Jérémy Sok

Ceremade

Existence of Ground State of an Electron in the BDF Approximation

The Bogoliubov-Dirac-Fock (BDF) model allows to describe relativistic electrons interacting with the Dirac sea. It can be seen as a mean-field approximation of Quantum Electro-dynamics (QED) where photons are neglected.

This paper treats the case of an electron together with the Dirac sea in the absence of any external field. Such a system is described by its one-body density matrix, an infinite rank, self-adjoint operator which is a compact pertubation of the negative spectral projector of the free Dirac operator.

The parameters of the model are the coupling constant α > 0 and the ultraviolet cut-off Λ > 0: we consider the subspace of squared integrable functions made of the functions whose Fourier transform vanishes outside the ball B(0, Λ). We prove the existence of minimizers of the BDF-energy under the charge constraint of one electron and no external field provided that α, Λ -1 and α log(Λ) are sufficiently small. The interpretation is the following: in this regime the electron creates a polarization in the Dirac vacuum which allows it to bind.

We then study the non-relativistic limit of such a system in which the speed of light tends to infinity (or equivalently α tends to zero) with α log(Λ) fixed: after rescaling the electronic solution tends to the Choquard-Pekar ground state. 4.3.4 Estimates around the fixed point method . . . . . . .

Introduction

The relativistic quantum theory of electrons is based on the free Dirac operator D 0 = -i cα • ∇ + mc 2 β. Here β and α k are the C 4 × C 4 matrices:

β := Id2 0 0 -Id2 , α k = 0 σj σ k 0 , σ1 = 0 1 1 0 , σ2 = 0 -i i 0 , σ3 = 1 0 0 -1 .
The free Dirac operator D 0 acts on 4-spinors, that is on H = L 2 (R 3 , C 4 ) which is the Hilbert space of one relativistic electron. It is self-adjoint with domain H 1 (R 3 , C 4 ) and form domain H 1/2 (R 3 , C 4 ). Moreover (D 0 ) 2 = m 2 c 4 -2 c 2 ∆. We write: P 0 -= 1 -P 0 + := χ (-∞,0) (D 0 ).

It is well known that its spectrum is σ(D 0 ) = (-∞, -mc 2 ] ∪ [mc 2 , +∞) leading to difficulties in relativistic quantum mechanics. This operator was introduced by Dirac to describe the energy of a free particle with spin 1 2 (e.g. an electron). To explain why electrons with negative energies are not observed, Dirac postulated that all the negative energy states are already occupied by virtual electrons, the so-called Dirac sea. By the Pauli principle, a real electron cannot have a negative energy.

We study an approximation of no-photon Quantum Electrodynamics (QED) allowing to describe the behavior of relativistic electrons in an external field interacting with the virtual electrons of the Dirac sea via the electrostatic potential in a mean-field type theory. This so-called Bogoliubov-Dirac-Fock (BDF) model was introduced by Chaix and Iracane [START_REF] Chaix | From quantum electrodynamics to mean-field theory: I. The Bogoliubov-Dirac-Fock formalism[END_REF] and then studied by Bach et al. in [START_REF] Bach | On the Stability of the Relativistic Electron-Positron Field[END_REF], by Hainzl et al. in [START_REF] Hainzl | Non-perturbative mass and charge renormalization in relativistic no-photon quantum electrodynamics[END_REF][START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF][START_REF] Hainzl | Self-consistent solution for the polarized vacuum in a no-photon QED model[END_REF][START_REF] Hainzl | The Mean-Field Approximation in Quantum Electrodynamics. The no-photon case[END_REF][START_REF] Hainzl | Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics[END_REF] and by Lewin et al. in [START_REF] Gravejat | Ground state and charge renormalization in a nonlinear model of relativistic atoms[END_REF]. In particular in those last papers, the authors are interested in the existence of ground states for this variational model.

Let us sketch how the BDF model is derived from full QED. We use relativistic units = c = 1 and set the bare particle mass equal to 1 and α = e 2 /(4π). When photons are neglected, the (formal) Hamiltonian H φ of QED acts on the Fock space F of H [START_REF] Thaller | The Dirac Equation[END_REF]:

H φ = Ψ * (x)D 0 Ψ(x)dx -φ(x)ρ(x)dx + α 2 ρ(x)ρ(y) |x -y| dxdy. (2)
Here Ψ(x) is the second-quantized field operator, φ is the external field and ρ(x) is the density operator:

ρ(x) = 4 σ=1
Ψ * (x)σΨ(x)σ -Ψ(x)σΨ * (x)σ .

(3)

2

In the presence of an external density ν, the corresponding external field is φ = αν * 1 |•| . This Hamiltonian is not bounded from below and it is not possible to solve the corresponding minimization problems.

The BDF variational model is obtained from this Hamiltonian by making several approximations.

The first one consists in restricting the energy to special states in F, the socalled Bogoliubov-Dirac-Fock (BDF) states. They are states ΩP which are fully described by their one-body density matrix P : P (x, y)σ,τ = ΩP |Ψ * (x)σΨ(y)τ |ΩP F .

(4)

For instance the vacuum state Ω0 (no electron and no positron) in F is a BDF state with one-body density matrix P 0 -. One must consider them as an infinite Slater determinant f1 ∧ f2 ∧ • • • where (fi) i≥1 is an orthonormal basis of the range Ran(P ) of P . We will write P instead of ΩP for a BDF state: the QED energy can be written in terms of P .

In [START_REF] Hainzl | The Mean-Field Approximation in Quantum Electrodynamics. The no-photon case[END_REF], Hainzl et al. study the corresponding minimization problem of H 0 in the space H L Λ of functions in L 2 ([-L/2, L/2) 3 , C 4 ) (with periodic boundary conditions) whose Fourier transform vanishes outside the ball B(0, Λ); the constant Λ > 0 is the so-called ultraviolet cut-off. This space has finite dimension and the corresponding Hamiltonian H 0 L is well-defined. It is then shown that, for each L > 0 and 0 < α < 4/π, there exists a minimizer PL = γL + 1 2 among BDF states (with energy EL(0)) and that in the thermodynamic limit L → +∞, ΓL tends in some sense to a self-adjoint, translation-invariant operator Γ0 of HΛ:

HΛ := {f ∈ H, supp f ⊂ B(0, Λ)}. (5)
Moreover Γ0 satisfies the following self-consistent equations:

     Γ0 = - sign(D 0 ) 2 , D 0 = D 0 -α Γ0(x, y) |x -y| . (6)
The operator P 0 -= Γ0 + 1 2 is the orthogonal projection χ (-∞,0) (D 0 ) and we write P 0 + = 1 -P 0 -. The operator D 0 has been previously introduced in [START_REF] Lieb | Renormalization of the regularized relativistic electron-positron field[END_REF] but in another context.

We will now take P 0 -as reference state. For a one-body density matrix P , the formal difference between the QED energies E ν QED (P - 1 2 ) and E 0 QED (P 0 --1 2 ) gives the following function of Q := P -P 0 -:

     E ν BDF (Q) = Tr D 0 (P 0 -QP 0 -+ P 0 + QP 0 + ) -α φ(x)ρQ(x)dx + α 2 D(ρQ, ρQ) -Ex[Q] , D(ρQ, ρQ) := ρQ(x) * ρQ(y) |x -y| dxdy, Ex[Q] := |Q(x, y)| 2 |x -y| dxdy.
The function E ν BDF is the BDF energy we will deal with in this paper. Notation 1. Throughout this paper we write P 0 ε QP 0 ε ′ = Q ε ε ′ where ε, ε ′ ∈ {+/-}. For an operator Q with integral kernel Q(x, y), we define RQ by its integral kernel: RQ(x, y) := Q(x,y) |x-y| . There holds:

Q 2 Ex := Ex[Q] = Tr(R * Q Q).
We write C the Hilbert space of densities with finite Coulomb energy:

C := ζ ∈ S ′ (R 3 ), ζ 2 C := 4π | ζ(k)| 2 |k| 2 dk < +∞ . (7)
The squared norm ζ 2 C coincides with

R 3 ×R 3
ζ * (x)ζ(y) dxdy |x-y| whenever this last integral converges.

A justification to study the BDF energy -stated in [START_REF] Hainzl | The Mean-Field Approximation in Quantum Electrodynamics. The no-photon case[END_REF] -is the following. In the presence of an external charge density ν such that D(ν, ν) < +∞ and that ν continuous is in B(0, Λ), one can consider the corresponding minimization problem of H φ in H L Λ . There also exists a minimizer with energy EL(φ) and in the thermodynamic limit:

lim L→+∞ (EL(φ) -EL(0)) = inf Q∈Q Λ E ν BDF (Q),
where QΛ := {Q ∈ S2(HΛ), -P 0 -≤ Q ≤ P 0 + , Q ++ , Q --∈ S1(HΛ)}.

Notation 2. We recall that for each 1 ≤ p ≤ +∞, Sp(HΛ) is the subspace of compact operators A ∈ B(HΛ) with Tr|A| p < +∞. The case p = 1 gives trace-class operators and p = 2 gives Hilbert-Schmidt operators. We recall Q is Hilbert-Schmidt if and only if its integral kernel is in L 2 (HΛ × HΛ).

Instead of minimizing over all states in QΛ, we may minimize over sector charge QΛ(q), q ∈ R : QΛ(q) := {Q ∈ QΛ,

Tr(Q ++ + Q --) = q}. (8)
The number q is interpreted as the number of electrons (if q ∈ N * ) or the number of positrons (if q ∈ Z\N). In the presence of an external field ν, the energy function is then defined as

E ν BDF (q) := inf E ν BDF (Q), Q ∈ QΛ(q) . (9)
In [START_REF] Hainzl | Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics[END_REF], Hainzl et al. have shown that for any q0 ∈ R, the problem E ν BDF (q0) admits a minimizer as soon as there hold binding inequalities:

∀ q ∈ R\{0}, E ν BDF (q0) < E ν BDF (q0 -q) + E 0 BDF (q). (10)
A more difficult task is to check these inequalities hold for some q0. In [START_REF] Hainzl | Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics[END_REF], by this method it is proved that for any ν ∈ L 1 (R 3 , R+) ∩ C and any integer M such that 0 ≤ M < ν + 1, the problem E ν BDF (M ) admits a minimizer (a so-called ground state) close to the limit α → 0 with Λ = Λ0 > 0 kept fixed.

