REF][START_REF] Herbst | Thermodynamics of QCD at vanishing density[END_REF][START_REF] Haas | Improved Polyakov-loop potential for effective models from functional calculations[END_REF][START_REF] Fu | Chiral criticality and glue dynamics[END_REF]. Note in this context that in the large density regime with µ B /T 2 -3 with baryon chemical potential µ B = 3µ q , regime lattice simulations are obstructed by the sign problem. In turn, functional approaches have to deal with the increasingly complex structure of the dynamical low energy degrees of freedom. In the presence of a chemical potential the pole-and cut-structure in the complex frequency plane gets relevant, leading to exciting phenomena such as the Silver-Blaze property of QCD. For FRG-works see [START_REF] Strodthoff | Quark-meson-diquark model for two-color QCD[END_REF][START_REF] Kamikado | Fluctuations in the quark-meson model for QCD with isospin chemical potential[END_REF][START_REF] Strodthoff | Polyakov-Quark-Meson-Diquark Model for two-color QCD[END_REF][START_REF] Khan | The Phase Diagram of QC2D from Functional Methods[END_REF]. In heavy ion collisions one often investigates fluctuation observables, for FRG-works see [START_REF] Schaefer | Susceptibilities near the QCD (tri)critical point[END_REF][START_REF] Nakano | Fluctuations and isentropes near the chiral critical endpoint[END_REF][START_REF] Skokov | Quark number fluctuations in the Polyakov loop-extended quark-meson model at finite baryon density[END_REF][START_REF] Skokov | Non-perturbative dynamics and charge fluctuations in effective chiral models[END_REF][START_REF] Skokov | Volume Fluctuations and Higher Order Cumulants of the Net Baryon Number[END_REF][START_REF] Morita | Net quark number probability distribution near the chiral crossover transition[END_REF][START_REF] Fu | On the relevance of matter and glue dynamics for baryon number fluctuations[END_REF][START_REF] Fu | Correlating the skewness and kurtosis of baryon number distributions[END_REF][START_REF] Fu | Baryon number fluctuations at finite temperature and density[END_REF][START_REF] Rennecke | Fluctuation-induced modifications of the phase structure in (2+1)-flavor QCD[END_REF][START_REF] Almasi | Volume dependence of baryon number cumulants and their ratios[END_REF][START_REF] Almasi | Baryon number fluctuations in chiral effective models and their phenomenological implications[END_REF][START_REF] K.-X. Sun | Baryon number probability distribution at finite temperature[END_REF][START_REF] Fu | Strangeness neutrality and baryon-strangeness correlations[END_REF][START_REF] Fu | Strangeness Neutrality and QCD Thermodynamics[END_REF][START_REF] Wen | Baryon number fluctuations in the 2+1 flavor low energy effective model[END_REF][START_REF] Wen | Correlations of conserved charges and QCD phase structure[END_REF]. In heavy ion collisions we also expect strong (chromo-) magnetic and electric fields, FRG-work in this direction has been done in [START_REF] Aoki | Functional renormalization group study of the Nambu-Jona-Lasinio model at finite temperature and density in an external magnetic field[END_REF][START_REF] Skokov | Phase diagram in an external magnetic field beyond a mean-field approximation[END_REF][START_REF] Fukushima | Magnetic catalysis in hot and dense quark matter and quantum fluctuations[END_REF][START_REF] Braun | Delayed Magnetic Catalysis[END_REF][START_REF] Mueller | Magnetic catalysis and inverse magnetic catalysis in QCD[END_REF][START_REF] Fu | Four-fermion interactions and the chiral symmetry breaking in an external magnetic field[END_REF][START_REF] Li | Thermodynamics of 2+1 Flavor Polyakov-Loop Quark-Meson Model under External Magnetic Field[END_REF].

A very exciting development has taken place in the past decade concerning the access of real-time properties of QCD with the FRG. Naturally, the real-time approach to QCD correlation functions has not only applications to the QCD phase structure but also to the hadron spectrum, see the discussion at the end of Sec. 5.3.3. Applications to heavy ion collisions range from dilepton rates, see e.g. [START_REF] Tripolt | In-medium spectral functions and dilepton rates with the Functional Renormalization Group[END_REF][START_REF] Tripolt | Electromagnetic and weak probes: theory[END_REF] or computing transport processes and transport coefficients relevant for the hydrodynamical phase of the collision, see e.g. [START_REF] Haas | Gluon spectral functions and transport coefficients in Yang-Mills theory[END_REF][START_REF] Christiansen | Transport Coefficients in Yang-Mills Theory and QCD[END_REF][START_REF] Bluhm | Time-evolution of fluctuations as signal of the phase transition dynamics in a QCD-assisted transport approach[END_REF].

Within the FRG, real-time correlation functions are either worked out with numerical analytic continuation, see e.g. [START_REF] Cyrol | Reconstructing the gluon[END_REF][START_REF] Li | The generalised infrared structure of the gluon propagator[END_REF][START_REF] Binosi | Spectral functions of confined particles[END_REF], or computed directly. Real-time FRGs have been developed in different areas of physics, see e.g. [START_REF] Gasenzer | Towards far-from-equilibrium quantum field dynamics: A functional renormalisation-group approach[END_REF][START_REF] Pietroni | Flowing with Time: a New Approach to Nonlinear Cosmological Perturbations[END_REF][START_REF] Gasenzer | Far-from-equilibrium quantum many-body dynamics[END_REF][START_REF] Corell | Flowing with the Temporal Renormalisation Group[END_REF][START_REF] Lesgourgues | Non-linear Power Spectrum including Massive Neutrinos: the Time-RG Flow Approach[END_REF][START_REF] Bartolo | Signatures of Primordial non-Gaussianities in the Matter Power-Spectrum and Bispectrum: the Time-RG Approach[END_REF], typically also using specific properties of the situation under consideration, for related work see Secs. 3, 6 .The following works are based on ideas developed in [START_REF] Floerchinger | Analytic Continuation of Functional Renormalization Group Equations[END_REF][START_REF] Kamikado | Real-Time Correlation Functions in the O(N) Model from the Functional Renormalization Group[END_REF][START_REF] Pawlowski | Real time correlation functions and the functional renormalization group[END_REF][START_REF] Steib | Renormalization in Minkowski space-time[END_REF]. The promising developments, that started with [START_REF] Kamikado | Real-Time Correlation Functions in the O(N) Model from the Functional Renormalization Group[END_REF], have been called aFRG (analytically-continued FRG), for works on meson (a) Chiral phase structure of QCD for 2+1-flavor QCD. The black dashed line is the chiral crossover line from the FRGcomputation in [START_REF] Fu | QCD phase structure at finite temperature and density[END_REF]. The blue dashed line is from an FRGassisted DSE-computation in [START_REF] Gao | QCD phase structure from functional methods[END_REF][START_REF] Gao | Chiral phase structure and critical end point in QCD[END_REF]. The transition temperature is determined by the peak of the thermal susceptibility of the chiral condensate of the light quarks. The black circle indicates the critical end point of the crossover line, by now corroborated by the most recent DSE result, [START_REF] Gao | Chiral phase structure and critical end point in QCD[END_REF]. The hatched red area depicts a regime with a sizable chiral condensate and a minimum of the meson dispersion at non-vanishing spatial momentum. This indicates a potentially inhomogeneous regime. We also provide lines for µ B /T = 2, 3 related to reliability bounds for lattice results, [START_REF] Bellwied | The QCD phase diagram from analytic continuation[END_REF][START_REF] Bazavov | Chiral crossover in QCD at zero and non-zero chemical potentials[END_REF][START_REF] Braguta | Finite-density QCD transition in a magnetic background field[END_REF], and µ B /T = 3, 4 for results from functional methods, [

(S + P)

(S + P) adj (V + A) (V + A) (V A) (V A) (V + A) adj (V A) adj (b)
Strength of all four-quark couplings two-flavor QCD at the crossover temperature in terms of quark chemical potential from [START_REF] Braun | Fierz-complete NJL model study III: Emergence from quark-gluon dynamics[END_REF]. This Fierz-complete computation shows dominance of the (σ -π)channel (scalar-pseudoscalar) for small chemical potentials. The transition temperature T trans from dominance of the (σ -π)-channel to the (scs)-channel (diquark) happens at T trans ≈ 0.8T c , for the details see [START_REF] Braun | Fierz-complete NJL model study III: Emergence from quark-gluon dynamics[END_REF]. In [START_REF] Fu | QCD phase structure at finite temperature and density[END_REF] it has been shown that relative temperatures and chemical potentials agree well for 2-and 2+1-flavor QCD. For 2 + 1 flavor QCD T trans agrees well with the onset of the potentially inhomogeneous regime at the crossover line: the intersection point of the hatched red area in Fig. 25(a) with the crossover line. In this regime the FRG-computations so far lack full reliability and have to be systematically improved. This requires in particular a Fierz-complete basis combined with higher-order mesonic scatterings.

Figure 25: QCD phase diagram from QCD flows, Data taken from [START_REF] Fu | QCD phase structure at finite temperature and density[END_REF][START_REF] Gao | QCD phase structure from functional methods[END_REF][START_REF] Gao | Chiral phase structure and critical end point in QCD[END_REF][START_REF] Braun | Fierz-complete NJL model study III: Emergence from quark-gluon dynamics[END_REF], more details can be found there.

spectral functions in quark-meson and meson effective theories in the vacuum and at finite temperatures and densities see [START_REF] Floerchinger | Analytic Continuation of Functional Renormalization Group Equations[END_REF][START_REF] Tripolt | Spectral Functions for the Quark-Meson Model Phase Diagram from the Functional Renormalization Group[END_REF][START_REF] Kamikado | Real-Time Correlation Functions in the O(N) Model from the Functional Renormalization Group[END_REF][START_REF] Pawlowski | Real time correlation functions and the functional renormalization group[END_REF][START_REF] Jung | In-Medium Spectral Functions of Vector-and Axial-Vector Mesons from the Functional Renormalization Group[END_REF][START_REF] Jung | Fluctuating vector mesons in analytically continued functional RG flow equations[END_REF][START_REF] Helmboldt | Towards quantitative precision in the chiral crossover: masses and fluctuation scales[END_REF][START_REF] Tripolt | Flow equations for spectral functions at finite external momenta[END_REF][START_REF] Strodthoff | Self-consistent spectral functions in the O(N) model from the functional renormalization group[END_REF][START_REF] Yokota | Functional renormalization group analysis of the soft mode at the QCD critical point[END_REF][START_REF] Yokota | Tachyonic instability of the scalar mode prior to the QCD critical point based on the functional renormalization-group method in the two-flavor case[END_REF][START_REF] Wang | Meson spectral functions at finite temperature and isospin density with the functional renormalization group[END_REF], for quark spectral functions see [START_REF] Tripolt | Fermionic excitations at finite temperature and density[END_REF][START_REF] Tripolt | Fermionic spectral functions with the Functional Renormalization Group[END_REF][START_REF] Wang | Fermion spectral function in hot strongly interacting matter from the functional renormalization group[END_REF]. In Figs. 24(a), 24(b) we have depicted aFRG results for the nonperturbative spectral functions for vector mesons and quarks obtained in [START_REF] Jung | In-Medium Spectral Functions of Vector-and Axial-Vector Mesons from the Functional Renormalization Group[END_REF][START_REF] Tripolt | Fermionic excitations at finite temperature and density[END_REF]. These real-time results allow to study the emergence and dissolution of the hadronic low-energy degrees of freedom onshell. This is important for an interpretation of HIC results. Moreover, within Euclidean numerical computations the respective physics information is diluted or even lost due to the necessity of analytically continuing numerical data with a non-trivial systematic error and finite numerical accuracy.

Towards the QCD phase structure from first principles

The advances in the FRG-approach to QCD concerning quantitative computations in extended truncations in the vacuum, see Sections 5.3.1, 5.3.2 and 5.3.3, and the plethora of results for LEFTs for finite temperature and density discussed in the Sections 5.4.1 and 5.4.2, prepare the stage for FRG-applications to QCD at finite temperature and density. While these investigations have not reached the quantitative level of the vacuum applications, results on two-flavor QCD at finite temperature and imaginary chemical potential, [START_REF] Braun | Phase Structure of Two-Flavor QCD at Finite Chemical Potential[END_REF], as well as one-, two-and 2+1 flavor QCD at finite temperature and chemical potential have been obtained, [START_REF] Fu | QCD phase structure at finite temperature and density[END_REF][START_REF] Braun | Fierz-complete NJL model study III: Emergence from quark-gluon dynamics[END_REF][START_REF] Braun | The QCD Phase Boundary from Quark-Gluon Dynamics[END_REF], see Fig 25 for some results. The nuclear equation of state (EoS) at low temperatures and high densities has been studied in [START_REF] Leonhardt | Symmetric nuclear matter from the strong interaction[END_REF]. In the recent work [START_REF] Braun | Chiral Susceptibility in (2+1)-flavour QCD[END_REF], the pion mass dependence of the chiral phase transition temperature (magnetic EoS) has been studied. For lattice results we refer the reader to [981-983, 995, 996] and references therein, for results from other functional methods we refer the reader to [766, 814-816, 818, 821, 979, 980, 984, 997-1000] and references therein, for a recent review see [START_REF] Fischer | QCD at finite temperature and chemical potential from Dyson-Schwinger equations[END_REF].

The phase boundary obtained in [START_REF] Fu | QCD phase structure at finite temperature and density[END_REF] for N f = 2 + 1 flavors, see Fig 25(a), agrees quantitatively with respective 2+1 flavor lattice results at small densities. It also agrees well with the most recent DSE-computations (FRG-assisted DSE), [START_REF] Gao | QCD phase structure from functional methods[END_REF][START_REF] Gao | Chiral phase structure and critical end point in QCD[END_REF]. In these works, the DSEs for the correlation functions are expanded about the quantitative N f = 2-flavour vacuum correlation functions obtained with the FRG in [START_REF] Cyrol | Nonperturbative quark, gluon, and meson correlators of unquenched QCD[END_REF]. In turn, at large densities no lattice results are present due to the notorious sign problem, and lattice result at large densities such as those in [START_REF] Braguta | Finite-density QCD transition in a magnetic background field[END_REF] depicted in Fig 25(a) are extrapolations. Interestingly, [START_REF] Fu | QCD phase structure at finite temperature and density[END_REF] predicts a critical end point at large baryon chemical potential µ B , as well as a potentially inhomogeneous regime. The location of the critical end is by now corroborated by the FRGassisted DSE results in [START_REF] Gao | Chiral phase structure and critical end point in QCD[END_REF]. However, for obtaining quantitative predictivity in this regime, functional approaches still require systematic improvements: The onset of the latter agrees well with the change of the dominant four-quark interaction channel from the scalar-pseudoscalar channel to a diquark channel found in [START_REF] Braun | Fierz-complete NJL model study III: Emergence from quark-gluon dynamics[END_REF], see the discussion below Fig 25(b). This coincidence is even more remarkable and non-trivial, as [START_REF] Fu | QCD phase structure at finite temperature and density[END_REF] works with dynamical hadronization of the scalar-pseudoscalar channel which gives access to higher order scattering of mesonic degrees of freedom, while [START_REF] Braun | Fierz-complete NJL model study III: Emergence from quark-gluon dynamics[END_REF] works with a Fierz-complete basis. This indicates the onset of new physics as well as additional resonant channels. Consequently, in this regime the respective systematic error of the present truncations grows large, and the results have to be corroborated within systematically improved truncations. For example, multi-scattering of potentially resonant diquark and density channels have to be taken into account. These systematic improvements pose solely technical challenges in comparison to the conceptual intricacies of the sign problem. We emphasize that the intricacies of the interaction-structure of QCD at large densities may still pose an insurmountable challenge despite being of technical nature. Whether or not apparent convergence can be achieved in the large density region of QCD within a first principles FRG-study remains to be seen in the future.

We close the QCD-part with a brief account of FRG-investigations in many-flavor QCD. This already connects QCD to high-energy physics at the high-energy frontier. Asymptotic freedom in QCD comes from the negative sign of the QCD β-function. At one loop it is given by β α s = -1/(8π 2 )(11/3N c -2/3N f ). For many flavors the sign of the β-function changes and asymptotic safety is lost, at one loop this happens at N af f = 11/2N c . The theory may also feature an infrared fixed point (triggered by the two-loop coefficient) for

N f > N CBZ f , the Caswell-Banks-Zaks fixed point with N CBZ f < N af
f . Finally, QCD may also have a conformal regime for

N CBZ f ≤ N cr f < N f < N af f
, for an FRG-discussion of all these features see [START_REF] Braun | Beyond Miransky Scaling[END_REF].

These fixed point structures and scaling properties call for FRG-applications. Indeed, the smallness of the βfunction in this regime supports the reliability of semi-perturbative approximations, while the nonperturbative FRGsetup is required and well-suited to unravel the fixed points and non-trivial scalings, for FRG-works see [START_REF] Braun | Beyond Miransky Scaling[END_REF][START_REF] Gies | Chiral phase structure of QCD with many flavors[END_REF][START_REF] Braun | Running coupling at finite temperature and chiral symmetry restoration in QCD[END_REF][START_REF] Braun | Chiral phase boundary of QCD at finite temperature[END_REF][START_REF] Terao | Conformal dynamics in gauge theories via non-perturbative renormalization group[END_REF][1007], for a QED 3 analogue see [START_REF] Braun | Phase structure of many-flavor QED 3[END_REF]. These works contain many results on the fixed point locations, but in particular on the scaling pattern in the regime, e.g. the 'beyond-Miransky scaling' at the quantum critical point [START_REF] Braun | Beyond Miransky Scaling[END_REF].

Electroweak phase transition, BSM physics & Supersymmetry

The interesting regime of QCD for small number of flavors is the low energy limit. The high energy regime of the Standard Model and its ultraviolet closure also harbour many fascinating phenomena and conceptual questions, ranging from the details of the electroweak phase transition, see [START_REF] Reuter | Running gauge coupling in three-dimensions and the electroweak phase transition[END_REF]1009], over potential beyond the Standard Model scenarios up to the question of a unification with (quantum) gravity, which is explained in more detail in Section 6.

An exciting link to the UV-completion of high energy physics including gravity are asymptotically safe matter systems, which includes asymptotically free systems as a specific case. Such systems may allow for a UV-completion of the Standard model in terms of stable interacting or free ultraviolet fixed points with a finite number of UV-relevant directions. In this setting the Standard Model emerges from a specific UV-IR trajectory. Note that such a embedding of the Standard Model in an asymptotically safe UV-completion of matter should not be seen as fundamental theory. Most likely quantum gravity effects deform or even change the UV completion beyond the Planck scale. This happens both within asymptotically safe gravity, see the review [START_REF] Eichhorn | An asymptotically safe guide to quantum gravity and matter[END_REF], and beyond, see [START_REF] De Alwis | Asymptotic safety, string theory and the weak gravity conjecture[END_REF]. Still, already the existence of such a fixed point constrains predictions in the presence of a physical UV-cutoff beyond which the low energy theory is changed by new physics, [START_REF] Held | Effective asymptotic safety and its predictive power: Gauge-Yukawa theories[END_REF].

Prior to the discovery of the Higgs particle, various aspects of Higgs physics were explored with FRG techniques in [1013,[START_REF] Gies | Towards a renormalizable standard model without fundamental Higgs scalar[END_REF], with a particular focus on asymptotic safety. Asymptotic safety in nonlinear SU(N) sigma models without propagating Higgs mode was investigated in [START_REF] Percacci | One loop beta functions and fixed points in Higher Derivative Sigma Models[END_REF][START_REF] Fabbrichesi | Asymptotic safety and the gauged SU(N) nonlinear sigma-model[END_REF][START_REF] Bazzocchi | Fermions and Goldstone bosons in an asymptotically safe model[END_REF][START_REF] Fabbrichesi | The Electroweak S and T parameters from a fixed point condition[END_REF]. Asymptotic safety has also been searched for in simple and chiral Yukawa models [START_REF] Gies | Asymptotic safety of simple Yukawa systems[END_REF][START_REF] Gies | Towards an Asymptotic-Safety Scenario for Chiral Yukawa Systems[END_REF][START_REF] Scherer | An Asymptotic-safety mechanism for chiral Yukawa systems[END_REF][START_REF] Vacca | Multimeson Yukawa interactions at criticality[END_REF] and in gauged chiral Yukawa models in [START_REF] Gies | An asymptotic safety scenario for gauged chiral Higgs-Yukawa models[END_REF]. Recently, asymptotic safety in QED with a Pauli term was explored in [1024], whereas a search for asymptotic safety in axion electrodynamics can be found in [START_REF] Eichhorn | Renormalization Flow of Axion Electrodynamics[END_REF]. Beyond the Standard Model, asymptotic safety in the dark-matter sector was explored in [START_REF] Eichhorn | Asymptotic safety in the dark[END_REF]; for studies including gravity we refer to Sec. 6.

Initiated in [START_REF] Litim | Asymptotic safety guaranteed[END_REF], there is by now a growing class of asymptotically safe gauge-matter models, where the UV fixed point can be accessed perturbatively in the Veneziano limit, see e.g. [START_REF] Litim | Asymptotic safety guaranteed[END_REF][START_REF] Litim | Vacuum stability of asymptotically safe gauge-Yukawa theories[END_REF][START_REF] Mann | Asymptotically Safe Standard Model via Vectorlike Fermions[END_REF][START_REF] Bond | Price of Asymptotic Safety[END_REF][START_REF] Dondi | Analytic Coupling Structure of Large N f (Super) QED and QCD[END_REF][START_REF] Bond | Asymptotic safety with Majorana fermions and new large N equivalences[END_REF][START_REF] Dondi | Towards the QED beta function and renormalons at 1/N 2 f and 1/N 3 f[END_REF]. The stability of these models in the presence of higher order couplings has been explored with the FRG for a gauge-Yukawa system in [START_REF] Buyukbese | Asymptotic safety of gauge theories beyond marginal interactions[END_REF].

The triviality problem in the Higgs sector was tackled in [START_REF] Gies | Asymptotically free scaling solutions in non-Abelian Higgs models[END_REF][START_REF] Gies | Non-Abelian Higgs models: Paving the way for asymptotic freedom[END_REF][START_REF] Gies | Asymptotic freedom in Z 2 -Yukawa-QCD models[END_REF][START_REF] Gies | Scheme dependence of asymptotically free solutions[END_REF], where asymptotically free scaling solutions were uncovered by imposing generalized boundary conditions on the correlation functions. With the Higgs mass of about 125 GeV and the top mass of about 172 GeV, the Standard Model lies on the border of stability: Specifically, for the present central value of the top mass, the Higgs quartic coupling must be negative at the Planck scale. Moreover, it only crosses into the positive values at a momentum scale of about 10 10 GeV. This is typically viewed as an indication of a metastable potential, i.e., the electroweak vacuum is only a local, but not a global minimum of the potential. The FRG is tailor-made for an investigation of this question, which has started in a simple Yukawa model [START_REF] Gies | Higgs Mass Bounds from Renormalization Flow for a simple Yukawa model[END_REF], also explored in [START_REF] Jakovac | Scalar mass stability bound in a simple Yukawa-theory from renormalization group equations[END_REF][START_REF] Sondenheimer | Nonpolynomial Higgs interactions and vacuum stability[END_REF]. This model was upgraded to a chiral one with the two heaviest quark flavors in [START_REF] Gies | Higgs Mass Bounds from Renormalization Flow for a Higgs-top-bottom model[END_REF]. The effect of gauge interactions was added in [START_REF] Eichhorn | The Higgs Mass and the Scale of New Physics[END_REF]. It was shown, that higher-order operators in the Higgs potential, as they arise from new physics, can lower the Higgs mass bound, see also [START_REF] Gies | Impact of generalized Yukawa interactions on the lower Higgs mass bound[END_REF] for studies of higher-order Yukawa interactions. In [START_REF] Borchardt | Global flow of the Higgs potential in a Yukawa model[END_REF] the vacuum stability was studied with spectral methods, that allow to explore the global form of the potential. Studies of vacuum stability with new degrees of freedom, specifically scalar and fermionic dark matter, coupled through a Higgs portal, can be found in [START_REF] Eichhorn | Planck scale, Higgs mass, and scalar dark matter[END_REF][START_REF] Held | Higgs stability-bound and fermionic dark matter[END_REF]. For a summary and references on this question with other techniques, see [START_REF] Gies | Renormalization Group Flow of the Higgs Potential[END_REF].

In the context of vacuum stability, new physics at a relatively high scales is of central interest. In contrast, new physics at scales close to LHC-scales is in the focus of a study of a first-order electroweak phase transition [START_REF] Reichert | Probing baryogenesis through the Higgs boson self-coupling[END_REF]. As a key advantage of FRG techniques, non-perturbative corrections to the Higgs potential, such as, e.g., operators of the form φ 4 exp-Λ 2 /φ 2 , where Λ is the scale of new physics, can be explored in a well-controlled way, using a numerical grid to evaluate the evolution of the potential as a function of field φ, temperature T and RG scale k.

