Hongguang Liu
email: liu.hongguang@cpt.univ-mrs.fr

Karim Noui
email: karim.noui@lmpt.univ-tours.fr

Gravity as an SU(1,1) gauge theory in four dimensions

We start with the Hamiltonian formulation of the first order action of pure gravity with a full sl(2, C) internal gauge symmetry. We make a partial gauge-fixing which reduces sl(2, C) to its sub-algebra su(1, 1). This case corresponds to a splitting of the space-time M = Σ × R where Σ inherits an arbitrary Lorentzian metric of signature (-, +, +). Then, we find a parametrization of the phase space in terms of an su(1, 1) commutative connection and its associated conjugate electric field. Following the techniques of Loop Quantum Gravity, we start the quantization of the theory and we consider the kinematical Hilbert space on a given fixed graph Γ whose edges are colored with unitary representations of su(1, 1). We compute the spectrum of area operators acting of the kinematical Hilbert space: we show that space-like areas have discrete spectra, in agreement with usual su(2) Loop Quantum Gravity, whereas time-like areas have continuous spectra. We conclude on the possibility to make use of this formulation of gravity to construct a holographic description of black holes in the framework of Loop Quantum Gravity.

I. INTRODUCTION

Loop quantum gravity was founded on the observation by Ashtekar [START_REF] Ashtekar | New variables for classical and quantum gravity[END_REF] that working only with the self-dual part (or equivalently the anti-self-dual part) of the Hilbert-Palatini action leads to a simplified parametrization of the phase space of pure gravity. Indeed, the canonical variables are very similar to those of Yang-Mills gauge theory, there is no second class constraints and the first class constraints associated to the local symmetries are polynomial functionals of the the canonical variables. The drawback of the original Ashtekar's approach is that the phase space becomes complex and then one requires the imposition of reality conditions in order to recover the phase space of real general relativity. Of course, if one imposes the reality conditions at the classical level, prior to quantization, one looses all the beauty of the Ashtekar formulation, and recovers the standard Palatini formulation of general relativity, which we do not know how to quantize. Unfortunatelly, so far no one knows how to go the other way around, and implement the reality conditions after quantization of the Ashtekar theory. This difficulty motivated the work of Barbero [START_REF] Barbero | Real Ashtekar variables for Lorentzian signature space-times[END_REF] and, later on, Immirzi [START_REF] Immirzi | Real and complex connections for canonical gravity[END_REF], who introduced a family of canonical transformations, parametrized by the so-called Barbero-Immirzi parameter γ, and leading to a canonical theory in terms of a real su(2) connection kown as the Ashtekar-Barbero connection. The action that leads to this canonical formulation was finally found by Holst [START_REF] Holst | Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action[END_REF].

In fact, the Holst action is a first order formulation of gravity with a full sl(2, C) internal symmetry and an explicit dependency on the parameter γ which appears as a coupling constant for a topological term. One uses a partial gauge fixing in this action in order to derive a canonical theory in terms of the Ashtekar-Barbero. This choice of gauge is referred to as the time gauge, and, by doing so, the Lorentz gauge algebra in the internal space is reduced to its rotational su(2) subalgebra. Finally, Loop Quantum Gravity is a canonical quantization of this gauge fixed first order formulation of gravity which lead to a beautiful construction of the space of quantum geometry states at the kinematical level. At this stage, one can naturally ask the question whether the construction of Loop Quantum Gravity deeply relies on the time gauge or not. A related question would be whether the physical predictions of Loop Quantum Gravity are changed or not when one makes another partial gauge fixing or no gauge fixing at all in the Holst action prior to quantization. Indeed, the discreteness of the quantum geometry at the Planck scale predicted in Loop Quantum Gravity can be interpreted as a direct consequence of the compactness (via Harmonic analysis) of the residual symmetry group SU (2) in the time gauge. These important problems have been studied quite a lot the last twenty years but it is fair to say that no definitive conclusion have closed the debates so far.

Most of the approaches to address this issue are based on attempts to quantize the Holst action without any partial gauge fixing, and then keeping the full Lorentz internal invariance of the theory. Now if one performs the canonical analysis of the sl(2, C) Holst action, second class constraints appear simply because the connection has more components than the tetrad field. The appearance of second class constraints makes the classical analysis and then the quantization of the theory much more involved. In the analysis of constrained systems, there are two ways of dealing with second class constraints: one can either solve them explicitly, or implement them in the symplectic structure by working with the Dirac bracket. These two methods are totally equivalent. Using the Dirac bracket, Alexandrov and collaborators [START_REF] Alexandrov | Path integral for the Hilbert-Palatini and Ashtekar gravity[END_REF][START_REF] Alexandrov | SO(4,C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter[END_REF][START_REF] Alexandrov | Area spectrum in Lorentz-covariant loop gravity[END_REF][START_REF] Alexandrov | SU(2) loop quantum gravity seen from covariant theory[END_REF] were able to construct a two-parameters family of Lorentz-covariant connections (which are diagonal under the action of the area operator, and transform properly under the action of spatial diffeomorphisms). Generically, these connections are non-commutative and therefore the theory becomes very difficult to quantize. The alternative route to deal with covariant connections was initiated by Barros e Sa in [START_REF] Barros E Sa | Hamiltonian analysis of general relativity with the Immirzi parameter[END_REF] who solved explicitly the second class constraints. In this approach, the phase space is parametrized by two pairs of canonical variables: the generalization (A, E) of the usual Ashtekar-Barbero connection and its conjugate densitized triad E; and a new pair of canonically conjugated fields (χ, ζ), where χ and ζ both take values in R 3 . Then, Barros e Sa expressed the remaining boost, rotation, diffeomorphism and scalar constraints in terms of these variables. The elegance of this approach is that it enables one to have a simple symplectic structure with commutative variables, and a tractable expression for the boost, rotation and diffeomorphism generators. Although the scalar constraint becomes more complicated, this structure is enough to study the kinematical structure of loop quantum gravity with a fully Lorentz invariance. This has precisely been done in [START_REF] Geiller | A Lorentz-Covariant Connection for Canonical Gravity[END_REF][START_REF] Geiller | A new look at Lorentz-Covariant Loop Quantum Gravity[END_REF] where one constructed the unique spatial connection which is not only commutative but also transforms covariantly under the action of boosts and rotations. In fact, this connection coincides with the commutative Lorentz connection studied earlier in [START_REF] Alexandrov | SU(2) loop quantum gravity seen from covariant theory[END_REF] and the one found in [START_REF] Cianfrani | Towards Loop Quantum Gravity without the time gauge[END_REF]. Furthermore, it has been shown to be gauge related to the Ashtekar-Barbero connection via a pure boost parametrized by the vector χ viewed as a velocity. Hence, the construction proposed in [START_REF] Geiller | A Lorentz-Covariant Connection for Canonical Gravity[END_REF][START_REF] Geiller | A new look at Lorentz-Covariant Loop Quantum Gravity[END_REF] works only when χ 2 < 1. Thus, the pairs of canonical variables formed with the sl(2, C) connection and its conjugate electric field parametrize only a part of the fully covariant phase space of the Holst action.

This paper enables us to explore the sector χ 2 > 1 while studying a partial gauge fixing of the Holst action that reduces sl(2, C) to su(1, 1). Hence, we start with the Lorentz covariant parametrization of the Holst action found by Barros e Sa [START_REF] Barros E Sa | Hamiltonian analysis of general relativity with the Immirzi parameter[END_REF]. We find a partial gauge fixing which breaks the sl(2, C) internal symmetry into su(1, 1) and this is possible if and only if χ 2 > 1. Such a partial gauge fixing corresponds to a canonical splitting of the space-time M = Σ × R where Σ is no more space-like (as it is the case in the usual Ashtekar-Barbero parametrization) but inherits a Lorentzian metric of signature is (-, +, +). As a consequence, only three out of the initial six first class constraints remain after the partial gauge fixing, and they generate as expected the local su(1, 1) gauge transformations. The other three constraints form with the three gauge fixing conditions a set of second class constraints that we solve explicitly. Then, we construct an su(1, 1) connection which appears to be commutative in the sense of the Poisson bracket. This remarkable construction allows us to investigate the loop quantization of the theory and to build the kinematical Hilbert space on a given graph Γ whose edges are associated to SU (1, 1) holonomies. It is wellknown that [START_REF] Livine | Spin networks for noncompact groups[END_REF] the non-compactness of the gauge group prevents us from defining the projective limit of spin-networks and then the sum over all graphs of kinematical Hilbert space is ill-defined. Nonetheless, if one restricts the study to one given graph Γ, it is possible to define the action of the area operator and one easily finds that a space-like area has a discrete spectrum whereas the spectrum of a time-like area is continuous. In other words, if one considers a spin-network defined on a graph Γ dual to a discretization ∆ = Γ * of a (2 + 1)-dimensional manifold, edges e of Γ are colored with representations in the discrete series (resp. in the continuous series) if the dual face f = e * of ∆ is space-like (resp. time-like). The spectrum of space-like areas is in total agreement with the one obtained in the usual Ashtekar-Barbero formalism for space-like surfaces.

The paper is organized as follows. After the introduction, we start in Section II with a brief summary of the canonical analysis à la Barros e sa of the fully Lorentz invariant Holst action. In Section III, we present the partial gauge fixing that breaks sl(2, C) into su(1, 1) before constructing the su(1, 1) connection and its associated electric field. In Section IV we explore the kinematical quantization of the theory on a given graph and we compute the spectra of area operators which act unitarily in the kinematical Hilbert space. We conclude in Section V with a brief summary of the most important results and a discussion on the consequences of this new parametrization for the description of black holes in Loop Quantum Gravity.

II. FIRST ORDER LORENTZ-COVARIANT GRAVITY

In this section, we summarize the main results of the Hamiltonian analysis of the fully Lorentz invariant Holst action. We start recalling the main steps of the constraints analysis and present the solutions of the second class constraints proposed by Barros e Sa [START_REF] Barros E Sa | Hamiltonian analysis of general relativity with the Immirzi parameter[END_REF]. Then, we describe the parametrization of the Lorentz covariant phase space that will serve to build the su(1, 1) connection in the next Section. Finally, we discuss the structure of the first class constraints focussing mainly on the generators of the internal Lorentz symmetry.