In this paper we show there exists a minimizer for E 0 BDF (1), provided α, Λ -1 and α log(Λ) are sufficiently small. It is remarkable that the system of one electron in the Dirac sea can bind in the absence of any external field: this answers an open question stated in [START_REF] Hainzl | The Mean-Field Approximation in Quantum Electrodynamics. The no-photon case[END_REF] (page 19). The presence of the electron induces the polarization of the Dirac sea: it is locally repelled in the neighbourhood of the particle. This fact is illustrated by the inequality E 0 BDF (1) < m(α) where m(α) is the infimum of the BDF energy among configurations where the Dirac sea, represented by P 0 -, is not polarized:

m(α) = inf φ∈Ker P 0 - E 0 BDF (|φ φ|) = inf σ(|D 0 |).
We are then interested in the non-relativistic limit α → 0 with α log(Λ) kept fixed to a small value (it may not be 0). The wave function ψ of the real electron has a specific behaviour. There exists c(α, Λ) > 0 with c = O(α -2 log(Λ) -1 ) such that up to translation and up to scaling by c > 0, the upper spinor of the wave function ψ tends to a minimizer of the Choquard-Pekar energy ECP [START_REF] Lieb | Existence and uniqueness of the minimizing solution to Choquard's nonlinear equation[END_REF]:

ECP := inf φ∈H 1 (R 3 ): φ L 2 =1 ECP(φ) := |∇φ| 2 dx -D(|φ| 2 , |φ| 2 ) < 0. (11)
More precisely the Choquard-Pekar energy ECP of ψ(x) := c 3/2 ψ(cx) tends to ECP.

The link with a model of polaron is natural: the Dirac sea is a polarizable system and like a lattice of ions reacts to the presence of an electron. The smallness of α log(Λ) corresponds to a small charge renormalisation. As explained in [8, part 4], the physical coupling constant α phys is different from its "bare" value α. More precisely in the reduced BDF model, where the exchange term is neglected, a minimizer of E ν BDF with ν ≥ 0 radial (interpreted for instance as ν = Z protons) and D(ν, ν) small enough has radial density ργ [START_REF] Gravejat | Ground state and charge renormalization in a nonlinear model of relativistic atoms[END_REF], the potential induced by ν at infinity is not αZ 1 |x| as it should be but rather

(ν -ργ ) * 1 |•| (x) ∼ |x|→+∞ α phys Z 1 |x|
where

α phys = αZ3, Z3 = 1 1 + αB 0 Λ (0) and B 0 Λ (0) = 2 3π log(Λ) + O(1). (12)
The quantity BΛ(0) 0 is the value at k = 0 of the function defined in Notation 5 below and Z3 is the charge renormalization constant. If we assume the charge renormalization in the full model to be a perturbation of [START_REF] Lewin | Derivation of Pekar's Polarons from a Microscopic of Quantum Crystals[END_REF], fixing 0

< α log(Λ) = L0 ≪ 1 corresponds to considering 0 < 1 -Z3 ≪ 1.
In this paper we have chosen the model of [START_REF] Hainzl | The Mean-Field Approximation in Quantum Electrodynamics. The no-photon case[END_REF] with P 0 -as reference state instead of that of [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF][START_REF] Hainzl | Self-consistent solution for the polarized vacuum in a no-photon QED model[END_REF] with P 0 -as reference state, but all the results proved here are also true in this last model with the same proofs.

The paper is organized as follows: in the next section we properly state the variational problem E 0 BDF (1) and state the main theorems. Subsections 3.1 and 3.2 are devoted to introduce the Banach spaces and the inequalities used throughout the paper. Theorem 1 gives an upper bound of E 0 BDF (1) which is the BDF energy of a test function Γ. This test function is defined by adapting the fixed point scheme in [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF]: the method is explained in Subsection 3.3 and the needed estimates in Appendices B.2 and B.3. Then Proposition 1 states that the binding inequalities at level 1 are true for E 0 BDF , as a consequence there exists a minimizer for E 0 BDF (1). Theorem 2 gives a lower bound of E 0 BDF (1) by computing the BDF energy of a minimizer. The two theorems and the proposition are proved in Section 4. At last we look at the nonrelativistic limit in Theorem 3. Appendix A is devoted to prove estimates linked to the use of the operator D 0 .

Description of the model and main results

We start with some definitions and notations. Our convention for the Fourier transform F is:

∀f ∈ HΛ ∩ L 1 (R 3 , C 4 ), f (p) := 1 (2π) 3/2 f (x)e -ix•p dx.
In Fourier space D 0 takes the following form

D 0 (p) = α • ωpg1(|p|) + g0(|p|)β, ωp = p |p| , (13)
where g0, g1 : [0, Λ) → R+ are real and smooth functions satisfying

x ≤ g1(x) ≤ xg0(x). (14)
It is possible to improve estimations of [START_REF] Lieb | Renormalization of the regularized relativistic electron-positron field[END_REF] in the regime L := α log(Λ) = O(1): we get estimates of the derivatives of g0, g1 by using their self-consistent equation (cf Appendix A). We write m(α) for the bottom of σ(|D 0 |):

m(α) := g0(0) = min(σ(|D 0 |)). (15)
We introduce the following notations concerning the Dirac operator:

Notation 3. We write E (p) := g0(p) 2 + g1(p) 2 = |D 0 (p)| and E (p) := 1 + |p| 2 = |D 0 (p)|.
We write g0 (respectively g1) for both functions g⋆ :

x ∈ [0, Λ] → g⋆(x) ∈ R + and g⋆ : p ∈ B(0, Λ) → g⋆(|p|) ∈ R + . The (g0)'s are in C ∞ while g1 ∈ C 1 (B(0, Λ)) (cf Appendix A). At last we write    g1 : p ∈ B(0, Λ) → g1(|p|)ωp ∈ R 3 g : p ∈ B(0, Λ) → g0(p) g1(p) ∈ R 4 .
Notation 4. C1 ≥ 1 denotes a constant satisfying g1(r) ≤ C1|r| and |g0|∞ ≤ C1. Notation 5. A recurrent function of this problem is

BΛ(k) := 1 π 2 |k| 2 |p=l+ k 2 |,|q=l- k 2 |≤Λ E (p) E (q) -g(p) • g(q) E (p) E (q) ( E (p) + E (q)) dl. (16)
If we replace E (•) by E(•) we get the function B 0 Λ of [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF][START_REF] Gravejat | Ground state and charge renormalization in a nonlinear model of relativistic atoms[END_REF]. We define the function bΛ(k) by the formula bΛ(k

) := αB Λ (k) 1+αB Λ (k) . (17)
In Appendix A it is shown that BΛ(k) = O(log(Λ)) and that for L ≪ 1 there holds BΛ(0) = 2 3π log(Λ) + O(L log(Λ) + 1). We consider then the P 0 --trace (P 0 -is defined in the introduction):

Tr0(Q) := Tr(P 0 -QP 0 -) + Tr(P 0 + QP 0 + ), P 0 + := 1 -P 0 -. (18)
As shown in [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF] we know the operators Q --= P 0 -QP 0 -and Q ++ = P 0 + QP 0 + are trace-class when Q ∈ S2(HΛ) is a difference of two orthogonal projectors of the form Q = P -P 0 -. In this case:

|Q| 2 = Q 2 = Q ++ -Q --.
We introduce the set of P 0 --trace class operators:

S P 0 - 1 (HΛ) = Q ∈ S2(HΛ) : Q ++ , Q --∈ S1(HΛ) .
The variational set QΛ (cf introduction) is a convex set of S P 0 -1 (HΛ) and its extremal points are that of the form Q = P -P 0 -where P is an orthogonal projector. The density of an operator Q ∈ QΛ is ρQ(x) = Tr C 4 (Q(x, x)). It is mathematically well defined since Q is locally trace-class (thanks to the cut-off). The Fourier transform of ρQ is:

ρQ(k) := 1 (2π) 3/2 |u+ k 2 |,|u- k 2 |≤Λ Tr C 4 ( Q(u + k 2 , u -k 2 ))du, (19)
In the absence of external field, the energy functional defined on QΛ is

E 0 BDF (Q) = Tr0(D 0 Q) + α 2 D(ρQ, ρQ) -Q 2 Ex .
The trace part is the kinetic energy while the two others are respectively the direct term and the exchange term. Moreover the following inequalities hold [1, 7, 10]

Tr0(D 0 Q) = Tr(|D 0 |(Q ++ -Q --)) ≥ Tr(|D 0 |Q 2 ), (20a)
|Q(x, y)| 2 |x -y| dxdy ≤ π 2 Tr(|D 0 |Q 2 ). (20b)
Inequality (20b) is due to Kato's inequality(37b). We assume that α < 4 π : in this case E 0 BDF is bounded from below [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF]. We study the variational problem E 0 BDF (1). To ensure the existence of a minimizer for E 0 BDF (1), it suffices to prove the following binding inequalities [START_REF] Hainzl | Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics[END_REF]. Proposition 1. There exist three constants α0, L0, Λ0 > 0 such that if 0 < α ≤ α0, 0 < L ≤ L0 and Λ ≥ Λ0, then:

∀ q ∈ R\{0, 1} : E 0 BDF (1) < E 0 BDF (1 -q) + E 0 BDF (q). (21)
This Proposition comes as a corollary of the following Theorem.

Theorem 1. There exist three constants α0, L0, Λ0 > 0 such that if α ≤ α0, L ≤ L0, Λ ≥ Λ0 then:

E 0 BDF (1) ≤ m(α) + (αbΛ(0)) 2 m(α) 2g ′ 1 (0) 2 ECP + o((αbΛ(0)) 2 ) < m(α), ( 22
)
where ECP is the Choquard-Pekar energy (see [START_REF] Hainzl | Non-perturbative mass and charge renormalization in relativistic no-photon quantum electrodynamics[END_REF]).