Even though the minimal supersymmetric extension of the Standard Model fails to meet experimental bounds, supersymmetric theories are still viable extensions of the Standard Model, and are specifically relevant in the context of low-energy effective theories of string theory. While full supersymmetry is typically broken in the presence of an infrared regulator as used in the FRG, it can be monitored similarly to gauge symmetries by modified STIs. FRG-applications to supersymmetric models can be found in [START_REF] Hellwig | Scaling and superscaling solutions from the functional renormalization group[END_REF][START_REF] Feldmann | Critical Wess-Zumino models with four supercharges in the functional renormalization group approach[END_REF][1050][START_REF] Sonoda | Construction of a Wilson action for the Wess-Zumino model[END_REF][START_REF] Rosten | On the Renormalization of Theories of a Scalar Chiral Superfield[END_REF][START_REF] Sonoda | An Elementary proof of the non-renormalization theorem for the Wess-Zumino model[END_REF][START_REF] Synatschke | Phase Diagram and Fixed-Point Structure of two dimensional N=1 Wess-Zumino Models[END_REF][START_REF] Heilmann | Convergence of Derivative Expansion in Supersymmetric Functional RG Flows[END_REF], for supersymmetry at finite temperature see [START_REF] Synatschke | N=1 Wess Zumino Model in d=3 at zero and finite temperature[END_REF] (N = 1 Wess-Zumino model). Supersymmetric gauge theories have been studied with the FRG in [START_REF] Granda | Exact renormalization group for O(4) gauged supergravity[END_REF][START_REF] Falkenberg | Effective average action in N=1 superYang-Mills theory[END_REF][START_REF] Arnone | Exact renormalization group equation in presence of rescaling anomaly[END_REF][START_REF] Bonini | Wilson renormalization group for supersymmetric gauge theories and gauge anomalies[END_REF][START_REF] Arnone | Applications of exact renormalization group techniques to the nonperturbative study of supersymmetric gauge field theory[END_REF]. The critical behaviour and the phase structure of supersymmetric O(N)-models is discussed in [START_REF] Litim | Critical behavior of supersymmetric O(N) models in the large-N limit[END_REF][START_REF] Heilmann | Phases of supersymmetric O(N) theories[END_REF], and the breaking of supersymmetry as a phase transition has been described in [START_REF] Gies | Supersymmetry breaking as a quantum phase transition[END_REF], for emergent supersymmetry see [START_REF] Gies | A functional perspective on emergent supersymmetry[END_REF].

Summary

We have discussed the FRG-setup for gauge theories, that are relevant for the formulation of Particle Physics and Gravity. By far the most FRG-applications are concerned with the strongly-correlated infrared sector of QCD at vanishing and finite temperature and density. These applications have been discussed in detail in Sec. 5.3 and Sec. 5.4. The phase structure of QCD features spontaneous symmetry breaking (chiral symmetry breaking and confinement), competing order effects (color superconducting phases at large densities), as well as the emergent dynamical degrees of freedom (from fundamental quarks & gluons to hadrons). These applications as well as technical developments hence also works as the showcase example for applications in and beyond the Standard Model. The respective works have been briefly discussed in Section 5.5. There, we have listed the FRG-works on the question of vacuum stability of the Higgs sector, investigations of asymptotically safe matter systems, and that of supersymmetric theories. This part already is already tightly linked with the quantum gravity applications reviewed in Section 6.

Gravity

Introduction: Quantum gravity -why and what?

A model of quantum gravity brings together the key aspects of quantum physics -the superposition principle, quantum interference and the uncertainty principle -with the insight that gravity is encoded in dynamical spacetime geometry. Thus, a quantum theory of gravity allows us to explore the consequences of quantum superpositions of spacetime. Why do we need such a theory? A key motivation comes from the observation of gravitational waves by the LIGO/VIRGO collaboration [START_REF] Abbott | Observation of Gravitational Waves from a Binary Black Hole Merger[END_REF] and the first image of a black hole captured by the EHT collaboration [START_REF] Akiyama | First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole[END_REF]. Within the uncertainties of the measurements, these are in agreement with the predictions from General Relativity (GR). Nevertheless, these observations reinforce the need to go beyond GR. The reason is that within the theoretical description by General Relativity the observed objects, black holes, contain curvature singularities -corresponding to diverging tidal forces -where the curvature exceeds the Planck scale. Briefly reinstating and c, the Planck scale reads

M Planck = c G N ≈ 10 19 GeV, l Planck = G N c 3 ≈ 10 -35 m. ( 95
)
This scale is characteristic of a relativistic (due to c), gravitational (due to G N ), quantum theory (due to ). Hence, it is expected to be the typical scale of quantum gravity. The resolution of spacetime-singularities is thus expected to be one of the hallmarks of a successful model of quantum gravity.

The key challenge of quantum gravity is to understand the consequences of quantum fluctuations of spacetime at all scales. An additional complication is that there is no fixed notion of scales, or more generally spacetime geometry, in quantum gravity. Nevertheless, (Wilsonian) RG concepts can be put to good use in this setting.

Due to the huge ratio of the Planck scale to experimentally accessible energy scales, e.g., at the LHC, E LHC /M Planck ≈ 10 -15 , direct experimental tests of quantum gravity remain out of reach (with the notable exception of tests of the breaking of Lorentz invariance at (trans)planckian energies [START_REF] Amelino-Camelia | Tests of quantum gravity from observations of gamma-ray bursts[END_REF][START_REF] Abdo | Fermi Observations of High-Energy Gamma-Ray Emission from GRB 080916C[END_REF][START_REF] Ackermann | A limit on the variation of the speed of light arising from quantum gravity effects[END_REF][1071]). As a consequence, a diversity of theoretical ideas about Planck-scale physics is being explored, see, e.g., [START_REF] Oriti | Approaches to quantum gravity: Toward a new understanding of space, time and matter[END_REF][1073][START_REF] Carlip | Quantum Gravity: A Brief History of Ideas and Some Prospects[END_REF] for overviews over several approaches. Several settings, e.g., asymptotically safe gravity, tensor models, dynamical triangulations, spin foams and causal sets are based on the gravitational path integral, schematically

Z = ˆspacetimes e i S [spacetime] . (96)
With the FRG, one can tackle such path integrals both in the continuum as well as in discrete settings, see Subsec. 6.5.2. As we will see, the notion of the RG scale is slightly different in these two settings. Nevertheless, a one-loop flow equation can be derived both in the continuum and in the discrete setting by introducing a partition of the configuration space into "microscopic" and "macroscopic" configurations. Before introducing the FRG for quantum gravity, let us review the problem of quantum gravity from the perspective of a perturbative quantization of General Relativity: To define what is meant by Eq. ( 96), one needs to specify several things: Firstly, a configuration space to be summed over needs to be defined. The minimalistic choice is a summation over geometries at fixed topology and dimensionality, for instance expressed as a sum over metrics 35 . Secondly, a microscopic dynamics is needed. The Einstein-Hilbert action

S EH = 1 16πG N ˆdd x √ -g -2 Λ + R , (97)
features the curvature scalar R, as well as a cosmological constant term, where g is the determinant of the metric and Λ is the cosmological constant. This action is not a good starting point within perturbation theory: Loop divergences require additional curvature invariants beyond the Einstein-Hilbert action to be added as counterterms, e.g., in d = 4, √ -gR 2 and √ -gR µν R µν at the one-loop level with matter [1075-1077], and √ -gR µνκλ R κλ ρσ R ρσ µν at the two-loop level [START_REF] Goroff | Quantum Gravity at two Loops[END_REF][START_REF] Van De Ven | Two loop quantum gravity[END_REF]. This follows, as the action in Eq. ( 97), when expanded in fluctuations of the metric around a flat background metric, features derivative interactions with a coupling with negative mass dimension, G N . This renders the theory power-counting nonrenormalizable and is expected to result in infinitely many counterterms at infinitely high loop order. Each counterterm comes with an independent coupling that is a free parameter, leading to a breakdown of predictivity of the perturbative approach based on the Einstein-Hilbert dynamics. As long as one focuses on energies sufficiently below the Planck scale, the additional terms are negligible, and predictions can be derived within an effective field theory approach to quantum gravity [1080]. As one approaches the Planck scale, all higherorder terms become important and the effective field theory approach breaks down. Thus, a different microscopic dynamics and/or nonperturbative techniques appear indicated36 .

Path integral for quantum gravity and asymptotic safety

A Wilsonian approach to gravity, like, e.g., the FRG, allows us to shift the focus away from a specific choice of S . Instead the theory space takes center stage. It is defined once the field content, i.e., the configuration space, and symmetries of the theory are specified. The theory space is spanned by the dimensionless counterparts of the couplings of all quasilocal (i.e., containing only non-negative powers of derivatives) symmetry invariants 37 . For instance, for a diffeomorphism invariant configuration space of the metric, the theory space is spanned by the dimensionless versions of the couplings of the curvature operators, i.e., √ g, √ gR, √ gR 2 , √ gR µν R µν , etc. which are written in the Euclidean setting, as required for the application of the FRG. This expansion is closely related to the derivative expansion widely used within many FRG applications, as presented, e.g., in Sect. 3. 38 .

The search for asymptotic safety proceeds in dimensionless couplings, akin to the previously discussed fixedpoint searches in statistical physics. For instance, for the first two terms in the curvature expansion, the dimensionless couplings in d spacetime dimensions are

λ = Λk -2 , G = G N k d-2 . ( 98
)
As discussed in Sec. 5, the gauge fixing and the regularization require the presence of a background field, and a breaking of diffeomorphism symmetry. This leads to a considerable enlargement of the theory space by the corresponding couplings. We will get back to this question below.

The FRG provides a vector field in the space of all couplings, denoted g i , which corresponds to the scale-derivative of the couplings at each point, i.e., the beta functions β i . The integral curves of this vector field are the RG trajectories. This is in full analogy to the setting described in all previous sections. The key difference lies in the fact that in statistical physics, condensed matter and high-energy particle physics one is typically interested in the RG flow starting from a given initial condition that describes the microscopic physics. In the asymptotic-safety approach to quantum gravity, the focus is different. Instead of using the Wetterich equation together with an initial condition to obtain Γ k→0 , first we are interested in discovering whether UV complete trajectories exist. One can view this as the search for a consistent microscopic initial condition, i.e., the search for a microscopic action S that will lead to a predictive theory 39 . The derivation of Γ k→0 which encodes the full physics of the theory is then a second step, to be tackled once the existence of a well-defined microscopic starting point Γ k→∞ for the dynamics has been found. In order to provide a predictive UV completion, a point in theory space has to satisfy two conditions: Firstly, it must be a fixed point of the RG flow, i.e., a zero of the vector field 40 , i.e., β i = 0 ∀i at g i = g i * . This generates quantum scale-invariance [1087] at this point, just as scale-invariance close to second-order phase transitions in statistical physics is associated to fixed points. Scale-invariance allows the RG flow to spend an infinite amount of RG "time" at the fixed point before the flow away from it breaks scale-invariance, making new physics at some high scale an option instead of a theoretical necessity. Figure 26: A fixed point (purple point) comes with a critical hypersurface, spanned by the relevant directions. RG trajectories emanating out of the fixed point must lie within the critical hypersurface, as it is IR attractive (cf. cyan trajectory). This results in predictions for the values of irrelevant couplings; e.g., in this case, g 3 is determined at all scales once experimental input fixes one UV-complete RG trajectory by determining the values of g 1 and g 2 at some scale.

m J a g K m H l V f U r m 3 n j R E D Y = " > A A A B 6 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l E U G 9 F L x 4 r G l t o Q 9 l s N + 3 S z S b s T o Q S + h O 8 e F D x 6 j / y 5 r 9 x 2 + a g r Q 8 G H u / N M D M v T K U w 6 L r f T m l l d W 1 9 o 7 x Z 2 d r e 2 d 2 r 7 h 8 8 m i T T j P s s k Y l u h 9 R w K R T 3 U a D k 7 V R z G o e S t 8 L R z d R v P X F t R K I e c J z y I K Y D J S L B K F r p f t D z e t W a W 3 d n I M v E K 0 g N C j R 7 1 a 9 u P 2 F Z z B U y S Y 3 p e G 6 K Q U 4 1 C i b 5 p N L N D E 8 p G 9 E B 7 1 i q a M x N k M 9 O n Z A T q / R J l G h b C s l M / T 2 R 0 9 i Y c R z a z p j i 0 C x 6 U / E / r 5 N h d B n k Q q U Z c s X m i 6 J M E k z I 9 G / S F 5 o z l G N L K N P C 3 k r Y k G r K 0 K Z T s S F 4 i y 8 v E / + s f l V 3
m J a g K m H l V f U r m 3 n j R E D Y = " > A A A B 6 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l E U G 9 F L x 4 r G l t o Q 9 l s N + 3 S z S b s T o Q S + h O 8 e F D x 6 j / y 5 r 9 x 2 + a g r Q 8 G H u / N M D M v T K U w 6 L r f T m l l d W 1 9 o 7 x Z 2 d r e 2 d 2 r 7 h 8 8 m i T T j P s s k Y l u h 9 R w K R T 3 U a D k 7 V R z G o e S t 8 L R z d R v P X F t R K I e c J z y I K Y D J S L B K F r p f t D z e t W a W 3 d n I M v E K 0 g N C j R 7 1 a 9 u P 2 F Z z B U y S Y 3 p e G 6 K Q U 4 1 C i b 5 p N L N D E 8 p G 9 E B 7 1 i q a M x N k M 9 O n Z A T q / R J l G h b C s l M / T 2 R 0 9 i Y c R z a z p j i 0 C x 6 U / E / r 5 N h d B n k Q q U Z c s X m i 6 J M E k z I 9 G / S F 5 o z l G N L K N P C 3 k r Y k G r K 0 K Z T s S F 4 i y 8 v E / + s f l V 3
m J a g K m H l V f U r m 3 n j R E D Y = " > A A A B 6 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l E U G 9 F L x 4 r G l t o Q 9 l s N + 3 S z S b s T o Q S + h O 8 e F D x 6 j / y 5 r 9 x 2 + a g r Q 8 G H u / N M D M v T K U w 6 L r f T m l l d W 1 9 o 7 x Z 2 d r e 2 d 2 r 7 h 8 8 m i T T j P s s k Y l u h 9 R w K R T 3 U a D k 7 V R z G o e S t 8 L R z d R v P X F t R K I e c J z y I K Y D J S L B K F r p f t D z e t W a W 3 d n I M v E K 0 g N C j R 7 1 a 9 u P 2 F Z z B U y S Y 3 p e G 6 K Q U 4 1 C i b 5 p N L N D E 8 p G 9 E B 7 1 i q a M x N k M 9 O n Z A T q / R J l G h b C s l M / T 2 R 0 9 i Y c R z a z p j i 0 C x 6 U / E / r 5 N h d B n k Q q U Z c s X m i 6 J M E k z I 9 G / S F 5 o z l G N L K N P C 3 k r Y k G r K 0 K Z T s S F 4 i y 8 v E / + s f l V 3
U w = " > A A A B 6 X i c b V B N S 8 N A E J 3 4 W e t X 1 a O X x S J 4 K k k R 1 F v R i 8 e K x h b a U D b b T b p 0 s w m 7 E 6 G U / g Q v H l S 8 + o + 8 + W / c t j l o 6 4 O B x 3 s z z M w L M y k M u u 6 3 s 7 K 6 t r 6 x W d o q b + / s 7 u 1 X D g 4 f T Z p r x n 2 W y l S 3 Q 2 q 4 F I r 7 K F D y d q Y 5 T U L J W + H w Z u q 3 n r g 2 I l U P O M p 4 k N B Y i U g w i l a 6 j 3 v 1 X q X q 1 t w Z y D L x C l K F A s 1 e 5 a v b T 1 m e c I V M U m M 6 n p t h M K Y a B Z N 8 U u 7 m h m e U D W n M O 5 Y q m n A T j G e n T s i p V f o k S r U t h W S m / p 4 Y 0 8 S Y U R L a z o T i w C x 6 U / E / r 5 N j d B m M h c p y 5 I r N F 0 W 5 J J i S 6 d + k L z R n K E e W U K a F v Z W w A d W U o U
Secondly, the vector field -β i should only have finitely many independent directions along which it points away from the fixed point, i.e., a finite number of relevant directions, corresponding to positive critical exponents 41 , cf. Fig. 26. These are the directions along which the RG flow towards the IR can leave the scale-invariant fixedpoint regime. The low-energy values of the corresponding relevant couplings cannot be predicted and are the free parameters of the theory. In contrast, all IR-attractive, i.e., irrelevant couplings are determined at all scales on a trajectory that emanates from the fixed point. If a theory space contains a fixed point with the above properties at nonzero values for (some of) the couplings, then a model with the corresponding field content and symmetries can be asymptotically safe. This idea was first put forward as a potential UV completion for gravity in [START_REF] Weinberg | ULTRAVIOLET DIVERGENCES IN QUANTUM THEORIES OF GRAVITATION, General Relativity: An Einstein centenary survey[END_REF]. For several fixed points in the same theory space, each one in principle defines a UV completion. Whether one of those is realized in nature can be tested by measuring whether the low-energy values of the couplings lie in its critical hypersurface. Note the similarity to fixed points in statistical physics: The relevant directions determine the deviation from scale invariance. In statistical physics they can be translated into experimentally tunable quantities, such as, e.g., the temperature, which allow one to tune a system to criticality. In gravity and high-energy physics, the relevant parameters encode how far the low-energy dynamics has deviated from scale-invariance and constitute the quantities that need measurements and cannot be calculated within the theory. To stress the similarity between fixed points in (often lower-dimensional) systems and gravity, let us highlight that it is a misconception that a fixed point is either UV or IR: Unless it features only IR repulsive or only IR attractive directions, a fixed point can actually be both; depending on the choice of RG trajectory, the fixed point is approached by the trajectory in the IR or serves as a repulsor of the trajectory in the UV, cf. the discussion in [START_REF] Rosten | Fundamentals of the Exact Renormalization Group[END_REF]. Therefore, the fixed points of relevance in statistical physics and the tentative gravitational fixed point are actually structurally similar in many aspects and can in particular be searched for and investigated with the same set of tools.

The FRG for quantum gravity: a brief manual

The construction in Sec. 2 can be generalized to a gravitational setting, as pioneered by Martin Reuter [START_REF] Reuter | Nonperturbative Evolution Equation for Quantum Gravity[END_REF], see also [START_REF] Dou | The running gravitational couplings[END_REF]. This entails working with Euclidean signature, i.e., the underlying path integral is for quantum space, not spacetime. The analytical continuation to real time is an outstanding challenge which is far more complicated in gravity than in theories on a flat background, see, e.g., [START_REF] Demmel | Connections and geodesics in the space of metrics[END_REF]1092]. To set up the Wetterich equation, one must distinguish the "fast" from the "slow" modes, i.e., the UV from the IR. The natural generalization of the flat-space construction of the regulator would at a first glance appear to be of the form g µν R µνκλ k (∆)g κλ , where ∆ = -g µν D µ D ν , and D µ is the covariant derivative. Yet, this has two problems: Firstly, because of metric compatibility, D µ g κλ = 0, and secondly, D µ itself depends on the metric, thus rendering the regulator term higher-order in the field. Yet, the quadratic nature of the regulator is key to the one-loop structure of the flow equation. Accordingly, setting up the Wetterich equation for gravity typically requires the introduction of an auxiliary background metric ḡµν which is also used to set up a background-invariant gauge fixing, see also Sec. 5; see also [START_REF] Falls | Background independent exact renormalisation[END_REF] for an alternative development. The regulator depends on the background-covariant Laplacian ∆ = -D2 , taking the form

∆S k [ḡ µν , h µν ] = 1 2 ˆdd x ḡ h µν R µνκλ k ( ∆)h κλ , (99)
where

h µν = g µν -ḡµν (100)
is the fluctuation field. Introducing the fluctuation field simply corresponds to a shift in the integration variable in the path integral; Eq. ( 100) should not be read in a perturbative sense. For the special choice ḡµν = δ µν , one recovers the construction familiar from previous sections, where the cutoff is implemented in momentum space. In the presence of a non-flat background metric ḡµν , the momentum space is generalized, as the eigenfunctions of the Laplacian ∆ are no longer plane waves. Thus, the regulator in the Wetterich equation is set up with respect to the eigenvalues λ ∆ of the Laplacian ∆: Eigenmodes with eigenvalues λ ∆ > k 2 are the UV-modes, and those with λ ∆ < k 2 are the IR-modes. Intuitively speaking, the background metric provides a notion of locality, allowing to define local patches in which the quantum fluctuations are averaged over, thus enabling a coarse-graining procedure. However, there is an added complication in quantum gravity, as all configurations of spacetime are to be treated on an equal footing and no metric should play a distinguished role. The background metric is therefore not to be understood as a physical background, but as an auxiliary field. Accordingly, in addition to the flow equation, there is a Ward identity that relates the background-field dependence of the average effective action to its fluctuation-field dependence, also cf. Sec. 5 for the discussion of such symmetry-identities and the flow equation.

The background field can also be used to gauge-fix the fluctuations, as their unregularized inverse propagator, Γ (2) k , is not invertible without a gauge-fixing. This is in direct analogy to the case of Yang-Mills theories discussed in the previous section, and entails similar technical challenges pertaining to the simultaneous solution of the Wetterich equation and symmetry-identities. The gauge fixing is often chosen to be of the form

S gf = 1 2α 1 16πG N ˆd4 x ḡ ḡµν Dκ h κµ - 1 + β 4 Dµ h Dρ h ρν - 1 + β 4 Dν h , (101)
where h = h µ µ , and indices are raised and lowered with the background metric. α and β are gauge-parameters, and α → 0 implements the Landau-gauge limit of the gauge condition. The corresponding Faddeev-Popov ghost term also contributes to the flow.

The trace in the Wetterich equation ( 7) can be evaluated by summing over the eigenvalues for backgrounds ḡµν for which the full spectrum of the operator ∆ is known, e.g., [START_REF] Eichhorn | Asymptotically free scalar curvature-ghost coupling in Quantum Einstein Gravity[END_REF][START_REF] Benedetti | The Local potential approximation in quantum gravity[END_REF][START_REF] Alkofer | Asymptotically safe f(R)-gravity coupled to matter I: the polynomial case[END_REF], or by using heat-kernel techniques, which only requires knowledge of the heat kernel instead of the full spectrum of the operator, see, e.g., [START_REF] Lauscher | Ultraviolet fixed point and generalized flow equation of quantum gravity[END_REF][START_REF] Reuter | Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation[END_REF][START_REF] Codello | Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation[END_REF][START_REF] Benedetti | The Universal RG Machine[END_REF][START_REF] Kluth | Heat kernel coefficients on the sphere in any dimension[END_REF]. This provides the machinery to extract beta functions for gravity from the Wetterich equation.

The last important aspect for applications of the Wetterich equation is the choice of truncation. Here, we will provide a brief overview of truncations that have mainly been used. It is generally assumed -and by now supported by many explicit results -that the canonical dimension is a robust guide to determine which couplings are relevant at the asymptotically safe fixed point. This is motivated firstly by the possibility that the Reuter fixed point could be connected continuously to the perturbative fixed point in d = 2+ dimensions [START_REF] Gastmans | Quantum Gravity Near Two-Dimensions[END_REF][START_REF] Christensen | QUANTUM GRAVITY IN TWO + epsilon DIMENSIONS[END_REF]. Secondly, explicit studies of truncations of the form Γ k = n i=0 ´d4 x √ ga i R i , with n = 70 [START_REF] Falls | A bootstrap towards asymptotic safety[END_REF][START_REF] Falls | Further evidence for asymptotic safety of quantum gravity[END_REF][START_REF] Falls | Aspects of asymptotic safety for quantum gravity[END_REF], exhibit a near-canonical scaling for the higher-order operators, see also [START_REF] Machado | On the renormalization group flow of f(R)-gravity[END_REF][START_REF] Codello | Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation[END_REF][START_REF] Falls | A bootstrap towards asymptotic safety[END_REF][START_REF] Falls | Further evidence for asymptotic safety of quantum gravity[END_REF][START_REF] Falls | Aspects of asymptotic safety for quantum gravity[END_REF][START_REF] Lauscher | Flow equation of quantum Einstein gravity in a higher-derivative truncation[END_REF][START_REF] Benedetti | Asymptotic safety in higher-derivative gravity[END_REF][START_REF] Ohta | Higher Derivative Gravity and Asymptotic Safety in Diverse Dimensions[END_REF][START_REF] Ohta | Renormalization Group Equation and scaling solutions for f(R) gravity in exponential parametrization[END_REF][START_REF] Ohta | Flow equation for f (R) gravity and some of its exact solutions[END_REF][START_REF] Gies | Gravitational Two-Loop Counterterm Is Asymptotically Safe[END_REF][START_REF] Falls | Asymptotic safety of quantum gravity beyond Ricci scalars[END_REF][START_REF] De Brito | Asymptotic safety and field parametrization dependence in the f (R) truncation[END_REF]; however note that a number of dimension-6 operators (and beyond) remain to be explored. Third, studies of the magnitude of diffeomorphism-symmetry breaking, expected to be large and connected to highly nontrivial Slavnov-Taylor identities in a nonperturbative regime, yield results compatible with a near-perturbative nature of the Reuter fixed point [START_REF] Eichhorn | Effective universality in quantum gravity[END_REF][START_REF] Eichhorn | How perturbative is quantum gravity?[END_REF][START_REF] Eichhorn | Zooming in on fermions and quantum gravity[END_REF]. Fourth, there are also hints for the Reuter fixed point from perturbative studies [START_REF] Codello | Fixed Points of Higher Derivative Gravity[END_REF][START_REF] Niedermaier | Gravitational Fixed Points from Perturbation Theory[END_REF][START_REF] Niedermaier | Gravitational fixed points and asymptotic safety from perturbation theory[END_REF]. For such a fixed point, near-canonical scaling of the critical exponents is expected. Specifically, this means that shifts O(1) of the critical exponents, compared to the canonical scaling dimension, can be expected, but not shifts O [START_REF] Wegner | Renormalization Group Equation for Critical Phenomena[END_REF]. This supports a truncation scheme based on the canonical dimension of couplings, where higher orders in the curvature and derivatives are expected to be irrelevant. It is worth to mention that, at odds with scalar theories where operators with high canonical dimension can be constructed without introducing high number of derivatives, in the quantum-gravity context the classification along canonical dimension in terms of diffeomorphism-invariant operators is more closely related to the derivative expansion. In principle, nearcanonical scaling might even provide a small parameter with which to achieve systematic error estimates, namely the deviation of quantum scaling at the fixed point from canonical scaling, see also Sec. 3 for the discussion of a small parameter in the derivation expansion.

A considerable body of literature employs the single-metric approximation, in which the distinction between background metric and full metric is only made in the gauge-fixing, ghost and regulator term. Based on canonical power counting as a guiding principle, one then sets up truncations in a curvature expansion, i.e.,

Γ k = ˆd4 x √ g 1 16π G N 2 Λ -R + a R 2 + b R µν R µν + ... + S gf + S gh . (102)
The ghost-sector and gauge-fixing are typically only accounted for by including the corresponding classical actions S gf and S gh . Explicit studies indicate that additional terms might be present for the ghost sector at a nontrivial fixed point [START_REF] Eichhorn | Faddeev-Popov ghosts in quantum gravity beyond perturbation theory[END_REF].