A. Action and constraints analysis

The Holst action [START_REF] Holst | Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action[END_REF] is a generalization of the Hilbert-Palatini first order action with a Barbero-Immirzi parameter γ. In terms of the co-tetrad e I α (x) and the Lorentz connection one-form ω IJ α (x), the corresponding Lagrangian density is

L[e, ω] = 1 2 IJKL e I ∧ e J ∧ F KL + 1 γ e I ∧ e J ∧ F IJ ,
where F [ω] = dω +ω ∧ω is the curvature two-form of the connection ω, IJKL the fully antisymmetric symbol which defines an invariant non-degenerate bilinear form on sl(2, C), and internal indices are lowered and raised with the flat metric η IJ and its inverse η IJ . It is well known that the Holst action is equivalent to the Hilbert-Palatini action. Indeed, if the co-tetrad is not degenerated (i.e. if its determinant is not vanishing), one can uniquely solve ω in terms of e (from the torsionless equation for ω) and find that ω IJ µ are nothing but the components of the Levi-Civita connection.

Plugging back this solution into the action eliminates the Barbero-Immirzi parameter γ by virtue of the Bianchi identities and leads to the second order Einstein-Hilbert action. Now, we recall basic results on the canonical analysis of the Holst Lagrangian. For this purpose, it is convenient to introduce the notation

γ ξ IJ = ξ IJ - 1 2γ IJKL ξ KL , (2.1)
for any element ξ ∈ sl(2, C). After performing a 3+1 decomposition (based on a splitting M = Σ×R of the space-time) in order to distinguish between temporal and spatial coordinates (0 is the time label and small latin letters from the beginning of the alphabet a, b, c, • • • hold for spacial indices), a straightforward calculation leads to the following canonical expression of the Lagrangian density

L[e, ω] = γ π a IJ ωIJ a -g IJ G IJ -N H -N a H a , (2.2)
where we have introduced the notations ω = ∂ 0 ω for the time derivative of ω, g IJ for -ω IJ 0 , N for the lapse function N , and N a for the shift vector. All these functions are Lagrange multipliers which enforce respectively the Gauss, Hamiltonian, and diffeomorphism constraints

G IJ = D a γ π a IJ , H = π a IK π bK J γ F IJ ab , H a = π b IJ γ F IJ ab . (2.3)
These constraints are expressed in terms of the spatial connection components ω IJ a , and the canonical momenta defined by

π a IJ ≡ IJKL abc e K b e L c . (2.4)
Since π a IJ = -π a JI contains 18 components, and the co-tetrad has only 12 independent components, we need to impose 6 primary constraints often called the simplicity constraints

C ab = IJKL π a IJ π b KL ≈ 0, (2.5)
in order to parametrize the space of momenta in terms of the π variables instead of the co-tetrad variables. Classically, it is equivalent to work with the 12 components e I a or with the 18 components π a IJ constrained to satisfy the 6 relations C ab ≈ 0. Hence, at this stage, the non-physical Hamiltonian phase space is parametrized by the 18 pairs of canonically conjugated variables (ω IJ a , π a IJ ), with the set of 10 constraints (2.3) to which we add the 6 constraints C ab ≈ 0.

Studying the stability under time evolution of these "primary" constraints is rather standard and has been performed first for the Hilbert-Palatini action in [START_REF] Peldan | Actions for gravity, with generalizations: A review[END_REF] and for the Holst action in [START_REF] Barros E Sa | Hamiltonian analysis of general relativity with the Immirzi parameter[END_REF]. Here we will not reproduce all the steps of this analysis, but only focus on the structure of the second class constraints and their resolution. Details with our notations can be found in [START_REF] Geiller | A Lorentz-Covariant Connection for Canonical Gravity[END_REF]. Notice first that in order to recover the 4 phase space degrees of freedom (per space-time points) of gravity, the theory needs to have secondary constraints, which in addition have to be second class. This is indeed the case. Technically, this comes from the fact that the algebra of constraints fails to close because the scalar constraint H does not commute weakly with the simplicity constraint C ab . Hence, requiring their stability under time evolution generates the following 6 additional secondary constraints

D ab = IJM N π cM N π aIK D c π bJ K + π bIK D c π aJ K ≈ 0.
The Dirac algorithm closes here with 18×2 phase space variables (parametrized by the components of π and ω), and 22 constraints H, H a , G IJ , C ab and D ab . Among these constraints, the first 10 are first class (up to adding second class constraints) as expected, and the remaining 12 are second class. One can check explicitly that C ab ≈ 0 and D ab ≈ 0 form a set of second class constraints (their associated Dirac matrix is invertible), and that the first class constraints generate the symmetries of the theory, namely the space-time diffeomorphisms and the Lorentz gauge symmetry. Finally, we are left with the expected 4 phase space degrees of freedom per spatial point:

18 × 2(dynamical variables) -10(first class constraints) × 2 -12(second class constraints).

We recover the two gravitational modes.

B. Parametrization of the phase space

Now that we have clarified the Hamiltonian structure of the theory, we are going to show how to solve the second class constraints following [START_REF] Barros E Sa | Hamiltonian analysis of general relativity with the Immirzi parameter[END_REF]. First, one writes the 18 components of π IJ a as

π a 0i = 2E a i , π a ij = 2(E a i χ j -E a j χ i ) , (2.6)
where χ i = e a i e 0 a (which encodes the deviation of the normal to the hypersurfaces from the time direction) and E a i (which corresponds to the usual densitized triad of loop gravity) are now twelve independent variables. Note that e a i is the inverse of e i a viewed as a 3 × 3 matrix. This is trivially a solution of the simplicity constraints (2.5) because somehow we have returned to the co-tetrad parametrization (2.4).

Then, we plug the solution (2.6) into the canonical term of the Lagrangian (2.2) which gives

γ π a IJ ωIJ a = E a i Ȧi a + ζ i χi where A i a = γ ω 0i a + γ ω ij a χ j and ζ i = γ ω ij a E a j . (2.7)
This result strongly suggests that the 18 components of the connection could be expressed in terms of the 12 independent variables (A i a , ζ i ) when one solves the 6 secondary second class constraints. This is indeed the case and it can be seen by inverting the relation (2.7) as follows

γ ω 0i a = A i a -γ ω ij a χ j , γ ω ij a = 1 2 Q ij a -E i a ζ j -E j a ζ i , (2.8)
where E i a is the inverse of E a i , and Q ij a = Q ji a has a vanishing action on E a i . The explicit form of Q ij a can be obtained from D ab ≈ 0 as shown in [START_REF] Barros E Sa | Hamiltonian analysis of general relativity with the Immirzi parameter[END_REF]. Furthermore, when γ 2 = 1, one can uniquely express ω in terms of γ ω using the inverse of the map (2.1).

As a consequence, the phase space can be parametrized by the twelve pairs of canonical variables (A i a , E a i ) and (χ i , ζ i ) with the (non-trivial) Poisson brackets given by

A i a (x), E b j (y) = δ i j δ b a δ 3 (x -y) and χ i (x), ζ j (y) = δ j i δ 3 (x -y).
(2.9)

Remark that if we work in the time gauge (i.e. χ = 0), the variable A i a coincides exactly with the usual Ashtekar-Barbero connection.

C. First class constraints

It remains to express the first class constraints (2.3) in terms of the new phase space variables (2.9). This is an easy task using the defining relations (2.6) and (2.8). This was done by Barros e Sa. The constraints have quite a simple form except the Hamiltonian constraint whose expression is more involved: it can be found in [START_REF] Barros E Sa | Hamiltonian analysis of general relativity with the Immirzi parameter[END_REF] and we will not consider this constraint in this paper. The vector constraint H a takes the form

H a = E b • (∂ a A b -∂ b A a ) + ζ • ∂ a χ + γ 2 1 + γ 2 (E b • A b )(A a • χ) -(E b • A a )(A b • χ) +(A a • χ)(ζ • χ) -(A a • ζ) + 1 γ E b • (A b × A a ) + ζ • (χ × A a ) , (2.10)
where • denotes the scalar product λ • µ = λ i µ i and × denotes the cross product (λ × µ) i = ijk λ j µ k for any two pairs of vectors λ and µ in R 3 . Concerning, the Lorentz constraints G IJ , they can be split into its boost part B i ≡ G 0i , and its rotational part

R i ≡ (1/2) jk i G jk whose expressions are B = ∂ a E a - 1 γ χ × E a -(χ × E a ) ∧ A a + ζ -(ζ • χ)χ , (2.11a) R = -∂ a χ × E a + 1 γ E a + A a × E a -ζ × χ . (2.11b)
One can check that these constraints satisfy indeed the Lorentz algebra

{B • u, B • v} = -R • u × v, {R • u, R • v} = R • u × v, {B • u, R • v} = B • u × v , (2.12)
where u and v are arbitrary vectors.

In the time gauge, one immediately recovers the constraints structure of the formulation of gravity in terms of the Astekar-Barbero connection. In that case, χ ≈ 0 drastically simplifies the boost constraints which become equivalent to ζ -∂ a E a ≈ 0. The conditions χ ≈ 0 and ζ -∂ a E a ≈ 0 form a set of second class constraints that can be solved explicitly for χ and ζ. By doing so, the variables (χ, ζ) are eliminated from the theory, and the vectorial, the rotational and also the Hamiltonian constraints are those of Loop Quantum Gravity. Now our task is to use the phase space variables (2.9) to make a partial gauge fixing which reduces the original Lorentz algebra to su(1, 1).

III. GRAVITY AS AN SU(1,1) GAUGE THEORY

In this section, we first show how to make a partial gauge fixing of the full Lorentz invariant Holst action which reduces the internal sl(2, C) gauge symmetry to su(1, 1). At the same time, we keep the invariance under diffeomorphisms on Σ. In that case, we will see that the splitting of the space-time M = Σ × R is such that Σ is no more a space-like hypersurface as it is the case in the time gauge but inherits instead a Lorentzian structure. Then, we construct a parametrization of the phase space in terms of an su(1, 1) connection and its conjugate electric field which transforms in the adjoint representation of su(1, 1). Furthermore, we show that these variables are Darboux coordinates for the phase space, which paves the way towards a quantization of the theory explored in the following Section.