Remark 1. For sufficiently small α log(Λ) we have g ′ 1 (0) > ε > 0. More generally all the results we need about g0 and g1 are proved in Appendix A. Notation 6. Throughout this paper we work in the regime

α ≪ 1, Λ ≫ 1, α log(Λ) = L ≤ ε0, (23)
γ + P 0 - = χ (-∞,0) (DQ), DQ := D 0 + α ρQ * | • | -1 -Q(x,y) |x-y| , |ψ ψ| = χ [0,µ] (DQ). (24)
The number 0 < µ < m(α) can be chosen such that DQψ = µψ.

Thanks to Proposition 1 of [START_REF] Hainzl | Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics[END_REF], there only remains to prove χ [0,µ] (DQ) has rank 1: as γ + P 0 -is a compact perturbation of P 0 -, its essential spectrum is the same and necessarily 0 ≤ µ < m(α) and χ [0,µ] (DQ) is the projection onto an eigenspace of DQ. It suffices to prove γ S 2 = o(1) to get:

Tr χ [0,µ] (DQ) = Tr0 χ [0,µ] (DQ) = Tr0(γ ′ ) -Tr0(γ) = 1.
The strategy for Theorem 1 is to take a test function Γ which satisfies an equation similar to (24). To this end let us first take φ ′ 1 the unique positive radial minimizer of the Choquard-Pekar energy (cf Introduction) and consider φ1 :=

P H Λ φ ′ 1 P H Λ φ ′ 1 L 2
where P H Λ is the projector onto HΛ. We consider the spinor: ψ1 := φ1 0

. For λ -1 := αb Λ (0)m(α)

g ′ 1 (0) 2
we write

ψ λ := λ -3/2 ψ1(λ -1 (•)), N = N λ := |ψ λ ψ λ | and n λ := |ψ λ | 2 = ρN . ( 25
)
It is possible to adapt the fixed point method of [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF] to define γ as the solution to

γ = χ (-∞,0) D 0 + α((ργ + n) * | • | -1 -R[γ + N ]) -P 0 -, (26)
provided α and α log(Λ) are small enough. In fact this paper [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF] treats the case of D 0 but in Appendix B it is shown that replacing it by D 0 is harmless (cf Lemmas 7 and 8). We chose the test function Γ defined by the formulae

Γ := γ + N ′ , π = γ + P 0 -, and
N ′ = |(1 -π)ψ λ (1 -π)ψ λ | 1 -πψ λ 2 L 2 . ( 27
)
We then compute compute E 0 BDF (Γ) using that an electron does not see its own field (that is here D(|ψ| 2 , |ψ| 2 ) -Ex |ψ ψ| = 0). Lemma 2. Let Γ be as above (26), (27). Then the following estimate holds:

E 0 BDF (Γ) = m(α) + αbΛ(0) 2λ ECP + o αbΛ(0) λ . ( 28
)
More precisely, writing

I = ρΓ 2 C -ρ N ′ 2 C and J = Ex[Γ] -ρ N ′ 2 C we have Tr0(D 0 N ′ ) = m(α) + g ′ 1 (0) 2 2λ 2 m |∇ψ1| 2 dx + o(λ -2 ), Tr0(Dγ) = α(b Λ (0)-b Λ (0) 2 ) 2λ D(n1, n1) + o αb Λ (0) λ , α 2 I = -α(2b Λ (0)-b Λ (0) 2 ) 2λ D(n1, n1) + o αb Λ (0) λ , αJ = o αb Λ (0) λ .
Lemma 2 is proved in Section 4.1. Theorem 1 is an obvious corollary.

At this point we know there exists a minimizer γ ′ = γ + |ψ ψ| for E 0 BDF (1) and it satisfies Eq. ( 24). The computation of its energy in terms of ψ gives a lower bound of E 0 BDF (1) of the same form as the right hand side of (28). Theorem 2. There exist three constants α1, L1, Λ1 > 0 such that for α ≤ α1, L ≤ L1, Λ ≥ Λ1, there holds α) in the regime (23) we have:

E 0 BDF (1) = m(α) + (αbΛ(0)) 2 m(α) 2(g ′ 1 (0)) 2 ECP + o (αbΛ(0)) 2 . ( 29
) Theorem 3. Writing C 2 0 := 2g ′ 1 (0) 2 (αb Λ (0)) 2 m(
lim inf α,Λ -1 →0 C 2 0 (E 0 BDF (1) -m(α)) = lim sup α,Λ -1 →0 C 2 0 (E 0 BDF (1) -m(α)) = ECP. ( 30
)
Assume Q is a minimizer for E 0 BDF (1): as in (24) we can write: Q = γ + |ψ ψ|. In the limit α → 0 where α log(Λ) = L ′ is kept fixed and for L ′ small enough the following holds:

Up to translation, the upper spinor ϕ ∈ H 1 (R 3 , C 2 ) of ψ(x) := c 3/2 ψ(cx) tends to a minimizer of the Choquar-Pekar energy ECP.

Remark 2. This paper uses heavily estimates and proofs of [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF]. For convenience Lemma 17 is not fully proved: it is an adaptation of [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF], the whole proof is in the thesis [START_REF] Sok | Modèle de champ moyen en électrodynamique quantique[END_REF] of the author.

Preliminary results

Banach spaces

In this paper several Banach spaces are used.

As usual

Q 2 Kin := Tr(|D 0 ||Q| 2 ) (31)
is linked to the kinetic energy of Q.

In [7] Hainzl et al. introduce the following norms for (Q, ρ) ∈ S2(HΛ) × C ∩ L 2 :      Q 2 Q := E (p -q) 2 E (p + q) | Q(p, q)| 2 dpdq, ρ 2 C := E (k) 2 | ρ(k)| 2 |k| 2 dk ρ 2 C + ρ 2 L 2 . ( 32
)
Strictly speaking, the authors use E(•) instead of E (•). However thanks to ( 14) and ( 15) these norms are equivalent:

∃K > 0, ∀ p ∈ B(0, Λ), 1 K E(p) ≤ E (p) ≤ KE(p).
Moreover we write for an operator R(x, y):

R 2 R := E(p-q) 2 E(p+q) | R(p, q)| 2 dpdq. (33)
As in [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF], we will estimate the above norm of RQ(x, y) = Q(x,y) |x-y| . Unfortunately this is not sufficient and intermediate norms between

• Kin and • Q (respectively • C and • C ) are necessary:            Q 2 q 1 := E (p -q) E (p + q) | Q(p, q)| 2 dpdq, Q 2 q 0 := E (p + q) | Q(p, q)| 2 dpdq, ρ 2
C 1 := E (k) | ρ(k)| 2 |k| 2 dk. (34)
The numbers 0 and 1 refer to the exponent of E (p -q) and E (k).

We also introduce:

Q 2 E := max E (p) , E (p -q) 2 , E (p -q) E (p + q) | Q(p, q)| 2 dpdq. ( 35
)
For any operator Q ∈ S2 we have:

2 π Q Ex ≤ Q Kin ≤ Q E ≤ Q Q. ( 36
)
For some function f : R 3 → [1, +∞), we write:

Q 2 Q f := f (p -q) E (p + q) | Q(p, q)| 2 dpdq, ρ 2 C f := f (k) |k| 2 | ρ(k)| 2 dk.

Some inequalities

Let us recall Hardy's and Kato's inequalities we will use throughout this paper. For φ ∈ L 2 (R 3 ), the following inequalities hold:

|φ(x)| 2 |x| 2 dx ≤ 4 |∇| 2 φ , φ , (37a) |φ
(x)| 2 |x| dx ≤ π 2 |∇|φ , φ , (37b)
Another recurrent inequality is Kato-Seiler-Simon's inequality (K.-S.-S.) [START_REF] Simon | Trace Ideals and their Applications[END_REF]: for any f, g ∈ B(R 3 , C 4 ) (Borelian functions), we have:

f (x)g(i∇) Sp ≤ (2π) - 3 p f L p g L p , 2 ≤ p < ∞. (38)
We use the following Sobolev inequalities in this paper (cf

[2] Theorem 1.38 p.29): for suitable f (f ∈ H 1 (R 3 ) for instance) f L 6 ∇f L 2 , f L 4 |∇| 3/4 f L 2 , f L 3 |∇| 1/2 f L 2 . ( 39
)
An immediate result of (38) (p = 6) and (39) (p = 3) is the following Lemma.

Lemma 3. Let ρ ∈ C and ϕρ := ρ * | • | -1 . For any t > 1/2 there exists Kt > 0 such that ϕρ|D 0 | -t S 6 ≤ Kt ρ C . Moreover we have: ϕρ|D 0 | -1 2 S 6 (log(Λ)) 1 6 ρ C , ϕρ|∇| -1 2 B ρ C Remark 3.
The notation ϕρ is used throughout the paper. Let us consider R = RQ with Q ∈ QΛ. The Lemma 8 of [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF] states that:

RQ R Q Q. ( 40
)
The following Lemma generalizes this result:

Lemma 4. Let t ≥ 0. Then we have:

|∇| -1/2 RQ S 2 Q Ex, (41a)
E (p -q) t E (q) 2 | R(p, q)| 2 dpdq E (p -q) t E (p + q) | Q(p, q)| 2 dpdq, (41b) | R(p, q)| 2 E (q) dpdq E (p -q) E (p + q) | Q(p, q)| 2 dpdq. (41c)
Proof: Ineq. (41c) is a consequence of (40) for E (q) -1

E(p-q)
E(p+q) . Ineq. (41b) can be proved by adapting the proof of Lemma 8. [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF] (see Lemma 15). This gives:

E (p -q) t E (q) 2 | R(p, q)| 2 dpdq ≤ 8 E (2v) t E (2ℓ) w(ℓ, v)| Q(ℓ + v, ℓ -v)| 2 dℓdv,
where w(ℓ, v) is a weight lesser than

E (2ℓ) (2π 2 ) -2 dudℓ ′ E (u -v) 2 E 2ℓ ′ 1+1 |ℓ -u| 2 |ℓ ′ -u| 2 -1 1.
Ineq. (41a) is proved as follows: up to a constant the operator |∇| -1 acts in Direct space as a convolution by

1 |•| 2 (cf [14], p.130). The operator R * Q 1
|∇| RQ is nonnegative and by Cauchy-Schwartz inequality:

Tr R * Q 1 |∇| RQ ≤ (R 3 ) 3 |Q(x, y)| |x -y| dxdydz |y -z| 2 |Q(z, x)| |z -x| ≤ (R 3 ) 3 |Q(x, y)| 2 dxdydz |y -z| 2 |z -x| 2 2 |Q(x, y)| 2
|x -y| dxdy.