Finally let us note that alternative gravitational theory spaces are also accessible to the FRG. Formulations of gravity which have been explored in this setup are based, e.g., on a unimodular theory space [START_REF] Eichhorn | On unimodular quantum gravity[END_REF][START_REF] Eichhorn | The Renormalization Group flow of unimodular f(R) gravity[END_REF][START_REF] Benedetti | Essential nature of Newton's constant in unimodular gravity[END_REF] in which the determinant of the metric is held fixed such that the cosmological constant is no longer a term in the action and therefore not subject to the usual fine-tuning. Further studies focussed on a formulation based on the tetrad and the spin connection [START_REF] Daum | Renormalization Group Flow of the Holst Action[END_REF][START_REF] Daum | Einstein-Cartan gravity, Asymptotic Safety, and the running Immirzi parameter[END_REF][START_REF] Harst | A new functional flow equation for Einstei-Cartan quantum gravity[END_REF][START_REF] Harst | On selfdual spin-connections and Asymptotic Safety[END_REF], the tetrad with a dependent connection [START_REF] Harst | The 'Tetrad only' theory space: Nonperturbative renormalization flow and Asymptotic Safety[END_REF], or including dynamical torsion degrees of freedom [START_REF] Pagani | Quantum gravity with torsion and non-metricity[END_REF][START_REF] Reuter | The metric on field space, functional renormalization, and metric-torsion quantum gravity[END_REF]. Topologically massive 3d-(super)gravity has been studied in [START_REF] Percacci | One Loop Beta Functions in Topologically Massive Gravity[END_REF][START_REF] Percacci | Beta Functions of Topologically Massive Supergravity[END_REF]; for a study of the massive Pauli-Fierz action see [START_REF] Binder | Functional Renormalization Group Flow of Massive Gravity[END_REF]. Conformal gravity has been explored in [START_REF] Ohta | Ultraviolet Fixed Points in Conformal Gravity and General Quadratic Theories[END_REF]. Intriguingly, results in truncations in many of these settings hint at asymptotically safe fixed points in these settings; in particular, the results in unimodular gravity are close to those for the Reuter fixed point, except for the fact that there is one relevant direction less in truncations, since the cosmological context is not a part of the theory space. This result would support the hypothesis that the spin-2 modes are most relevant to generate an asymptotically safe fixed point. Interestingly, the loosely speaking "opposite" approximation, where only the conformal mode is kept, also features an asymptotically safe fixed point in the truncations explored in [START_REF] Dietz | Background independent exact renormalization group for conformally reduced gravity[END_REF][START_REF] Labus | Background independence in a background dependent renormalization group[END_REF][START_REF] Reuter | Conformal sector of Quantum Einstein Gravity in the local potential approximation: non-Gaussian fixed point and a phase of diffeomorphism invariance[END_REF][START_REF] Reuter | Background Independence and Asymptotic Safety in Conformally Reduced Gravity[END_REF][START_REF] Daum | Effective Potential of the Conformal Factor: Gravitational Average Action and Dynamical Triangulations[END_REF][START_REF] Machado | Conformally reduced quantum gravity revisited[END_REF][START_REF] Bonanno | Universality and Symmetry Breaking in Conformally Reduced Quantum Gravity[END_REF][START_REF] Dietz | Fixed point structure of the conformal factor field in quantum gravity[END_REF].

The development of a flow equation based on a single metric can be found in [START_REF] Wetterich | Gauge invariant flow equation[END_REF] and a proper-time flow equation for gravity in [START_REF] Bonanno | Proper time flow equation for gravity[END_REF]. The Polchinski equation has been applied to test the asymptotic-safety paradigm in quantum gravity in [START_REF] De Alwis | Exact RG Flow Equations and Quantum Gravity[END_REF][START_REF] De Alwis | Higher Derivative Corrections to Lower Order RG Flow Equations[END_REF].

Status and open questions of asymptotically safe gravity 6.4.1. Indications for the Reuter fixed point

The Reuter fixed point is the tentative interacting, predictive fixed point in gravity with full diffeomorphism invariance 42 . Indications for this fixed point have robustly been found in all quasilocal truncations in the literature, see Fig. 27 for an illustration, including studies of the Einstein-Hilbert truncation [START_REF] Lauscher | Ultraviolet fixed point and generalized flow equation of quantum gravity[END_REF][START_REF] Reuter | Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation[END_REF][START_REF] Souma | Nontrivial ultraviolet fixed point in quantum gravity[END_REF][START_REF] Litim | Fixed points of quantum gravity[END_REF][START_REF] Groh | Ghost wave-function renormalization in Asymptotically Safe Quantum Gravity[END_REF][START_REF] Eichhorn | Ghost anomalous dimension in asymptotically safe quantum gravity[END_REF][START_REF] Nagy | Critical exponents in quantum Einstein gravity[END_REF][START_REF] Falls | Asymptotic safety and the cosmological constant[END_REF][START_REF] Gies | Generalized Parametrization Dependence in Quantum Gravity[END_REF] and beyond [1082, 1098, 1103-1108, 1110-1112, 1156], see [START_REF] Pawlowski | Quantum gravity: a fluctuating point of view[END_REF][START_REF] Eichhorn | An asymptotically safe guide to quantum gravity and matter[END_REF][START_REF] Reuter | Quantum Einstein Gravity[END_REF][START_REF] Percacci | An Introduction to Covariant Quantum Gravity and Asymptotic Safety[END_REF][START_REF] Reuter | Quantum Gravity and the Functional Renormalization Group[END_REF] for reviews and [1160,[START_REF] Reichert | Lecture notes: Functional Renormalisation Group and Asymptotically Safe Quantum Gravity[END_REF] for recent introductory lectures. Extensions to higher orders in R n can be found in [START_REF] Machado | On the renormalization group flow of f(R)-gravity[END_REF][START_REF] Benedetti | The Local potential approximation in quantum gravity[END_REF][START_REF] Codello | Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation[END_REF][START_REF] Falls | A bootstrap towards asymptotic safety[END_REF][START_REF] Falls | Further evidence for asymptotic safety of quantum gravity[END_REF][START_REF] Falls | Aspects of asymptotic safety for quantum gravity[END_REF][START_REF] Lauscher | Flow equation of quantum Einstein gravity in a higher-derivative truncation[END_REF][START_REF] Codello | Ultraviolet properties of f(R)-gravity[END_REF][START_REF] Demmel | Fixed-Functionals of three-dimensional Quantum Einstein Gravity[END_REF][START_REF] Dietz | Asymptotic safety in the f(R) approximation[END_REF][START_REF] Dietz | Redundant operators in the exact renormalisation group and in the f(R) approximation to asymptotic safety[END_REF][START_REF] Demmel | A proper fixed functional for four-dimensional Quantum Einstein Gravity[END_REF][START_REF] Gonzalez-Martin | Asymptotic solutions in asymptotic safety[END_REF], and inclusions of Ricci tensor invariants and Riemann tensor invariants in [START_REF] Benedetti | Asymptotic safety in higher-derivative gravity[END_REF][START_REF] Gies | Gravitational Two-Loop Counterterm Is Asymptotically Safe[END_REF][START_REF] Falls | Asymptotic safety of quantum gravity beyond Ricci scalars[END_REF][START_REF] Benedetti | Taming perturbative divergences in asymptotically safe gravity[END_REF]. A complete study at order curvature-squared can be found in [START_REF] Falls | Towards the determination of the dimension of the critical surface in asymptotically safe gravity[END_REF]. These terms cannot consistently be set to zero at an interacting fixed point in the Einstein-Hilbert truncation; they are generated by the flow and it constitutes a nontrivial check of the fixed-point hypothesis to explore whether an extension of the fixed point in the Einstein-Hilbert truncation exists

• • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ○ ○ ○ □ □ □ □ □ □ ◇ ◇ ◇ ◇ -15 -10 -5 0 5 θ i • 1 ■ 2 ◆ 3 ▲ 4 ▼ 5 ○ 6 □ 7 ◇ 8
Figure 28: Real part of critical exponents sorted from most relevant towards irrelevant from the following references in single-metric calculations: 1 = [1097], 2= [START_REF] Falls | A bootstrap towards asymptotic safety[END_REF], 3= [START_REF] Litim | Fixed points of quantum gravity[END_REF],4 =[1082], 5= [START_REF] Codello | Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation[END_REF], 6= [START_REF] Benedetti | Asymptotic safety in higher-derivative gravity[END_REF], 7= [START_REF] Falls | Asymptotic safety of quantum gravity beyond Ricci scalars[END_REF], 8= [START_REF] Falls | Towards the determination of the dimension of the critical surface in asymptotically safe gravity[END_REF]. In [START_REF] Falls | A bootstrap towards asymptotic safety[END_REF][START_REF] Falls | Asymptotic safety of quantum gravity beyond Ricci scalars[END_REF], critical exponents are calculated to significantly higher orders in the expansion than is shown here and clearly show a near-Gaussian scaling. The results are consistent with a threedimensional critical hypersurface.

once these terms are included in the truncation. On manifolds with boundaries, boundary terms have to be added to the action. First steps to explore their flows, focussing on the Gibbons-Hawking term, can be found in [START_REF] Becker | Running boundary actions, asymptotic safety, and black hole thermodynamics[END_REF][START_REF] Falls | Physical renormalization schemes and asymptotic safety in quantum gravity[END_REF].

A key difference to the perturbative renormalization of gravity is that higher-order terms, such as, e.g., R µνκλ R κλ ρσ R ρσµν add new free parameters in perturbation theory that cannot be neglected at the Planck scale. At the asymptotically safe fixed point, e.g., the so-called Goroff-Sagnotti term R µνκλ R κλ ρσ R ρσµν does not lead to the appearance of a new relevant direction within a truncation where it is coupled to the Einstein-Hilbert term [START_REF] Gies | Gravitational Two-Loop Counterterm Is Asymptotically Safe[END_REF]. This constitutes an example of how the demand that RG trajectories emanate out of a UV fixed point in the flow towards the IR (and conversely, can be traced back into the fixed point if the flow is reversed, which can be done in any finite-dimensional truncation), is a strong principle inducing predictivity, i.e., fixing free parameters that are present in an effective-field-theory setting. In the above truncations the Reuter fixed point features three relevant directions, although it is important to keep in mind that significant systematic uncertainties still affect the calculation of the critical exponents, cf. Fig. 28.

As has been discussed, e.g., in Sec. 3, the FRG can be applied with the dimensionality as a free parameter, allowing to track fixed points across different dimensionalities. This has been used to explore the Reuter universality class also away from four dimensions, see, e.g., [START_REF] Falls | Physical renormalization schemes and asymptotic safety in quantum gravity[END_REF][START_REF] Fischer | Fixed points of quantum gravity in extra dimensions[END_REF][START_REF] Nink | On the physical mechanism underlying Asymptotic Safety[END_REF][START_REF] Falls | Renormalization of Newton's constant[END_REF][START_REF] Biemans | Quantum gravity on foliated spacetimes: Asymptotically safe and sound[END_REF]. One should note that as d increases, one expects the same truncation to perform worse than in low d, since increasingly many higher-order curvature invariants become canonically relevant. With this caveat in mind, results in the above references indicate that the fixed point in pure gravity can be extended beyond d = 4, although it remains unclear whether there is an upper critical dimension.

The robustness of results is tested by i) extending the truncation and ii) studying the dependence of universal quantities such as critical exponents on unphysical parameters of the setup: Just as any other approximation scheme, including perturbation theory, physical quantities acquire a dependence on unphysical choices that define a scheme in a broad sense. In the FRG, these include the choice of regulator shape function, e.g., [START_REF] Reuter | Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation[END_REF][START_REF] Groh | Ghost wave-function renormalization in Asymptotically Safe Quantum Gravity[END_REF], the choice of gauge fixing [START_REF] Gies | Generalized Parametrization Dependence in Quantum Gravity[END_REF][START_REF] Ohta | Gauges and functional measures in quantum gravity I: Einstein theory[END_REF][START_REF] Ohta | Gauges and functional measures in quantum gravity I: Einstein theory[END_REF], as well as the choice of parameterization [START_REF] De Brito | Asymptotic safety and field parametrization dependence in the f (R) truncation[END_REF][START_REF] Gies | Generalized Parametrization Dependence in Quantum Gravity[END_REF][START_REF] Ohta | Gauges and functional measures in quantum gravity I: Einstein theory[END_REF] linked to the introduction of the background field.

Similarly to many systems discussed in previous sections, momentum-dependence can play an important role. Physically, this is clearly relevant for gravitational scattering processes, including graviton-mediated scattering processes as well as graviton-scattering. Despite the outstanding challenge to access such scattering amplitudes experimentally, their behavior at high energies is a useful theoretical check since it carries information on the unitarity of the theory. To study the momentum-dependence in the gravitational case, one can for instance introduce form-factors [START_REF] Bosma | Resolving Spacetime Singularities within Asymptotic Safety[END_REF][START_REF] Knorr | Form Factors in Asymptotic Safety: conceptual ideas and computational toolbox[END_REF], e.g.,

Γ k = Γ EH + ˆd4 x √ g R f 1 (-D 2 )R + R µν f 2 (-D 2 )R µν + R µνκλ f 3 (-D 2 )R µνκλ + ... + S gf + S gh . (103)
This is simply a way of rearranging the couplings discussed previously into a different expansion; the couplings are given by the Taylor coefficients of the form factors. In a flat expansion, corresponding terms have been explored starting in [START_REF] Christiansen | Fixed points and infrared completion of quantum gravity[END_REF][START_REF] Christiansen | Local Quantum Gravity[END_REF]. Such form factors could become important to achieve a control of the behavior of the theory not only at high k, but at high, physical momentum scales; see also, e.g., [START_REF] Denz | Towards apparent convergence in asymptotically safe quantum gravity[END_REF][START_REF] Christiansen | Curvature dependence of quantum gravity[END_REF][START_REF] Eichhorn | Zooming in on fermions and quantum gravity[END_REF][START_REF] Knorr | Correlation functions on a curved background[END_REF] for studies of the momentum dependence of metric correlation functions. In particular, correlation functions for the metric can also be studied in a curvature-dependent fashion [START_REF] Christiansen | Curvature dependence of quantum gravity[END_REF][START_REF] Knorr | Correlation functions on a curved background[END_REF][START_REF] Bürger | Curvature dependence of quantum gravity with scalars[END_REF].

Background field and dynamical field

The fixed point exists in the single-metric approximation, where one equates g µν = ḡµν after determining Γ (0,2) k [ḡ µν ; h µν ] in a given truncation. This approximation neglects the difference between background couplings and fluctuation couplings that exist due to the different appearance of the two fields in the regulator and gauge-fixing terms. Their difference is controlled by the shift Ward identity [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Reuter | Gluon condensation in nonperturbative flow equations[END_REF][START_REF] Safari | Splitting Ward identity[END_REF][START_REF] Litim | Wilsonian flows and background fields[END_REF][START_REF] Bridle | The local potential approximation in the background field formalism[END_REF], a. k. a. split Ward identity which takes the following form for the gravitational case

δΓ k δḡ µν - δΓ k δh µν = 1 2 Tr 1 √ ḡ δ ḡR k [ḡ] δḡ µν Γ (0,2) k [ḡ µν ; h µν ] + R k [ḡ] -1 + δS gf δḡ µν - δS gf δh µν + δS gh δḡ µν - δS gh δh µν . (104)
The left hand-side is the difference between the background-field dependence and the fluctuation field dependence. The right-hand side encodes the sources of this difference, which are given by the background-field dependence of the regulator (first term on the lhs) as well as the gauge-fixing and ghost terms. Since fluctuations in h µν are gaugefixed with respect to the background field, the gauge-fixing and ghost terms treat ḡµν differently from h µν and must accordingly appear on the right-hand-side of this symmetry identity. Regarding the first term, one can see that it is structurally similar to the right-hand-side of the flow equation itself. This is a consequence of the fact that this term encodes the dependence of the regulator on an external field (ḡ µν ) or parameter (k); therefore, except for the substitution ∂ k R k → δ ḡR k [ḡ]/δḡ µν , these terms must be the same. The Ward-identity is explicitly studied in truncations in [START_REF] Donkin | The phase diagram of quantum gravity from diffeomorphism-invariant RG-flows[END_REF][START_REF] Dietz | Background independent exact renormalization group for conformally reduced gravity[END_REF][START_REF] Labus | Background independence in a background dependent renormalization group[END_REF][START_REF] Eichhorn | Effective universality in quantum gravity[END_REF][START_REF] Morris | Large curvature and background scale independence in single-metric approximations to asymptotic safety[END_REF][START_REF] Percacci | The background scale Ward identity in quantum gravity[END_REF][START_REF] Nieto | Split Weyl transformations in quantum gravity[END_REF][START_REF] Ohta | Background Scale Independence in Quantum Gravity[END_REF].

Note that the physics of asymptotic safety is encoded in the full effective action at g µν = ḡµν , i.e., Γ k→0 [ḡ µν ; 0]. Yet, in order to calculate this quantity, it is important to distinguish

Γ (0,2) k [ḡ µν ; h µν ] and Γ (2,0) k [ḡ µν ; h µν ].
It is the former, not the latter, that drives the flow; and they differ as described by Eq. ( 104). Accordingly, although the physics is encoded in background quantities, intermediate steps in their calculation require a clean distinction of background and fluctuation quantities. In particular, in [1187], the importance of keeping track of a self-consistent choice of background has been emphasized. The bimetric approximation distinguishes between the flow of background quantities and dynamical quantities. It can equivalently be implemented by tracking the dependence of the average effective action on ḡµν and g µν or that on ḡµν and h µν . Studies in these various setups hint at the qualitative reliability of the single-metric approximation in pure gravity [START_REF] Denz | Towards apparent convergence in asymptotically safe quantum gravity[END_REF][START_REF] Christiansen | Curvature dependence of quantum gravity[END_REF][START_REF] Christiansen | Global Flows in Quantum Gravity[END_REF][START_REF] Christiansen | Fixed points and infrared completion of quantum gravity[END_REF][START_REF] Christiansen | Local Quantum Gravity[END_REF][START_REF] Knorr | Correlation functions on a curved background[END_REF][START_REF] Manrique | Bimetric Truncations for Quantum Einstein Gravity and Asymptotic Safety[END_REF][START_REF] Manrique | Bimetric Renormalization Group Flows in Quantum Einstein Gravity[END_REF][START_REF] Becker | En route to Background Independence: Broken split-symmetry, and how to restore it with bi-metric average actions[END_REF][START_REF] Codello | Consistent closure of renormalization group flow equations in quantum gravity[END_REF][START_REF] Knorr | Infinite order quantum-gravitational correlations[END_REF], although the background cosmological constant appears to deviate significantly from its fluctuation counterpart under the impact of quantum fluctuations of matter [START_REF] Eichhorn | Effective universality in quantum gravity[END_REF]. Note that there are various ways to split the full metric g µν into a background ḡµν and fluctuation field h µν . Most studies to date employ a linear split as in Eq. ( 100), but alternative parameterizations have also been explored, see, e.g., [START_REF] Ohta | Flow equation for f (R) gravity and some of its exact solutions[END_REF][START_REF] De Brito | Asymptotic safety and field parametrization dependence in the f (R) truncation[END_REF][START_REF] Eichhorn | On unimodular quantum gravity[END_REF][START_REF] Eichhorn | The Renormalization Group flow of unimodular f(R) gravity[END_REF][START_REF] Gies | Generalized Parametrization Dependence in Quantum Gravity[END_REF][START_REF] Falls | Renormalization of Newton's constant[END_REF][START_REF] Ohta | Gauges and functional measures in quantum gravity I: Einstein theory[END_REF][START_REF] Nink | Field Parametrization Dependence in Asymptotically Safe Quantum Gravity[END_REF][START_REF] Percacci | Search of scaling solutions in scalar-tensor gravity[END_REF]. A Vilkovisky-DeWitt approach in the FRG framework [START_REF] Pawlowski | Geometrical effective action and Wilsonian flows[END_REF] for gravity has been explored in [START_REF] Donkin | The phase diagram of quantum gravity from diffeomorphism-invariant RG-flows[END_REF]. When changing from one parameterization to another, a Jacobian arises in the path integral. Further, the domain of integration can in general change, see, e.g., [START_REF] Nink | Field Parametrization Dependence in Asymptotically Safe Quantum Gravity[END_REF]. Note that the studies that compare the results within different parameterizations up to date neglect the Jacobian and change of domain of integration.

In summary, there are compelling indications for the existence of the Reuter fixed point in four-dimensional Euclidean gravity, although the implementation of background independence currently remains an outstanding problem, see also [START_REF] Bonanno | Critical reflections on asymptotically safe gravity[END_REF] for a discussion of current open questions in asymptotically safe gravity.

Key challenges for asymptotically safe gravity

Additional key points for future research include an understanding of Lorentzian gravity, see [START_REF] Manrique | Asymptotically Safe Lorentzian Gravity[END_REF]. Unlike in all other sections of this review, a Wick rotation is in general not available in quantum gravity [1092,[START_REF] Visser | How to Wick rotate generic curved spacetime[END_REF]. For instance, the existence of horizons, which are null hypersurfaces, is tied to the existence of causal structure, i.e., Lorentzian nature. Further, the configuration space of Riemannian as compared to Lorentzian metrics has significantly different properties [START_REF] Demmel | Connections and geodesics in the space of metrics[END_REF]. Therefore a scale-invariant fixed-point regime in Euclidean gravity does not automatically imply the same for Lorentzian gravity, and Lorentzian gravity must be explored separately, see also [START_REF] Donoghue | A Critique of the Asymptotic Safety Program[END_REF] for a discussion of this point. Since a momentum cutoff is most straightforwardly imposed in Euclidean signature, the study of Lorentzian asymptotic safety is a challenge. The impact of a foliation structure on the Reuter fixed point has been explored in [START_REF] Biemans | Quantum gravity on foliated spacetimes: Asymptotically safe and sound[END_REF][START_REF] Biemans | Renormalization group fixed points of foliated gravity-matter systems[END_REF][START_REF] Houthoff | Impact of topology in foliated Quantum Einstein Gravity[END_REF], also bringing the FRG studies closer to Monte Carlo simulations based on a configuration space where each configuration admits a foliation [START_REF] Ambjorn | A Nonperturbative Lorentzian path integral for gravity[END_REF][START_REF] Ambjorn | Emergence of a 4-D world from causal quantum gravity[END_REF][START_REF] Ambjorn | A Second-order phase transition in CDT[END_REF][START_REF] Ambjorn | Nonperturbative Quantum Gravity[END_REF].

A further central open question is to determine which degrees of freedom propagate. This is not fixed uniquely once the field is specified. For instance, an action including ´d4 x √ g R 2 in addition to the Einstein-Hilbert term propagates an additional scalar, as one can most easily see by performing a conformal transformation which isolates the kinetic term for the conformal mode. Given that the action in asymptotic safety is expected to contain higherderivative terms, there might be further modes that propagate beyond a massless spin-2 graviton. This is not only potentially phenomenologically relevant, as an additional scalar might drive an inflationary regime [START_REF] Bonanno | Asymptotically safe inflation from quadratic gravity[END_REF], but such modes could threaten the consistency of the theory, as unitarity/kinematic stability could be violated [START_REF] Arici | Reflection positivity in higher derivative scalar theories[END_REF][START_REF] Becker | On avoiding Ostrogradski instabilities within Asymptotic Safety[END_REF]. In two-dimensional gravity, a unitary theory underlying asymptotic safety has been identified, see [START_REF] Nink | The unitary conformal field theory behind 2D Asymptotic Safety[END_REF]. Additionally, it is worthwhile noting that Causal Dynamical Triangulations (CDTs) feature a self-adjoint and bounded transfer matrix. In CDTs, each configuration can be Wick-rotated, implying that a well-defined Hamiltonian exists in the discrete setting [START_REF] Ambjorn | A Nonperturbative Lorentzian path integral for gravity[END_REF][START_REF] Ambjorn | Dynamically triangulating Lorentzian quantum gravity[END_REF]. If this property persists in the continuum limit, and the universality class of that continuum limit is the Reuter fixed point, then asymptotically safe gravity would be expected to be unitary according to this notion of unitarity.

It should be noted that in general the question of unitarity is significantly more subtle in quantum gravity than in flat-space QFTs without gauge symmetry. These subtleties include but are not limited to the points that i) flat space does not necessarily have to be stable to agree with observations (since we live in a universe with a nonvanishing cosmological constant) and therefore (tachyonic) instabilities around flat space are not necessarily a problem for the viability of the theory; ii) the definition of an S-matrix on a generic non-trivial backgrounds is an outstanding challenge; iii) spectral representations for the propagators in a gauge theory can feature negative parts without posing a problem for the unitarity of the theory -in fact, Yang-Mills theory is an example, since the spectral function of the gluon features negative parts. Therefore the notion of unitarity that is used in flat-space QFTs without gauge symmetries might be applicable in quantum gravity in a perturbative regime, but might not suffice to analyze the theory in the fully non-perturbative regime -or even in a pre-geometric one, where no nontrivial background metric is available.