A. Breaking the internal symmetry: from sl(2, C) to su(1, 1)

As we have already underlined in the previous section, imposing the time gauge χ ≈ 0 in the fully covariant Holst action breaks the boost invariance and only the rotational parts of the constraints remain first class among the original 6 internal symmetries. Hence, we get an su(2) invariant theory of gravity. In fact, we proceed in a very similar way to construct an su(1, 1) invariant theory from the Holst action: we find a partial gauge fixing such that two components of the boosts constraints and one of the rotational constraints remain first class whereas the three others form with the gauge fixing conditions a second class system. Naturally, we consider a gauge fixing condition of the form

X ≡ χ -χ 0 ≈ 0 (3.1)
where χ 0 is a fixed non-dynamical vector. Inspiring ourselves with what happens in the time gauge, we expect (3.1) to form a second class system with three out of the six constraints (2.11). These three second class components of the Lorentz generators are supposed to be

R • u ≈ 0 , R • v ≈ 0 , B • n ≈ 0 , (3.2)
where u and v are two given normalized orthogonal vectors and n = v × u. The reason is that we are left with two boosts and one rotations which are expected to reproduce (up to the addition of second class constraints) an su(1, 1) Poisson algebra. To derive the conditions for this to happen, we start rewriting (3.2) as a linear system of equations for ζ:

M ζ =   ζ • U ζ • V ζ • W   ≈   R • u| ζ=0 R • v| ζ=0 B • n| ζ=0   with M ≡   t U t V t W   and    U ≡ χ × u V ≡ χ × v W ≡ -n + (χ • n)χ (3.3)
The system admits an unique solution for ζ if and only if

detM = U × V • W = (1 -χ 2 )(χ • n) 2 = 0 , (3.4)
which implies that χ 2 = 1 and χ • n = 0. When we assume this is the case, the solution ζ 0 can be easily expressed in terms of the components of χ 0 , E and A inverting (3.3) as follows

ζ 0 = M -1   R • u| ζ=0 R • v| ζ=0 B • n| ζ=0   = (B • n + R • χ × n) χ -(1 -χ 2 )R × n (1 -χ 2 )n • χ | ζ=0 , (3.5)
where we used the expression

M -1 = 1 U × V • W V × W , W × U , U × V . (3.6)
Hence, the three constraints (3.2) are equivalent to the three conditions

Z ≡ ζ -ζ 0 (χ 0 , E, A) ≈ 0 . (3.7)
Now, it becomes clear that the gauge fixing conditions X ≈ 0 (3.1) and the three constraints Z ≈ 0 form a second class system because their associated 6

× 6 Dirac matrix ∆ ∆(x, y) ≡ X(x, y) Y (x, y) -t Y (x, y) Z(x, y) with    X i j (x, y) ≡ {χ i (x), χ j (y)} = 0 Y i j (x, y) ≡ {X i (x), Z j (y)} = δ j i δ 3 (x -y) Z i j (x, y) ≡ {Z i (x), Z j (y)} (3.8)
is invertible whatever Z is. These two constraints allow to eliminate the variables χ and ζ from the phase space provided that one introduces the external non dynamical field χ 0 .

We are left with three constraints from (2.11) which are required to satisfy an su(1, 1) Poisson algebra once one replaces χ by χ 0 and ζ by ζ 0 . These constraints are denoted

J u ≡ B • u| χ 0 ,ζ 0 , J v ≡ B • v| χ 0 ,ζ 0 , J n ≡ R • n| χ 0 ,ζ 0 .
(3.9)

From now on, we will omit to mention the index 0 for χ to lighten the notations. However, χ has to be understood as an external non dynamical field, and not as the initial dynamical variable in the fully Lorentz invariant Holst action.

A long but standard calculation shows that the three constraints (3.9) form a closed Poisson algebra only when

u • χ = v • χ = 0 . (3.10)
This is equivalent to the condition that χ = ±|χ|n where |χ| ≡ √ χ • χ is the norm of χ. Without loss of generality, we choose χ = |χ|n. As a consequence, the partial gauge fixing (3.1) leaves the remaining three constraints (3.9) first class only when (3.10) is satisfied. In that case, the expressions of (3.9) simplify a lot and they can be written as

J 0 ≡ J n = n • J , J 1 ≡ CJ v = Cu • J , J 2 ≡ CJ u = -Cv • J , (3.11)
where C = 1/ |χ 2 -1| is a normalization function and we introduced the vector field

J ≡ - 1 γ ∂ a E a + ∂ a (E a × χ) × χ + Ãa × E a (3.12)
given in terms of the su(2)-valued one form à defined by

Ãa = A a -(A a • χ)χ -∂ a χ . (3.13)
Finally, one shows that the constraints algebra reduces to the simple form

{J 0 , J 1 } = J 2 , {J 0 , J 2 } = -J 1 , {J 1 , J 2 } = σJ 0 , (3.14)
where

σ ≡ 1 -χ 2 |1 -χ 2 | = sg(1 -χ 2 ). (3.15)
The function sg(x) denotes the sign of x = 0. As a consequence, the remaining three constraints form an su(2) Poisson algebra when χ 2 < 1 and an su(1, 1) Poisson algebra when χ 2 > 1 (the case χ 2 = 1 is excluded from the scope of our method and should be studied in a different way1 ). We can write the constraints algebra in the more compact form

{J α , J β } = αβ τ J τ (3.16)
where α, β, τ ∈ (0, 1, 2) and αβτ is the totally antisymmetric symbol with 012 = +1. Furthermore, the indices are lowered and raised with the flat metric and its inverse diag(σ, +1, +1): it is the flat Euclidean metric δ αβ when σ = +1 and the flat Minkowski metric η αβ ≡ diag(-1, +1, +1) when σ = -1. Hence, as announced above, one recognizes respectively the su(2) and the su(1, 1) Lie algebras.

Let us close this analysis with one remark. The gauge fixing condition (3.1) makes the three constraints (3.2) (which are first class in the full Lorentz invariant Holst action) second class. Hence, we have left two boosts and one rotation first class in order to get an su(1, 1) gauge symmetry at the end of the process. This is what we arrive at when χ 2 > 1 but we obtain an su(2) gauge symmetry when χ 2 < 1 even though we kept two boosts among the remaining first class constraints. The reason is that, at the end of the gauge fixing process, the remaining first class constraints are nontrivial linear combinations of the six initial first class constraints and the gauge fixing conditions. Hence, they could form either an su(1, 1) or an su(2) algebra. The two most important ingredients in our construction is that, first, we replace three out of the initial six first class constraints by constraints of the type (3.7) which fix ζ, and second we impose that the remaining constraints (when ζ and χ are replaced from X ≈ 0 and Z ≈ 0) form a closed Poisson algebra. In that respect, we could have considered the conditions B.u ≈ B.v ≈ B.n ≈ 0 instead of (3.2): we would have obtained another set of conditions fixing ζ and then, following the same strategy, we would have shown that the remaining three constraints are generators of a closed algebra provided that (3.10) is satisfied. The remaining symmetry would have been su(2) or su(1, 1) depending on the sign of σ exactly as in the previous analysis.

B. On the space-time foliation

Let us discuss the reason why the sign σ of (χ 2 -1) determines the signature of the symmetry algebra su(2) or su(1, 1). For that purpose, it is very instructive to study the properties of the metric g ab induced on the hypersurface Σ whose expression is

g ab ≡ e I a η IJ e J b = e i a γ ij e j b with γ ij ≡ δ ij -χ i χ j (3.17)
where we inverted the defining relation χ i = e a i e 0 a to replace e 0 a by e i a χ i . It is immediate to notice that this formula is compatible with the expression of the inverse metric given in [START_REF] Barros E Sa | Hamiltonian analysis of general relativity with the Immirzi parameter[END_REF][START_REF] Geiller | A new look at Lorentz-Covariant Loop Quantum Gravity[END_REF] det(g)

g ab = (1 -χ 2 ) E a i γ ij E b j , γ ij ≡ δ ij - χ i χ j 1 -χ 2 , (3.18)
due to the properties

E a i = det(e)e a i , det(g) = (1 -χ 2 )det(e) 2 , γ ij γ jk = δ i k . (3.19)
Thus, the identity (3.17) implies immediately that the metric induced on Σ has the same signature as γ ij . This latter metric can be easily diagonalized and its eigenvalues/eigenvectors are easily obtained from

γ ij u j = u i when u • χ = 0 , and γ ij χ j = (1 -χ 2 )χ i . (3.20)
Therefore, the signature of the metric depends on the sign of (χ 2 -1): Σ is spacelike when χ 2 < 1 whereas it inherits a Lorentzian metric when χ 2 > 1. This clearly explains the presence of σ in the constraints algebra (3.15) and the nature of the gauge symmetry. When the symmetry algebra is su(2), the space-time is foliated as usual into hypersurfaces orthogonal to a timelike vector whereas it is foliated in a space-like direction when the symmetry algebra is su(1, 1). This latest case is not conventional but it is the one we are interested in.

C. Phase space parametrization

From now on, we will mainly focus on the case χ 2 > 1 which has never been studied so far (we will shortly discuss the case χ 2 < 1 at the end of this Section). As the theory admits su(1, 1) as a gauge symmetry algebra, it is natural to look for a parametrization of the phase space adapted to this symmetry. More precisely, we look for conjugate variables which transform in a covariant way under the Poisson action of the su(1, 1) generators. In a first part, we exhibit an unique su(1, 1)valued connection which is commutative in the sense of the Poisson bracket. This connection is the su(1, 1) analogous of the generalized Ashtekar-Barbero connection defined for χ = 0 in [START_REF] Geiller | A Lorentz-Covariant Connection for Canonical Gravity[END_REF][START_REF] Geiller | A new look at Lorentz-Covariant Loop Quantum Gravity[END_REF] for instance. In a second part, we show that it is canonically conjugate to an electric field which transforms as a vector under the action of the first class constraints. Hence, the su(1, 1)-connection together with its conjugate electric field provide us with a very useful and natural parametrization of the phase space. We finish with computing the action of the vectorial constraints on these variables which transform as expected under the action of the generators of diffeomorphisms.

The connection

Now, we address the problem of finding an su(1, 1) connection defined by

A = A 0 J 0 + A 1 J 1 + A 2 J 2 with [J α , J β ] = αβ τ J τ (3.21)
which satisfies the following requirements. First, is constructed from the components of A (such that it is commutative in the sense of the Poisson bracket) and the non-dynamical vectors (χ, u and v) only. Second it transforms as

δ ε A = dε + [A, ε] , (3.22)
under the action of the gauge transformations where ε = ε α (x)J α is an arbitrary su(1, 1)-valued function on Σ. For this relation to make sense, we have to precise the definition of δ ε in terms of the gauge generators. In particular, we have to establish the link between the parameter ∈ R 3 entering in the smeared constraint J ( ) and the parameter ε defining the su(1, 1) infinitesimal gauge transformations of A. From (3.11), it is natural to expect that

δ ε A = { J ( ), A} with ε 0 = • n, ε 1 = c 1 • u, ε 2 = c 2 • v , (3.23)
where c 1 and c 2 are functions of χ. Now, the problem consists in finding the components of A and the functions c 1 and c 2 such that A transforms as an su(1, 1) connection under the action of the first class constraints.