Lemma 5. There exist 0 < ε < 1 and K0 > 0 such that, for all

(Q, ρ) ∈ Ex × C, if α( Q Ex + ρ C ) < ε, then |D 0 | 1-αK0( Q Ex + ρ C ) ≤ |D 0 +α(ϕρ -RQ)| ≤ |D 0 | 1+αK0( Q Ex + ρ C ) . (42)
Proof: We have

RQ|D 0 | -1 B ≤ RQ|D 0 | -1 S 2 Q Ex and ϕρ|D 0 | -1 B ρ C .
As shown in [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF], it suffices to take the square root of

|D 0 | 1-2αK( Q Ex + ρ C ) ≤ |D 0 +α(ϕρ-RQ)| 2 ≤ |D 0 | 2 1+αK( Q Ex + ρ C ) 2 .

The fixed point method

In [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF] the authors prove the existence of a global minimizer of E ν BDF under some assumptions on α, Λ, ν C . The authors show there exists a solution to the selfconsistent equation that should satisfy a minimizer Q0 of E ν BDF (when P 0 -is taken as reference state). This equation is:

Q0 + P 0 -= χ (-∞,0) D 0 + α((ρQ 0 -ν) * 1 |•| ) -RQ 0 .
To this end a fixed-point scheme based on this equation is used: let us adapt this proof to our problem.

As shown in [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF] we can use the Cauchy's expansion to write (at least formally)

Q = χ (-∞,0) (D 0 + α(ϕQ -RQ)) -χ (-∞,0) (D 0 ) = ∞ k=1 α k Q k , (43a)
Q k = - 1 2π +∞ -∞ dη 1 D 0 + iη (RQ -ϕQ) 1 D 0 + iη k . ( 43b
)
We also expand

(R -ϕ) k , Q k := k j=0 Q j,j-k : the function Q j,j-k (•,
•) is polynomial of degree j in RQ and polynomial of degree (j -k) in ϕQ. Thanks to Lemmas 3 and 4 we know that each integral converges at least in S6(HΛ). If we take the density of each Q k , we also obtain a (formal) expansion of ρ[ Q]:

ρ[ Q] = +∞ k=1 α k ρ k = +∞ k=1 k j=0 α k ρ j,j-k . (44)
In [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF] it is proved that provided α(

Q Q + ρQ C ) is small enough, those sums converge in Q for Q and in C for ρ[ Q].
In fact the authors show: Proposition 2. For any k ∈ N * and any 0 ≤ j ≤ k, the function

F k,j : Q × C → Q × C (Q, ρ) → Q j,k-j [Q, ρ], ρ j,k-j [Q, ρ]
is a continuous polynomial operator (with estimates of the norm precised in Lemmas 16 and 17 in Appendix B.2).

We prove a similar result in the cited Lemmas.

It is necessary to precise the particular form of ρ0,1[ρ]. A direct computation in Fourier space gives the following formula [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF]. Lemma 6. For ρ ∈ C we have:

ρ0,1(ρ; k) = -BΛ(k) ρ(k) ∈ C. If ρ is in C (respectively C1) then so is ρ0,1[ρ].
The last statement follows from the fact that |BΛ(k)| log(Λ), proved in Appendix A.

Let us describe a fixed-point scheme adapted to our problem in the spirit of [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF]. Given the projector N that corresponds to the "real" electrons and n = ρN its density, we try to define the dressed vacuum Q surrounding it. We seek a solution to

Q + P 0 -= χ (-∞,0) D 0 + α(ϕQ+N -R(Q + N )) . ( 45
)
For convenience we write 45) can be rewritten:

ρ ′ = ρ ′ γ := ρ + n, Q ′ = Q + N , ϕ ′ Q = ϕ Q ′ ; Eq. (
FQ(Q ′ , ρ ′ ) := χ (-∞,0) (D 0 +α(ϕ ′ Q -R ′ Q ))-χ (-∞,0) (D 0 )+N = N + ∞ k=1 α k Q k (Q ′ , ρ ′ ). (46)
Taking the density ρ of both sides and using Lemma 6 we get ρ

Q ′ = Fρ(Q ′ , ρ ′ ) with: Fρ(k) := 1 1 + αBΛ(k) n(k) + α ρ1,0(Q ′ ; k) + ℓ≥2 α ℓ ρ ℓ (Q ′ , ρ ′ ; k) . (47)
We must precise the domain of the function

F := FQ × Fρ. (48)
Following [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF] we first consider the Banach space X = Q × C with the norm

(Q, ρ) X = 2C 3/2 1 (2 √ 2 ρ C + CR √ 2 Q Q),
where CR > 0 is defined in [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF] and C1 ≥ 1 is defined in Notation 4.

Lemma 7. There exist RΛ, ε1, ε2 > 0 such that if

√ Lα ≤ ε1, α (N, n) X ≤ ε2 then BX (0, RΛ) is F -invariant. The number RΛ is O( log(Λ)). Moreover in this ball F is Lipschitz with constant ν0 = O( √ Lα).
In other words the fixed point theorem can be applied to F on BX (0, RΛ). This lemma and the next one are proved in Appendix B.2.

Remark 4. As explained and proved in Appendix B.2, by adapting the estimates of [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF] we realize that another choice of norms for F is possible and so another choice of Banach space on which applying the Banach fixed point theorem. Indeed let us take a radial function f : R 3 → [1, +∞): as long as there exists a constant

C(f ) ≥ 1 such that f (p -q) ≤ C(f )( f (p -p1) + f (p1 -q)), (49)
we can apply the fixed point theorem with the norms

Q 2 Q f := f (p -q) E (p + q) | Q(p, q)| 2 dpdq, ρ 2 C f := f (k)| ρ(k)| 2 |k| 2 dk.
Here we are interested in the case f (p -q) = E (p -q) and f (p

-q) = 1. Let X f ⊂ S2(HΛ) × C be the Banach space with norm (Q, ρ) X f := K(f )( Q q 1 + ρ X 1 ), for some K(f ) > 0 depending on f (Appendix B.2). Lemma 8. There exist R ′ Λ , ε ′ 1 , ε ′ 2 > 0 such that if L ≤ ε ′ 1 , α (N, n) X 1 ≤ ε ′ 2 then BX f (0, R ′ Λ ) is F -invariant. The number R ′ Λ is O( log(Λ)). Moreover in this ball F is Lipschitz with constant ν ′ 0 = O( √ Lα).

Proofs

We will use the following Lemma, proved in Appendix B (Subsection B.3).

Lemma 9. Let ψ λ , γ, ργ defined in (25) and (26). Then the following estimates hold:

γ Q α γ E Lα, γ S 2 α √ Lα, ργ C L √ Lα, ργ C L √ Lα. (50)
Moreover:

γ|D 0 |ψ λ L 2 + γψ λ L 2 α √ Lα and |D 0 |, γ S 2 Lα. (51)

Proof of Lemma 2

We recall N and n are defined in (25).

Notation 7. For convenience we write

φ λ := (1 -π)ψ λ (1 -π)ψ λ L 2 = (1 -π)ψ λ 1 -πψ λ 2 L 2
.

So we have N ′ = |φ λ φ λ |. Moreover we write

τ := αbΛ(0). ( 52
)
Remark 5. Here λ -1 and τ are of the same order Lα. A direct calculation shows that P 0

-|D 0 |ψ λ L 2 = O(λ -1 ) and |D 0 |ψ λ L 2 = O(1)
. We will often use

πψ λ L 2 ≤ γψ λ L 2 + P 0 -ψ λ L 2 (o(τ ) + λ -1 ). (53) 1. Estimation of J Lemma 9 gives γ 2 Ex γ 2 E = O(τ 2 )
. By Cauchy-Schwarz inequality and Ineq. (37a): for any

G = |f g| with f, g ∈ H 1 |Tr G * Rγ | ≤ min( γ Ex G Ex , 2 γ S 2 ∇f L 2 g L 2 ).
Now thanks to Ineq. (37b) and Lemma 9:

|πψ λ | 2 2 C πψ λ 2 L 2 |D 0 |πψ λ , πψ λ |D 0 |πψ λ , πψ λ ≤ 2 ψ λ L 2 |D 0 |, γ S 2 + γ|D 0 |ψ λ L 2 = O((Lα) 2 ).
Similarly we have:

D |ψ λ | 2 , |πψ λ | 2 πψ λ 2 L 2 |∇|ψ λ , ψ λ (τ +λ -1 ) λ , and : |Tr(R * γ N ′ )| ≤ 2 γ S 2 ∇ψ λ L 2 ψ λ L 2 τ λ -1 . Thus J = O(τ 2 + λ -2 ) = O((Lα) 2 ).