Understanding spacetime structure

A central question of (asymptotically safe) quantum gravity concerns the properties of the effective quantum geometries that arise. Most importantly, these pertain to physically relevant spacetimes, such as that of black holes or in cosmology. Using the idea of RG-improvement, quantum-gravity inspired models have been explored. Their starting point is the RG scale dependence of the gravitational couplings: The dimensionful Newton coupling is constant in the classical gravity regime below the transition scale 1γ to the fixed-point regime. In the fixed-point regime, the dimensionless Newton coupling G takes its constant fixed-point value, which is actually given by G * = 1/γ, such that G N = Gk -2 = 1/γk -2 falls off like k-2. The simplest form of the dimensionful Newton coupling G N (k) that models this RG flow between the scale-invariant fixed-point regime and the classical regime in the infrared is given by

G N (k) = G 0 1 + γ G 0 k 2 , ( 105
)
where γ is the inverse dimensionless transition scale between the two regimes. G 0 is the value of the dimensionful Newton couplings at low scales, i.e., G N (k → 0) → G 0 , i.e., it is the value of G N that is achieved once all quantum fluctuations have been integrated out. As one explores higher k, the fixed-point scaling stops the quadratic growth of the dimensionless Newton coupling G = G N k 2 , which instead levels off to a constant. Conversely, this implies that the dimensionful Newton coupling G N that is constant at low k, scales quadratically, G N ∼ G * k -2 , at high k. Eq. ( 105) exhibits a relatively sharp transition between the classical and the fixed-point regime, which it shares with the scale dependence of G N (k) obtained by integrating the RG flow in a given truncation, see, e.g., [START_REF] Denz | Towards apparent convergence in asymptotically safe quantum gravity[END_REF][START_REF] Reuter | Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation[END_REF][START_REF] Litim | Fixed points of quantum gravity[END_REF].

One might be tempted to interpret this very directly in terms of the dynamics of gravity at high energy/momentum/curvature scales. Yet, one should be rather careful with an identification of k with a physical scale, see, e.g., an example in the Lorentzian case [START_REF] Anber | On the running of the gravitational constant[END_REF]. Models of spacetime obtained from an identification of k with physical scales should thus be considered as "quantum-gravity inspired" models that could provide a first qualitative understanding of potential quantum gravitational effects. With this in mind, the weakening of the dimensionfull Newton coupling towards high momentum scales could rise to the expectation that classically singular black-hole spacetimes should become regular. This has been explored using RG-improvement, i.e., upgrading the classical solution/dynamics by a scale-dependent coupling. The Schwarzschild case, first analyzed in [START_REF] Bonanno | Quantum gravity effects near the null black hole singularity[END_REF][START_REF] Bonanno | Renormalization group improved black hole spacetimes[END_REF], provides the "blueprint" for later works [START_REF] Bonanno | Spacetime structure of an evaporating black hole in quantum gravity[END_REF][START_REF] Cai | Black holes in an asymptotically safe gravity theory with higher derivatives[END_REF][START_REF] Reuter | Quantum Gravity Effects in the Kerr Spacetime[END_REF][START_REF] Falls | Black Holes and Asymptotically Safe Gravity[END_REF][START_REF] Falls | Black hole thermodynamics under the microscope[END_REF][START_REF] Torres | Evaporation of (quantum) black holes and energy conservation[END_REF][START_REF] Litim | Quantum gravity effects in Myers-Perry space-times[END_REF][START_REF] Koch | Structural aspects of asymptotically safe black holes[END_REF][START_REF] Kofinas | Avoidance of singularities in asymptotically safe Quantum Einstein Gravity[END_REF][START_REF] Torres | Nonsingular black holes, the cosmological constant, and asymptotic safety[END_REF][START_REF] Pawlowski | Quantum-improved Schwarzschild-(A)dS and Kerr-(A)dS spacetimes[END_REF][START_REF] Adeifeoba | Towards conditions for black-hole singularity-resolution in asymptotically safe quantum gravity[END_REF]. For black holes, a "natural" scale identification is to use the curvature scale, as this provides a coordinateindependent notion of scale. As black hole spacetimes are vacuum solutions of the Einstein equations, R = 0 and R µν = 0. Instead, one uses the Kretschmann scalar, which is given by K

= R µνκλ R µνκλ = 48G 2 0 M 2 r 6
for the spherically symmetric case, where G 0 is the classical value of the dimensionful Newton coupling, M is the mass of the black hole and r is the radial coordinate in Schwarzschild coordinates. Due to the spherical symmetry and static nature of the Schwarzschild spacetime, all other curvature invariants are proportional to the Kretschmann scalar and therefore yield the same result for the RG-improved metric. Once K 1/4 exceeds the Planck mass, one expects quantum-gravity effects to kick in. This motivates the scale identification

k = α K 1/4 , (106)
where the power is fixed on dimensional grounds and α is a dimensionless number of order one. Inserting G N (k) = G N (K 1/4 ) from Eq. ( 105) in the classical form of the metric leads to an upgraded metric, for which the Kretschmann scalar is no longer singular. This regular black-hole metric should be understood as an asymptotic-safety inspired model for a black hole, and has been explored further in the literature. In [START_REF] Platania | Dynamical renormalization of black-hole spacetimes[END_REF], the RG improvement procedure is iterated using the Kretschmann scalar of the improved metric, in which case the RG improved black-hole spacetime converges to the so-called Dymnikova metric [1226]. In [START_REF] Held | Asymptotic safety casts its shadow[END_REF], the size and shape of black-hole shadows for the corresponding modified spacetimes are explored, see also [START_REF] Kumar | Rotating black hole shadow in asymptotically safe gravity[END_REF]. RG improved gravitational collapse has been explored in [1229][START_REF] Fayos | A quantum improvement to the gravitational collapse of radiating stars[END_REF][START_REF] Torres | Singularity-free gravitational collapse and asymptotic safety[END_REF][START_REF] Torres | Singularity free gravitational collapse in an effective dynamical quantum spacetime[END_REF][START_REF] Bonanno | Cosmic Censorship in Quantum Einstein Gravity[END_REF][START_REF] Bonanno | Gravitational collapse in Quantum Einstein Gravity[END_REF][START_REF] Bonanno | Asymptotically Safe gravitational collapse: Kuroda-Papapetrou RG-improved model[END_REF] and RG improved stellar interiors have been studied in [START_REF] Bonanno | Gravitational antiscreening in stellar interiors[END_REF]. It should be emphasized that the RG improvement procedure could be implemented at the level of a given spacetime geometry, at the level of the equations of motion, or at the level of the action, without a guarantee of agreement between the outcomes. Besides black holes, another phenomenologically relevant class of spacetimes in which quantum gravity is expected to play a role is cosmological spacetimes. In particular, one expects effects of quantum gravity in the early universe. This provides a potential testing ground for asymptotic safety, as well as other models of quantum gravity. Various aspects of cosmology have been explored in the context of asymptotic safety, resulting in asymptotic-safety inspired cosmological models. For recent reviews of potential imprints of asymptotic safety in cosmology, as suggested by exploring RG improved models, see [START_REF] Bonanno | Asymptotically safe cosmology -A status report[END_REF][START_REF] Platania | From renormalization group flows to cosmology[END_REF]. Cosmological singularity resolution and the suppression of anisotropies and inhomogeneities based on the presence of higher-order interactions in asymptotically safe gravity has been suggested in [START_REF] Lehners | A Safe Beginning for the Universe?[END_REF]. This work is based on an argument regarding the gravitational path integral, where early-universe spacetimes interfere destructively, if they are associated to a diverging action. This is the case for a curvature-squared action, as one might expect to play a role in asymptotic safety, when applied to early-universe spacetimes with anisotropies and inhomogeneities, which are disfavored also observationally. Further, asymptotic safety might play a role in the context of inflation [START_REF] Weinberg | Asymptotically Safe Inflation[END_REF].

Here, the first main question is whether asymptotic safety could result in an inflationary phase without the need for an additional scalar field that is introduced ad-hoc. Given that, e.g., f (R) theories come with an additional propagating scalar, as one can see by performing an appropriate conformal transformation on the metric, the inflaton might arise as an automatic consequence of asymptotically safe gravity. The second main question is whether an inflationary phase results in agreement with the observed spectrum of the cosmic microwave background. The deviation from scale invariance, tensor-to-scalar-ratio and energy scale associated to inflation, encoded in the amplitude of scalar fluctuations, are measured/constrained by Planck and therefore constitute potential checks of the quantum-gravity theory. RG improvement has been used to derive quantum-gravity inspired models of inflation, see [START_REF] Bonanno | Cosmology of the Planck era from a renormalization group for quantum gravity[END_REF][START_REF] Reuter | From big bang to asymptotic de Sitter: Complete cosmologies in a quantum gravity framework[END_REF][START_REF] Bonanno | Inflationary solutions in asymptotically safe f(R) theories[END_REF][START_REF] Bonanno | Entropy Production during Asymptotically Safe Inflation[END_REF][1245][START_REF] Bonanno | An effective action for asymptotically safe gravity[END_REF][START_REF] Copeland | Asymptotically Safe Starobinsky Inflation[END_REF][START_REF] Tronconi | Asymptotically Safe Non-Minimal Inflation[END_REF][START_REF] Bonanno | Cosmological bounds on the field content of asymptotically safe gravity-matter models[END_REF][START_REF] Liu | Inflation in an effective gravitational model and asymptotic safety[END_REF][START_REF] Platania | The inflationary mechanism in Asymptotically Safe Gravity[END_REF] and the early universe [START_REF] Hindmarsh | Asymptotically Safe Cosmology[END_REF][START_REF] Kofinas | Asymptotically Safe gravity and non-singular inflationary Big Bang with vacuum birth[END_REF]. Cosmological perturbation theory in an RG improved context has been studied in [START_REF] Bonanno | Cosmological perturbations in renormalization group derived cosmologies[END_REF][START_REF] Contillo | Evolution of cosmological perturbations in an RG-driven inflationary scenario[END_REF]. Late-time modifications of cosmology that might arise due to infrared effects of quantum gravity in the context of RG fixed points, see [START_REF] Dou | The running gravitational couplings[END_REF][START_REF] Bonanno | Cosmology with self-adjusting vacuum energy density from a renormalization group fixed point[END_REF][START_REF] Babic | Renormalization-group running cosmologies. A Scale-setting procedure[END_REF][START_REF] Ahn | From Asymptotic Safety to Dark Energy[END_REF][START_REF] Bonanno | Dynamical System Analysis of Cosmologies with Running Cosmological Constant from Quantum Einstein Gravity[END_REF][START_REF] Wetterich | Infrared limit of quantum gravity[END_REF][START_REF] Anagnostopoulos | Constraining the Asymptotically Safe Cosmology: cosmic acceleration without dark energy[END_REF][START_REF] Gubitosi | Consistent early and late time cosmology from the RG flow of gravity[END_REF].

Another central task for any approach to quantum gravity is to characterize the properties of quantum spacetime in the vicinity of the Planck scale. Indications for a reduction from d = 4 to d = 2 exist in the ultraviolet in the spectral dimension which is extracted from an RG improved diffusion equation [START_REF] Lauscher | Fractal spacetime structure in asymptotically safe gravity[END_REF][START_REF] Reuter | Fractal space-times under the microscope: A Renormalization Group view on Monte Carlo data[END_REF][START_REF] Rechenberger | The R 2 phase-diagram of QEG and its spectral dimension[END_REF][START_REF] Calcagni | Probing the quantum nature of spacetime by diffusion[END_REF]. The underlying idea is that a diffusion process, i.e., a random walk, probes the properties of the effective spacetime manifold at a given scale. For instance, the return probability to the starting point can give direct insight into the underlying dimensionality, providing what is known as the spectral dimension. Here, some care is required when performing the RG improvement, as only the use of diffusion time as inverse RG scale results in a well-defined diffusion equation [START_REF] Calcagni | Probing the quantum nature of spacetime by diffusion[END_REF], highlighting the potential pitfalls of the RG improvement procedure. The observed reduction in the spectral dimension does not imply a reduction in the topological dimension of the manifold: In common with fractals, dimensionality can become a resolution-scale dependent quantity that further depends on the "prescription" with which it is measured. For instance, the effects of quantum fluctuations of spacetime that slow down the random walker underlying the measurement of the spectral dimension do not automatically impact the scaling of the volume of an d-sphere with its radius. The latter provides the Hausdorff dimension which actually stays constant at all scales in asymptotic safety [START_REF] Reuter | Fractal space-times under the microscope: A Renormalization Group view on Monte Carlo data[END_REF]. A dimensional reduction is also familiar from some other quantum-gravity approaches, see [START_REF] Carlip | Dimension and Dimensional Reduction in Quantum Gravity[END_REF] for an overview.

To further characterize the geometry of spacetime, one can for instance analyze the behavior of geodesic curves, or various hypersurfaces. Progress towards understanding these has been made in terms of a flow equation for composite operators [START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Pagani | Products of composite operators in the exact renormalization group formalism[END_REF][START_REF] Igarashi | Realization of Symmetry in the ERG Approach to Quantum Field Theory[END_REF][START_REF] Pagani | Composite Operators in Asymptotic Safety[END_REF], which provides hints for dimensional reduction from the scaling behavior of geodesics and areas [START_REF] Houthoff | On the scaling of composite operators in Asymptotic Safety[END_REF][START_REF] Kurov | On characterizing the Quantum Geometry underlying Asymptotic Safety[END_REF][START_REF] Becker | Geometric operators in the asymptotic safety scenario for quantum gravity[END_REF], see also [START_REF] Becker | Fractal geometry of higher derivative gravity[END_REF] for a comparison to higher-derivative gravity.

Asymptotically safe gravity and particle physics

Although a substantial part of quantum-gravity research is focused on a purely gravitational setting, it is neither necessary nor sufficient for a quantum description of spacetime to be viable in our universe to exist as a purely gravitational theory, since we know from observations that matter exists. Accordingly, the existence of a fixed point in gravity-matter systems is actually key for a phenomenologically viable asymptotic-safety paradigm. The exploration of asymptotically safe gravity-matter models started in [START_REF] Percacci | Constraints on matter from asymptotic safety[END_REF][START_REF] Percacci | Asymptotic safety of gravity coupled to matter[END_REF]. Indications for the existence of the Reuter fixed point in the presence of minimally coupled Standard Model matter have been found in [START_REF] Wetterich | Variable Planck mass from the gauge invariant flow equation[END_REF][START_REF] Alkofer | Asymptotically safe f(R)-gravity coupled to matter I: the polynomial case[END_REF][START_REF] Donà | Matter matters in asymptotically safe quantum gravity[END_REF], see [START_REF] Donà | Consistency of matter models with asymptotically safe quantum gravity[END_REF] for an extension including gravitinos. In [START_REF] Donà | Matter matters in asymptotically safe quantum gravity[END_REF] and the follow-up works, an asymptotically safe fixed point has been found when the matter content of the Standard Model is added. This fixed point can be connected continuously to the pure-gravity fixed point, if the numbers of matter fields are treated as external parameters. On the technical side, the treatment of fermions in the single-metric approximation has been discussed in [START_REF] Donà | Functional renormalization with fermions and tetrads[END_REF] and the spin-base invariance formalism has been developed in [START_REF] Gies | Fermions in gravity with local spin-base invariance[END_REF][START_REF] Gies | Global surpluses of spin-base invariant fermions[END_REF][START_REF] Lippoldt | Spin-base invariance of Fermions in arbitrary dimensions[END_REF]. The impact of fermions, scalar and vectors has separately been studied in various truncations in [START_REF] Eichhorn | Effective universality in quantum gravity[END_REF][START_REF] Eichhorn | Zooming in on fermions and quantum gravity[END_REF][START_REF] Bürger | Curvature dependence of quantum gravity with scalars[END_REF][START_REF] Meibohm | Asymptotic safety of gravity-matter systems[END_REF][START_REF] Donà | Asymptotic safety in an interacting system of gravity and scalar matter[END_REF][START_REF] Eichhorn | Quantum gravity and Standard-Model-like fermions[END_REF][START_REF] Eichhorn | Nonminimal hints for asymptotic safety[END_REF][START_REF] Christiansen | Asymptotic safety of gravity with matter[END_REF][START_REF] Hamada | Asymptotic safety of higher derivative quantum gravity non-minimally coupled with a matter system[END_REF], some of which go beyond minimal coupling.

It should be stressed that extended truncations are required in order to settle whether the Reuter fixed point can be extended continuously to a large numbers of matter fields. Alternatively, gravity-matter systems with large numbers of matter fields might be a new universality class, or not feature a fixed point at all.

Conversely, the impact of quantum gravity on matter is key to generate an asymptotically safe model and bridge the gap from quantum gravity to phenomenology. The Standard Model can be consistently extended all the way up to the Planck scale 43 , yet, beyond the Planck scale, the Landau pole problems from Abelian gauge theories and scalar theories are expected to render the Abelian hypercharge and the Higgs-Yukawa sector of the Standard Model inconsistent. Further, the Standard Model has 19 free parameters that lack a fundamental dynamical explanation. These include the ratio of the Higgs mass to the Planck mass, the tiny value of which is argued to pose a particularly severe problem. Yet, the values of the fermion masses, related to Yukawa couplings, as well as the strength of the Higgs self-interaction and the gauge interactions are also unexplained, with the Yukawa sector also containing a significant hierarchy. Moreover, the Standard Model only includes three of the four currently known fundamental interactions, i.e., lacks gravity. The asymptotically safe perspective motivates the idea that all three challenges (UV completion, increase in predictivity, inclusion of gravity) can be tackled simultaneously, see [START_REF] Eichhorn | An asymptotically safe guide to quantum gravity and matter[END_REF][START_REF] Eichhorn | Status of the asymptotic safety paradigm for quantum gravity and matter[END_REF] for reviews. Quantum-gravity effects have been explored on the scalar sector [START_REF] Labus | Asymptotic safety in O(N) scalar models coupled to gravity[END_REF][START_REF] Pawlowski | Higgs scalar potential in asymptotically safe quantum gravity[END_REF][START_REF] Percacci | Search of scaling solutions in scalar-tensor gravity[END_REF][START_REF] Eichhorn | Nonminimal hints for asymptotic safety[END_REF][START_REF] Narain | Renormalization Group Flow in Scalar-Tensor Theories. I[END_REF][START_REF] Narain | Renormalization Group Flow in Scalar-Tensor Theories[END_REF][START_REF] Eichhorn | Quantum-gravity-induced matter self-interactions in the asymptotic-safety scenario[END_REF][1290][START_REF] Henz | Scaling solutions for Dilaton Quantum Gravity[END_REF][START_REF] Eichhorn | Quantum gravity fluctuations flatten the Planck-scale Higgs potential[END_REF][START_REF] Wetterich | Effective scalar potential in asymptotically safe quantum gravity[END_REF], where the scalar mass parameter remains relevant unless quantum-gravity effects are very strong, and the quartic scalar coupling and higher-order couplings are irrelevant. Therefore, an asymptotically safe gravity-scalar fixed point results in predictive power in the scalar sector. The Yukawa sector has been explored in [START_REF] Hamada | Asymptotic safety of higher derivative quantum gravity non-minimally coupled with a matter system[END_REF][START_REF] Zanusso | Gravitational corrections to Yukawa systems[END_REF][START_REF] Vacca | Asymptotic Safety in Einstein Gravity and Scalar-Fermion Matter[END_REF][START_REF] Oda | Non-minimal coupling in Higgs-Yukawa model with asymptotically safe gravity[END_REF][START_REF] Eichhorn | Quantum-gravity effects on a Higgs-Yukawa model[END_REF][START_REF] Eichhorn | Viability of quantum-gravity induced ultraviolet completions for matter[END_REF][START_REF] Eichhorn | Top mass from asymptotic safety[END_REF][START_REF] De Brito | On the impact of Majorana masses in gravity-matter systems[END_REF], where a predictive fixed point for the Yukawa coupling appears to be possible in a restricted part of the gravitational parameter space. Additionally, fermions can remain light 44 despite quantum-gravity fluctuations [START_REF] Eichhorn | Light fermions in quantum gravity[END_REF][START_REF] Meibohm | Chiral fermions in asymptotically safe quantum gravity[END_REF][START_REF] Gies | Curvature bound from gravitational catalysis[END_REF] (in contrast to the analogous setting in non Abelian gauge theories, see, e.g., [START_REF] Braun | Fermion Interactions and Universal Behavior in Strongly Interacting Theories[END_REF][START_REF] Braun | Running coupling at finite temperature and chiral symmetry restoration in QCD[END_REF][START_REF] Braun | Chiral phase boundary of QCD at finite temperature[END_REF]). Finally, the gauge sector has been studied in [START_REF] Folkerts | Asymptotic freedom of Yang-Mills theory with gravity[END_REF][START_REF] Christiansen | Asymptotic safety of gravity with matter[END_REF][START_REF] Daum | Running Gauge Coupling in Asymptotically Safe Quantum Gravity[END_REF][START_REF] Daum | Non-perturbative QEG Corrections to the Yang-Mills Beta Function[END_REF][START_REF] Harst | QED coupled to QEG[END_REF][START_REF] Christiansen | An asymptotically safe solution to the U(1) triviality problem[END_REF][START_REF] Eichhorn | Upper bound on the Abelian gauge coupling from asymptotic safety[END_REF][START_REF] Eichhorn | Quantum-gravity predictions for the fine-structure constant[END_REF], finding indications that asymptotic freedom of non-Abelian gauge theories is compatible with an asymptotically safe fixed point, while Abelian gauge theories could be UV completed [START_REF] Harst | QED coupled to QEG[END_REF][START_REF] Eichhorn | Upper bound on the Abelian gauge coupling from asymptotic safety[END_REF]. For studies of these effects in a unimodular setting, see [START_REF] De Brito | A link that matters: Towards phenomenological tests of unimodular asymptotic safety[END_REF]. Demanding UV complete matter sectors restricts the values of curvature-squared couplings in the unimodular setting, analogously to the standard setting [START_REF] Eichhorn | Viability of quantum-gravity induced ultraviolet completions for matter[END_REF].

In truncations, which to date only include canonically relevant and marginal couplings, as well as a small subset of the canonically irrelevant couplings in some cases, the following is observed: Quantum gravity triggers an asymptotically safe UV completion of Standard Model-like theories with novel, quantum-gravity generated interactions at high scales, [START_REF] Eichhorn | Viability of quantum-gravity induced ultraviolet completions for matter[END_REF]. At the same time, marginally irrelevant couplings of the Standard Model, corresponding to the free parameters of the Standard Model, could be irrelevant at a joint, interacting matter-gravity fixed point, cf. Fig. 29. As such, their values at all scales would be predictable. Once the Higgs vacuum expectation value, which remains a relevant parameter, unless quantum-gravity effects are very strong [START_REF] Eichhorn | Quantum gravity fluctuations flatten the Planck-scale Higgs potential[END_REF][START_REF] Wetterich | Gauge hierarchy problem in asymptotically safe gravity-the resurgence mechanism[END_REF], is set, the Higgs mass [START_REF] Shaposhnikov | Asymptotic safety of gravity and the Higgs boson mass[END_REF], top quark mass [START_REF] Eichhorn | Top mass from asymptotic safety[END_REF], bottom quark mass [START_REF] Eichhorn | Mass difference for charged quarks from asymptotically safe quantum gravity[END_REF] and Abelian gauge coupling [START_REF] Harst | QED coupled to QEG[END_REF][START_REF] Eichhorn | Upper bound on the Abelian gauge coupling from asymptotic safety[END_REF][START_REF] Eichhorn | Mass difference for charged quarks from asymptotically safe quantum gravity[END_REF] become calculable within the toy models defined by the truncated RG flows of subsectors of the Standard Model. This is a first hint for an enhancement of predictive power in the parameters that determine the dynamics of the model. Intriguingly, demanding that asymptotically safe gravity provides a consistent UV completion for Standard-Model like matter theories might potentially even set geometric parameters, and e.g., fix the spacetime dimensionality to 4 [START_REF] Eichhorn | d = 4 as the critical dimensionality of asymptotically safe interactions[END_REF]. If such findings persist in extended truncations and the universal quantities exhibit apparent convergence, this will allow to either obtain a model with a higher predictive power than the Standard Model which additionally is UV complete and includes gravity, or to rule out this model, as the reduced number of free parameters compared to the Standard Model could lead to a conflict with observational data. The latter point is noteworthy, as it implies that experimental constraints can be imposed on quantum gravity using data at scales much below the Planck scale, such as, e.g., the electroweak scale.

Constraints that could arise on the number of fields in a dark sector have been discussed in [START_REF] Eichhorn | Top mass from asymptotic safety[END_REF] and the inclusion of a Higgs portal in asymptotic safety has been explored in [START_REF] Eichhorn | Quantum gravity fluctuations flatten the Planck-scale Higgs potential[END_REF][START_REF] Reichert | Dark Matter meets Quantum Gravity[END_REF][START_REF] Hamada | Scalegenesis and fermionic dark matters in the flatland scenario[END_REF][START_REF] Eichhorn | Safety in darkness: Higgs portal to simple Yukawa systems[END_REF]. Grand unified theories are expected to be constrained [START_REF] Eichhorn | Quantum-gravity predictions for the fine-structure constant[END_REF][START_REF] Eichhorn | Predictive power of grand unification from quantum gravity[END_REF] if indeed they can be consistently coupled to asymptotically safe gravity. For studies of further BSM settings coupled to asymptotically safe quantum gravity, see [START_REF] Kwapisz | Asymptotic safety, the Higgs boson mass, and beyond the standard model physics[END_REF][START_REF] Grabowski | Asymptotic safety and Conformal Standard Model[END_REF]. In summary, these exploratory first studies suggest that asymptotically safe gravity could have an enhanced predictive power in BSM settings, and therefore restrict the theoretically viable extensions of the Standard Model.

0 g Y* 0 G * M Pl g Y* RG scale k in GeV
We briefly highlight that the gravitational contributions to beta functions in the matter sector is not universal, due to the nonvanishing canonical dimension of the Newton coupling. The universal statement therefore consists in the number of positive critical exponents that determines how many free parameters remain, and how many relations between various couplings can be imposed. While the actual flows are therefore expected to differ in different schemes, the values of actual physical observables in the IR only exhibit a scheme-dependence within approximations, but should converge to the same values as the approximation is improved.