We are going to propose an ansatz for A. As the expressions of the gauge generators are simpler with à instead of A itself, we also look for an su(1, 1) connection A written in terms of Ã. This is possible because, when χ 2 = 1, à can be uniquely expressed in terms of A and χ inverting the relation (3.13) as follows:

A a = Ãa + ∂ a χ + χ • ( Ãa + ∂ a χ) χ 1 -χ 2 .
(3.24)

Inspiring ourselves from the decomposition (3.11) of the first class constraints into su(1, 1) gauge generators, we propose the following form for the components of A:

A 0 = p 0 ( Ã • n) + q 0 , A 1 = p 1 ( Ã • u) + q 1 , A 2 = p 2 ( Ã • v) + q 2 , (3.25)
where (p 0 , p 1 , p 2 ) are functions of χ whereas (q 0 , q 1 , q 2 ) are one-forms constructed from dχ, du and dv only.

Hence, the problem reduces now in finding the functions (c 1 , c 2 ) and (p 0 , p 1 , p 2 ) together with the one-forms (q 0 , q 1 , q 2 ) which solve the equations (3.23). These equations can be more explicitly written as

p 0 { J ( ), ( Ã • n)} = d( • n) + c 1 (p 2 ( Ã • v) + q 2 ) • u -c 2 (p 1 ( Ã • u) + q 1 ) • v , (3.26) p 1 { J ( ), ( Ã • u)} = d(c 1 • u) + (p 2 ( Ã • v) + q 2 ) • n -c 2 (p 0 ( Ã • n) + q 0 ) • v , (3.27) p 2 { J ( ), ( Ã • v)} = d(c 2 • v) -(p 1 ( Ã • u) + q 1 ) • n + c 1 (p 0 ( Ã • n) + q 0 ) • u , (3.28)
where each Poisson brackets on the l.h.s. are easily deduced from

{ J ( ), Ã} = - 1 γ (1 -χ 2 )d + Ã × - 1 γ χ × (dχ × ) + ( Ã • χ × )χ . (3.29)
A straightforward calculations show that the previous system reduces to the following three sets of equations:

p 0 (1 -χ 2 ) = c 1 p 2 = c 2 p 1 = -γ , dn + c 1 q 2 u -c 2 q 1 v = 0 , p 1 = -p 2 = -c 2 p 0 = γc 1 /(χ 2 -1) , d(c 1 u) + q 2 n -c 2 q 0 v + p 1 [(u • dχ)χ -(χ • dχ)u]/γ = 0 , p 1 = -p 2 = c 1 p 0 = -γc 2 /(χ 2 -1) , d(c 2 v) -q 1 n + c 1 q 0 u + p 2 [(v • dχ)χ -(χ • dχ)v]/γ = 0 .
This is clearly an overcomplete set of conditions for the unkowns of the problem. However, an immediate analysis shows that (up to a simple sign ambiguity), the system admits an unique solution given by

p 0 = γ χ 2 -1 , p 1 = γ χ 2 -1 , p 2 = - γ χ 2 -1 , (3.30)
q 0 = dv • u , q 1 = - 1 χ 2 -1 v • dn , q 2 = - 1 χ 2 -1 u • dn , (3.31)
with

c 1 = -c 2 = χ 2 -1.
As a conclusion, let us summarize the main results of this part. The theory admits an su(1, 1) gauge connection A = A 0 J 0 + A 1 J 1 + A 2 J 2 whose components are

A 0 = γ χ 2 -1 Ã • n + u • dv , (3.32) A 1 = 1 χ 2 -1 γ Ã • u -v • dn , (3.33)
A 2 = - 1 χ 2 -1 γ Ã • v + u • dn . (3.34)
We have just proved that it transforms as follows

δ ε A = { J ( ), A} = dε + [A, ε] with = ε 0 n + ε 1 u -ε 2 v χ 2 -1 (3.35)
under the action of the first class constraints. Note that this transformation law is totally consistent with the fact that

J ( ) = J 0 (ε 0 ) + J 1 (ε 1 ) + J 2 (ε 2 ) , (3.36)
where the components of J are the smeared su(1, 1) generators introduced in (3.11).

Let us close this analysis with two remarks. First, one can reproduce exactly the same analysis when χ 2 < 1. In that case, one obtains an su(2) connection whose expression is very similar to the previous one obtained for su(1, 1):

everything happens as if one makes the replacement χ 2 -1 → -1 -χ 2 in the components of the connection. The su(2)-valued connection is certainly related to the generalized Ashtekar-Barbero connection obtained in different ways [START_REF] Alexandrov | SU(2) loop quantum gravity seen from covariant theory[END_REF][START_REF] Geiller | A new look at Lorentz-Covariant Loop Quantum Gravity[END_REF][START_REF] Cianfrani | Towards Loop Quantum Gravity without the time gauge[END_REF]. In the limit χ → 0 with n constant, one recovers the usual Ashtekar-Barbero connection in the time-gauge written in the orthonormal basis (n, -u, v):

A = A 0 n -A 1 u + A 2 v = γA . (3.37)
Second, by construction, the limit χ → 0 does not exist for the su(1, 1)-valued connection. The analogous of the time gauge is defined by the limit |χ| → ∞ where the direction n tends to a constant. Let us study this limit, and for simplicity, we assume that the direction n is constant.

Starting from the relation

Ãi a = γ ω ij a χ j + γ ω 0i a -γ ω 0i a χ j χ i , (3.38)
we obtain the following limits for the components of

A A 0 a → -γ γ ω 0i a n i , A 1 a → γ γ ω ij a n j u i , A 2 a → -γ γ ω ij a n j v i . (3.39)
One recognizes the components of the spin-connection in what we could call the "space-gauge" which would be defined by the choice e a i n i = 0 (instead of e a 0 = 0 for the usual time gauge). As a consequence, the limit |χ| → ∞ with n constant is well-defined and consists in a foliation of the space-time M = Σ × R where the slices Σ are orthogonal to the space-like vector (0, n).

The electric field

We follow the same strategy to construct an electric field E which transforms as an su(1, 1) under the gauge transformations. More precisely, we are looking for E = E 0 J 0 + E 1 J 1 + E 2 J 2 which satisfies two conditions. First we require its components to be constructed from E, χ, u and v only and we consider the natural ansatz

E 0 = r 0 (E • n) , E 1 = r 1 (E • u) , E 2 = r 2 (E • v) , (3.40)
where (r 0 , r 1 , r 2 ) are functions of χ only. Second we require E to transform as a vector

δ ε E ≡ { J ( ), E} = [E, ε] with = ε 0 n + ε 1 u -ε 2 v χ 2 -1 , (3.41)
in adequacy with what has been done in the previous part for the connection. A simple calculation shows that these conditions implies necessarily

r 1 = χ 2 -1 r 0 , r 2 = -χ 2 -1 r 0 , (3.42)
where, at this point, r 0 is free because equations (3.41) form a linear system for the unknowns (r 0 , r 1 , r 2 ).

Let us close this analysis with three remarks. First, the free parameter r 0 can be fixed requiring in addition that E is canonically conjugate to A according to

{A 1 , E 1 } = {A 2 , E 2 } = 1 and {A 0 , E 0 } = -1, (3.43)
which easily leads to r 0 = 1/γ.

Second, it will be useful to express the (inverse of the) induced metric q ab on Σ in terms of the su(1, 1)-covariant electric field. A direct calculation shows that det(g)

g ab = -γ 2 E αa η αβ E βb . (3.44)
Note that this formula makes very clear that the metric g ab is Lorentzian and its signature is (-1, +1, +1) as we have already seen in a previous analysis (3.20).

The final remark concerns the su(1, 1) gauge generators J α . It is immediate to see that one can express them in terms of A and E only as follows

J α (x) J α = ∂ a E a (x) + [A a (x) , E a (x)] . (3.45)
We recover the usual Gauss-like form of the constraints, and this expression makes very clear that A and E transforms respectively as a connection and a vector under the action of the gauge generators.

Transformations under diffeomophisms

As for the Ashtekar-Barbero connection (or its generalization), we do not expect A to be a fully space-time connection on M. However, it must transform correctly under diffeomorphisms induced on the hypersurface Σ. To see this is indeed the case, we first need to identify the generators of diffeomorphisms on Σ. A direct calculation shows that they are given by the following linear combination of the su(1, 1) gauge generators and the vectorial constraints:

H(N a ) ≡ H(N a ) - γ (1 + γ 2 )χ 2 J (N a Ω a ) with Ω a ≡ γχ × A a -(A a • χ)χ , (3.46)
which, after some calculations, reduces to

H(N a ) = d 3 x N a E b • (∂ a A b -∂ b A a ) -A a • ∂ b E b + ζ 0 • ∂ a χ = d 3 x N a η αβ E αb • (∂ a A α b -∂ b A α a ) -A α a • ∂ b E αb . (3.47)
Hence, it is immediate to see from this last expression that the constraints H(N a ) form the algebra of diffeomorphisms. Furthermore, their actions on A and E is exactly the lie derivative along the vector field N a :

{ H(N a ), A b } = -L N a A b , { H(N a ), E b } = -L N a E b . (3.48)
Thus, as announced above, A is an su(1, 1)-valued connection on Σ.

IV. ON THE QUANTIZATION

We have now all the ingredients to start the quantization of gravity formulated in terms of the su(1, 1) gauge connection. Following the standard construction of Loop Quantum Gravity, we assume that quantum states are polymer states, and then we build the kinematical Hilbert space from holonomies of the connection along edges on Σ.

A. Quantum states on a fixed graph

As usual, to any closed graph Γ ⊂ Σ with N nodes and E edges, one associates a kinematical Hilbert space H kin (Γ) which is isomorphic to

H kin (Γ) Fun[SU (1, 1) ⊗E ]/SU (1, 1) ⊗N ; dµ ⊗E , (4.1)
where dµ is the Haar measure on SU (1, 1). Due to the non-compactness of the gauge group, such a Hilbert space needs a regularization to be well-defined (which consists basically in "dividing" by the infinite volume of the group). The details of the regularization of non-compact spin-networks has been well studied in [START_REF] Livine | Spin networks for noncompact groups[END_REF]. However, it is well-known that the "projective sum" ⊕ Γ H kin (Γ) on the space of all graphs on Σ is ill-defined and, up to our knowledge, no one knows how to construct a non-compact Ashtekar-Lewandowski measure. Thus, only the kinematical Hilbert space on a fixed graph Γ is mathematically well-defined and we limit the study of quantum states as elements of H kin (Γ) only. Hence, a quantum state is a function

ψ Γ [A] ≡ f (U 1 , • • • , U E ) of the holonomies U e ≡ P exp e A ∈ SU (1, 1) (4.2)
along the edges e of Γ. The electric field E is promoted as an operator whose action on ψ Γ is formally given by Êa

i (x) ψ Γ [A] = i 2 p δ δA i a (x) ψ Γ [A] , (4.3)
where p is the Planck length. Note that the flux of E across a surface is a well-defined operator on H kin (Γ): it acts as a vector field on the space of SU (1, 1) functions. The Peter-Weyl theorem implies that ψ Γ can be formally decomposed as follows

Ψ Γ [A] = s 1 ,••• ,s E tr f (s e ) E e=1 π se (U e ) (4.4)
where

π s : SU (1, 1) → End(V s ) and f ∈ N e=1 V * se . (4.5)
The sum runs over unitary irreducible representations of SU (1, 1) labelled generically by s e . We used the notation V se for the modulus of the representation, V * se for its dual, and tr denotes the pairing between ⊗ e V se and its dual ⊗ e V * se . Due to the gauge invariance of ψ Γ , the Fourier modes f are in fact SU (1, 1) intertwiners and the expression of ψ[A] needs a regularization to be well-defined [START_REF] Livine | Spin networks for noncompact groups[END_REF]. Furthermore, unitary irreducible representations of SU (1, 1), which are classified into the two discrete series (both labelled with integers) and the continuous series (labelled with real numbers), are infinite dimensional (see [START_REF] Ruhl | The Lorentz group and harmonic analysis[END_REF] for a review on representations theory of su(1, 1)).