Estimation of I

According to the self-consistent equation satisfied by ργ , we write

ρ(γ; k) = -bΛ(k) n(k) + (1 -bΛ(k)) ρ1,0(γ; k) + (1 -bΛ(k)) ∞ ℓ=2 α ℓ ρ ℓ (γ; k) (54)
where we recall that bΛ(p) = αB Λ (p) 1+αB Λ (p) . We write ρ ℓ := ρ ℓ (γ) and := +∞ ℓ=2 α ℓ ρ ℓ for short. There holds:

D(ργ, ργ ) = 4π k dk |k| 2 bΛ(k) 2 | n(k)| 2 + (1 -bΛ(k)) 2 |α ρ1,0(k)| 2 + (1 -bΛ(k)) 2 2 +2R bΛ(k)(1 -bΛ(k)) n(k) α ρ1,0(k) + + (1 -bΛ(k)) 2 α ρ1,0(k) .
By Cauchy-Schwarz inequality it suffices to study | ρ(k)| 2 |k| 2 dk for ρ ∈ {n, ρ1,0, }. We recall bΛ L ∞ L < 2 -1 for sufficiently small L.

Lemma 10. Let i ∈ {1, 2}, then there holds:

4π p bΛ(p) i | n λ (p)| 2 |p| 2 dp = bΛ(0) i D(n1, n1) λ + o λ→∞ (L i λ -1 ). ( 55
)
Moreover: α ρ1,0(γ) C √ Lα γ E (Lα) -3/2 , C α 2 . ( 56
)
Before proving this Lemma, we show the estimation of I. First there holds:

ργ 2 C = bΛ(0) 2 λ D(n1, n1) + o λ→∞ L λ .
Then

|φ λ | 2 (x) = 1 1-πψ λ 2 L 2 (|ψ λ (x)| 2 + |πψ λ (x)| 2 -2R ψ * λ (x)(πψ λ )(x))
. By Cauchy-Schwarz and Kato inequalities the two last terms are O(L(Lα) 2 ). In fact:

|πψ λ | 2 2 C πψ λ 2 L 2 |∇|πψ λ , πψ λ (Lα) 4 |πψ λ ||ψ λ | 2 C ≤ 2 πψ λ 2 L 2 ∇ψ λ L 2 ψ λ L 2 (Lα) 3 , so D ργ, |πψ λ | 2 -2R{ψ * λ (πψ λ )} L √ Lα(Lα) 3/2 . Then D(ργ, n λ ) = -4π bΛ(k)| n λ (k)| 2 dk |k| 2 + O (α ρ1,0 C + C ) n λ C .
In the same way:

D(ργ , |φ λ | 2 ) = -bΛ(0)D(n λ , n λ ) + o λ→∞ ( L λ ).
Since

1 1-πψ λ 2 L 2 = 1 + O((τ + λ -1 )
2 ), we finally obtain:

I = - 2bΛ(0) + bΛ(0) 2 λ D(n1, n1) + o λ→∞ L λ (57)
Proof of Lemma 10. We use Proposition 6 (Appendix A). In the regime (23) and for ε = 1 6 , in a neighbourhood B(0, rε) of 0 independent of α, Λ we have:

∀ k ∈ B(0, rε)\{0}, |BΛ(|k|) -BΛ(0)| |k| (Λ -1 + |k| 1/2 ) =: z(|k|). (58)
Then

k bΛ(k) 2 | n λ (k)| 2 |k| 2 dk = 1 λ k bΛ( k λ ) 2 | n1(k)| 2 |k| 2 dk, For λ ≥ r -4 ε and k ∈ B(0, λ 3/4 ): |BΛ(k/λ) -BΛ(0)| ≤ |k| λ (z(λ -1/4 ) + KΛ -1 ). As f1 : t ∈ R + → t
1+t and f2 = f 2 1 have bounded derivatives (by 1 and 2 respectively), for k with BΛ(p) = BΛ(0),

|bΛ(k) -bΛ(0)| ≤ α|BΛ(k) -BΛ(0)|, |bΛ(k) 2 -bΛ(0) 2 | ≤ 2α|BΛ(k) -BΛ(0)|, so |k|≤λ 3/4 |fi(αBΛ(k)) -fi(αBΛ(0))| | n λ (k)| 2 dk |k| 2 ≤ 2α z(λ -1/4 ) + KΛ -1 λ | n1(k)| 2 dk |k| α z(λ -1/4 ) + Λ -1 λ n1 C ψ1 2 L 4 .
As f1(t), f2(t) ≤ t 2 then

|k|>λ 3/4 bΛ(k) i | n1(k)| 2 |k| 2 dk λ -3/2 L i | n1(k)| 2 dk λ -3/2 L i ψ1 2 H 1 = O(L i λ -3/2 ) and k bΛ(k) i | n λ (k)| 2 |k| 2 dk = bΛ(0) i D(n1, n1) λ + o λ→∞ (L i λ -1 ).
There holds

k α 2 (1 -bΛ(k)) 2 | ρ 1,0 (k)| 2 |k| 2 dk α 2 ρ1,0 2 C . Then estimates of ρ1,0 C and C are proved in Appendix B.3. 3. Estimation of Tr0(D 0 N ′ )
We emphasize that ψ λ has no lower part as a spinor.

There holds

D 0 πψ λ , πψ λ = -|D 0 |P 0 -ψ λ , P 0 -ψ λ + D 0 γψ λ , γψ λ + 2R D 0 P 0 -ψ λ , γψ λ = -|D 0 |ψ λ , ψ λ + O γψ λ L 2 |D 0 |γψ λ L 2 + |D 0 |P 0 -ψ λ L 2 = -|D 0 |ψ λ , ψ λ + o((Lα) 2 ).
Then we have:

D 0 ψ λ , πψ λ = D 0 ψ λ , γψ λ -|D 0 |P 0 + ψ λ , ψ λ = |D 0 |ψ λ , P 0 + γP 0 + ψ λ + |D 0 |ψ λ , P 0 + γP 0 -ψ λ -|D 0 |P 0 + ψ λ , ψ λ = -|D 0 |P 0 + ψ λ , ψ λ + O( D 0 ψ λ L 2 γ 2 S 2 + D 0 ψ λ L 2 γ B P 0 -ψ λ L 2 ) = -|D 0 |P 0 + ψ λ , ψ λ + o((Lα) 2 )
.

Hence D 0 φ λ , φ λ = D 0 ψ λ , ψ λ 1-πψ λ 2 L 2 + |D 0 |P 0 -ψ λ , ψ λ + o((Lα) 2
). Notation 8. We write g⋆ψ , ψ for g⋆(-i∇)ψ , ψ for ⋆ ∈ {0, 1}.

As g ′ 0 (0) = 0 and g ′′ 0 ∞

α and the (g ′ 1 )α,Λ's are uniformly continuous in a neighbourhood of 0 (cf Proposition 3 in Appendix A)

D 0 ψ λ , ψ λ 1-πψ λ 2 L 2 = g0ψ λ , ψ λ (1 + P 0 -ψ λ , ψ λ ) + o((Lα) 2 ) =g0(0) + g 0 (0) 4 g 2 1 g 2 0 ψ λ , ψ λ + o((Lα) 2 ) =g0(0) + g ′ 1 (0) 4g 0 (0)λ 2 |∇| 2 ψ1 , ψ1 + o((Lα) 2 ).
Furthermore

|D 0 |P 0 -ψ λ , ψ λ = 1 2 (|D 0 | -g0)ψ λ , ψ λ = 1 4g 0 (0) g 2 1 ψ λ , ψ λ + o(λ -2 ).
Finally

Tr0(D 0 N ′ ) = D 0 φ λ , φ λ = g0(0) + g ′ 1 (0) 2
= R ′ γ -ϕ ′ γ = R(γ + N ) -(ργ + n) * | • | -1 . Remark 6.
Let us recall Lemma 1. [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF]: if P, Π are two projectors such that:

P -Π ∈ S2 then Q ∈ S P 1 ⇐⇒ Q ∈ S Π
1 and then TrP (Q) = TrΠ(Q). We apply this Lemma for P = P 0 -and Π := χ (-∞,0) (D 0 + αB): formally

Tr0((D 0 + αB)γ) = Tr(|D 0 |γ 2 ) + αTr0(Bγ) (60a) Tr0((D 0 + αB)γ) = -Tr(|D 0 + αB|γ 2 ) = -Tr(|D 0 |γ 2 ) + o(Tr(|D 0 |γ 2 )). ( 60b
)
So we would like to show that

Tr(|D 0 |γ 2 ) = -α 2 Tr0(Bγ) + o((Lα) 2 ), = -α 2 D(ργ + n, ργ) -Tr(R ′ γ γ) + o((Lα) 2 ), = -α 2 D(ργ + n, ργ ) + o((Lα) 2 ). (61)
We have to prove that Bγ in S ). We use (41c):

R ′ γ γ S 1 ≤ R(γ)|D 0 | -1/2 S 2 |D 0 | 1/2 γ S 2 + R(N ) S 2 γ S 2 (τ + λ -1 )τ.
First let us prove that Tr(|D 0 + αB|γ 2 ) = Tr(|D 0 |γ 2 ) + O(α(Lα) 2 ).

Thanks to Lemma 5, there holds:

|D 0 + αB| ≥ |D 0 | 1 -αK( γ Kin + ργ C + |∇| 1/2 ψ λ L 2 ) , |D 0 + αB| ≤ |D 0 | 1 + αK( γ Kin + ργ C + |∇| 1/2 ψ λ L 2 ) .
Then we multiply by γ * = γ on the left and by γ on the right: this does not change the inequalities. To conclude it suffices to take the trace.