Finally, we highlight that an interacting RG fixed point in (matter)-gravity systems is also of interest in a setting where a UV cutoff Λ fund exists, beyond which some more fundamental description of the degrees of freedom is applicable. In that setting, the RG fixed point can actually serve as an intermediate fixed point: The fundamental theory determines the values of all couplings in the effective description at Λ fund . From there, the RG flow can approach the interacting fixed point arbitrarily close, if the values of the relevant couplings at Λ fund are appropriate. Then, the RG flow "washes out" many of the microscopic details of the fundamental theory and the fixed point generates universality, in much the same way as it does in the previous sections, in particular for statistical physics. In particular, if the Reuter universality class is characterized by only a small number of free parameters, this high degree of predictivity is effectively imposed on the underlying, more fundamental theory as well. For this scenario to be realistic, the RG trajectory cannot be a true fixed-point trajectory, i.e., it must not end in the Reuter fixed point in the IR, as this does not yield a classical gravity regime. Instead, the RG flow should pass very close to the Reuter fixed point and exhibit a long range of scales over which it is nearly scale-invariant. At a transition scale, the flow should then leave the near-fixed-point regime to flow towards a classical-gravity regime in the IR, as trajectories that emanate out of the Reuter fixed point exhibit [START_REF] Reuter | Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation[END_REF][START_REF] Reuter | From big bang to asymptotic de Sitter: Complete cosmologies in a quantum gravity framework[END_REF]. For the first discussion of effective asymptotic safety in the gravitational context see [START_REF] Percacci | Asymptotic Safety, Emergence and Minimal Length[END_REF], for a proposal in the context of string theory see [START_REF] De Alwis | Asymptotic safety, string theory and the weak gravity conjecture[END_REF], and for a quantitative measure of predictivity in effective asymptotic safety, see [START_REF] Held | Effective asymptotic safety and its predictive power: Gauge-Yukawa theories[END_REF].

Let us also mention that the Wetterich equation can be applied to other forms of matter-gravity models, e.g., a study in the context of the spectral action can be found in [START_REF] Estrada | Asymptotic safety, hypergeometric functions, and the Higgs mass in spectral action models[END_REF].

The FRG in other approaches to quantum gravity

Many quantum-gravity approaches face one or both of the following two questions:

• Given a proposal for a microscopic dynamics, is the emergent effective low-energy theory compatible with observations?

• Defining the gravitational path integral through a discretization (typically not in metric variables), can the continuum limit be taken?

The FRG is a suitable tool to tackle both questions, and an example of each is introduced below.

Lorentz-symmetry violating quantum gravity

Lorentz symmetry is a cornerstone in the Standard Model of particle physics. This goes hand in hand with strong observational constraints on violations of Lorentz symmetry in high-energy physics. Yet, these constraints typically only cover a low-energy (in comparison to the Planck scale) range of scales. This motivates giving up Lorentz symmetry in quantum gravity, which in turn opens the door for perturbative renormalizability [START_REF] Horava | Quantum Gravity at a Lifshitz Point[END_REF], and in particular asymptotic freedom [START_REF] Odorico | Asymptotic Freedom in Horava-Lifshitz Gravity[END_REF][START_REF] Barvinsky | Hořava Gravity is Asymptotically Free in 2 + 1 Dimensions[END_REF]. A key question in this setting is whether violations of Lorentz symmetry at and beyond the Planck scale are compatible with the strong observational constraints at lower scales [START_REF] Yagi | Strong Binary Pulsar Constraints on Lorentz Violation in Gravity[END_REF][START_REF] Ramos | Constraints on Hořava gravity from binary black hole observations[END_REF]. The FRG is a powerful tool to tackle this question [START_REF] Contillo | Renormalization group flow of Hořava-Lifshitz gravity at low energies[END_REF]. Its application to this setting, pioneered in [START_REF] Rechenberger | A functional renormalization group equation for foliated spacetimes[END_REF] relies on a 3+1 split of the metric, such that a (Euclidean) time-direction as well as spatial hypersurfaces can be distinguished. It allows to follow the RG flow in Horava-Lifshitz gravity [START_REF] Horava | Quantum Gravity at a Lifshitz Point[END_REF] to low energies [START_REF] Contillo | Renormalization group flow of Hořava-Lifshitz gravity at low energies[END_REF][START_REF] Odorico | Covariant computation of effective actions in Hořava-Lifshitz gravity[END_REF] The flow of Lorentzsymmetry violating operators, such as, e.g., couplings of monomials build out of the extrinsic curvature of spatial hypersurfaces, has been explored, showing that such operators are RG relevant [START_REF] Knorr | Lorentz symmetry is relevant[END_REF]. The proliferation of Lorentzinvariance violation from the gravitational to the matter sector with this tool has been explored in [START_REF] Eichhorn | Lorentz invariance violations in the interplay of quantum gravity with matter[END_REF].

Discrete quantum-gravity models

In discrete quantum-gravity models, the FRG can be used to search for the existence of a continuum limit, as it is related to a second-order phase transition in the space of couplings, its universal properties in turn being encoded in a fixed point. A noteworthy aspect of these developments is a reinterpretation of the coarse-graining scale. Formally, an analogous derivation to that of Eq. ( 7) goes through if the regulator is not a function of a momentum scale, but of some other external variable that allows to sort fluctuations in the path integral. This technical observation is the basis for a flow equation, derived in [START_REF] Eichhorn | Continuum limit in matrix models for quantum gravity from the Functional Renormalization Group[END_REF] and following similar developments for fuzzy spaces [START_REF] Sfondrini | Functional Renormalization of Noncommutative Scalar Field Theory[END_REF], that takes the number of degrees of freedom as a (non-local) coarse graining scale. In the UV, many degrees of freedom are present, whereas in the IR only few (effective) degrees of freedom are present. This reinterpretation of the notion of scale is particularly suited to fully background-independent quantum gravity models where a local coarse-graining procedure cannot be defined, as it relies on a background metric with respect to which one can define local "patches" to be integrated over. Instead, these models are typically formulated as spacetime-free (or pre-geometric) matrix or tensor models, where the number of degrees of freedom is encoded in the tensor size N. Acccordingly, N can be used as a cutoff scale and a background-independent version of the flow equation ( 7) can be derived taking the form

N∂ N Γ N [T a 1 ...a d ] = 1 2 Tr        δ 2 Γ N δT a 1 ...a d δT b 1 ...b d + R N (a 1 , ..., b d ) -1 N∂ N R N        . (107)
It encodes how the effective dynamics change, as the outermost "layers" of a tensor (rows and columns in the case of matrix models) are integrated out, following the intuition of [START_REF] Brezin | Renormalization group approach to matrix models[END_REF]. For a version of the Polchinski equation following similar ideas see [START_REF] Krajewski | Polchinski's exact renormalisation group for tensorial theories: Gaussian universality and power counting[END_REF][START_REF] Krajewski | Exact Renormalisation Group Equations and Loop Equations for Tensor Models[END_REF]. This tool has been benchmarked for the case of 2d gravity, which corresponds to a matrix model. The leading critical exponent in that model is known to be θ = 0.8, enabling a check of the tool, cf. Fig. 30.

The connection to quantum gravity arises as the Feynman diagrams of such rank-d models are dual to triangulations of d dimensional space, see [START_REF] Ambjorn | Three-dimensional simplicial quantum gravity and generalized matrix models[END_REF][START_REF] Sasakura | Tensor model for gravity and orientability of manifold[END_REF][START_REF] Godfrey | Simplicial quantum gravity in more than two-dimensions[END_REF][START_REF] Gross | Tensor models and simplicial quantum gravity in > 2-D[END_REF][START_REF] Gurau | The 1/N expansion of colored tensor models[END_REF][START_REF] Bonzom | Random tensor models in the large N limit: Uncoloring the colored tensor models[END_REF], see [START_REF] Rivasseau | Quantum Gravity and Renormalization: The Tensor Track[END_REF][START_REF] Gurau | Colored Tensor Models -a review[END_REF][START_REF] Rivasseau | The Tensor Track: an Update[END_REF][START_REF] Rivasseau | The Tensor Track, III[END_REF][START_REF] Rivasseau | Random Tensors and Quantum Gravity[END_REF][START_REF] Rivasseau | The Tensor Track, IV[END_REF][START_REF] Gurau | Invitation to Random Tensors[END_REF] for reviews. To that end, the interactions of the model are tensor invariants, such that the identification of two indices of two tensors can be interpreted as dual to the gluing of two building blocks of d -1 dimensional space. For instance,

T a 1 a 2 a 3 T a 1 b 2 b 3 T b 1 b 2 a 3 T b 1 a 2 b 3
is dual to four triangles, one for each tensor, which are glued along adjacent edges to form a tetrahedron. Note that the combinatorial nonlocality (i.e., the indices are only contracted pairwise between tensors, there is no index common to all four tensors) of these interactions is crucial to allow for the dual interpretation in terms of building blocks of space. A universal continuum limit of these is related to a universal N → ∞ limit -which in turn is conjectured to be discoverable as a fixed point of the background-independent flow equation. Tentative indications for such fixed-point candidates have been discovered in [START_REF] Eichhorn | Flowing to the continuum in discrete tensor models for quantum gravity[END_REF][START_REF] Eichhorn | Towards background independent quantum gravity with tensor models[END_REF][START_REF] Eichhorn | Universal critical behavior in tensor models for four-dimensional quantum gravity[END_REF], for reviews see [START_REF] Eichhorn | Status of background-independent coarse-graining in tensor models for quantum gravity[END_REF][START_REF] Pereira | Quantum spacetime and the renormalization group: Progress and visions[END_REF]. In related models, so-called group-field theories, which are combinatorically nonlocal 45 i=1 g i TrT i , as in [START_REF] Eichhorn | Status of background-independent coarse-graining in tensor models for quantum gravity[END_REF]. The critical exponent is evaluated at fixed anomalous dimension, and can be fit as θ = 0.907 -1.540 exp(-0.292 n), extrapolating to θ = 0.91, compared to the exact result θ = 0.8. quantum gravity defined on a group manifold, where an extension of the FRG-equation ( 107) developed in [START_REF] Benedetti | Functional Renormalisation Group Approach for Tensorial Group Field Theory: a Rank-3 Model[END_REF] has been used to search for infrared fixed points [START_REF] Ben Geloun | Functional Renormalization Group analysis of a Tensorial Group Field Theory on R 3[END_REF][START_REF] Benedetti | Functional Renormalization Group Approach for Tensorial Group Field Theory: A Rank-6 Model with Closure Constraint[END_REF][START_REF] Ben Geloun | Functional Renormalisation Group analysis of Tensorial Group Field Theories on R d[END_REF][START_REF] Ben Geloun | Nontrivial UV behavior of rank-4 tensor field models for quantum gravity[END_REF][START_REF] Lahoche | Unitary symmetry constraints on tensorial group field theory renormalization group flow[END_REF][START_REF] Ben Geloun | Functional Renormalization Group analysis of rank 3 tensorial group field theory: The full quartic invariant truncation[END_REF][START_REF] Lahoche | Nonperturbative renormalization group beyond the melonic sector: The effective vertex expansion method for group fields theories[END_REF][START_REF] Lahoche | Ward identity violation for melonic T 4 -truncation[END_REF], as well as to support indications for asymptotic safety [START_REF] Carrozza | Asymptotic safety in three-dimensional SU(2) Group Field Theory: evidence in the local potential approximation[END_REF]. Studies of the modified Ward-identity that arises due to symmetry-breaking by the regulator, cf. Sec. 5 can be found in [START_REF] Lahoche | Unitary symmetry constraints on tensorial group field theory renormalization group flow[END_REF][START_REF] Lahoche | Ward identity violation for melonic T 4 -truncation[END_REF][START_REF] Lahoche | Ward-constrained melonic renormalization group flow[END_REF]; a first solution to the Ward-identity within a truncation has been put forward in [START_REF] Eichhorn | Towards phase transitions between discrete and continuum quantum spacetime from the Renormalization Group[END_REF]. One easily finds that the generating functional of connected correlation functions, W k [J] = ln Z k [J], satisfies the RG equation

∂ t W k [J] = - 1 2 ˆr,r ∂ t R k (r -r ) δ 2 W k [J] δJ i (r)δJ i (r ) + δW k [J] δJ i (r) δW k [J] δJ i (
∂ t Γ k [φ] = 1 2 ˆr,r ∂ t R k (r -r ) δ 2 W k [J] δJ i (r)δJ i (r ) = 1 2
Tr ∂ t R k (Γ (2) [φ] + R k ) -1 , (B.5)

where Tr denotes a trace wrt space and the O(N) index of the field. The last result in (B.5) is obtained by noting that the propagator G k,i j [r, r ; J] = δ 2 W k [J]/δJ i (r)δJ j (r ) is the inverse (in a matrix sense) of Γ (2) k [φ] + R k . 49 Substitution of Γ (2) k [φ] by S (2) [φ] in (B.5) gives a flow equation that can be easily integrated out and yields the one-loop result (A. [START_REF] Feldman | Construction and Borel Summability of Infrared φ 4 in Four-dimensions by a Phase Space Expansion[END_REF]).

Appendix B.2. The local potential approximation (LPA)

In the LPA the effective action ( 8) is entirely determined by the effective potential U k (ρ). To derive the flow equation ∂ t U k (ρ), it is sufficient to consider a uniform field configuration φ(r) = φ. In that case the 2-point vertex is diagonal in Fourier space and, owing to the O(N) symmetry, can be written as Γ (2) k,i j (p, φ) = 2) k,L (p, ρ) + δ i, j - (with r ≡ r(y) and r ≡ r (y)), where we use the dimensionless variable y = p 2 /k 2 and v -1 d = 2 d+1 π d/2 Γ(d/2). 50 Equation (B.9) can be integrated with the initial condition U Λ (ρ) = r 0 ρ + (u 0 /6)ρ 2 at scale k = Λ corresponding to the classical (or mean-field) effective potential of the O(N) model [START_REF] Gell-Mann | Quantum electrodynamics at small distances[END_REF]. U Λ (ρ) exhibits a minimum at ρ 0,Λ defined by (A.16). Since fluctuations tend to disorder the system, ρ 0,k can only decrease during the flow. The equilibrium state depends on the value of ρ 0 = lim k→0 ρ 0,k . When r 0 is larger than a critical value r 0c < 0, related to the critical temperature via r 0c = r0 (T c -T 0 ) (see Sec. 2), ρ 0 = 0 and the system is in the symmetric (high-temperature) phase. When r 0 < r 0c , the system is in the ordered (low-temperature) phase and the O(N) symmetry is spontaneously broken; the uniform field in the equilibrium state has a fixed length |φ eq | = 2ρ 0 but its direction is arbitrary. Note that when r 0 = r 0c , ρ 0,k remains nonzero for all k > 0 and vanishes only when k = 0. Typical flows of the effective potential are shown in Fig. B.31. 49 The relation G -1 k = Γ (2) k + R k (rather than G -1 k = Γ (2) k ) is due to the fact that the true Legendre transform of W k [J] is Γ k [φ] + ∆S k [φ] and not Γ k [φ]; see Eq. (B.3). 50 With the theta regulator R k (q) = (k 2q 2 )Θ(k 2q 2 ), i.e., r(y) = ((1y)/y)Θ(1y), the integral in the rhs of (B.9) can be carried out analytically [START_REF] Litim | Optimized renormalization group flows[END_REF]. Since we are using the variable ρ (rather than φ), the usual picture of the Mexican-hat shape potential when ρ 0,k > 0 is partially lost (in particular because U k (0) 0).

φ i φ j 2ρ Γ (
φ i φ j 2ρ Γ (
Appendix B.2.1. Scaling form of the LPA flow equation For r 0 = r 0c the system is scale invariant (the correlation length diverges) and the only length scale in the problem, k -1 , comes from the infrared regulator. Thus we expect that the effective potential U k (ρ) will exhibit scale invariance, i.e., reach a fixed point of the flow equation ∂ t U k , if we measure all quantities in units of k. In practice, this is achieved by introducing dimensionless quantities, of the dimensionless effective potential Ũk ( ρ) follows from (B.9) and (B.10) once the derivative wrt t at fixed ρ in (B.9) has been changed for a derivative at fixed ρ:

∂ t | ρ = ∂ t | ρ + ∂ t ρ| ρ∂ ρ .
To eliminate a field-independent, relevant, constant it is often more convenient to consider the RG equation This equation admits the trivial solution Ũ * ( ρ) = 0 corresponding to the (noninteracting) Gaussian fixed point with all n-point vertices Γ (n) (n > 3) vanishing while Γ (2) (p) = p 2 . For simplicity we now consider the case N = 1 and the theta regulator for which l d 1 (w) = (2/d)(1 + w) -2 . To obtain the critical exponents associated with the Gaussian fixed point, we must linearize the flow equation about Ũ * ( ρ) = 0. Taking φ as the variable, This equation is known to have polynomial solutions, 52 given by the Hermite polynomials h(y) = Ĥ2n-1 (y) = where n = 1, 2, 3... If one considers symmetric perturbations, even degree Hermite polynomials are not allowed since in that case the functions Ũ k (φ) and f k (x) are odd. Note that the λ n 's coincide with the scaling dimension [v 2n ] of the vertex v 2n ´r ϕ 2n at the Gaussian fixed point.

∂ t Ũ k = -2 Ũ k + (d -2) ρ Ũ k -2v d [(3 Ũ k + 2 ρ Ũ k )l d 1 ( Ũ k + 2 ρ Ũ k ) + (N -1) Ũ k l d 1 ( Ũ k )] (B.
∂ t Ũ k = - d 2 + 1 Ũ k + d 2 -1 φ Ũ k -4
When d > 4 all eigenvalues λ n are negative except λ 1 which determines the correlation-length critical exponent ν = 1/λ 1 = 1/2. The associated eigenvector is given by Ĥ1 (y) ∝ y, which corresponds to a φ 2 term in U k (φ). The less negative eigenvalue λ 2 determines the correction-to-scaling exponent ω = -λ 2 = d -4, i.e. the speed at which Ũ k approaches the fixed-point solution Ũ * when the system is critical. When d < 4,53 λ 2 becomes positive and we expect the phase transition to be described by a nontrivial fixed point with a single relevant direction (i.e. the Wilson-Fisher fixed point for the O(N) model [START_REF] Wilson | Critical Exponents in 3.99 Dimensions[END_REF]). The Gaussian fixed point then becomes a tricritical fixed point.

Nontrivial fixed points cannot be found analytically and one must rely on a numerical solution. Since Eq. (B.15) is a second-order differential equation, a solution is a priori parameterized by two arbitrary constants. However, if we require Ũ * ( ρ) to be regular at the origin, setting ρ = 0 in (B.15) yields Ũ * (0) + v d (N + 2) Ũ * (0)l d 1 ( Ũ * (0)) = 0. Moreover, if Ũ * ( ρ) does not vanish for ρ → ∞, it must behave as Ũ * ( ρ) ∼ ρ2/(d-2) . 54 Thus we now have a secondorder differential equation with two boundary conditions. Most solutions are found to be singular at some ρc and must be discarded, so that we end up with a discrete set of acceptable solutions. In practice, the fixed-point solution Ũ * can be determined by the shooting method, i.e., by fine tuning Ũ * (0) and Ũ * (0) until a regular solution satisfying the above-mentioned boundary conditions for ρ → 0 and ρ → ∞ is found [START_REF] Morris | On truncations of the exact renormalization group[END_REF][START_REF] Morris | Derivative expansion of the exact renormalization group[END_REF][START_REF] Morris | The renormalization group and two-dimensional multicritical effective scalar field theory[END_REF]. For d ≥ 4 only the Gaussian fixed point is found. For simplicity we consider here perturbations respecting the O(N) symmetry (non-symmetric perturbations can be considered in a similar way). Again the solutions of this second-order differential equation are labeled by two parameters. However, one can choose g(0) = 1 (arbitrary normalization) while g(0) and g (0) are not independent if g( ρ) is regular at the origin. The solution is then unique for a given λ. Regular solutions are obtained only for a countable set of λ's and behave as g( ρ) ∼ ρ(2-λ)/(d-2) for ρ → ∞; 54 they can be determined by the shooting method. For N = 1 and d = 3, the LPA with the exponential regulator function R k (q) = q 2 /(e q 2 /k 2 -1) gives ν = 0.6589 and ω = 0.6440 [START_REF] Polsi | Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group[END_REF], to be compared with the values shown in Table 1. Since the anomalous dimension η obviously vanishes in the LPA, all other critical exponents (α, β, γ and δ) can be deduced from the value of ν and the usual scaling laws [START_REF] Ma | Modern theory of critical phenomena, Advanced book classics[END_REF]. For a more detailed discussion of the shooting method, we refer to Refs. [13, 92, 97-101, 109, 146]. The shooting method suffers from an important drawback. Beyond the LPA, there are other functions to be considered in addition to the effective potential, e.g. Z k (ρ) and Y k (ρ) in the derivative expansion to second order (Appendix B.4), which may require to fine tune additional parameters. As an alternative to the shooting method, which is not restricted to the LPA, one can numerically solve the flow equation for a system near criticality. In the O(N) model, the initial effective potential Ũ Λ ( ρ) = r0 + (ũ 0 /3) ρ depends on two parameters; the critical point can be reached by fine tuning r0 (with ũ0 fixed). When r0 is near the critical value r0c , the solution of the flow equation takes the form Ũ k ( ρ) Ũ * ( ρ) + g 1 ( ρ)e ωt + g 2 ( ρ)e -t/ν (B.21) for large |t|. For r0 slightly detuned from the critical value r0c , Ũ k first moves closer to Ũ * with a speed controlled by the correction-to-scaling exponent ω, remains nearly equal to Ũ * for a long time, before eventually running away with a rate 1/ν given by the inverse of the correlation-length exponent. This allows us to determine both Ũ * and the critical exponents ν and ω. 55 A typical flow for a system near criticality is shown in Fig. B.32. Note that the vanishing of Ũ * at a nonzero ρ * 0 , corresponding to the minimum of Ũ * , is not in contradiction with the system being critical; going back to dimensionful variables one finds that U k (ρ) exhibits a minimum at ρ 0,k = k d-2 ρ * 0 which vanishes for k → 0.

A third possible procedure to determine the critical exponents is the following. The (approximate) fixed-point potential Ũ * found from the numerical solution of the flow equation can also be used as an initial guess of the solution of Eq. (B.15). Discretizing the ρ variable, i.e. ρi = i∆ ρ (i = 0, • • • , M -1), one is then left with a system of M equations for the M variables Ũ * ( ρi ), which can be solved with standard numerical algorithms. The initial guess, being close to the exact solution, ensures convergence to the physical solution, the accuracy being limited by the finite number M of ρi variables and the maximum value ρM-1 . To compute the correlation-length exponent ν, one writes Ũ k ( ρi ) = Ũ * ( ρi ) + e -λt g( ρi ), see Eq. (B. [START_REF] Feldman | Construction and Borel Summability of Infrared φ 4 in Four-dimensions by a Phase Space Expansion[END_REF], and linearize the flow equation ∂ t Ũ k about Ũ * . The possible values of λ are then given by the eigenvalues of the stability matrix

L i j = - δ∂ t Ũ k ( ρi ) δ Ũ k ( ρ j ) Ũ k = Ũ * . (B.22)
The largest eigenvalue (the only positive one for a standard bicritical point) determines 1/ν. The absolute value of the less negative eigenvalue gives the correction-to-scaling exponent ω. Finally, we note that it is possible to determine the critical exponents directly from physical quantities computed at k = 0. For instance, in the disordered phase G k=0,ii (p, φ = 0) = (p 2 + U k=0 (0)) -1 and the correlation length is given From a practical point of view, the approach to convexity of the potential makes its numerical determination difficult. Indeed, a tiny numerical error can lead to negative values of q 2 + R k (q 2 ) + U k (ρ), even in cases where we know that conditions (B.27) should remain fulfilled. This difficulty is more pronounced for N = 1, mainly because U k (ρ) becomes discontinuous at ρ = ρ 0,k in the limit k → 0: U k (ρ) → 0 for ρ < ρ 0,k due to the approach to convexity whereas U k (ρ 0,k + 0 + ) remains nonzero. 57 Nevertheless, analytical solutions of the potential in the internal region can be exploited to set up efficient algorithms for the broken-symmetry phase [START_REF] Peláez | Ordered phase of the O(N) model within the nonperturbative renormalization group[END_REF][START_REF] Caillol | The non-perturbative renormalization group in the ordered phase[END_REF]. 58 Appendix B.3. Improving the LPA: the LPA The LPA , defined by the effective action [START_REF] Fisher | Renormalization group theory: Its basis and formulation in statistical physics[END_REF], is the minimal improvement of the LPA that allows one to compute the anomalous dimension η. In addition to the effective potential U k (ρ), the effective action includes a field renormalization factor Z k which diverges as Z k ∼ k -η when the system is critical.