B. Area operators

Thus, edges of SU (1, 1) spin-networks can be colored with discrete or real numbers. The geometrical interpretation is clear: these two different types of colors label edges which are normal to either time-like or space-like surfaces. To see how to link the representations to the time-like or space-like natures of the surfaces, we have to compute the spectrum of the area operators in terms of the quadratic Casimir of su [START_REF] Ashtekar | New variables for classical and quantum gravity[END_REF][START_REF] Ashtekar | New variables for classical and quantum gravity[END_REF]. For that purpose, we start with the expression (3.44) of the inverse metric g ab that we contract twice with the normal n a to a given surface S. This leads to the formula det(g)

n 2 = -γ 2 (n a E aα ) η αβ (E bβ n b ) , (4.6)
where n 2 = n a n b g ab . Hence, the determinant of the induced metric h on the surface S is given by

det(h) = -γ 2 (n a E aα ) η αβ (E bβ n b ) . (4.7)
As a consequence, the action of the area operator Ŝ, punctured by an edge e of the graph Γ colored by a representation s e , on H kin (Σ) is diagonal and its eigenvalue S(s) is given by the equation

S(e) 2 = -i 2 γ 2 4 p π e (J 2 1 + J 2 2 -J 2 0 ) = γ 2 4 p π e (C) (4.8)
where π e (C) is identified with the unique eigenvalue of the Casimir tensor C ≡ -J 2 0 +J 2 1 +J 2 2 in the representation s e . Obviously, the evaluation π e (C) depends on the nature discrete (s e = j e ∈ N) or continuous (s e ∈ R) of the representation according to π je (C) = j e (j e + 1) and π se

(C) = -(s 2 e + 1 4
) . (4.9)

We deduce immediately that S(e) 2 is positive when e is colored with a discrete representation whereas S(e) 2 is negative when e is colored with a representation in the continuous series. As a consequence, the area operator of any space-like surface has a discrete spectrum and the area operator of any time-like surface has a continuous spectrum. Furthermore, the spectrum of spacelike areas is in total agreement of the usual spectrum in Loop Quantum Gravity. Note that a very similar result has been recently derived in the context of twisted geometries [START_REF] Rennert | Timelike twisted geometries[END_REF].

V. DISCUSSION

In this paper, we have formulated gravity as an SU (1, 1) gauge theory. We have started with the Hamiltonian formulation of the fully Lorentz invariant Holst action on a space-time manifold of the form M = Σ×R. Then we have considered a partial gauge fixing which reduces the internal sl(2, C) gauge symmetry to su(1, 1). The 3-dimensional slice Σ inherits a Lorentzian metric of signature (-, +, +). The partial gauge fixing relies on the introduction on an external non-dynamical vector field χ which measures the normal of the hypersurface Σ but it plays in fact no physical role at the end of the process.

Next we found that the phase space of the partially gauge fixed theory is well-parametrized by a pair (A, E) formed with an su(1, 1)-valued connection on Σ and its canonically conjugate electric field whose components can be identified to vectors in the flat (2+1) Minkowski space-time. The phase space comes with first class constraints: the Gauss constraints which generate su(1, 1) gauge transformations, the vectorial constraints which have been shown to generate diffeomorphisms on Σ and the usual scalar constraint that we have not studied in this paper.

Finally, we have explored the quantization of the theory studying some aspects of the kinematical Hilbert space H kin (Γ) on a fixed given graph Γ which lies on Σ. Due to the non-compactness of the gauge group SU (1, 1), H kin (Γ) needs a regularization to be well-defined and the projective sum over all possible graphs is not under control. This is why we restrict our study to the quantization on a fixed graph only. We compute the spectrum of the area operators acting on H kin (Γ) and found that the spectrum is discrete for space-like surfaces and continuous for time-like surfaces. In that case, the horizon appears as a boundary of Σ. In the picture (a), we have represented two slicings of the black hole space-time where Σ are Lorentzian hypersurfaces: these gauge choices would lead to new descriptions of black holes in Loop Quantum Gravity. In particular, the slicing which does not cross the horizon is interesting in view of a holographic description of black holes in the frame of Loop Quantum Gravity.

Furthermore, the usual quantization of the Holst action in the time-gauge (χ = 0) and the new quantization presented here and based to another totally inequivalent partial gauge fixing (χ 2 > 1) lead to exactly the same spectrum of the area operator (on space-like surfaces) at the kinematical level. This strongly suggests that the time gauge introduces no anomaly in the quantization of gravity, at least at the kinematical level, as it was already underlined in [START_REF] Geiller | A new look at Lorentz-Covariant Loop Quantum Gravity[END_REF] in a different situation. This formulation of gravity seems very interesting because it offers another point of view on the quantization of gravity in four dimensions. Now, we have a description of the kinematical quantum states of gravity not only on space-like surfaces Σ but also on time-like surfaces (only remains the description of the quantum states on null-surfaces, what we hope to study in the future). Hence, with those space-like and time-like kinematical quantum states, we are not far from having a fully covariant description of quantum gravity. In that respect, it would be very instructive to make a contact between these two canonical quantizations and spin-foam models for covariant quantum gravity. Furthermore, if we understand how to "connect" the time-like and the space-like kinematical quantum states, we could open a new and promising way towards a better understanding of the dynamics in Loop Quantum Gravity.

It is also interesting to notice that the Hamiltonian constraint in the formalism where Σ is space-like becomes a component of the vectorial constraints in the formalism where Σ is timelike. The reverse is also true. As we know very well how to quantize the vectorial constraints on the kinematical Hilbert space, we think again that understanding the relation between these two Hamiltonian quantizations could lead us to a solution of the Hamiltonian constraint. We hope to study these questions related to the quantum dynamics in the future.

Beside, we deeply think that this new formulation will allow us to understand better the physics of quantum black holes in Loop Quantum Gravity. In the usual treatment [START_REF] Rovelli | Black hole entropy from loop quantum gravity[END_REF][START_REF] Ashtekar | Quantum geometry of isolated horizons and black hole entropy[END_REF][START_REF] Meissner | Black hole entropy in loop quantum gravity[END_REF][START_REF] Agullo | Black hole state counting in loop quantum gravity: A number theoretical approach[END_REF][START_REF] Engle | Black hole entropy and SU(2) Chern-Simons theory[END_REF][START_REF] Engle | Black hole entropy from an SU(2)-invariant formulation of type I isolated horizons[END_REF][START_REF] Engle | The SU (2) black hole entropy revisited[END_REF][START_REF] Heidmann | Semi-classical analysis of black holes in Loop Quantum Gravity: Modeling Hawking radiation with volume fluctuations[END_REF], black holes are considered as isolated horizons and they appear as boundary of a 3 dimensional space-like hypersurface Σ. Their effective dynamics has been shown to be governed by an SU (2) Chern-Simons theory whose quantization leads to the construction and the counting of the quantum microstates for the black holes. With the su(1, 1) formulation of gravity, it is now possible to start a Hamiltonian quantization of gravity where Σ is time-like. Naturally, one would expect that quantizing black holes with space-like or time-like slices would lead to two equivalent descriptions of the black hole microstates. At first sight, we would say that, starting with a time-like slicing, one would get an SU (1, 1) Chern-Simons theory as an effective dynamics for the spherical black hole for instance. Thus, we can ask the question how an SU (1, 1) and an SU (2) Chern-Simons theories could provide two equivalent Hilbert spaces when they are quantized. This may be possible when γ becomes complex and equal to ±i because, in that case, we expect the two gauge group of the Chern-Simons theories to become the same Lorentz group. This would give one more argument in favor of the analytic continuation procedure introduced and studied in [START_REF] Frodden | Black hole entropy from complex Ashtekar variables[END_REF][START_REF] Bodendorfer | Loop quantum gravity without the Hamiltonian constraint Class[END_REF][START_REF] Pranzetti | Black hole entropy from KMS-states of quantum isolated horizons[END_REF][START_REF] Bodendorfer | Imaginary action, spinfoam asymptotics and the 'transplanckian' regime of loop quantum gravity[END_REF][START_REF] Ben Achour | Analytic continuation of Black Hole Entropy in Loop Quantum Gravity[END_REF]. However, this idea might be too naive because, on a time-like slicing, the black hole does not appear as a boundary anymore and a particle leaving on the slice Σ now cross the horizon and does not see any border. To finish, this new formulation of Loop Quantum Gravity opens the possibility to define a kind of "holographic" description for black holes in the framework of Loop Quantum Gravity as shown in the picture Fig. 1 above. We hope to study all these very intriguing aspects related to black holes in a future work [31].

α ∈ (0, 1, 2) and a ∈ (0, 1, 2) also) according to the formula

γ π a IJ ∂ 3 ω IJ a = E a α ∂ 3 A α a ,
where A α a ≡ γ ω α3 a .

(A1)

Hence, the connection A is clearly the variable canonically conjugate to E. Finally, one shows that the resolution of the second class constrains D ab ≈ 0 leads to the following expression for the gauge generators

J 0 = - 1 γ ∂ a E a0 -A 1 a E a2 + A 2 a E a1 , ( A2
)

J 1 = - 1 γ ∂ a E a2 + A 0 E 1 -A 1 E 0 , (A3)
J 2 = 1 γ ∂ a E a 1 -A 0 E 2 + A 2 E 0 . (A4)
They satisfy the constraints algebra

{J 0 , J 1 } = J 2 , {J 0 , J 2 } = -J 1 , {J 1 , J 2 } = -J 0 , (A5)
which is nothing by the su(1, 1) algebra. At this point, it is not difficult to see that the associated covariant connection has the following components

A 0 a = -γ γ ω 03 a , A 1 a = γ γ ω 23 a , A 2 a = -γ γ ω 13 a . (A6)

FIG. 1 :

 1 FIG.1: Different Hamiltonian slicings of a spherical black hole space-time. The picture (b) represents the usual slicing in terms of space-like hypersurfaces which leads to the effective SU (2) Chern-Simons description of the black hole: In that case, the horizon appears as a boundary of Σ. In the picture (a), we have represented two slicings of the black hole space-time where Σ are Lorentzian hypersurfaces: these gauge choices would lead to new descriptions of black holes in Loop Quantum Gravity. In particular, the slicing which does not cross the horizon is interesting in view of a holographic description of black holes in the frame of Loop Quantum Gravity.