Let us prove Tr0(ϕ ′ γ γ) = D(ργ + n λ , ργ ). In fact if Q ∈ S

(2π) -3/2 |p|,|q|<Λ ϕ ′ γ (p -q)(Tr( γ(p, q))) * dpdq = (2π) -3/2 |u+ k 2 |,|u- k 2 |<Λ ϕ ′ γ (k)(Tr( γ(u + k/2, u -k/2))) * dudk = k ϕ ′ γ (k) ργ(k) * dk = 4π k ρ ′ γ (k) ργ(k) * |k| 2 dk = D(ργ, ρ ′ γ ).
As shown in the estimation of I, there holds

D(ργ , ργ + n λ ) = b Λ (0) 2 -b Λ (0) λ D(n1, n1) + o L λ , so we get Tr(|D 0 |γ 2 ) = α b Λ (0) 2 -b Λ (0) 2λ D(n1, n1) + o L λ . (62)
Remark 7. The calculation above is correct if γ(p, q) ∈ C 0 (B(0, Λ) 2 ):

|u± k 2 |<Λ | ρ(k)| |k| 2 | γ(u + k 2 , u -k 2 )|dudk Λ 3/2 ρ C (Λ 3/2 γ L ∞ + γ S 2 ).
We conclude by continuity of Q ∈ S P 0 -1 → ρQ ∈ C shown in [START_REF] Hainzl | Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics[END_REF], that of

Q ∈ S P 0 - 1 → Tr0(ϕ ′ γ Q) and the density of C 0 (B(0, Λ) 2 ) in F (S P 0 - 1 (HΛ)). Let us prove ϕ ′ γ Q ∈ S P 0 - 1 .
We have:

(ϕ ′ γ Q) --= (P 0 -[ϕ ′ γ , P 0 + ]|D 0 | -1/2 ) ∈S 2 (H Λ ) |D 0 | 1/2 Q +- ∈S 2 (H Λ ) + (ϕ ′ γ |D 0 | -1/2 ) -- ∈B(H Λ ) |D 0 | 1/2 Q -- ∈S 1 (H Λ ) ∈ S1(HΛ) (63) and so |Tr0(ϕ ′ γ Q)| ≤ (Λ 1/2 + log(Λ)) ρ ′ γ C Q S 1,P 0 - with Q S 1,P 0 - := Q -- S 1 + Q ++ S 1 + Q -+ S 2 + Q +- S 2 . ( 64
)
To see

P 0 -[ϕ ′ γ , P 0 + ]|D 0 | -1/2
is Hilbert-Schmidt, it suffices to prove the kernel of its Fourier transform is in L 2 (B(0, Λ) 2 ): this is easy with the help of Lemma 14. To conclude this section there remains to deal with R ′ γ γ, we recall this operator is trace-class (cf Lemma 4):

R ′ γ γ = (R ′ γ )|D 0 | -1/2 ∈S 2 (H Λ ) |D 0 | 1/2 γ ∈S 2 (H Λ )
and

Tr R ′ γ γ = O( RN S 2 γ S 2 + Rγ |D 0 | -1/2 S 2 |D 0 | 1/2 γ S 2 ) = O α √ Lα λ +(Lα) 2 . ( 65
)
Remark 8. As Λ → +∞ there holds |D 0 | 2 ψ1 , ψ1 -D(n1, n1) = ECP + o(1).

In fact ψ1 = (φ1, 0) T where φ1 = P H Λ φ ′ 1 / P H Λ φ ′ 1 L 2 and φ ′ 1 is the minimizer of Choquard-Pekar energy. P H Λ is the projector onto HΛ. So we have φ1

H 1 → Λ→+∞ φ ′ 1 . Writing n ′ = |φ ′ 1 | 2 there holds by Kato's inequality (37b) n1 C -n ′ C ≤ n1 -n ′ C |∇|ψ1 , ψ1 + |∇|φ ′ 1 , φ ′ 1 ψ1 2 L 2 -φ ′ 1 2 L 2 |∇|φ ′ 1 , φ ′ 1 ψ1 2 L 2 -φ ′ 1 2 L 2 → Λ→∞ 0.

Proof of Proposition 1

In this part we write E(•) for E 0 BDF (•). Let us prove now the binding inequalities for 0 < q < 1. According to Lieb's principle (Proposition 3. [START_REF] Hainzl | Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics[END_REF]) for each q we can take minimizing sequences for E(q) of the form

Q (k) = P (k) -P 0 -+ q|ψ k ψ k |, k ∈ N with (P (k) -P 0 -) ∈ QΛ(0) and P 2 k = P k , P k ψ k = 0. (66)
We write as before

γ k = P k -P 0 -, n k = |ψ k | 2 , N k = |ψ k ψ k |.
We will forget to emphasize the dependence in k.

Writing Iγ(N ) = αR D(ργ, n) -Tr(R * N γ) , E 0 BDF (Q) can be written: E 0 BDF (Q) = E 0 BDF (γ) + q D 0 ψ , ψ + qIγ (N ) = (1 -q)E 0 BDF (γ) + qE 0 BDF (γ + N ).
Taking the lim inf, we obtain As P k ψ k = 0, there holds P 0 -ψ k = γψ k , in particular

E(q) = lim inf k→∞ ((1 -q)E 0 BDF (γ) + qE 0 BDF (γ + N )) ≥ (1 -q) lim inf k→∞ E 0 BDF (γ) + qE (1)
P 0 + ψ 2 = ψ 2 -P 0 -ψ 2 = 1 -γψ 2 → 1 and D 0 ψ , ψ = |D 0 |ψ + , ψ + -|D 0 |γψ , γψ where ψ ε = P 0 ε ψ. As |D 0 | 1/2 γψ 2 L 2 ≤ Tr(|D 0 |γ 2 ) ψ 2 L 2 and ψ 2 = 1: up to extraction we have lim k→∞ D 0 ψ , ψ = lim k→∞ |D 0 |ψ + , ψ + ≥ m(α).
The sequence ( D 0 ψ k , ψ k ) k is bounded, else by Cauchy-Schwarz and Kato's inequality

E 0 BDF (γ + N ) ≥ E 0 BDF (γ) + D 0 ψ , ψ - 1 2 ργ 2 C + γ 2 Ex + πα 2 |∇|ψ , ψ -→ k→+∞ +∞.
By Cauchy-Schwartz inequality Iγ (N ) → 0 and

lim inf k→∞ E 0 BDF (Q k ) = E(q) ≥ lim inf k→∞ E 0 BDF (γ)+q lim inf k→∞ Iγ (N )+q lim inf k→∞ D 0 ψ , ψ ≥ qm(α).
It implies E(q) = qm(α), but we can use the method of Section 4.1. to prove that E(q) < qm(α) for sufficiently small α and L in regard with q: we define Q by the formulae  

 Π := γ + P 0 -= χ (-∞,0) D 0 + α ϕγ + qn * | • | -1 -R(γ + qN ) , Q := γ + q 1-Πψ λ 2 L 2 |(1 -Π)ψ λ (1 -Π)ψ λ |.
If we assume that E(q) = qm(α) once E(1) < m(α) has been proven, we also obtain E(q) > qE [START_REF] Bach | On the Stability of the Relativistic Electron-Positron Field[END_REF]. We thus get E(q) + E(1 -q) > qE(1) + (1 -q)E(1) = E(1).

There remains the case q > 1. However it has been proved in [START_REF] Hainzl | Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics[END_REF] that for each integer M , EBDF(•) is concave on [M, M + 1]. Besides thanks to (20) there holds

E(q) ≥ q(1 -α π 4 )m(α). So it suffices that 2(1 -α π 4 )m(α) > E(1)
to get E(q) > E(1) for q > 1. For α < 2 π it is true and as E(q) > 0 for q = 0 the binding inequalities for q > 1 are proved.

4.3 Proof of Theorems 2 and 3

Notations

Let Q = γ ′ = γ + N be a minimizer written with the notation of (24). As before we write n := ρN .

We have N = χ (0,µ] (DQ) with DQ :

= D 0 + α(R ′ γ -ϕ ′ γ ).
We have to show that N = |ψ ψ|, then we can choose µ such that DQψ = |DQ|ψ = µψ with µ ≤ m(α).

We split ψ in two: ψ = ϕ χ . The wave function ϕ ∈ L 2 (R 3 , C 2 ) is the upper spinor and χ ∈ L 2 (R 3 , C 2 ) the lower spinor.

We write C 2 0 :=

2g ′ 1 (0) 2
(αb Λ (0)) 2 m(α) and c :=

(g ′ 1 (0)) 2
αb Λ (0)m(α) . As (R(N ) -ψ |ψ| 2 )ψ = 0, there holds

(D 0 + α(ϕγ -Rγ ))ψ = µψ = |D 0 + α(ϕγ -Rγ )|ψ. (67)
We write

v ⋆ γ := ϕ ⋆ γ , b ⋆ γ := v ⋆ γ -R ⋆ γ ,
where ⋆ is a prime symbol or no prime. Moreover we write d := D 0 . We recall:

vγ ψ , ψ = D(ργ , n) and | Rγ ψ , ψ | ≤ γ Ex n C , (68)
We recall the notation g⋆ψ , ψ := g⋆(-i∇)ψ , ψ with ⋆ ∈ {0, 1}.

Strategy of the proof

The proof of Theorem 2 relies on bootsrap arguments enabling us to get appropriate estimates of |∇| s ψ L 2 for s = 1 2 , 1, 3 2 . The starting point is a priori estimates of |∇| 1/2 ψ L 2 , ∇ψ L 2 , Tr(|D 0 |γ 2 ). It is possible to use an adaptation of the fixed point method of [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF] to get estimates of

E (p -q) 2s E (p + q) | g(p, q)| 2 dpdq and E (k) 2s | ργ(k)| 2 |k| 2 dk
in terms of the Sobolev norms ψ H s+1/2 at least for s = 0, 1 2 , 1. Then the second part of Eq. ( 24) enables to get estimates of |∇| s+1 ψ L 2 in terms of |∇| s+1/2 ψ L 2 and the (squared) norms above. It is possible to keep going as explained in the thesis of the author [START_REF] Sok | Modèle de champ moyen en électrodynamique quantique[END_REF], provided α, L are small enough.

More precisely the steps are the following. 1. We first prove a priori estimates and get ρQ C , Q Kin are O(1) and then show that γ Kin = o(1). As a consequence Lemma 1 holds and we get ργ C , γ q 0 , |D 0 |ψ , ψ are O(1). This enables us to show that we can apply the fixed point method (Lemma 8, f = 1) and that the minimizer γ + N and its density ργ + n form a fixed point (at least in the space associated to • q 0 and • C ). 2. We then prove

ψ H 3/2 , n C = O(1). (69)
Thus we can apply the fixed-point method (Lemma 7) with n = |ψ| 2 and N = |ψ ψ| and so to construct (γ + N ; ργ + ρN ) as a fixed point in (a ball of) X .