The flow equation of the effective potential is given by ( 9) where the longitudinal and transverse propagators G k,L (q, ρ) = [Γ (2) k,L (q, ρ) + R k (q)] -1 = [Z k q 2 + U k (ρ) + 2ρU k (ρ) + R k (q)] -1 , G k,T (q, ρ) = [Γ (2) k,T (q, ρ) + R k (q)] -1 = [Z k q 2 + U k (ρ) + R k (q)] -1 G k,i 1 ,i 2 (q, φ)Γ (4) k,i ji 2 i 1 (p, -p, q, -q, φ)

-G k,i 1 ,i 2 (q, φ)Γ (3) k,ii 2 i 3 (p, q, -pq, φ)G k,i 3 ,i 4 (p + q, φ)Γ (3) k, ji 4 i 1 (-p, p + q, -q, φ) , (B. [START_REF] Berezinskii | Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group i. classical systems[END_REF] where ∂t acts only on the k dependence of R k : ∂t G k,i 1 i 2 (q, φ) = -G k,i 1 i 3 (q, φ)∂ t R k (q)G k,i 3 i 2 (q, φ). The reason for defining Z k from the transverse 2-point vertex is that in the full derivative expansion to order O(∂ where G k (p, ρ 0,k ) = ∂ p 2 G k (p, ρ 0,k ). 59 The LPA flow equations can be written in dimensionless form by making use of the dimensionless variables (B.10) provided that we define φ(r) = √ Z k k -(d-2)/2 φ(r) with a factor √ Z k and write the regulator function in the form R k (q) = Z k q 2 r(q 2 /k 2 ): ηr -(η + 4)yr -2y 2 r , (B.33) 57 In the O(N > 1) model, U k (ρ 0,k + 0 + ) vanishes for k → 0 (this is due to the divergence of the longitudinal susceptibility discussed in Appendix B.3.2) and U k (ρ) remains a regular function around ρ 0,k . 58 We refer to Ref. [START_REF] Peláez | Ordered phase of the O(N) model within the nonperturbative renormalization group[END_REF] for a discussion of the approach to convexity in the derivative expansion beyond the LPA. 59 For a derivation of (B.31) see, e.g., Ref. [START_REF] Tetradis | Critical exponents from the effective average action[END_REF]. where P(w) = y(1 + r) + w. To find the fixed-point solution Ũ * and the critical exponents, we can use the same methods as in the LPA. When the fixed point is reached by fine tuning r 0 , the anomalous dimension η can be estimated from the value of η k on the plateau ξ ). The LPA can be simplified by expanding the effective potential about ρ 0,k as in Eq. ( 13). The flow equations then reduce to coupled ordinary differential equations for the dimensionless variables ρ0,k = Z k k 2-d ρ 0,k , δk = (Z k k 2 ) -1 δ k , λk = Z -2 k k d-4 λ k and η k . In spite of its simplicity the truncated LPA turns out to be highly nontrivial and fundamentally different from a perturbative RG based on a loop expansion (the flow equations are still nonperturbative since they are of infinite order in the coupling constant λk ). Although it is not reliable for an accurate estimate of the critical exponents, it yields a rather complete picture of the long-distance physics at and near criticality, as well as in the low-temperature phase, regardless of the value of N and for all dimensions d ≥ 2. 60 It also captures some features of the Berezinskii-Kosterliz-Thouless transition when d = 2 and N = 2 [START_REF] Gräter | Kosterlitz-Thouless Phase Transition in the Two Dimensional Linear σ Model[END_REF] although a more complete analysis requires the DE to second order [START_REF] Gersdorff | Nonperturbative renormalization flow and essential scaling for the Kosterlitz-Thouless transition[END_REF][START_REF] Jakubczyk | Reexamination of the nonperturbative renormalization-group approach to the Kosterlitz-Thouless transition[END_REF]. 2), in agreement with the one-loop perturbative RG [START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF][START_REF] Ma | Modern theory of critical phenomena, Advanced book classics[END_REF]. More surprisingly, one also recovers the critical exponents obtained from the nonlinear sigma model near two dimensions, i.e. for d = 2 + : ν = 1/ + O( 0 ) and η = /(N -2) + O( 2 ) [START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF]. Since ρ * 0 = O( -1 ) and λ * = O( 0 ), in the critical regime ξ -1 , ξ -1 J k ξ -1 G the longitudinal mode has a large (square) dimensionless mass 2 λk ρ0,k ∼ -1 which suppresses the threshold functions. We thus recover the fact that near two dimensions the critical behavior is entirely determined by the Goldstone modes, which is the basic assumption when studying the critical behavior of the O(N) model in the framework of the nonlinear sigma model. The relation to the nonlinear sigma model can be further understood [START_REF] Delamotte | Nonperturbative renormalization-group approach to frustrated magnets[END_REF] which is identical, to order 2 , to the RG equation obtained from the nonlinear sigma model to one-loop order [START_REF] Polyakov | Interaction of goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills fields[END_REF][START_REF] Nelson | Momentum-shell recursion relations, anisotropic spins, and liquid crystals in 2 + dimensions[END_REF][START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF]. There is nevertheless an important difference between the usual perturbative RG approach to the nonlinear sigma model and the nonperturbative FRG approach to the linear O(N) model near two dimensions. While in the former case the RG equation (B.37) is not valid anymore when the coupling constant gk becomes of order one, in the latter case there is no difficulty to continue the LPA flow in the strong-coupling regime and describe the disordered phase with a finite correlation length. Finally we note that the truncated LPA flow equations are also exact to leading order in the large-N limit where ν = 1/(d -2) + O(N -1 ) and η = O(N -1 ).

∂ t Ũ k = (η k -2) Ũ k + (d -2 + η k ) ρ Ũ k -2v d [(3 Ũ k + 2 ρ Ũ k )l d 1 ( Ũ k + 2 ρ Ũ k , η k ) + (N -1) Ũ k l d 1 ( Ũ k , η k )], η k = 16

Appendix B.3.2. Low-temperature phase

The physics of the O(N) model remains nontrivial in the whole ordered phase when N ≥ 2 because of the Goldstone modes associated with the spontaneously broken symmetry. Mean-field (or Gaussian theory) predicts G L (p) = 1/(p 2 + 2|r 0 |) and G T (p) = 1/p 2 when r 0 < 0 [START_REF] Ma | Modern theory of critical phenomena, Advanced book classics[END_REF]. The transverse propagator is gapless, in agreement with Goldstone's theorem while longitudinal fluctuations have a finite correlation length ξ = (2|r 0 |) -1/2 . The last result is however an artifact of mean-field theory [START_REF] Dupuis | Infrared behavior in systems with a broken continuous symmetry: Classical O(N) model versus interacting bosons[END_REF][START_REF] Patasinskij | Longitudinal susceptibility and correlations in degenerate systems[END_REF]. The coupling between transverse and longitudinal fluctuations implies that G L (r) ∼ (N -1)[G T (r)] 2 ∼ 1/|r| 2d-4 when d ≤ 4, i.e. G L (p) ∼ 1/|p| 4-d (the divergence for p → 0 is logarithmic when d = 4). The presence of a singularity in the longitudinal channel, driven by transverse fluctuations, is a general phenomenon in systems with a continuous broken symmetry [START_REF] Patasinskij | Longitudinal susceptibility and correlations in degenerate systems[END_REF].

In the LPA the divergence of the longitudinal susceptibility G L (p = 0) can be traced back to the infrared behavior of the coupling constant λk → λ * for d < 4 (here λ * should not be confused with the critical value λ * crit discussed in Appendix B. LPA results do not appear within the narrow ranges of values chosen here (see Table 1). Reprinted from Ref. [START_REF] Balog | Convergence of nonperturbative approximations to the renormalization group[END_REF].

where y = q 2 /k 2 and α is a variational parameter determined by the principle of minimum sensitivity (PMS), that is by demanding that locally critical exponents be independent of α, e.g. dν/dα| α opt = 0 (see Sec. For all studied values of N the anomalous dimension η alternates around the most precise estimates reported in the literature (whenever these estimates are more precise than the DE) when one considers higher and higher orders of the DE. This also occurs for the exponent ν within LPA and DE 2 and, for 1 ≤ N ≤ 5 (but not for any N), also at order DE 4 (see Fig. B.35). The alternating behavior is not seen between DE 4 and DE 6 when N = 1 (the only case that has been considered at order DE 6 ) and is also absent for the exponent ω. When the alternating behavior is observed, the PMS choice of α shows the fastest apparent convergence (in the sense that α opt is also the value obtained by minimizing the difference between two successive estimates of the critical exponent). As discussed in the next section, this observation can be exploited in order to improve the estimates and error bars of the exponents.

Appendix B.4.2. Estimate of the error

The small expansion parameter discussed in Sec. 2.3.3 [START_REF] Balog | Convergence of nonperturbative approximations to the renormalization group[END_REF] can be exploited in order to give a well-founded estimate of error bars in the DE [START_REF] Polsi | Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group[END_REF]. Since the radius of convergence of the DE is expected to be of the order of 4 -9 for |p i • p j |/k 2 and, at the same time, the derivative ∂ t R k (q) of the regulator function restricts the internal momentum to the range q 2 k 2 , one expects successive orders of the expansion in momentum to be suppressed by a factor 1/9-1/4 for a given family of regulators. As as consequence, for a quantity Q a typical (somewhat pessimistic 62 ) estimate of the error is

∆Q (s) = |Q (s) -Q (s-2) | 4 , (B.42)
where Q (s) is the value obtained from the PMS at order s of the DE. When one considers several families of regulators, a natural estimate is given by the middle value. This introduces another potential error, in addition to (B.42), which can be estimated by Q (s) max -Q (s) min where the maximum and minimum values are derived from the PMS values obtained among some reasonable families of regulators [START_REF] Polsi | Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group[END_REF].

In some cases, there are strong reasons to believe that successive orders of the DE give strict (upper or lower) bounds to a given critical exponent (at least up to a given order). This significantly improves the estimates of central values and error bars, as discussed in Ref. [START_REF] Polsi | Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group[END_REF].

These error estimates, whether or not one can use the improved ones based on strict bounds, have been tested numerically in Ref. [START_REF] Polsi | Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group[END_REF]. For the O(N) model up to order DE 4 , with 1 ≤ N ≤ 5, and up to order DE 6 for N = 1, the obtained values of the critical exponents, when compared with the most precise results available in the literature (when the latter are more precise than the DE), are always within the estimated error bars. Appendix B.5. Fourth-and sixth-order of the derivative expansion

The DE has been pushed to fourth order for the O(N) model with 0 ≤ N ≤ 4 [START_REF] Polsi | Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group[END_REF] and to sixth order for the Z 2 (Ising) universality class [START_REF] Balog | Convergence of nonperturbative approximations to the renormalization group[END_REF]. The "strict" DE has been used since it keeps equations to a more manageable size. The corresponding ansatz for the effective action reads to fourth order [START_REF] Polsi | Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group[END_REF], and

Γ ∂ 4 k [φ] = ˆr U k (ρ) + 1 2 Z k (ρ) ∂ µ φ a 2 + 1 4 Y k (ρ) ∂ µ ρ 2 + W 1,k (ρ) 2 ∂ µ ∂ ν φ a 2 + W 2,
Γ ∂ 6 ,Z 2 k [φ] = ˆr U k (φ) + 1 2 Z k (φ)(∂ µ φ) 2 + 1 2 W a k (φ)(∂ µ ∂ ν φ) 2 + 1 2 φW b k (φ)(∂ 2 φ)(∂ µ φ) 2 + 1 2 W c k (φ) (∂ µ φ) 2 2 + 1 2 Xa k (φ)(∂ µ ∂ ν ∂ ρ φ) 2 + 1 2 φ Xb k (φ)(∂ µ ∂ ν φ)(∂ ν ∂ ρ φ)(∂ µ ∂ ρ φ) + 1 2 φ Xc k (φ) ∂ 2 φ 3 + 1 2 Xd k (φ) ∂ 2 φ 2 (∂ µ φ) 2 + 1 2 Xe k (φ)(∂ ν φ) 2 (∂ µ φ)(∂ 2 ∂ µ φ) + 1 2 X f k (φ)(∂ ρ φ) 2 (∂ µ ∂ ν φ) 2 + 1 2 φ Xg k (φ) ∂ 2 φ (∂ µ φ) 2 2 + 1 96 Xh k (φ) (∂ µ φ) 2 3 (B.44)
to sixth order [START_REF] Balog | Convergence of nonperturbative approximations to the renormalization group[END_REF].

1 . 2 5 2 2 . 2 2 . 3

 1252223 's renormalization group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The functional renormalization group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Scope of the review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The FRG in a nutshell 6 2.1 The scale-dependent effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 The regulator function R k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.2 General properties of the effective action Γ k . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The exact flow equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 The derivative expansion (DE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.1 Local-potential approximation: LPA and LPA . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.2 Second order of the DE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.3 Validity of the DE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.4 Further results obtained from the DE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Computing momentum-dependent correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Lattice models and realistic microscopic actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.6 Quantum models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 Statistical mechanics 15 3.1 Why and when is the FRG useful? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Equilibrium statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2.1 Frustrated magnets as an example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Figure 1 :

 1 Figure 1: Typical shape of the regulator function R k (p).

Figure 3 :

 3 Figure 3: RG flow in the parameter space of the effective action. The solid lines show the exact RG flows obtained with two different regulator functions R k . The dashed lines show the RG flows obtained by solving the RG equation with the same approximation and two different regulator functions.

Figure 4 :

 4 Figure 4: Configuration minimizing the energy in the STA model (reprinted from Ref. [66]).

Figure 5 :

 5 Figure 5: Critical RG flow for N > N c (d) (left) and N < N c (d) (right) (reprinted from Ref. [66]).

Figure 6 :

 6 Figure 6: Critical value N c (d) below which the transition becomes first order in the O(N)×O(2) model. Solid line: results obtained from the three-loop calculation improved by the constraint N c (d = 2) = 2 [279]. Crosses: FRG results [66].

Figure 7 :

 7 Figure 7: Schematic phase diagram of the RFIM in the disorder strength ∆-temperature T plane above the lower critical dimension d lc = 2. At low disorder and low temperature, the system is ferromagnetic, and is paramagnetic otherwise. The arrows describe how the renormalized parameters evolve under the RG flow at long distance, and I and RF denote the critical fixed points of the pure and random-field Ising models, respectively.

  3.3.1 and obtain the exact flow equation for Γ k [{φ a }].

Figure 8 :

 8 Figure 8: Anomalous dimensions η and η of the RFIM as a function of space dimension. The large symbols for integer dimensions are lattice results from Refs. [386-389].

Figure 9 :

 9 Figure 9: Comparison of the scaling functions f (y) (left panel) and f (k) (right panel) obtained from FRG (red curves with dots)[START_REF] Canet | Nonperturbative Renormalization Group for the Kardar-Parisi-Zhang equation: General framework and first applications[END_REF][START_REF] Canet | Erratum: Nonperturbative Renormalization Group for the Kardar-Parisi-Zhang equation: General framework and first applications[END_REF] with the exact ones (black curves with squares)[START_REF] Prähofer | Exact scaling functions for one-dimensional stationary KPZ growth[END_REF]. The inset on the right panel shows the stretched exponential behavior of the tail with the superimposed oscillations, developing on the same scale k 3/2 . Note the vertical scale: this behavior develops with amplitudes below typically 10 -6 .

Figure 10 :

 10 Figure 10: Phase diagram of the BARW model[START_REF] Ringwald | Average action for the N-component ϕ 4 theory[END_REF] in dimensions 1 to 6 from FRG calculations (solid lines) and numerical simulations (symbols) from[START_REF] Canet | Quantitative phase diagrams of branching and annihilating random walks[END_REF]. For each dimension, the active phase lies on the left of the transition line, and the absorbing phase on its right.

Figure 11 :

 11 Figure 11: Flow diagram of the BARW model A σ -→ 2A and 3A λ -→ ∅ in d = 1 from FRG at LPA[START_REF] Canet | Optimization of field-dependent nonperturbative renormalization group flows[END_REF]. The PC fixed point drives a transition between the active state and the absorbing one controlled by the PA fixed point.

Figure 12 :

 12 Figure 12: (left) Phase diagram of the 3D Bose-Hubbard model showing the first Mott lobe (with a density n = 1) and the surrounding superfluid phase. The solid (red) line shows the FRG result and the (green) dashed line the mean-field one. The QMC and DMFT data are obtained from Refs. [529] and [530], respectively . (right) Phase diagram of the 2D Bose-Hubbard model. The QMC data are obtained from Ref. [531]. Reprinted from Ref. [232].

Figure 13 :

 13 Figure 13: Scaling function ϑ(x, 0) determining the singular part of the internal energy density, sing (δ, T ) = ϑ(x, 0)(k B T ) 3 /( c) 2 (c is the velocity of the critical fluctuations), in the two-dimensional quantum O(N) model near its quantum critical point (full lines); x ∝ δ/T 1/ν where δ measures the distance to the quantum critical point and ν is the correlation-length critical exponent of the T = 0 phase transition. The symbols show the results obtained from Monte Carlo simulations of three-dimensional classical spin models in a finite geometry where ϑ(x, 0) determines the universal Casimir force. The horizontal dashed line shows the (exact) limit -2(N-1)ζ(3)/2π for x → -∞. (left panel) Ising (N=1) universality class. Monte Carlo simulations are from Ref.[START_REF] Vasilyev | Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations[END_REF] (blue diamonds), Ref.[START_REF] Hucht | Aspect-ratio dependence of thermodynamic Casimir forces[END_REF] (green squares) and Ref.[START_REF] Cardozo | Finite size scaling and the critical Casimir force : Ising magnets and binary fluids[END_REF] (red circles). (middle panel) XY universality class (N=2). The Monte Carlo data are from Ref.[START_REF] Vasilyev | Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations[END_REF]. (right panel) Heisenberg universality class (N=3). The Monte Carlo data[START_REF] Dantchev | Critical Casimir force and its fluctuations in lattice spin models: Exact and Monte Carlo results[END_REF] have been rescaled so as to satisfy the correct asymptotic value for x→-∞, the bare data are shown in the inset. Reprinted from Ref.[START_REF] Ranc ¸on | Critical Casimir forces from the equation of state of quantum critical systems[END_REF].

Figure 15 :

 15 Figure 15: Momentum space region around the Fermi surface excluded by a sharp momentum cutoff for fermions with a tight-binding dispersion on a two-dimensional square lattice (lattice constant a = 1).

Figure 17 :

 17 Figure17: Left: Ground state phase diagram of the 2D Hubbard model near half-filling (marked by dashed line) at weak coupling (U/t ≤ 2) for fixed t = -0.01t and variable µ. The symbols indicate whether the dominant instability is magnetic (squares) or superconducting (circles); the solid line separates the magnetic from the pairing regime (from Ref.[START_REF] Halboth | d-Wave Superconductivity and Pomeranchuk Instability in the Two-Dimensional Hubbard Model[END_REF]). Right: Pseudocritical temperature obtained from the temperature flow for the 2D Hubbard model at van Hove filling as a function of t /t for U = 3t (from Ref.[START_REF] Honerkamp | Magnetic and Superconducting Instabilities of the Hubbard Model at the Van Hove Filling[END_REF]).

Figure 18 :

 18 Figure 18: Katanin truncation for the coupled flow of self-energy and two-particle vertex.

Figure 19 :

 19 Figure 19: Flow of the effective action of QCD. The first three diagrams arise from the gluon, ghost, and the quark degrees of freedom respectively. The last diagram is that of the mesonic contribution. The double line with the up-down arrows indicates the nature of the mesons as quark-antiquark composites. The crossed circles indicate the regulator insertion in the flow equation.

  Effective running couplings,[START_REF] Cyrol | Landau gauge Yang-Mills correlation functions[END_REF], as obtained from different Yang-Mills vertices: three-gluon vertex: α A 3 , four-gluon vertex: α At , ghost-gluon vertex: α Acc . At large momenta the couplings agree due to universality.

FRG Nf = 2 + 1 FRG Nf = 2 Lattice Nf = 2 + 1 Lattice Nf = 2 (c) 2 -

 12122 and 2 + 1-flavor gluon dressing functions 1/Z A (p 2 ) = p 2 G A (p 2 ) with the gluon propagator G A (p 2 ). FRG: 2-flavor[START_REF] Cyrol | Nonperturbative quark, gluon, and meson correlators of unquenched QCD[END_REF], 2 + 1-flavor[START_REF] Fu | QCD phase structure at finite temperature and density[END_REF]. Lattice: 2-flavor[START_REF] Sternbeck | Determination of LambdaMS from the gluon and ghost propagators in Landau gauge[END_REF], 2 + 1-flavor[START_REF] Zafeiropoulos | Strong Running Coupling from the Gauge Sector of Domain Wall Lattice QCD with Physical Quark Masses[END_REF][START_REF] Boucaud | Discretization effects on renormalized gauge-field Green's functions, scale setting, and the gluon mass[END_REF].

Figure 20 :

 20 Figure20: YM gluon propagator G A (p 2 ) (right) and couplings (middle) and 2-and 2 + 1-flavor gluon dressing function p 2 G A (p 2 ) (right) in comparison to lattice data. The results show the (apparent) convergence of the vertex expansion scheme, for more details see in particular[START_REF] Cyrol | Landau gauge Yang-Mills correlation functions[END_REF][START_REF] Cyrol | Nonperturbative quark, gluon, and meson correlators of unquenched QCD[END_REF].

<

  l a t e x i t s h a 1 _ b a s e 6 4 = " M R e t w z i 8 o g 2 t r I P I 4 e X a s y b

  r G S o 8 9 F M k I + / r G 7 Z c r b t W d g 6 w S L y c V y N H o l 7 9 6 g 5 i m E Z O G C t S 6 6 7 m J 8 T N U h l P B p q V e q l m C d I x D 1 r V U Y s S 0 n 8 0 v n p I z q w x I G C t b 0 p C 5 + n s i w 0 j r S R T Y z g j N S C 9 7 M / E / r 5 u a 8 M r P

1 +

 1 m2 π ) h2 /(1 + m2 σ ) (b) Cutoff-dependence of gluon-exchange coupling for the light u, d quarks (g 2 lAl ) and the strange quark (g 2 sAs ). Meson-exchange couplings ( h2 /(1 + m2 ) with m2 = m 2 /k 2 . Gluons, quarks, and mesons decouple sequentially from the matter dynamics. Figure taken from[START_REF] Fu | QCD phase structure at finite temperature and density[END_REF].

Figure 22 :

 22 Figure 22: Four-quark coupling with dynamical hadronization (left), strength of exchange couplings in physical N f = 2 + 1 flavor QCD (right).

Figure 23 :

 23 Figure 23: Chiral phase structure for 2+1 flavor QCD in the (m π , m K )-and (µ, m K )-planes (Columbia plot) obtained in[START_REF] Resch | Mass sensitivity of the three-flavor chiral phase transition[END_REF]. At µ = 0 there is a small first-order region around the chiral limit. The anomalous U A (1)-breaking is chiefly important for this results, without the anomaly the phase structure is significantly changed, see[START_REF] Resch | Mass sensitivity of the three-flavor chiral phase transition[END_REF].

  Vector meson spectral functions from[START_REF] Jung | In-Medium Spectral Functions of Vector-and Axial-Vector Mesons from the Functional Renormalization Group[END_REF][START_REF] Jung | Fluctuating vector mesons in analytically continued functional RG flow equations[END_REF] with the analytically continued FRG (aFRG). The inlay shows the temperature dependence of the pole masses. The different scattering thresholds are clearly visible.

Figure 24 :

 24 Figure 24: Spectral functions in the 2-flavor QM-model: Vector meson spectral functions (left), quark spectral functions (right).

  et al. DSE: Fischer et al. DSE: Gao et al. freezeout: STAR freezeout: Alba et al. freezeout: Andronic et al. freezeout: Becattini et al. freezeout: Vovchenko et al. freezeout: Sagun et al.