This case corresponds to a slicing of the space-time in a light like direction. Our analysis based on a partial gauge fixing can be adapted to that situation. Such a Hamiltonian description could provide us with a new formulation (eventually simpler) of gravity in the light front related to[START_REF] Alexandrov | First order gravity on the light front[END_REF].

Acknowledgments

We are particularly indebted to Alejandro Perez and Simone Speziale for numerous discussions on this project. K.N. want also to thank Jibril Ben Achour who initially collaborated on this project.

We recover as announced the same expression of the su(1, 1)-valued connection in the limit |χ| → ∞ (3.39) a part that we have interchanged the components 0 and 3 of space-time indices.

Appendix A: "Time" vs. "Space" gauge in the Holst action

The very well-known "time" gauge refers to the condition e 0 a which breaks sl(2, C) into su(2) in the Holst action. It corresponds to taking a slicing Σ × R of the space-time where the hypersurfaces Σ are space-like. In fact, one can easily generalize the time gauge by considering instead the condition e µ a n µ = 0 where n µ is a given fixed vector. When n µ is time-like, the slices Σ are spacelike (as for the time gauge where n µ = δ 0 µ ) whereas the slices are time-like when n µ is space-like. We want to study thus latter case in this appendix. To simplify the analysis, we assume (without loss of generality) that n µ = δ 3 µ . We are going to show that the Hamiltonian analysis of the Holst action such a gauge leads to a phase space which corresponds to the limit |χ| → ∞ and n i → δ 3 i . First, we notice that the only non vanishing components of π a IJ are E a α ≡ π a α3 with α ∈ (0, 1, 2). It is immediate to check that the simplicity constraints C ab ≈ 0 are satisfied. In this gauge, it is "natural" to choose the third direction to be the "time" parameter because of the slicing. Hence, the "symplectic" term (in the third direction) in the Holst action involves only the component ω α3 a of the spin-connection (with


published or not. The documents may come

Planck stars as observational probes of quantum gravity

Planck stars: new sources in radio and gamma astronomy?

Nature Astronomy 1 (2017) 0065

Carlo Rovelli

CPT, Aix-Marseille Université, Université de Toulon, CNRS, F-13288 Marseille, France.

A new phenomenon, recently studied in theoretical physics, may have considerable interest for astronomers: the explosive decay of old primordial black holes via quantum tunnelling. Models predict radio and gamma bursts with a characteristic frequency-distance relation making them identifiable. Their detection would be of major theoretical importance.

The expected signal may include two components [1]: (i) strong impulsive emission in the high-energy gamma spectrum (∼ T eV ), and (ii) strong impulsive signals in the radio, tantalisingly similar to the recently discovered and "very perplexing" [2] Fast Radio Bursts. Both the gamma and the radio components are expected to display a characteristic flattening of the cosmological wavelength-distance relation, which can make them identifiable [3,4].

The physics governing the decay is not exotic-in fact, it is conservative: just general relativity and quantum mechanics, physically reliable theories. However, lacking a consensual theory of quantum gravity, current models are hypothetical. Detection and identification of these signals would represent the first direct observation of a quantum-gravitational phenomenon.

A striking conclusion from the observations of the last decade is that our universe teems with black holes of widely different masses, spanning at least nine orders of magnitude-a conclusion reinforced by the recent gravitational-waves detection of the merger of two black holes of unexpected mass [5]. Black holes are sta- ble according to classical general relativity, but there is theoretical consensus that they decay via quantum processes. Until recently, the only decay channel studied was Hawking evaporation [6], a perturbative phenomenon too slow to have astrophysical interest: evaporation time of a stellar black hole is 10 50 Hubble times.

What can bring black hole decay within potential observable reach is a different, non-perturbative, quantum phenomenon: tunnelling, the same phenomenon that triggers nuclear decay in atoms. The explosion of a black hole out of its horizon is forbidden by the classical Einstein equations but classical equations are violated by quantum tunnelling. Violation in a finite spacetime region turns out to be sufficient for a black hole to tunnel into a white hole and explode [7]. The phenomenon does not violate causality, since it is the spacetime causal structure itself to tunnel.

In the process, the gravitationally collapsing matter falls inside its horizon until its density reaches the Planck density, namely the quantity with the dimension of a density determined by the Planck constant , the Newton constant G and the speed of light c-these are the quantities that set the scale of quantum-gravitational phenomena. At this stage, called "Planck star" [8], the star can still be much larger than the Planck length, but quantum effects are expected to make gravity strongly repulsive [9], triggering a bounce. The phenomenon is similar to the "quantum pressure" that prevents electrons from falling into an atomic nucleus.

Remarkably, because of the huge general-relativistic gravitational time dilation involved, collapse and bounce can be fast (milliseconds) in the proper time of the collapsing matter, and extremely slow (millions of years) in the external time. Thus the black holes we see in the sky can be bouncing Planck stars during their deep bounce phase, observed in extreme slow motion because of the very large gravitational time dilation.

This phenomenon is plausible on the basis of our current understanding of gravity and quantum theory; the theoretical uncertainty regards the scale of the decay time. If it is exponentially suppressed like generic macroscopic quantum tunnelling, it has no astrophysical consequences either. But the dumping exponential factor may be balanced by the phase-space factor due to the large black hole entropy, and arguments have been given [7] indicating that the decay time could be of the order of

τ ∼ m 2 m 2 Planck t Planck ( 1
)
where m is the mass of the hole, and t Planck and m Planck are the Planck time and mass. A detailed calculation of arXiv:1708.01789v1 [gr-qc] 5 Aug 2017 the hole decay time from first principles is undertaken in [10] using loop quantum gravity. For a black hole of planetary mass, equation (1) gives a lifetime of the order of the current Hubble time. This implies that primordial black holes -black holes formed by the large thermal fluctuations of the early universeof such mass may be exploding today. Such an explosion should release an energy of the order

E = mc 2 ∼ 10 47 erg, (2)
or the mass of a small planet, exploding suddenly from a region of millimetre size. See Figure 1 for an artistic illustration of the phenomenon. A low-energy component of the signal emitted at the explosion should be a powerful burst with wavelength of the order of the size of the hole, thus around the millimetre range [3]. This is the predicted signal tantalisingly close to the observed Fast Radio Bursts.

A second, high-energy, component is the photon gas originally collapsed in the early universe, liberated after the short-internal/long-external time. According to standard primordial black hole formation theory, the scale of the energy of these photons is determined by formation time, which in turn depends on the mass, and for millimetres black holes is expected in the T eV range.

The interesting aspect of the predicted signals is that their frequency-distance relation is expected to be different from the standard cosmological redshift. This is because holes we see exploding at cosmological distances have exploded in the past, therefore had a shorter life, and therefore, according to equation ( 1), should be smaller. For the radio component, smaller mass implies smaller size and therefore shorter emitted wavelength. For the gamma component, smaller mass implies, according to standard primordial black hole formation theory, a slightly earlier formation, when the plasma was hotter; this gives hotter photons trapped into the hole, and therefore, again, a shorter emission wavelength at explosion time (the internal proper time of the bounce is short, not allowing much internal evolution). Both emitted frequencies are thus higher for black holes we see exploding further away from us, partially compensating the cosmological redshift. The resulting flattened wavelengthdistance relation is [3]

λ obs ∼ 2Gm c 2 (1 + z) H -1 0 6 kΩ 1/2 Λ sinh -1 ΩΛ ΩM (z + 1) 3 (3)
(z is the redshift factor, H 0 , Ω Λ , Ω M are the Hubble constant and the cosmological-constant and matter densities) and is depicted in Figure 2. If observed, it would represent the smoking gun for identifying the Planck stars signals.

This modified redshift curve affects also the shape of the spectrum of the diffuse background due to the integrated emission of a population of bouncing black holes, opening up the possibility of revealing these signals as components of the cosmic ray background [4]. The resulting spectrum looks like a slightly distorted (by the redshift/distance integration) blackbody, depicted in Figure 3 for its high-energy component, a shape not expected from any other known astrophysical phenomenon.

Fast Radio Bursts observations are rapidly improving. CHIME is expected to observe dozens a day before the end of the year. Radio telescopes working in the radio, such as ALMA and SKA, could detect the low energy signal. Bursts are now receiving increasing attention, but the integrated emission may be easier to analyse. Theoretical models favour signals with shorter wavelengths around λ obs ∼ .2mm. There are detectors operating at these wavelengths, such as the Herschel instruments. The 200 micron range can be observed both by PACS and SPIRE. The predicted signal falls in between PACS and SPIRE sensitivity zones. A problem is that the bolometer technology makes detecting short black-hole bursts difficult: they are likely to be mistaken for cosmic ray noise. For the high energy component, the Fermi-LAT data could be particularly relevant.

Theoretical research in quantum gravity has been moving increasingly closer to phenomenology. The possibility of observing a quantum gravitational phenomenon is not anymore considered remote by theoreticians, and a number of possibilities have been suggested (see for instance [11]). Because of the smallness of the Planck scale, the observation of a quantum gravitational phenomenon requires a large multiplicative factor. For a Planck star, the large ratio of the Hubble time t Hubble to the Planck time t Planck provides such a factor, scaling up the Planck length L Planck to millimetres. Equation (1) gives indeed

t Hubble t Planck L Planck ∼ 1 mm. (4)
This is how a Planckian phenomenon can yield an effect at a macroscopic wavelength. Exploding black holes, or "Planck stars", represent a speculative but realistic possibility to observe quantum gravity effects. Detecting and identifying their signal in the sky would be of immense scientific value.

FIG. 1 .

 1 FIG. 1. Illustration of the Planck star phenomenology: matter collapses in the early universe forming a black hole that undergoes a rapid bounce. Because of the huge gravitational redshift, the subsequent explosion happens a cosmological external time later, producing signals we may observe.

FIG. 2 .

 2 FIG. 2. The flattened wavelength-distance relation equation (3) (continuous line) for the radio component of the Planck star signal, compared with the strandard redshift (dotted).