3. Using the estimates that we deduce from the fixed-point method and Eq. ( 24) we then prove that |∇| 2 ψ , ψ = O((αbΛ(0)) 2 ). 4. Following [START_REF] Hainzl | Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics[END_REF], we apply a scaling transform to the minimizer with the scaling factor c = O(αbΛ(0)) defined in Subsection 4.3.1: we get ψ(x) := c 3/2 ψ(cx) ∈ H 1 (C 4 ). The previous results will give

ψ H 3/2 = O(1), χ H 1 = O(Lα),
where χ ∈ H 1 (R 3 , C 2 ) is the lower spinor of ψ. 5. At last we compute the energy and show the asymptotic expansion.

A priori estimates

The first step is the following result.

Lemma 11. For Q = γ + N a minimizer of E 0 BDF (1), then N has rank 1 and there holds the following a priori estimates:

Tr(|D 0 |γ 2 ) + αD(ργ , ργ) + |D 0 |ψ , ψ 1.
The decomposition γ + N is the same as in (24), Section 2 with N = |ψ ψ|.

Assuming this result is true we can go further: we know that F (Q, ρQ) = (Q, ρQ) where F is the function defined in ( 46) and (47). Using the estimates of Appendix B.3 we get that:

ργ C L n C + √ Lα Q q 0 + +∞ j=2 (αK( ρQ C + Q q 0 )) j L = O(1).
We then apply Lemma 8 (with f = 1): we get that (Q, ρQ) is in fact the unique fixed point of F in a ball of X0. Proof of Lemma 11: As Q is a minimizer and that E 0 BDF (1) ≤ m(α) then there holds:

m(α) ≥ E 0 BDF (Q) ≥ 1 -α π 4 Tr(|D 0 |Q 2 ) + α 2 D(ρQ, ρQ), (70)
and Q Kin, √ α ρQ C = O(1). As γ = χ (-∞,0) (D 0 Q ) -P 0 -, using estimates of Lemmas 16 and 17 we get:

γ S 2 α( ρQ C + Q Ex) √ α. Thus |Tr0(γ)| ≤ γ 2 S 2
α < 1, as a consequence Tr0(γ) = 0 and N has rank 1. Thanks to (68) and Kato's inequality there holds

D 0 ψ , ψ = DQψ , ψ -α bγψ , ψ = |DQ|ψ , ψ + O(α ψ H 1/2 ( γ Ex + ργ C )).
(71) We apply Lemma 5 on |DQ|:

|DQ|ψ , ψ ≥ 1 -K(α Q Ex + α 1/2 × α 1/2 ρQ C ) |D 0 |ψ , ψ , (72)
and:

D 0 ψ , ψ ≥ (1 -K √ α) |D 0 |ψ , ψ + O α( ρQ C + Q Kin) ψ H 1/2 . ( 73
)
By Cauchy-Schwartz inequality and Kato's inequality:

E 0 BDF (Q) = E 0 BDF (γ) + D 0 ψ , ψ + αR D(ργ , n) -Tr(Rγ N ) ≥ (1 -α π 4 )Tr(|D 0 |γ 2 ) + α 2 D(ργ , ργ ) + (1 -C2 √ α) |D 0 |ψ , ψ -α ψ H 1/2 ( γ Kin + ργ C ). As E 0 BDF (Q) ≤ m(α) we have Tr(|D 0 |γ 2 ) + αD(ργ , ργ ) + |D 0 |ψ , ψ = O(1). (74)

Estimates around the fixed point method

Let us prove that we can construct (Q, ρQ) as a fixed point in X . We have to show n C , N Q = O(1) and as N Q ψ 2 H 3/2 it suffices to prove (69). By Sobolev inequality (39):

n L 2 = ψ 2 L 4 |∇| 3/4 ψ 2 L 2 ∇ψ 3/2 L 2 ψ 1/2 L 2 = O( √ α).
Moreover there holds D(n, n) ≤ π 2 |∇|ψ , ψ 1 and n C = O(1). At this point we have: n C 1 , N q 1 1: we can apply Lemma 8 with f (p -q) = E (p -q) and construct (Q, ρQ) as a fixed point in X1. As shown in Appendix B.3, there holds γ

C 1 + ργ C 1 1.
Let us now prove that ψ H 3/2 1. By (67) we have |d| 2 ψ = µdψ -αdbγ ψ, therefore:

|d| 3 ψ , ψ = µ |d|dψ , ψ + α |d| 1/2 (Rγ -vγ )|d| -3/2 |d| 3/2 ψ , d|d| 1/2 ψ .
Then thanks to (41b) and Lemma 12 below, writing

|d| 1/2 bγ |d| -3/2 = [ |d| 1/2 , bγ ]|d| -3/2 + bγ |d| -1 we get |d| 1/2 bγ |d| -3/2 B ( γ Ex + ργ C ) + γ q 1 .
We obtain at last ψ H 3/2

1. In particular we can apply Lemma 7 and construct (Q, ρQ) as a fixed point in X and get γ Q, ργ C 1.

Lemma 12. Let (γ ′ 0 , ρ ′ 0 ) be in Q × C and b0 := ρ ′ 0 * 1 |•| -γ ′ 0 , v ′ 0 := ρ ′ 0 * 1 |•| . Then there holds: |D 0 | -3 2 b0, |D 0 | 1 2 B + |D 0 | -1 b0, |D 0 | 1 4 B ( γ ′ 0 Q + ρ ′ 0 C ).
Proof : The estimation for the term R(γ ′ 0 ) comes from (41b) in Lemma 4: indeed we have

| E (p) s -E (q) s | ≤ K |p -q| E (p) 1-s + E (q) 1-s for s = 1 2 k , k ∈ N * .
We write f ∈ HΛ and

Φ = |D 0 | -3 2 v ′ 0 , |D 0 | 1 2
, the following holds:

p | Φf (p)| 2 dp ≤ K dpdq E (p) 3 | E (p) -E (q) | 2 |p -q| 4 | ρ ′ 0 (p -q)| 2 E (p) + E (q) | f (q)| 2 dq.
To deal with last term we use the same method. Let us prove |∇| 2 ψ , ψ = O((αbΛ(0)) 2 ). We write x = x(N ) = g1(-i∇)ψ L 2 . By Lemma 19 we have:

ργ C Lx 1/2 + αx + Lα, (75a)
γ Ex √ Lαx 1/2 + αx + Lα. ( 75b
)
Taking • L 2 -norm of dψ = µψ -αbγ ψ, we have (cf Proposition 3 for g ′′ 0 ∞ ):

d 2 ψ , ψ = x 2 + m(α) 2 + O( g ′′ 0 ∞ x 2 ) = x 2 + m(α) 2 + O(αx 2 ) α| bγ ψ , ψ | + α 2 bγ ψ 2 L 2 ≤ K1Lα 2 x 1/2 + K2Lαx + K3α 2 x 3/2 + K4(Lα 3 )x 2 + K6α 4 x 3 µ 2 ψ 2 L 2 ≤ m(α) 2 .
For the first equality we have used Taylor's Formula (order 2) and the fact that g ′ 0 (0) = 0. As x = O(1) we have α 4 x 3 = O(α 4 x 2 ) and

x 2 ≤ k1Lα 2 x 1/2 + k2(Lα)x + k3α 2 x 3/2 . ( 76
)
Finally we obtain

x 1/2 ≤ k 1/3 1 (Lα 2 ) 1/3 + k 1/2 2 (Lα) 1/2 + k3α 2 (Lα) 1/2 , ( 77
)
and there holds

x 2 ≤ K(Lα) 2 = O(c -2 ).
By Lemma 19 the following estimates hold for the minimizer:

γ Q α, ργ C (L + w(N )) √ Lα, γ E Lα, ργ C L √ Lα
where we recall:

w(N ) := |p -q| 2 |p + q|| N (p, q)| 2 dpdq 1/2 |∇| 3/2 ψ L 2 .

Scaling

We have considered so far the problem associated with Ec=1,α,Λ(BDF energy where the parameters are: speed of light 1, fine structure constant α and cut-off Λ). We link it to the BDF energy in another choice of parameters: speed of light c, fine structure constant αc and cut-off cΛ, with c > 0 defined in Subsection 4.3.1.

As in [START_REF] Hainzl | Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics[END_REF] we write

U * c : HΛ → HcΛ φ → c 3/2 φ(c(•)),
and so Ucφ(x) = c -3/2 φ(x/c). There holds a scaling correspondence between (1, α, Λ) and (c, cα, cΛ) :

Ec,cα,cΛ(U * c QUc) = c 2 E1,α,Λ(Q).
To distinguish the corresponding objects of (c, cα, cΛ) we underline them:

ψ(x) = U * c ψ(x) = c 3/2 ψ(cx), D 0 = c 2 U * c D 0 Uc = mc 2 β + cT, γ(x, y) = U * c γUc(x, y) = c 3 γ(cx, cy), m = g0(-i∇/c), ργ (x) = c 3 ργ (cx), v = ργ * | • | -1 , Tα = cg1(-i∇/c)α • -i∇ |∇| , R(x, y) = γ(x, y)|x -y| -1 , Tσ = cg1(-i∇/c)σ • -i∇ |∇| .
There holds |∇| ≤ |Tσ| ≤ C1|∇| and

γ Ex = √ c γ Ex ργ C = √ c ργ C , so R|D 0 | -1/2 B γ Ex = √ c γ Ex v|D 0 | -1/2 B ργ C = √ c ργ C
We have shown g 2 1 ψ , ψ = O((Lα) 2 ), so for c :=

g ′ 1 (0) 2
αb Λ (0) , ψ has uniformly bounded H 1 norm with respect to the parameters in the regime (23). Remark 9. Here the constant of scaling c corresponds to λ of the test function.