  -out data:[START_REF] Adamczyk | Bulk Properties of the Medium Produced in Relativistic Heavy-Ion Collisions from the Beam Energy Scan Program[END_REF][START_REF] Alba | Freeze-out conditions from net-proton and net-charge fluctuations at RHIC[END_REF][START_REF] Andronic | Decoding the phase structure of QCD via particle production at high energy[END_REF][START_REF] Becattini | Hadronization conditions in relativistic nuclear collisions and the QCD pseudo-critical line[END_REF][START_REF] Vovchenko | Hadron multiplicities and chemical freeze-out conditions in proton-proton and nucleusnucleus collisions[END_REF][START_REF] Sagun | Hadron Resonance Gas Model with Induced Surface Tension[END_REF].

g 1 <

 1 l a t e x i t s h a 1 _ b a s e 6 4 = " 8 e 9 / h Z h

  7 8 5 r j e s i j T I c w T G c g g c X 0 I B b a I I P D A b w D K / w 5 k j n x X l 3 P u a t J a e Y O Y Q / c D 5 / A F 9 O j W I = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 e 9 / h Z h

  7 8 5 r j e s i j T I c w T G c g g c X 0 I B b a I I P D A b w D K / w 5 k j n x X l 3 P u a t J a e Y O Y Q / c D 5 / A F 9 O j W I = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 e 9 / h Z h

2 <

 2 7 8 5 r j e s i j T I c w T G c g g c X 0 I B b a I I P D A b w D K / w 5 k j n x X l 3 P u a t J a e Y O Y Q / c D 5 / A F 9 O j W I = < / l a t e x i t > g l a t e x i t s h a 1 _ b a s e 6 4 = " I b S / G 7 l p S 7 t L h 3 E Q 2 y 5 Q u f 3 3 P

3 <

 3 2 n b E P w F l 9 e J n 6 9 d l V z 7 8 6 r j e s i j R I c w w m c g Q c X 0 I B b a I I P D G J 4 h l d 4 c 6 T z 4 r w 7 H / P W F a e Y O Y I / c D 5 / A G D R j W M = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " I b S / G 7 l p S 7 t L h 3 E Q 2 y 5 Q u f 3 3 P U w = " > A A A B 6 X i c b V B N S 8 N A E J 3 4 W e t X 1 a O X x S J 4 K k k R 1 F v R i 8 e K x h b a U D b b T b p 0 s w m 7 E 6 G U / g Q v H l S 8 + o + 8 + W / c t j l o 6 4 O B x 3 s z z M w L M y k M u u 6 3 s 7 K 6 t r 6 x W d o q b + / s 7 u 1 X D g 4 f T Z p r x n 2 W y l S 3 Q 2 q 4 F I r 7 K F D y d q Y 5 T U L J W + H w Z u q 3 n r g 2 I l U P O M p 4 k N B Y i U g w i l a 6 j 3 v 1 X q X q 1 t w Z y D L x C l K F A s 1 e 5 a v b T 1 m e c I V M U m M 6 n p t h M K Y a B Z N 8 U u 7 m h m e U D W n M O 5 Y q m n A T j G e n T s i p V f o k S r U t h W S m / p 4 Y 0 8 S Y U R L a z o T i w C x 6 U / E / r 5 N j d B m M h c p y 5 I r N F 0 W 5 J J i S 6 d + k L z R n K E e W U K a F v Z W w A d W U o U 2 n b E P w F l 9 e J n 6 9 d l V z 7 8 6 r j e s i j R I c w w m c g Q c X 0 I B b a I I P D G J 4 h l d 4 c 6 T z 4 r w 7 H / P W F a e Y O Y I / c D 5 / A G D R j W M = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " I b S / G 7 l p S 7 t L h 3 E Q 2 y 5 Q u f 3 3 PU w = " > A A A B 6 X i c b V B N S 8 N A E J 3 4 W e t X 1 a O X x S J 4 K k k R 1 F v R i 8 e K x h b a U D b b T b p 0 s w m 7 E 6 G U / g Q v H l S 8 + o + 8 + W / c t j l o 6 4 O B x 3 s z z M w L M y k M u u 6 3 s 7 K 6 t r 6 x W d o q b + / s 7 u 1 X D g 4 f T Z p r x n 2 W y l S 3 Q 2 q 4 F I r 7 K F D y d q Y 5 T U L J W + H w Z u q 3 n r g 2 I l U P O M p 4 k N B Y i U g w i l a 6 j 3 v 1 X q X q 1 t w Z y D L x C l K F A s 1 e 5 a v b T 1 m e c I V M U m M 6 n p t h M K Y a B Z N 8 U u 7 m h m e U D W n M O 5 Y q m n A T j G e n T s i p V f o k S r U t h W S m / p 4 Y 0 8 S Y U R L a z o T i w C x 6 U / E / r 5 N j d B m M h c p y 5 I r N F 0 W 5 J J i S 6 d + k L z R n K E e W U K a F v Z W w A d W U o U2 n b E P w F l 9 e J n 6 9 d l V z 7 8 6 r j e s i j R I c w w m c g Q c X 0 I B b a I I P D G J 4 h l d 4 c 6 T z 4 r w 7 H / P W F a e Y O Y I / c D 5 / A G D R j W M = < / l a t e x i t > g l a t e x i t s h a 1 _ b a s e 6 4 = " H 7 L I h A V s i u T Y 9 k v V p S e d q I 9 H c z M = " > A A A B 6 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l U U G 9 F L x 4 r G l t o Q 9 l s J + n S z S b s b o R S + h O 8 e F D x 6 j / y 5 r 9 x 2 + a g 1 Q c D j / d m m J k X Z o J r 4 7 p f T m l p e W V 1 r b x e 2 d j c 2 t 6 p 7 u 4 9 6 D R X D H 2 W i l S 1 Q 6 p R c I m + 4 U Z g O 1 N I k 1 B g K x x e T / 3 W I y r N U 3 l v R h k G C Y 0 l j z i j x k p 3 c e + 0 V 6 2 5 d X c G 8 p d 4 B a l B g W a v + t n t p y x P U B o m q N Y d z 8 1 M M K b K c C Z w U u n m G j P K h j T G j q W S J q i D 8 e z U C T m y S p 9 E q b I l D Z m p P y f G N N F 6 l I S 2 M 6 F m o B e 9 q f i f 1 8 l N d B G M u c x y g 5 L N F 0 W 5 I C Y l 0 7 9 J n y t k R o w s o U x x e y t h A 6 o o M z a d i g 3 B W 3 z 5 L / F P 6 p d 1 9 / a s 1 r g q 0 i j D A R z C M X h w D g 2 4 g S b 4 w C C G J 3 i B V 0 c 4 z 8 6 b 8 z 5 v L T n F z D 7 8 g v P x D W J U j W Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H 7 L I h A V s i u T Y 9 k v V p S e d q I 9 H c z M = " > A A A B 6 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l U U G 9 F L x 4 r G l t o Q 9 l s J + n S z S b s b o R S + h O 8 e F D x 6 j / y 5 r 9 x 2 + a g 1 Q c D j / d m m J k X Z o J r 4 7 p f T m l p e W V 1 r b x e 2 d j c 2 t 6 p 7 u 4 9 6 D R X D H 2 W i l S 1 Q 6 p R c I m + 4 U Z g O 1 N I k 1 B g K x x e T / 3 W I y r N U 3 l v R h k G C Y 0 l j z i j x k p 3 c e + 0 V 6 2 5 d X c G 8 p d 4 B a l B g W a v + t n t p y x P U B o m q N Y d z 8 1 M M K b K c C Z w U u n m G j P K h j T G j q W S J q i D 8 e z U C T m y S p 9 E q b I l D Z m p P y f G N N F 6 l I S 2 M 6 F m o B e 9 q f i f 1 8 l N d B G M u c x y g 5 L N F 0 W 5 I C Y l 0 7 9 J n y t k R o w s o U x x e y t h A 6 o o M z a d i g 3 B W 3 z 5 L / F P 6 p d 1 9 / a s 1 r g q 0 i j D A R z C M X h w D g 2 4 g S b 4 w C C G J 3 i B V 0 c 4 z 8 6 b 8 z 5 v L T n F z D 7 8 g v P x D W J U j W Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H 7 L I h A V s i u T Y 9 k v V p S e d q I 9 H c z M = " > A A A B 6 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l U U G 9 F L x 4 r G l t o Q 9 l s J + n S z S b s b o R S + h O 8 e F D x 6 j / y 5 r 9 x 2 + a g 1 Q c D j / d m m J k X Z o J r 4 7 p f T m l p e W V 1 r b x e 2 d j c 2 t 6 p 7 u 4 9 6 D R X D H 2 W i l S 1 Q 6 p R c I m + 4 U Z g O 1 N I k 1 B g K x x e T / 3 W I y r N U 3 l v R h k G C Y 0 l j z i j x k p 3 c e + 0 V 6 2 5 d X c G 8 p d 4 B a l B g W a v + t n t p y x P U B o m q N Y d z 8 1 M M K b K c C Z w U u n m G j P K h j T G j q W S J q i D 8 e z U C T m y S p 9 E q b I l D Z m p P y f G N N F 6 l I S 2 M 6 F m o B e 9 q f i f 1 8 l N d B G M u c x y g 5 L N F 0 W 5 I C Y l 0 7 9 J n y t k R o w s o U x x e y t h A 6 o o M z a d i g 3 B W 3 z 5 L / F P 6 p d 1 9 / a s 1 r g q 0 i j D A R z C M X h w D g 2 4 g S b 4 w C C G J 3 i B V 0 c 4 z 8 6 b 8 z 5 v L T n F z D 7 8 g v P x D W J U j W Q = < / l a t e x i t >

Figure 27 :

 27 Figure 27: The RG flow in a) the Einstein-Hilbert truncation from [1097], b) the Einstein-Hilbert truncation plus a curvature-cubed term from [1111] and c) in the reverse direction (i.e., flow trajectories shown towards the UV) in the Einstein-Hilbert truncation in the fluctuation-field approach (where µ = -2λ) from [1143]. A UV fixed point that is IR repulsive in two directions and IR attractive in one direction (panel b)) is marked by a purple dot. Perturbative nonrenormalizability of GR is linked to the fact that the free fixed point (marked in blue in plot a)) is IR attractive in the Newton coupling.

Figure 29 :

 29 Figure29: The RG flow in the plane spanned by the Newton coupling and an Abelian gauge coupling exhibits an interacting fixed point with one IR-attractive direction (left panel), cf.[START_REF] Harst | QED coupled to QEG[END_REF][START_REF] Eichhorn | Upper bound on the Abelian gauge coupling from asymptotic safety[END_REF][START_REF] Eichhorn | Mass difference for charged quarks from asymptotically safe quantum gravity[END_REF]. The corresponding flows exhibit a transition between the regime with gravity at k ≥ M Pl and the classical regime (without gravity), where G(k) ∼ k 2 and quantum-gravity effects are negligible. One single trajectory for the gauge coupling emanates from the interacting fixed point (right panel), resulting in a prediction of the IR value.

Figure 30 :

 30 Figure30: Leading critical exponent in the hermitian matrix model for 2d gravity as a function of the order n in an expansion Γ N = n i=1 g i TrT i , as in[START_REF] Eichhorn | Status of background-independent coarse-graining in tensor models for quantum gravity[END_REF]. The critical exponent is evaluated at fixed anomalous dimension, and can be fit as θ = 0.907 -1.540 exp(-0.292 n), extrapolating to θ = 0.91, compared to the exact result θ = 0.8.

Appendix B. 1 .

 1 The exact flow equation In the presence of the infrared regulator term ∆S k the scale-dependent partition function is defined byZ k [J] = ˆD[ϕ] e -S [ϕ]-∆S k [ϕ]+ ´r J•ϕ . (B.1)

  2) k,T (p, ρ),(B.6) where the longitudinal and transverse parts, Γ(2) k,L (p, ρ) and Γ(2) k,T (p, ρ), are functions of the O(N) invariant ρ. Within the LPA, one hasΓ (2) k,L (p, ρ) = p 2 + U k (ρ) + 2ρU k (ρ) + R k (p), Γ (2) k,T (p, ρ) = p 2 + U k (ρ) + R k (p). (B.7)An expression similar to (B.6) holds for the propagator G k,i j (p, φ), with G k,α (p, ρ) = 1/Γ(2) k,α (p, ρ) (α = L, T). From the definition (A.6) of the effective potential and Wetterich's equation[START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF], we then find the exact flow equation of the effective potential,∂ t U k (ρ) = 1 2 ˆp ∂ t R k (p)G k,ii (p, φ) = 1 2 ˆp ∂ t R k (p) G k,L (p, ρ) + (N -1)G k,T (p, ρ) . (B.8)Writing the regulator function as R k (p) = p 2 r(p 2 /k 2 ) and employing the LPA (B.7) we finally obtain∂ t U k = -2v d k d ˆ∞ 0 dy y d/2+1 r 1 y(r + 1) + k -2 (U k + 2ρU k ) + N -1 y(r + 1) + k -2 U k (B.9)

5 Figure B. 31 :

 531 Figure B.31: Effective potential U k (ρ) -U k (0) obtained from the LPA flow equation (B.9) and the initial condition U Λ (ρ) = r 0 ρ + (u 0 /6)ρ 2 for various values of the RG time t = ln(k/Λ) (N = 3 and d = 3). Left panel: disordered phase, ρ 0,Λ = 0.05 and ρ 0 = 0. Right panel: ordered phase, ρ 0,Λ = 0.1 and ρ 0 0.05. Note that the potential becomes flat for ρ ∈ [0, ρ 0 ] as k → 0, showing the approach to the convex effective potential. Since we are using the variable ρ (rather than φ), the usual picture of the Mexican-hat shape potential when ρ 0,k > 0 is partially lost (in particular because U k (0) 0).

r 2 (∇ r φ) 2 +

 22 = kr, φ(r) = k -(d-2)/2 φ(r), Ũk ρ(r) = k -d U k ρ(r) , (B.10) in terms of which the LPA effective action reads Γ k [φ] = ˆr 1 Ũk ( ρ) . (B.11)The flow equation∂ t Ũk = -d Ũk + (d -2) ρ Ũ k + 2v d [l d 0 ( Ũ k + 2 ρ Ũ k ) + (N -1)l d 0 ( Ũ k )] (B.12)

  [START_REF] Bagnuls | Exact renormalization equations: an introductory review[END_REF] satisfied by the derivative of the effective potential. In Eqs. (B.12,B.13) we have introduced the "threshold" functionsl d n (w) = -(n + δ n,0 ) ˆ∞ 0 dy y d/2+1 r [y(1 + r) + w] n+1 . (B.14) Appendix B.2.2. Fixed-point solutions and critical exponents The fixed-point solutions Ũ * ( ρ) of the flow equation are obtained from 0 = ∂ t Ũ * = -2 Ũ * + (d -2) ρ Ũ * -2v d [(3 Ũ * + 2 ρ Ũ * )l d 1 ( Ũ * + 2 ρ Ũ * ) + (N -1) Ũ * l d 1 ( Ũ * )]. (B.15)

  ) = f ( √ αφ) with α = d/4v d , f (x) = h(βx)e -λt and β = √ d -2/2 (assuming d > 2), we finally obtain 51 h (y) -2yh (y) + 2 d -2 (2 + d -2λ)h(y) = 0. (B.17

  )

2 n- 1 / 2 H

 12 2n-1 (y) of degree 2n -1, only for a set of discrete values of λ satisfying 2n -1 = d + 2 -2λ n d -2 , i.e. λ n = dn(d -2), (B.18)

  For 3 ≤ d < 4 a nontrivial fixed point (the Wilson-Fisher fixed point) appears. A new nontrivial fixed point emanates from the Gaussian fixed point each time that one of the eigenvalues λ n [Eq. (B.18)] becomes negative, which occurs at the dimensional thresholds d n = 2n/(n -1) (n ≥ 2). Once a fixed point is identified, one can determine the critical exponents by linearizing the flow about Ũ * . Setting Ũ k ( ρ) = Ũ * ( ρ) + e -λt g( ρ), (B.19) we find (λ -2)g + (d -2) ρg -2v d {(3g + 2 ρg )l d 1 ( Ũ * + 2 ρ Ũ * ) -(g + 2 ρg )(3 Ũ * + 2 ρ Ũ * )l d 2 ( Ũ * + 2 ρ Ũ * ) + (N -1)[g l d 1 ( Ũ * )g Ũ * l d 2 ( Ũ * )]} = 0. (B.20)

Figure B. 32 :

 32 Figure B.32: (Left) Derivative Ũ k ( ρ) of the effective potential in the LPA for a system near criticality and an RG time t = ln(k/Λ) between 0 and -20 (d = 3, N = 1). (Right) Ũ k (0) vs |t|. For |t| larger than the Ginzburg scale |t G | = ln(Λξ G ) 10 (see Appendix B.3.1 for the definition of the Ginzburg length ξ G ), Ũ k ( ρ) is nearly equal to the fixed-point value Ũ * ( ρ).

(B. 28 )

 28 now include the field renormalization factor Z k . SinceZ k = lim p→0 ∂ ∂p 2 Γ (2) k,T (p, ρ 0,k ), (B.29)the flow equation ∂ t Z k is deduced from that of the two-point vertex in a uniform field φ(r) = φ, ∂ t Γ(2) k,i j (p, φ)

  k ( ρ0,k ) 2 m d 22 (2 ρ0,k Ũ k ( ρ0,k ), η k ), (B.32)where η k = -∂ t ln Z k is the "running" anomalous dimension. The threshold functions are defined byl d n (w, η) = -n + δ n,02ˆ∞ 0 dy y d/2 ηr + 2yr P(w) n+1 , m d 22 (w, η) = -ˆ∞ 0 dy y d/2 1 + r + yr P(w) 2 P(0) 2 y(ηr + 2yr )(1 + r + yr )

  Figure B.33: ρ0,k , λk and η k vs ln(Λ/k) = -t near criticality in the LPA (d = 3 and N = 3). The vertical dotted lines show the Ginzburg scale |t G | = ln(Λξ G ) and the correlation length scale |t ξ | = ln(Λξ) (T > T c i.e. r 0 > r 0c ) or the Josephson scale |t J | = ln(Λξ J ) (T < T c i.e. r 0 < r 0c ).

-1 k ξ - 1 G

 1 (see Appendix B.3.1 and Fig. B.33

Appendix B. 3 . 1 . 1 G

 311 Critical behavior: the limits d → 4, d → 2 and N → ∞ Figure B.33 shows typical solutions of the truncated LPA flow equations for a nearly critical system. One clearly observes a critical regime where ρ0,k , λk and η k are very close to their fixed-point values ρ * 0 , λ * and η. The critical regime starts when k ∼ ξ -1 G is of the order of the inverse Ginzburg length ξ G ∼ u 1/(d-4) 0 (the Ginzburg scale k G = ξ -is defined by λk G ∼ 1 and signals the breakdown of perturbation theory). It ends when k ∼ ξ -1 (disordered phase) or k ∼ ξ -1 J (ordered phase) where ξ and ξ J are the correlation and Josephson lengths, respectively. Near four dimensions, i.e. d = 4 -, the truncated flow equations give ν = 1/2 + ( /4)(N + 2)/(N + 4) + O( 2 ) and ω = + O(

= φ2 / 2 .Figure B. 34 :

 234 Figure B.34: λk and 2 λk ρ0,k vs ln(Λ/k) in the ordered phase for d = 3 and N = 3 in the LPA . Left panel: deep in the ordered phase T c -T G T c -T (T G is the Ginzburg temperature). The dotted vertical lines show the hydrodynamic (k c ) and Ginzburg (k G ) scales. Right panel: critical regime T c -T T c -T G . The dotted vertical lines show the Ginzburg (k G ) and Josephson (k J ) scales. The first plateau, λk ∼ λ * crit , corresponds to the critical regime and the second one, λk ∼ λ * (visible for ln(Λ/k) 30), to the low-temperature fixed point.

  3.1 and shown in Fig. B.33); see Fig. B.34. Thus the mass of the longitudinal mode behaves as 2ρ k,0 λ k ∼ k 4-d since Z k → Z and ρ 0,k → ρ 0 when k → 0. If we approximate G k=0,L (p, ρ 0 ) by G k∼|p|,L (p, ρ 0 ), arguing that p acts as an effective infrared cutoff when |p| k, and use the LPA result for G k∼|p|,L (p, ρ 0 ), one finds G k=0,L (p, ρ 0 ) ∼ 1/|p| 4-d . For d = 4, λk vanishes logarithmically and G k=0,L (p, ρ 0 ) ∼ ln(1/|p|).The criterion 2 λk ρ0,k ∼ 1 defines a hydrodynamic momentum scale k c below which the contribution of the longitudinal mode to the flow equations is suppressed. Deep in the ordered phase, when T c -T G T c -T (with T G the Ginzburg temperature), we observe a crossover for k ∼ k G k c between a Gaussian regime, where mean-field

Figure B. 35 :

 35 Figure B.35: Exponent values ν(α) and η(α) in the three-dimensional Ising universality class at different orders of the DE with the regulator E k (q) [Eq. (B.41)]. Vertical lines indicate the value α opt obtained from the PMS and the horizontal dashed lines the conformal bootstrap results [1369].LPA results do not appear within the narrow ranges of values chosen here (see Table1). Reprinted from Ref.[START_REF] Balog | Convergence of nonperturbative approximations to the renormalization group[END_REF].

  2.3.2). The determination of the critical exponents ν and η from the PMS is illustrated in Fig. B.35.

Table 2 :

 2 Typical values of the pseudo-critical exponents associated with the weakly first-order transition in STA (obtained from the FRG for N = 3 and d = 3)[START_REF] Delamotte | Nonperturbative renormalization-group approach to frustrated magnets[END_REF], compared with critical exponents of resumed perturbative series at six loops[START_REF] Pelissetto | The Critical behavior of frustrated spin models with noncollinear order[END_REF].

	Method	α	β	γ	ν	η
	FRG	0.38	0.29	1.04	0.54	0.072
	6-loop	0.35(9) 0.30(2) 1.06(5) 0.55(3) 0.08

Table 3 :

 3 Critical exponents of Model A in d = 3 from the different FRG approximations (LPA, LPA

  r ) , (B.2) where t = ln(k/Λ) is the (negative) RG time introduced in Sec. 2.2 and a sum over repeated indices is implied. The derivative ∂ t in (B.2) is taken at fixed external source J. To obtain the flow equation of the scale-dependent effective action Γ k[φ] = -W k [J] + ˆr J • φ -∆S k [φ] (B.3) at fixed φ, we must consider J ≡ J k [φ] as a k-dependent functional of φ defined by φ i,k [r; J] = δW k [J]/δJ i (r). Using ∂ t W k [J] φ = ∂ t W k [J] J + ˆr δW k [J] δJ i (r) ∂ t J i (r)| φ = ∂ t W k [J] J + ˆr φ i (r)∂ t J i (r)| φ ,

	(B.4)
	we obtain Wetterich's equation [59, 61-63]

  2 ) the longitudinal vertex Γ(2) k,L includes a term Y k p 2 (see Appendix B.4). Although this term is neglected in the LPA , the equation ∂ t Γ(2) k,L yields contributions to both ∂ t Z k and ∂ t Y k whereas ∂ t Γ (2) k,T contributes only to ∂ t Z k . Equations (B.29) and (B.30) imply

	∂ t Z k = 16	v d d	ρ 0,k U k (ρ 0,k ) 2	∂t	ˆ∞ 0	d|p| |p| d+1 G k,L (p, ρ 0,k )G k,T (p, ρ 0,k ),	(B.31)

  k (ρ) 2 φ a ∂ µ ∂ ν φ a 2 + W 3,k (ρ)∂ µ ρ∂ ν φ a ∂ µ ∂ ν φ a + W 4,k (ρ) 2 φ b ∂ µ φ a ∂ ν φ a ∂ µ ∂ ν φ b + W 5,k (ρ) 2 ϕ a ∂ µ ρ∂ ν ρ∂ µ ∂ ν φ a +

								W 6,k (ρ) 4	∂ µ φ a 2 2
	+	W 7,k (ρ) 4	∂ µ φ a ∂ ν φ a 2 +	W 8,k (ρ) 2	∂ µ φ a ∂ ν φ a ∂ µ ρ∂ ν ρ +	W 9,k (ρ) 2	∂ µ φ a 2 ∂ ν ρ 2 +	W 10,k (ρ) 4	∂ µ ρ 2 2	(B.43)

The RG was pioneered in high-energy physics[START_REF] Stueckelberg De Breidenbach | Normalization of constants in the quanta theory[END_REF][START_REF] Gell-Mann | Quantum electrodynamics at small distances[END_REF][START_REF] Bogolyubov | QUANTUM FIELDS[END_REF][START_REF] Symanzik | Small distance behavior in field theory and power counting[END_REF][START_REF] Callan | Broken scale invariance in scalar field theory[END_REF].

High-order perturbative expansions are usually obtained within the field theoretical RG approach; see, e.g.,[START_REF] Guida | Critical exponents of the N-vector model[END_REF][START_REF] Pelissetto | Critical phenomena and renormalization-group theory[END_REF].

The RG has also been used as a mathematical tool for a rigorous non-perturbative construction of field theories[START_REF] Gawedzki | Massless lattice ϕ 4 4 theory: Rigorous control of a renormalizable asymptotically free model[END_REF][START_REF] Balaban | Convergent Renormalization Expansions for Lattice Gauge Theories[END_REF][START_REF] Brydges | Mayer expansions and the hamilton-jacobi equation[END_REF][START_REF] Feldman | Construction and Borel Summability of Infrared φ 4 in Four-dimensions by a Phase Space Expansion[END_REF][START_REF] Rivasseau | From perturbative to constructive renormalization[END_REF], and as a nonperturbative computational tool, such as Wilson's numerical RG for the Kondo problem[START_REF] Wilson | The Renormalization Group: Critical Phenomena and the Kondo Problem[END_REF] or the density matrix renormalization group (DMRG) for one-dimensional lattice models[START_REF] Schollwöck | The density-matrix renormalization group[END_REF].

[START_REF] Symanzik | Small distance behavior in field theory and power counting[END_REF] The BKT transition is often studied in the framework of the Coulomb gas, Villain or sine-Gordon models for which the perturbative RG is useful; see, e.g.,[START_REF] Chaikin | Principles of Condensed Matter Physics[END_REF]. A direct study in the O(2) model, i.e., without introducing explicitly the vortices, is much more

challenging.[START_REF] Callan | Broken scale invariance in scalar field theory[END_REF] In statistical field theory, the UV cutoff Λ of the theory has usually a well-defined physical meaning (e.g., the inverse of the lattice spacing of the original model) and the interactions at that scale are known from experiments or ab initio calculations based on microscopic (realistic) models. In quantum field theory, Λ stands for the highest momentum scale where the theory is valid.

Γ k [φ] is a scale-dependent generating functional of 1PI vertices (see Sec. 2)

The field φ(r, t) depends on time in out-of-equilibrium classical and quantum (be them at equilibrium or not) systems. In the Euclidean (quantum) formalism t = -iτ is an imaginary time. φ is an anticommuting Grassmann variable if the degrees of freedom are fermionic.

For previous general reviews on the nonperturbative FRG, see[START_REF] Bagnuls | Exact renormalization equations: an introductory review[END_REF][START_REF] Berges | Non-perturbative renormalization flow in quantum field theory and statistical physics[END_REF][START_REF] Aoki | Introduction to the non-perturbative renormalization group and its recent applications[END_REF][START_REF] Polonyi | Lectures on the functional renormalization group method[END_REF][START_REF] Delamotte | Nonperturbative renormalization-group approach to frustrated magnets[END_REF][START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Rosten | Fundamentals of the Exact Renormalization Group[END_REF][START_REF] Kopietz | Introduction to the Functional Renormalization Group[END_REF][START_REF] Braun | Fermion Interactions and Universal Behavior in Strongly Interacting Theories[END_REF][START_REF] Delamotte | An Introduction to the Nonperturbative Renormalization Group[END_REF][START_REF] Gies | Introduction to the Functional RG and Applications to Gauge Theories[END_REF].

We refer to Appendix A for a brief introduction to the effective action formalism.

For earlier versions of Eq.[START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF], see[START_REF] Symanzik | Small distance behavior in field theory and power counting[END_REF][START_REF] Chang | Differential renormalization-group generators for static and dynamic critical phenomena[END_REF][START_REF] Parola | Liquid state theory and critical phenomena[END_REF][START_REF] Nicoll | An exact one-particle-irreducible renormalization-group generator for critical phenomena[END_REF][START_REF] Parola | Liquid-State Theory for Critical Phenomena[END_REF].

For a study in the framework of general Wilsonian RG flows, see[START_REF] Latorre | Exact scheme independence[END_REF][START_REF] Latorre | Scheme Independence as an Inherent Redundancy in Quantum Field Theory[END_REF]. For a discussion at one-and two-loop order in the Wilson-Polchinski and effective-action formulations, see[START_REF] Arnone | Exact scheme independence at one loop[END_REF][START_REF] Arnone | Exact scheme independence at two loops[END_REF].

The R k dependence is similar to the scheme dependence in perturbative RG (physical results depend on the RG prescription, e.g. MS scheme, massive zero-momentum scheme, etc.), see e.g.[START_REF] Pawlowski | Aspects of the functional renormalisation group[END_REF][START_REF] Latorre | Exact scheme independence[END_REF][START_REF] Arnone | Exact scheme independence at one loop[END_REF][START_REF] Arnone | Exact scheme independence at two loops[END_REF][START_REF] Litim | Scheme independence at first order phase transitions and the renormalization group[END_REF][START_REF] Pernici | Hard -soft renormalization and the exact renormalization group[END_REF][START_REF] Ellwanger | The Running gauge coupling in the exact renormalization group approach[END_REF][START_REF] Pernici | Wilsonian flow and mass independent renormalization[END_REF][START_REF] Rosten | Universality From Very General Nonperturbative Flow Equations in QCD[END_REF][START_REF] Codello | Scheme dependence and universality in the functional renormalization group[END_REF].