FIG. 3 .

 3 FIG. 3. Spectrum of the integrated emission of the gamma component of the Planck star signal [4].


Eduardo Basso

Claude Bourrely

Roman Pasechnik

Jacques Soffer

The landscape of W ± and Z bosons produced in pp collisions up to LHC energies

Keywords: electroweak gauge bosons, Drell-Yan process, parton distribution functions

We consider a selection of recent experimental results on electroweak W ± , Z gauge boson production in pp collisions at BNL RHIC and CERN LHC energies in comparison to prediction of perturbative QCD calculations based on different sets of NLO parton distribution functions including the statistical PDF model known from fits to the DIS data. We show that the current statistical PDF parameterisation (fitted to the DIS data only) underestimates the LHC data on W ± , Z gauge boson production cross sections at the NLO by about 20%. This suggests that there is a need to refit the parameters of the statistical PDF including the latest LHC data.

Introduction

The ongoing measurements at particle colliders such as RHIC and the LHC continue precision tests of particle production mechanisms. In this respect, there is a growing demand for a better understanding of underlined QCD uncertainties, in particular, related to modeling of parton density functions (PDFs), the key ingredients of QCD collinear factorisation. A major effort of the Particle Physics community over past decades has been directed towards constraining the QCD parton (quark and gluon) dynamics at various momentum scales connected, via DGLAP evolution, to the universal nonperturbative parton densities at some low scale Q 0 . The latter are not fully predicted by the first QCD principles but are usually parametrized and extracted from the data.

In recent years, production of electroweak gauge bosons, both charged (W ± ) and neutral (Z 0 , γ * ), has attracted a lot of attention from theory and experiments as an important test of QCD (see e.g. Refs. [1,2,3]). In particular, these processes are traditionally considered as an ideal tool for probing PDFs at various x and Q 2 [4,5,6]. For example, by controlling the c.m.s. energy √ s, di-lepton rapidity Y and invariant mass M in the Drell-Yan (DY) process pp → (Z 0 /γ * → l + l -) + X one could access quark and gluon PDFs at both small and large x i = (M/ √ s)e ±Y with i = 1, 2 denoting the incoming protons. While contributions from gluon and seaquark PDFs to gauge boson production dominate presumably in kinematic regions of the LHC (except, probably, highly forward regions of the phase space), at lower energies of RHIC one expects an increased sensitivity to valence quark distributions.

In our previous study [7], several most recent PDF parametrizations at the next-to-leading order (NLO) including the statistical PDF (known as NLO BS15) model [8] were used for a description of the existing DY data for the normalised differential distributions available from Tevatron and LHC measurements. While a fairly good description of the DY data at high energies has been found for all the chosen PDF sets, at low energies the PDF models exhibit more substantial differences in shapes of the invariant mass and x F distributions. Provided that the BS15 model having much fewer free parameters results in as good data description as other popular models so it should be considered on the same footing as the current global PDF fits. In this paper, we extend our previous analysis [7] to RHIC and LHCb kinematic regions incorporating also W ± production observables and verify the predic-tions of the statistical PDF model at NLO accounting for resummation up to the Next-to-Leading Log (NLL) level against the corresponding experimental data.

The paper is organized as follows. In Section 2, we review the main features of two sets of PDFs we have used for our calculations. In Section 3, we consider a selection of recent W ± production data sets available from STAR, CMS and LHCb and the DY-pair Z/γ * → l l production data -from the STAR and LHCb measurements. We make a comparison of theoretical predictions for the selected PDFs with these data. We give our summary and final remarks in Section 4.

Selection of two key PDF sets

We will now summarize the essential properties of two sets of PDFs which will be tested in our further analysis of W ± and DY-pair production in pp collisions at various energies.

The basic features of the statistical PDF approach alternative to canonical polynomial parametrizations which are inspired by the Regge theory at small x and by counting rules at large x have been discussed in Ref. [8]. In particular, (anti)quark distributions at the input scale

Q 2 0 = 1 GeV 2 xq h (x, Q 2 0 ) = A q X h 0q x bq exp[(x -X h 0q )/x] + 1 + Ãq x bq exp(x/x) + 1 , (1)
xq h (x, Q 2 0 ) = Āq (X -h 0q ) -1 x bq exp[(x + X -h 0q )/x] + 1 + Ãq x bq exp(x/x) + 1 , (2)
are defined in terms a quasi Fermi-Dirac function (first terms) and a helicity independent diffractive component (second terms). Note, the latter does not enter the quark helicity ∆q and valence q -q distributions. The multipliers X h 0q and (X -h 0q ) -1 in the diffractive contributions have been justified in the statistical approach to transverse momentum dependent PDFs in Ref. [9]. In Eqs. ( 1) and (2) for a given quark q with fixed helicity h = ± the parameters x and X h 0q play the role of universal temperature and thermodynamical potential encoding the main characteristics of the model (for antiquarks the sign of helicity and potentials is changed). Remarkably, the statistical PDF approach enables to describe both upolarised observables and helicity asymmetries. In what follows, however, only spin-independent observables are considered.

The statistical (anti)quark distributions contain in total eight parameters1 for a given q, namely, Āq , A q , Ãq , X ± 0q , bq , b q , and bq . Then, the valence sum rule, (q(x) -q(x))dx = N q , N q = 2, 1, 0 for u, d, s quarks, respectively, reduces the parameter space to seven free parameters. The additional constraints apply for q = {u, d} [10]

Āu = Ād , A u = A d , b u = b d , Ãu = Ãd , bu = bd , bu = bd , (3)
thus, leading to eight free parameters in the sector of light quarks such that the diffractive contribution is flavor independent (for more details, see e.g. Ref. [11]). The expression for the statistical gluon PDF at µ = Q 0 is inspired by the black-body spectrum and has a form of a quasi Bose-Einstein function

xG(x, Q 2 0 ) = A G x b G exp(x/x) -1 , (4)
where A G is found by the momentum sum rule such that b G is the only additional free parameter.

To conclude, the statistical PDF sets2 contain seventeen free parameters in total. Besides the temperature x and the exponent of the gluon distribution b G , the light u, d and strange s quark PDFs are constructed in terms of eight and seven free parameters, respectively. These were fitted to a large set of accurate unpolarised and polarised DIS data only at the NLO QCD level and therefore are denoted as NLO BS15 from now on.

Let us discuss the second PDF set used in our calculations of the unpolarized cross sections below. Several versions were proposed by the CTEQ-TEA global analysis of QCD up to NNLO [12] including data from HERA, Tevatron and LHC. For each flavor the Regge-motivated parametrization is of the form

xf a (x, Q 2 0 ) = x a 1 (1 -x) a 2 P a (x) , (5)
with a slowly-varying polynomial factor P a (x). In the CT14 model [13] this factor for valence distributions is represented in terms of a linear combination of Bernstein polynomials

P qv = d 0 p 0 (y) + d 1 p 1 (y) + d 2 p 2 (y) + d 3 p 3 (y) + d 4 p 4 (y) , y = √ x , p 0 (y) = (1 -y) 4 , p 1 (y) = 4y(1 -y) 3 , p 2 (y) = 6y 2 (1 -y) 2 , p 3 (y) = 4y 3 (1 -y) , p 4 (y) = y 4 . (6)
By fixing d 1 = 1, d 3 = 1 + a 1 /2 and using the valence sum rule the number of free parameters for each flavor is reduced to four. Thus, eight parameters fully determine the valence u v and d v distributions. Due to fewer constraints on the gluon distribution, the CT14 gluon PDF is constructed in terms of a lower-order polynomial

P g (y ′ ) = g 0 [e 0 q 0 (y ′ ) + e 1 q 1 (y ′ ) + q 2 (y ′ )] , q 0 (y ′ ) = (1 -y ′ ) 2 , q 1 (y ′ ) = 2y ′ (1 -y ′ ) , q 2 (y ′ ) = y ′ 2 , (7)
where

y ′ = 1-(1- √ x) 2 = 2 √ x-x.
With an account for the momentum sum rule the gluon PDF is determined by five free parameters in total. Fourthorder polynomials in the same variable y ′ as for the gluon PDF were employed for building the sea d and ū distributions assuming ū(x)/ d(x) → 1 at x → 0. Altogether, in the CT14 model the sectors of valence and sea quark PDFs contain eight and thirteen free parameters, respectively, while the gluon PDF contains five parameters amounting to twenty six fitting parameters in total. The (N)NLO QCD global fits in this model have been performed at Q 0 =1.295 GeV.

In numerical analysis we employ the up-to-date computing tools available for high precision PDFs studies. All the numerical results presented below were obtained using the DYRes code [14] which computes the DY observables up to NNLO performing the resummation of logarithms that become large when the vector boson transverse momentum is much smaller than the boson mass. Such a resummed result is matched order by order, up to O(α 2 s ), with the fixed-order result at large transverse momenta which is obtained within the dipole formalism [15] and is implemented in the MCFM code [16,[START_REF] Campbell | MCFM -Monte Carlo for FeMtobarn processes[END_REF]. The divergences in the fixed-order calculation at small p T are subtracted resulting in a modification of the DYNNLO program [START_REF] Catani | [END_REF] which was used in our previous DY analysis [7].

Although the resummation may not improve the results at larger values of transverse momentum being probed in the present study, its results have shown to agree with those from fixed-order calculations [14]. Since both the resummed and the fixed order calculation are expected to agree at large p T , we will proceed using DYRes at the NLL level throughout all the predictions below unless there is a notable disagreement between DYRes and the fixedorder results provided by DYNNLO. We indeed observed an overall agreement between both frameworks for most of the results presented bellow, with an exception that DYNNLO results on the W + /W -ratio at NLO appear to describe the current LHCb data noticeably better than the DYRes ones.

3 Results for W ± and Z/γ * observables Our previous analysis of recent Z-boson production observables measured at different energies including recent LHC data in Ref. [7] (and normalised to the total cross section) has shown an overall very good consistency of the BS15 PDF model with the data. Does the same situation persist for W ± observables?

Note, in what follows we use the BS15 model that was fitted to the DIS data only and apply it in analysis of the data obtained at energies as high as the LHC ones. This step is needed to estimate the underlined theoretical uncertainties and to justify whether or not a global fit (including the LHC data) is needed for the statistical PDF parameterisation. In all the figures below, we have used DYRES to the NLL accuracy which is sufficient to see an overall consistency with the data. Clearly, when global fits with BS model at the NLO are performed in the future, one could go higher in the resummation accuracy if needed.