See Appendix B for a further discussion of the DE.

The prefactor Z k in the definition of R k is required by the Ward identities associated with scale invariance[START_REF] Delamotte | Scale invariance implies conformal invariance for the three-dimensional Ising model[END_REF]. It ensures that no intrinsic scale is introduced in the renormalized inverse propagator Γ[START_REF] Gell-Mann | Quantum electrodynamics at small distances[END_REF] k (p)/Z k = p 2 (1 + r(p 2 /k 2 )) when U k = 0.

Stricto sensu this is true only if the longitudinal propagator G k,L (q, ρ) is O(N 0 ) for any value of ρ (the N → ∞ limit of U k (ρ) is then a regular function), which is the case for the Wilson-Fisher fixed point but not for all fixed points[START_REF] Yabunaka | Surprises in o(n) models: Nonperturbative fixed points, large n limits, and multicriticality[END_REF][START_REF] Yabunaka | Why might the standard large n analysis fail in the O(n) model: The role of cusps in fixed point potentials[END_REF][START_REF] Katsis | Multicritical points of the O(N) scalar theory in 2 < d < 4 for large N[END_REF].

One could expect that it is possible to devise a function R k which yields a mass m k = k and is (at least) exponentially suppressed for momenta above p max k. Such a regulator would however introduce a singularity near the origin (implying large high-order derivatives) and spoil the analytic structure of the correlation functions and thus the convergence of the DE.

In this section we mainly focus on O(N)-like models. Many other applications of the DE are described in Secs.3-6.

Application of the FRG to lattice models is common in fermion systems, see Sec. 4.

In order to determine nonuniversal properties with the FRG, it is necessary to consider a more realistic microscopic Hamiltonian as discussed in Sec. 3.2.3.

In fact even sophisticated momentum-dependent approximations, such as the BMW approach, seem to have difficulties to quantitatively reproduce the quantum-mechanical tunneling[START_REF] Rulquin | Nonperturbative fluctuations and metastability in a simple model: from observables to microscopic theory and back[END_REF].

The precise knowledge of the microscopic action is not always necessary. In dilute Bose gases, for instance, the inter-particle interaction potential enters the low-temperature equation of state only through the s-wave scattering length so that the determination of the equation of state can be based on a simple interaction potential with a trivial UV behavior (e.g. a delta potential with a UV regularization).

Depending on the context, h is referred to as a random field, a random potential, etc.

In the first calculations of RFIM critical properties performed with the nonperturbative FRG[START_REF] Tissier | Two-loop functional renormalization group of the random field and random anisotropy O(N) models[END_REF][START_REF] Tissier | Unified picture of ferromagnetism, quasi-long-range order, and criticality in random-field models[END_REF][START_REF] Tarjus | Nonperturbative functional renormalization group for random field models and related disordered systems. i. effective average action formalism[END_REF][START_REF] Tissier | Nonperturbative functional renormalization group for random field models and related disordered systems. II. Results for the random field O(N) model[END_REF], the regulator term was breaking explicitly the supersymmetry. It incorrectly led to critical exponents that never fulfilled dimensional reduction.

The transformation with v(t) = v × t corresponds to the usual (non-gauged) Galilean symmetry.

In the fermionic case, the terminology functional RG originally stems from the momentum dependence of the vertices and not their field dependence.

In the literature on interacting fermion systems the greek letter Λ is the most commonly used symbol for the flow parameter. However, for the sake of a consistent notation within this review, we denote the flow parameter by k.

Note that high-temperature QCD still exhibits spatial confinement, so not all its properties are captured by perturbation theory.

The gauge[START_REF] Morris | Equivalence of local potential approximations[END_REF] and many other sufficiently smooth gauges are subject to the Gribov problem: the solution of (75) is not unique, there are Gribov copies, and their treatment is relevant for the infrared regime of correlation functions. The discussion of this highly interesting issue goes far beyond the scope of the present work, for reviews and recent work see e.g.[START_REF] Maas | Describing gauge bosons at zero and finite temperature[END_REF][START_REF] Vandersickel | The Gribov problem and QCD dynamics[END_REF][START_REF] Capri | Renormalizability of the refined Gribov-Zwanziger action in linear covariant gauges[END_REF].

If one gives up locality, even the Landau-gauge propagator is an observable as we can define a gauge-invariant gauge field A gi with A gi [A] = A U L[A] , where U L [A] is the gauge transformation that implements the Landau gauge on A. Then A gi A gi is gauge-invariant and equals the Landau gauge propagator. However, this 'observable' is not local.

This analysis also implies that strictly speaking the full breaking of center symmetry underlying confinement is signalled by tr P n ( x) = 0 for n = 1, ..., N 2 c -1, see e.g.[START_REF] Ford | Monopoles, Polyakov loops and gauge fixing on the torus[END_REF][START_REF] Van Baal | QCD in a finite volume[END_REF][START_REF] Reinosa | Deconfinement transition in SU(N) theories from perturbation theory[END_REF][START_REF] Herbst | Confinement order parameters and fluctuations[END_REF] and references therein.

In contradistinction to the effective potential of a the real scalar field, V eff [A 0 ] is singular at the maximally center-symmetry breaking points with |L[A 0 ]| = 1. These singluarities are already present at one-loop. Consequently the effective potential is convex within the Weyl chambers. This complicates its numerical solution for S U(N > 2) and requires advanced numerical techniques such as put forward in[START_REF] Grossi | Resolving phase transitions with Discontinuous Galerkin methods[END_REF].

Dynamical quarks break the center symmetry of the gauge group as they live in the fundamental representation. Hence for sufficiently small quark masses the phase transition in QCD is a crossover.

Alternatively, additional geometric degrees of freedom, such as, e.g., spacetime torsion can be taken into account. There are several choices for the fields to be included in the path integral, many of which lead to the identical classical dynamics, but which are expected to lead to different quantum theories of gravity.

An enhancement of the symmetry to maximal supersymmetry, i.e., N = 8 supergravity, is also being explored as a way to reduce the number of ultraviolet divergences, see [1081] and references therein.

The presence of nonlocalities of the effective action Γ k→0 does not imply that one needs to include inverse powers of derivatives among the operators whose couplings span the theory space. The increasing canonical relevance of operators with an increasing negative power of derivatives makes it unlikely that such a theory space can contain a fixed point with a finite number of relevant couplings, see, e.g.,[START_REF] Machado | On the renormalization group flow of f(R)-gravity[END_REF]. Indeed, nonlocalities in Γ k of order k -1 are expected to be parameterized by the couplings of quasi-local operators, see[START_REF] Codello | Polyakov Effective Action from Functional Renormalization Group Equation[END_REF] for a specific example.

[START_REF] Nicoll | Approximate Renormalization Group Based on the Wegner-Houghton Differential Generator[END_REF] To bridge the gap to quantum field theories on a flat background discussed in the previous sections, one can expand each invariant in perturbations of the metric about a flat background, i.e., in the graviton. Then, each term contributes to arbitrarily high n-point graviton vertices, and each successive power of the curvature generates two additional derivatives. Thus, all interactions of gravitons, except those from the cosmologicalconstant term Λ ´dd x √ g are derivative

interactions.[START_REF] Nicoll | An exact one-particle-irreducible renormalization-group generator for critical phenomena[END_REF] The relation between Γ k→∞ and the bare action S is addressed in[START_REF] Manrique | Bare Action and Regularized Functional Integral of Asymptotically Safe Quantum Gravity[END_REF][START_REF] Morris | Solutions to the reconstruction problem in asymptotic safety[END_REF]. Reconstructing the bare action is an intermediate step to bridge the gap to formulations of quantum gravity based on

the Hamiltonian.[START_REF] Parola | Liquid-State Theory for Critical Phenomena[END_REF] More exotic possibilities such as, e.g., limit cycles, see[1086], could also in principle be viable, or a fixed point at infinite value of the coupling, i.e., a flow that only leads to an infinite value of the coupling in the limit k → ∞, see[START_REF] Eichhorn | Renormalization Flow of Axion Electrodynamics[END_REF].

We use the following convention for critical exponents: θ I = -eig ∂β i ∂g j g * . The opposite sign convention can also sometimes be found in the literature.

It is thereby distinct from the asymptotically free fixed point in curvature squared gravity[1144][START_REF] De Berredo-Peixoto | Conformal quantum gravity with the Gauss-Bonnet term[END_REF][START_REF] De Berredo-Peixoto | Higher derivative quantum gravity with Gauss-Bonnet term[END_REF] and the free fixed point in Horava gravity[START_REF] Odorico | Asymptotic Freedom in Horava-Lifshitz Gravity[END_REF][START_REF] Barvinsky | Hořava Gravity is Asymptotically Free in 2 + 1 Dimensions[END_REF].

Whether or not the electroweak vacuum is absolutely stable or metastable sensitively depends[START_REF] Bezrukov | Why should we care about the top quark Yukawa coupling?[END_REF] on the value of the top quark mass[START_REF] Aad | Measurements of top-quark pair differential and double-differential cross-sections in the +jets channel with pp collisions at √ s = 13 TeV using the ATLAS detector[END_REF][START_REF] Sirunyan | Measurement of t t normalised multi-differential cross sections in pp collisions at √ s = 13 TeV, and simultaneous determination of the strong coupling strength, top quark pole mass, and parton distribution functions[END_REF].

These studies are based on the dynamics of four-fermion couplings. The latter actually provide a paradigmatic example of asymptotic safety in 3 dimensions[START_REF] Gies | UV fixed-point structure of the three-dimensional Thirring model[END_REF][START_REF] Braun | Asymptotic safety: a simple example[END_REF][START_REF] Gehring | Fixed-point structure of low-dimensional relativistic fermion field theories: Universality classes and emergent symmetry[END_REF][START_REF] Knorr | Ising and Gross-Neveu model in next-to-leading order[END_REF][START_REF] Classen | Competition of density waves and quantum multicritical behavior in Dirac materials from functional renormalization[END_REF].

This refers to a nonlocality of the interactions, which share the combinatorial structure with tensor models.

There are also physical situations corresponding to a nonzero source, e.g. a uniform external magnetic field J(r) = J in a spin system.

When extending the 1PI formalism to out-of-equilibrium systems (Sec. 3.4), the effective action may not be real but is nevertheless stationary in the absence of external sources.

To make the discussion self-contained, some basic equations appearing in the main text are reproduced in the Appendix.

The same linearized equation is obtained from the Wilson-Polchinski equation[START_REF] Polchinski | Renormalization and effective Lagrangians[END_REF] in the LPA. This follows from an exact map between the Wilson-Polchinski and FRG equations in the LPA when the theta regulator is used[START_REF] Morris | Equivalence of local potential approximations[END_REF].

Non-polynomial solutions imply a continuum of eigenvalues and can be discarded on physical grounds[START_REF] Morris | Derivative expansion of the exact renormalization group[END_REF][START_REF] Morris | The renormalization group and two-dimensional multicritical effective scalar field theory[END_REF].

To determine whether the field associated with the eigenvalue λ 2 = 4d is relevant or irrelevant in four dimensions, one must go beyond the linear approximation (B.16). We do not discuss the case d = 4 here.

The large-ρ behavior of Ũ * ( ρ) and g( ρ) is obtained by noting that the threshold functions l d n gives a subleading contribution when ρ → ∞.

To determine ν and ω it is sufficient to consider, e.g., the flow of Ũ k (0) and use (B.21).

The breakdown of the LPA for a two-dimensional critical system can be seen from the expected large-field behavior of the effective potential at the fixed point: Ũ * ( ρ) ∼ ρd/(d-2) . The DE to second order gives results in two dimensions for both N = 1 and N = 2[START_REF] Jakubczyk | Reexamination of the nonperturbative renormalization-group approach to the Kosterlitz-Thouless transition[END_REF].

Note however that a polynomial approximation of the effective potential, although stable to quadratic order in ρ, does not always converge to higher orders below three dimensions.

Equation (B.42) can be too optimistic if Q (s) and Q (s-2) accidentally cross when varying a certain parameter, e.g. N or d. In Ref.[START_REF] Polsi | Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group[END_REF] this issue is taken into account by estimating the error bar using a value of the parameter N away from the accidental crossing.

Acknowledgments

We have benefited from collaborations and discussions with many people. In particular we are grateful to Sabine Andergassen, Ivan Balog, Jean-Paul Blaizot, Federico Benitez, Jens Braun, Anton Cyrol, Bertrand Delamotte, Gonzalo De Polsi, Sebastian Diehl, Andreas Eberlein, Wei-jie Fu, Thomas Gasenzer, Holger Gies, Aaron Held, Carsten Honerkamp, Yuji Igarashi, Katsumi Itoh, Pawel Jakubczyk, Thomas Kloss, Tim Koslowski, Daniel Litim, Volker Meden, Mario Mitter, Dominique Mouhanna, Carlo Pagani, Marcela Pelàez, Roberto Percacci, Antonio Pereira, Alessia Platania, Adam Ranc ¸on, Manuel Reichert, Fabian Rennecke, Martin Reuter, Félix Rose, Manfred Salmhofer, Frank Saueressig, Bernd-Jochen Schaefer, Michael Scherer, Kurt Schönhammer, Lorenz von Smekal, Davide Squizzato, Nils Strodthoff, Gilles Tarjus, Malo Tarpin, Arno Tripolt, Demetrio Vilardi, Christof Wetterich, Nicolas Wink, Masatoshi Yamada and Hiroyuki Yamase. We also thank Bertrand Delamotte and D. Benedetti, the organizers of the conference ERG2018, who suggested the writing of this review. L. C. acknowledges support from Institut Universitaire de France and from ANR through the project NeqFluids (grant ANR-18-CE92-0019). A. E. is supported by the DFG under grant no. Ei-1037/1, by a research grant (29405) from VILLUM FONDEN, and also partially supported by a visiting fellowship at the Perimeter Institute for Theoretical Physics. L. C., M. T. and N. W. acknowledge support from the program ECOS Sud U17E01. N. W. is supported by Grant I+D number 412 of the CSIC (Ude-laR) Commission and Programa de Desarrollo de las Ciencias Básicas (PEDECIBA). J.M.P. acknowledges support by the DFG under Germany's Excellence Strategy EXC 2181/1 -390900948 (the Heidelberg STRUCTURES Excellence Cluster), the DFG Collaborative Research Centre SFB 1225 (ISOQUANT), and the BMBF grant 05P18VHFCA.

Introduction Appendix A. The effective action formalism

In this Appendix we briefly review the effective action formalism [START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF][START_REF] Bellac | Quantum and Statistical Field Theory[END_REF], commonly used in field theory and at the basis of the FRG approach. In statistical physics, the effective action is nothing but the Gibbs free energy and therefore contains all information about the thermodynamics of the system. It is also the generating functional of 1PI vertices, which are closely related to the correlation functions. For simplicity, we consider the O(N) model defined by the action (2).

Appendix A.1. Effective action

In the presence of an external source J = (J 1 , • • • , J N ), the partition function reads

is the generating functional of the connected n-point correlation functions (aka Green functions)

For the lowest orders, one has

where φ i (r) ≡ φ i [r; J] = ϕ i (r) is the expectation value of the field ϕ i in the presence of the external source. A nonzero value of φ(r) in the limit J(r) → 0 + implies that the O(N) symmetry of the action is spontaneously broken. This occurs if d > 1 (N = 1) or d > 2 (N ≥ 2) when r 0 is smaller than a critical value r 0c = r0 (T c -T 0 ). The effective action Γ[φ] is defined as the Legendre transform of W[J],

where J ≡ J[r; φ] must be considered as a functional of φ obtained by inverting the relation φ = φ[J]. The equation of state, relating the order parameter φ to the external source, can now be written as

The effective potential

(V is the volume of the system) is obtained by evaluating the effective action in a uniform field configuration φ(r) = φ and, owing to the O(N) symmetry, is a function of the O(N) invariant ρ = φ 2 /2. The field configurations φ eq in the equilibrium state (corresponding in general to a vanishing external source 46 ) are thus obtained from the minimum of U(ρ); they all have the same magnitude |φ eq | = 2ρ 0 (with ρ 0 the position of the minimum of U) and are related to one another by an O(N) transformation. 47 The free energy of the system at zero external source is given by

1PI vertices

The effective action is the generating functional of the n-point 1PI vertices

The correlation functions G (n) can be related to the Γ (n) 's. Taking the functional derivative of (A.5) wrt the external source, one obtains

(a summation over repeated indices is implied) or, in matrix form,

where we denote the 2-point correlation function (propagator) G (2) merely by G. The 2-point vertex Γ (2) [φ] =

By taking an additional functional derivative in (A.8), one obtains a relation between the 3-point correlation function and the 3-point vertex,

To alleviate the notations, we do not write the functional dependence on J and φ. Similarly we can relate the 4-point correlation function G (4) to the 4-and 3-point vertices,

More generally the knowledge of the 1PI vertices is sufficient to reconstruct all connected correlation functions. In diagrammatic representation, the latter can be obtained as the sum of tree diagrams whose vertices are the Γ (n) 's, as illustrated by Eqs. (A.10) and (A.11). This provides us with a simple interpretation of the Γ (n) 's as the effective interaction vertices; they are 1PI to the extent that they are represented by diagrams that cannot be split into two disconnected pieces by cutting only one line.

Appendix A.3. Loop expansion

Let us write the partition function and the effective action as

where φ(r) = lδ ln Z[J]/δJ(r). The real parameter l (which will eventually be set to unity), as in the path integral of a quantum-mechanical system, can be used to organize the perturbation expansion as a "loop expansion".

In the limit l → 0, the saddle-point approximation becomes exact,

where the "classical' field ϕ cl is defined by δS

where Γ cl is referred to as the classical (or mean-field) effective action. From (A.14) we deduce the 1PI vertices in a uniform field φ(r) = φ (for the O(N) model),

where V is the volume of the system. The classical effective action reproduces the mean-field (Landau) theory. From the equation of state (A.5) with J = 0, we find

assuming a uniform field φ eq . In the broken-symmetry phase (r 0 < 0) the 2-point vertex

cl,i j (p, φ eq ) = φ eq,i φ eq, j 2ρ 0 Γ (2) cl,L (p, φ eq ) + δ i j -φ eq,i φ eq, j 2ρ 0 Γ (2) cl,T (p, φ eq ) (A.17) is defined by its longitudinal and transverse parts,

cl,T (p, φ eq ) = p 2 , (A.18) while Γ (2) cl,i j (p, φ = 0) = δ i j (p 2 + r 0 ) in the symmetric phase. It is possible to compute the correction to Γ cl in a systematic expansion in l. To leading order,

Taking functional derivatives of this equation, we can obtain the one-loop corrections to the 1PI vertices Γ (n) cl (these corrections include all one-loop diagrams that are 1PI). We shall see in Appendix B.1 that the exact FRG flow equation ( 7) is closely related to (A. [START_REF] Feldman | Construction and Borel Summability of Infrared φ 4 in Four-dimensions by a Phase Space Expansion[END_REF]).

Appendix B. The FRG at work: the case of the derivative expansion 48 The aim of this Appendix is to show how the FRG works in practice, considering the derivative expansion as an example and emphasizing technical aspects that were omitted or only briefly mentioned in Sec. 2.3. We consider the O(N) model defined by the action [START_REF] Gell-Mann | Quantum electrodynamics at small distances[END_REF]. by ξ = 1/ U k=0 (0). This allows us to obtain ν using

Similarly, in the ordered phase one can compute the magnitude of the order parameter,

to obtain the exponent β.

Appendix B.2.3. Upper and lower critical dimensions -Consistency with the Mermin-Wagner theorem

The LPA recovers the well-known result that the upper critical dimension is d + c = 4 for the O(N) model: For d ≥ d + c , the fixed point controlling the transition is the (trivial) Gaussian fixed point and the critical exponents take their mean-field values ν = 1/2, η = 0, etc. For d ≤ 2 and N > 1 no fixed point is obtained, the RG flow always drives the system to the disordered phase. This identifies d - c = 2 as the lower critical dimension (for N > 1) [START_REF] Defenu | Truncation effects in the functional renormalization group study of spontaneous symmetry breaking[END_REF][START_REF] Codello | O(N)-Universality Classes and the Mermin-Wagner Theorem[END_REF]. This can be understood from the flow equation (B.9). Let us assume that the effective potential exhibits a minimum at ρ 0,k > 0. Differentiating U k (ρ 0,k ) = 0 wrt t we find

Considering the theta regulator for simplicity,

, which is incompatible with U k (ρ 0,k ) = 0 and therefore ρ 0,k > 0. It follows that ρ 0,k must vanish at a nonzero momentum scale k (the numerical solution of the LPA flow equation confirms this analysis) in agreement with the Mermin-Wagner theorem forbidding the spontaneous breaking of a continuous symmetry in twoand one-dimensional classical systems [START_REF] Mermin | Absence of Ferromagnetism or Antiferromagnetism in One-or Two-Dimensional Isotropic Heisenberg Models[END_REF]. In two dimensions the LPA fails when N = 1, since it does not allow for spontaneous symmetry breaking, and does not describe the Berezinskii-Kosterliz-Thouless transition when N = 2. 56 Appendix B.2.4. Spontaneous symmetry breaking and approach to convexity The stability of the LPA flow equation ( 9) or (B.9) requires

If conditions (B.27) are not fulfilled, a pole appears in the propagator G k,L or G k,T thus making the threshold functions

In the broken-symmetry phase, U k (ρ) is negative for 0 ≤ ρ ≤ ρ 0,k and there is no guarantee that Eqs. (B.27) are satisfied. In the LPA, and for some regulators, the singularity of the threshold function is approached but never reached. This is the case for the theta regulator or the exponential regulator (with a prefactor α > 2) [START_REF] Berges | Non-perturbative renormalization flow in quantum field theory and statistical physics[END_REF][START_REF] Tetradis | Scale dependence of the average potential around the maximum in φ 4 theory[END_REF][START_REF] Tetradis | Analytical solutions of exact renormalization group equations[END_REF][START_REF] Peláez | Ordered phase of the O(N) model within the nonperturbative renormalization group[END_REF]. A detailed analysis shows that for these regulators the potential behaves as U k (ρ) ∼ -k 2 in the internal region 0 ≤ ρ ρ 0,k but both stability conditions (B.27) remain fulfilled. This implies that the free energy U k=0 (ρ) is convex, with a flat part U k=0 (ρ) = const for 0 ≤ ρ ≤ ρ 0 = lim k→0 ρ 0,k , as shown in Fig. B.31. It should be emphasized that, in contrast to most perturbative approaches, the convexity of the thermodynamic potential is not obtained here from the Maxwell construction but is a genuine property of the effective potential in the LPA. theory is essentially correct, and a Goldstone regime characterized by the divergence of the longitudinal propagator. Sufficiently close to the critical point (T c -T T c -T G ), where k c < k G , there is first a crossover for k ∼ k G from the Gaussian to the critical regime, followed by a second crossover to the Goldstone regime at a characteristic scale k J ∼ ξ In the second order of the derivative expansion, the scale-dependent effective action is defined by

which gives the two-point vertices

in a uniform field. The potential U k (ρ) still satisfies Eq. ( 9), and the flow equations of Z k (ρ) and Y k (ρ) are obtained from

Inserting the vertices derived from (B.38) into the RG equations ∂ t U k (ρ), ∂ t Z k (ρ) and ∂ t Y k (ρ) yields the standard (or "complete") flow equations at order DE 2 . 61 However, the DE being a low-momentum expansion, one can also truncate the rhs of the flow equations to order p 2 , which is sometimes referred to as the "strict" DE 2 expansion. At order LPA or LPA both approximation schemes coincide. At the level of the DE 2 they differ by terms at least of order |p| 4 and, as expected, their relative difference at a given order is bounded by 1/4. The estimates of the standard and strict DE 2 for the critical exponents agree within error bars [START_REF] Polsi | Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group[END_REF] Appendix B.4.1. Choice of the regulator Let us now discuss the shape of the regulator function [START_REF] Balog | Convergence of nonperturbative approximations to the renormalization group[END_REF]. The DE, like any approximation scheme, introduces a dependence of the end results on the choice of R k . There exists some constraints and a priori guidelines to choose R k so that its influence stays minimal.

where R, typically between 4k 2 and 9k 2 , is the radius of convergence of the expansion (see Sec. 2.3.3). This implies that whenever the Γ (n) k 's are replaced in a flow equation by their DE, the momentum region beyond R must be efficiently cut off. This is achieved by the ∂ t R k (q) term in [START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF] provided that R k (q) nearly vanishes for |q| k. On the other hand, modes with momenta |q| k are almost frozen if R k (q) is of order k 2 in that momentum range. These characteristic features give the overall shape of the regulator function. Note also that if a nonanalytic regulator is chosen, one must make sure that the nonanalyticities introduced in the complex plane of q 2 are at least at a distance R from the origin. Finally, at order s of the DE (Appendix B.5), the flow equations involve ∂ t R k (q) and ∂ n q 2 R k (q) with n varying from 1 to s/2. Since the DE is performed around q = 0, it is important that these derivatives decrease monotonically; if not, a "bump" at a finite value q 2 = q 2 0 > 0 would magnify a region around q 0 which is less accurately described by the DE.

Possible choices of the regulator functions are [START_REF] Polsi | Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group[END_REF][START_REF] Balog | Convergence of nonperturbative approximations to the renormalization group[END_REF] W k (q) = αZ k k 2 y e y -1 ,

Θ n k (q) = αZ k k 2 (1y) n θ(1y) n ∈ N, E k (q) = αZ k k 2 e -y , (B.41) 61 The regulator function R k (p) = Z k p 2 r(p 2 /k 2 ) has the same form as in the LPA where the field renormalization factor Z k is defined by [START_REF] Pelissetto | Critical phenomena and renormalization-group theory[END_REF]. The expression of the "complete" flow equations can be found in Ref. [START_REF] Gersdorff | Nonperturbative renormalization flow and essential scaling for the Kosterlitz-Thouless transition[END_REF].