Consider first, as displayed in Fig. 1, a comparison of preliminary STAR data [19] on the W + to W -differential cross section ratio

W + /W -≡ dσ + (η e )/dη e dσ -(η e )/dη e
with theoretical predictions based on the BS15 and CT14 PDF models. It is clear that both PDFs are consistent with the trend of the data given its poor uncertainty, but the planned 2017 run at BNL RHIC will be able to increase substantially the precision on this measurement. We observe that both predictions are almost sitting on top of each other, but they would be more distinguishable for larger η e . However, this region is not accessible to STAR and the only way to find out which one agrees best with the data is to measure the individual cross sections near η e = 0. This is indeed what one sees clearly in Fig. 2, where the BS15 result remains below the CT14 one in both W + and W -cases. We urge the STAR Collaboration to perform this important test.

Figure 1: The ratio the W + and W -differential cross sections at √ s = 500 GeV, versus the pseudo-rapidity of the charged lepton η e , obtained with the BS15 and CT14 PDF models and compared to the preliminary data from the STAR experiment [19].

The individual W ± differential cross sections have been measured at the LHC at √ s = 8 TeV by the CMS Collaboration [20] and are shown in Fig. 3 together with the corresponding theoretical predictions using the BS15 and CT14 PDF models. While the general trends for the W + cross section are similar to that of the data for both BS15 and CT14 models, the shape of the BS15 prediction for the W -cross section somewhat deviates from the data in the high η µ region. As we shall see below, this implies a prediction for the charge asymmetry that misses the data by a few percent. Unlike the neutral-current Z-boson observables discussed earlier in Ref. [7], the CT14 and especially BS15 predictions fail to describe the charged-current precision data. The BS15 model prediction is off by about 20% while the CT14 is doing much better and is only 3% off the data in overall normalisation.

These potentially indicate a strong need in making a global fit of the BS15 NLO model parameters including the LHC data. This stricking issue with normalisation was not seen earlier in the Z 0 -boson production observables that were always normalised to the total cross section and strongly suggests future studies in this direction.

Another way to present the data is the charge asymmetry defined as

dσ + (η µ )/dη µ -dσ -(η µ )/dη µ dσ + (η µ )/dη µ + dσ -(η µ )/dη µ ,
which is related to W + /W -ratio considered above. This is shown in Fig. 4 (left), but the disagreement of the BS15 and CT14 predictions with the data persists for a limited η region. The shape for BS15 model in the case of W -, as seen in Fig. 3, is the main source of deviation in the asymmetry. In Fig. 4 (right) we present the predictions for W + /W -ratio at √ s = 13 TeV.

This ratio was also measured at √ s = 8 TeV in a large pseudo-rapidity region by the LHCb Collaboration [21] (see Fig. 5). In order to compare the fixed-order (NLO) vs resummed (NLO+NLL) results with the data in Fig. 5 we show both DYNNLO and DYRES results. We notice that DYNNLO works somewhat better against the data than DYRES in low η region for both PDF parameterisations. Even in the case of the fixed-order DYNNLO analysis, one again notices a larger deviation from data for BS15, which reinforces the importance of a detailed analysis of the statistical PDF model, including global fits to the LHC data, to investigate the reasons for the W -deviation in shape from data, and also to correct for the overall normalization when considering LHC data. Such an analysis is beyond the scope of the preset work, but should be carried out in the future to improve the reliability of the statistical model. The fixed-order DYNNLO (NLO) results are compared to the resummed DYRES (NLO+NLL) results.

Several important aspects of the available data on Z/γ * production have already been studied in our earlier paper [7]. In addition, in Fig. 6 we show the predictions using the BS15 and CT14 PDFs for the differential cross section for Z/γ * production, versus the dilepton rapidity, in view of a future data taking from STAR at BNL RHIC. Once again it is not possible to distinguish them in this limited kinematic region. Indeed, an insufficient precision of the current experimental data makes it difficult to verify the PDF models so further improvements toward a reduction of experimental uncertainties are needed. In Fig. 7, we display the normalized differential cross sections for the forward Z/γ * production from the LHCb experiment at two different energies ( √ s = 8 TeV [21] and √ s = 13 TeV [22]) which turn out to be in a very good agreement with the predictions using the BS15 and CT14 PDF models. Finally, a similar situation holds for BS15 and CT14 predictions for the dilepton transverse momentum distribution against most recent CMS data at √ s = 8 TeV as shown in Fig. 8.

Summary

As a natural continuation of our previous study on neutral-current Z-boson production observables [7], the main purpose of this analysis is to further test the statistical BS15 NLO model for parton density functions fitted only to DIS data versus the most recent RHIC and LHC data on charged-current W ± bosons production.

In this paper, we have studied the W ± charge asymmetry and the differential (in lepton pseudorapidity) W ± cross sections, as well as the differential Drell-Yan (DY) pair Z/γ * → l l production cross sections, at RHIC The analysis of neutral-current (DY) observables (normalised to the total cross section) results in a fairly good description of the latest data, in full consistency with our previous analysis in Ref. [7]. Now, we have looked into such observables as the dilepton rapidity distributions (normalised to the total DY Z/γ * cross section) at √ s = 8 and 13 TeV in the LHCb kinematic regions, as well as the dilepton transverse momentum distribution at √ s = 8

TeV. No noticeable deviations in shapes of the BS15 and CT14 predictions at NLO+NLL versus data have been observed. The normalisation of the DY differential cross sections is not reproduced by both PDF models although CT14 NLO predictions are closer to the data than the BS15 ones, in particular, due to the fact the DY LHC data were not included into the BS15 fits. Both BS15 and CT14 models work quite well also against the most recent data from RHIC on the W + /W -ratio in the electron pseudorapidity region -1.5 < η e < 1.5 at √ s = 500 GeV. For a more definite conclusion, one should have data on the individual W ± distributions since the corresponding Figure 8: The differential cross section for DY-pair Z/γ * → µ + µ -production, versus dilepton transverse momentum, against the data by the CMS Collaboration [23], with the cuts corresponding to the acceptance of the CMS detector. The predictions use the BS15 and CT14 PDF models, for comparison.

BS15 and CT14 predictions for the W + and W -distributions differ in overall normalisation by a few percent.

Notably, the BS15 model starts to exhibit larger discrepancies with respect to the W ± production data at the LHC energies. In particular, its predictions for both W + and W -total cross sections are off by about twenty percent with respect to the CMS data at √ s = 8 TeV. While the shape of W + pseudorapidity distribution, dσ + /dη µ , marginally reproduces that of the data, the shape of the W -pseudorapidity distribution, dσ -/dη µ , is flatter and somewhat deviates from that of the data leading to a few-percent discrepancy at η µ ≃ 1.5 -2.0. The latter discrepancy then translates into the corresponding deviations of the W charge asymmetry at mid-pseudorapidities as compared to the CMS data.

Interestingly enough, at forward pseudorapidities in the LHCb kinematic region, the shape of the lepton pseudorapidity distributions are in an overall consistency with our above conclusions at mid and central pseudorapidities. This is suggested by our analysis of W + /W -ratio against the LHCb data, with up to 20% deviation of the BS15 prediction versus data at η µ > 3.5 primarily caused by a deviating behavior of the W -pseudorapidity distribution. This is not surprising since (i) the BS15 fit has less free parameters than CT14, and (ii) the CT14 PDF fit includes also LHC (and Tevatron) W ± , Z 0 boson data. Such discrepancies in the shape of the W -pseudorapidity distribution and overall normalisation versus the CMS and LHCb data are most likely due to the fact that the starting BS15 parametrizations were fitted to the DIS data only and have not been included into the global fit yet. Indeed, the missing NNLO/NNLL corrections could not substantially change the normalisation of the cross sections although may, in principle, somewhat affect the shapes of the differential distributions. The effect from a global fit accounting for the LHC data is expected to provide the biggest impact on the overall normalisation where the discrepancy of the BS model and the data are the most pronounced. This analysis should incorporate heavy quark (in particular, charm and beauty) PDFs properly as the DIS was not particularly sensitive to those and could not constrain them well, in distinction with the LHC data. This is an important subject for a future work that is needed in order to further test the reliability of the statistical approach. Finally, the global analysis of the statistical PDF model accounting for all available data at various energies up to LHC will verify if the number of parameters in the model should remain the same or certain modifications will be required. Clearly, the precision data on the absolute cross sections and individual W ± distributions in lepton pseudorapidity at RHIC and LHC energies will be necessary for such a study.

Figure 2 :

 2 Figure 2: The differential cross sections for W + , dσ + (η e )/dη e (left), and W -, dσ -(η e )/dη e (right), production at √ s = 500 GeV, versus pseudo-rapidity of the charged lepton η e , with the cuts corresponding to the acceptance of the STAR detector, as predicted by the BS15 and CT14 PDF models.

Figure 3 :

 3 Figure 3: The differential cross sections for W + , dσ + (η µ )/dη µ (left), and W -, dσ -(η µ )/dη µ (right), production at √ s = 8 TeV, versus pseudo-rapidity of the charged lepton η µ , with the cuts corresponding to the acceptance of the CMS detector, as predicted by the BS15 and CT14 PDF models, in comparison to the data by CMS Collaboration [20].

Figure 4 :

 4 Figure 4: The W ± charge asymmetry at √ s = 8 TeV, versus pseudo-rapidity of the charged lepton η µ , with the cuts corresponding to the acceptance of the CMS detector, as predicted by the BS15 and CT14 PDF models, is shown in the left panel in comparison to the data by CMS Collaboration [20]. The prediction for W + /W -ratio based on BS15 and CT14 PDF models at √ s = 13 TeV are shown in the right panel.

Figure 5 :

 5 Figure 5: The W + /W -ratio based on BS15 and CT14 PDF models at √ s = 8 TeV in comparison to the data from the LHCb Collaboration [21].

Figure 6 :

 6 Figure 6: The differential cross section for DY-pair Z/γ * → l l production, versus dilepton rapidity, with the cuts corresponding to the acceptance of the STAR detector. The predictions use the BS15 and CT14 PDF models, for comparison.

Figure 7 :

 7 Figure 7: The normalized differential cross section for the forward DY-pair Z/γ * → l l production as a function of dilepton rapidity against the data by the LHCb Collaboration, at √ s = 8 TeV (left) [21] and at √ s = 13 TeV(right)[22]. The experimental data are compared to the predictions using the BS15 and CT14 PDF models.

It turns out that X - 0u and X - 0d were found almost identical.

2 In Ref.[8] we have also considered the helicity gluon distribution which is irrelevant in the present work.

Acknowledgments The authors would like to thank M. Posik for helpful correspondence. R. P. was partially supported by the Swedish Research Council, contract number 621-2013-428, and by CONICYT grant PIA ACT1406 (Chile). E. B. is supported by CAPES and CNPq (Brazil), contract numbers 2362/13-9 and 150674/2015-5.