First we we prove the following middle results.

Lemma 13. Let Y = Y (ψ) := g 3/2
1 ψ L 2 where ψ is defined as above. Then we have χ L 2 c -1 and ∇χ L 2 αY + c -1 .

Moreover µ = m(α) + O(c -2 ) and E 0 BDF (1) = E 0 BDF (γ ′ ) = m(α) + O(c -2
). Proof : Thanks to (67) we have

mc 2 βψ + cTαψ + αc(v -R)ψ = µc 2 ψ. ( 78
)
Considering the upper part ϕ and the lower part χ of ψ:

mc 2 ϕ + cTσχ + αcvϕ -αc(Rψ)1 = µc 2 ϕ (79a) -mc 2 χ + cTσϕ + αcvχ -αc(Rψ)2 = µc 2 χ (79b)
From (79b) we obtain

χ = Tσ mc + µc ϕ + α mc + µc ((Rψ)2 -vχ).
We take the L 2 -norm:

χ L 2 ψ H 1 c + α √ c ( ργ C + γ Ex ) 1 c + αL √ Lα √ c + αLα √ c 1 c .
In particular we have

χ L 2 = χ L 2 = O(c -1
).

We write Sx = g 1 (-i∇) • x with x either σ or α. As Tα exchanges upper and lower spinors, by Cauchy-Schwarz inequality the following holds:

D 0 ψ , ψ = g0ϕ , ϕ -g0χ , χ + 2R Sσϕ , χ = m(α) ϕ 2 L 2 + O(c -2 ) = m(α) + O(c -2 ).

It enables us to estimate

µ = m(α) + O(c -2 ) and E 0 BDF (1) = E 0 BDF (γ ′ ) = m(α) + O(c -2 ). ( 80
)
From Eq. (79a) we get

Tσχ = (µc 2 -mc 2 )ϕ c + α[(Rψ)1 -V ϕ]. As µ = m(α) + O(c -2
), the L 2 -norm of Tσχ has the following upper bound:

Tσχ L 2 α + c -1 + α √ c(Lα + L √ Lα) α,
writing Y 2 = Y (ψ) 2 := g 3 1 ψ , ψ , we get the middle estimates

χ H 1 α (81a) χ H 1 (αY + c -1 ). ( 81b
)
Indeed writing µ = m(α) + δm, c 2 × δm c ϕ has L 2 -norm lesser than Kc -1 . Then:

g0(p/c) -g0(0) =        1 0 g ′ 0 (tp/c)dt |p| c ≤ Kα |p| c , 1 0 g ′′ 0 (tp/c)(1 -t)dt |p| 2 c 2 ≤ Kα |p| 2 c 2 ,
and |g0(p/c) -g0(0)| α|p| 3 /c 3 . In particular

g1χ , χ ≤ χ , χ g 2 1 χ , χ = O(c -1 × (αY + c -1 )c -1 ) = O(αY c -2 + c -3 ) (82)
and there also holds the middle estimate:

∇χ L 2 αc -1 . Let us prove that U * c ψ H 3/2 = O(1)
. The method is the following: we take the scalar product of |∇|ψ with each part of the equation |D 0 | 2 ψ = D 0 (µ-αbγ )ψ. Then we cancel the leading terms in order to get an inequality involving Y 2 = g 3 1 ψ , ψ of the form:

Y 2 ≤ O(c -3 + Y c -3/2 + Y 3/2 c -3/4 ).
As a consequence we get

Y 2 = O(c -3 ). Let us first deal with |D 0 | 2 ψ , |∇|ψ .
Thanks to estimate (82) there holds

µ g1α • -i∇ |i∇| ψ , |∇|ψ |∇| 3/2 ϕ L 2 |∇| 1/2 χ L 2 = O(Y c -3/2 + Y 3/2 √ αc -1 ). We recall that |g0(p) -m(α)| ≤ min( g ′ 0 L ∞ |p|, 2 g0 L ∞ ): it is O(min(1, α|p|)
). Then we have:

g 2 0 ψ , |∇|ψ = m(α) 2 |∇|ψ , ψ + 2m(α) (g0 -m(α))ψ , |∇|ψ + (g0 -m(α)) 2 ψ , |∇|ψ = m(α) 2 + O(αY 2 ),
Thus we have:

|D 0 | 2 ψ , |∇|ψ = m(α) 2 |∇|ψ , ψ + g 2 1 |∇|ψ , ψ + O(αY 2 ).
Let us now treat the term D 0 (µ -αbγ )ψ , |∇|ψ and first the term µ D 0 ψ , |∇|ψ .

g0βψ , |∇|ψ = g0ψ , |∇|ψ -2 g0χ , |∇|χ = g0ψ , |∇|ψ + O(αY c -2 + c -3 ) = m(α) |∇|ψ , ψ + O(αY 2 + αY c -2 + c -3 ), D 0 ψ , |∇|ψ = g0βψ , |∇|ψ + 2R( Sσϕ , |∇|χ ) = m(α) |∇|ψ , ψ + O(αY 2 + αY c -2 + c -3 + Y c -3/2 + Y 3/2 √ αc -1 ), µ D 0 ψ , |∇|ψ = m(α) 2 |∇|ψ , ψ + O(αY 2 + Y 3/2 √ αc -1 + Y c -3/2 + c -3 ).
We write:

|d| 1/2 Rγ ψ = |d| 1/2 , Rγ |d| -1 |d|ψ + Rγ |d| 1/2 ψ,
and thanks to Lemma 4 we have:

|d| 1/2 , Rγ |d| -1 2 S 2 γ 2 q 1 γ 2 E c -2 .
By adapting the proof of Lemma 12 we can prove the follwing estimates:

[|∇|, vγ ] 1 |∇||D 0 | 1/2 B , [|∇| 1/2 , vγ ] 1 |∇| 1/2 |D 0 | 1/2 B ργ C log(Λ).
We use Lemmas 3 and 4 to get estimates of bγ ψ L 2 . First we deal with the terms with Sα:

| Rγ ψ , Sα|∇|ψ | ≤ |∇| 1/2 , Rγ |d| -1 |d|ψ, Sα|∇| 1/2 ψ + | Rγ |∇| 1/2 ψ , Sα|∇| 1/2 ψ | Y γ Ex (1 + ∇ψ L 2 ) Y (Lα).
The operator Sα exchanges upper and lower spinors, so we get:

| Sσvγ ϕ , |∇|χ | = |∇|vγ ϕ , Sσχ ≤ |∇|vγ ϕ L 2 Sχ L 2 ≤ αc -1 |∇|, vγ ϕ L 2 + vρ|∇|ϕ L 2 αc -1 log(Λ) ργ C × |∇||d| 1/2 ϕ L 2 + ργ C Y ≤ Lc -5/2 ( ∇ϕ L 2 + |∇| 3/2 ϕ L 2 ) Lc -7/2 + Lc -5/2 Y.
Similarly the following holds:

| Sσvγ χ , |∇|ϕ | = |∇| 1/2 vγ χ , |∇| 1/2 Sσϕ ≤ |∇| 1/2 , vγ χ L 2 + vγ |∇| 1/2 χ L 2 |∇| 3/2 ϕ L 2 log(Λ) ργ C |∇| 1/2 |d| 1/2 χ L 2 + ργ C |∇| 3/2 χ L 2 Y (L log(Λ)c -1/2 (c -3/2 + c -1 √ αY + αc -2 Y ))Y.
We treat now the terms with g0(-i∇):

| vγ ϕ , |∇|g0ϕ | ργ C |∇| 1/2 ϕ L 2 |∇|g0ϕ L 2 Lc -2 | vγ χ , |∇|g0χ | ργ C |∇| 1/2 χ L 2 |∇|g0χ L 2 Lc -2 , | Rγ ψ , |∇|g0ψ | γ Ex |∇| 1/2 ψ L 2 |∇|g0ψ L 2 c -5/2 .
It is clear that α(Lc -2 + c -5/2 ) = O(c -3 ). At last:

D 0 (µ -αbγ)ψ , |∇|ψ = m(α) 2 |∇|ψ , ψ + O(αY 2 + Y 3/2 √ αc -1 + Y c -3/2 + c -3 ),
and:

Y 2 (1 -Kα) ≤ K0c -3 + K1(Lα 2 )Y + K3 √ αc -1 Y 3/2 . As √ αc -1 = O(c -3/4 ) (because α(log(Λ)) 1/4 = o(1)
in the regime (23)), we deduce

|∇| 3 ψ , ψ = O(c -3 ), equivalently ψ H 3/2 = O(1)
.

We now improve estimate (81a) as written before:

g0(p/c) -g0(0) = 1 0 g ′ 0 (tp/c) |p| c dt = 1 0 (1 -t)g ′′ 0 (tp/c) |p| 2 c 2 dt |g0(p/c) -g0(0)| 2 = 1 0 g ′ 0 (tp/c)dt 1 0 (1 -u)g ′′ 0 (up/c)du |p| 3 c 3 ,
and therefore

(m(α) -m)cψ L 2 ≤ K g ′ 0 ∞ g ′′ 0 ∞ c = Kα √ Lα = o(c -1 ). ( 83
) So χ H 1 = O(c -1 ) and |∇|χ L 2 = O(c -2 ). (84) 4.3.6 Estimation of E 0 BDF (1).
Thanks to Eq. (79b)

χ = Sσ g0 + µ ϕ + α g0 + µ (Rγψ)2 -vγ χ = Sσ g0 + µ ϕ + δχ,
where the remainder δχ is such that δχ L 2 is lesser than

Kα( γ Ex |∇| 1/2 ψ L 2 + ρ C |∇| 1/2 χ L 2 ) = O(αc -3/2 ) = o(c -1
). Thanks to Proposition 3, as g1ψ L 2 = O(c -1 ), we have the following asymptotic expansion:

E 0 BDF (1) + αb Λ (0) 2c D(n, n) = g0ϕ , ϕ -g 0 g 0 +µ Sσϕ , 1 g 0 +µ Sσϕ + 2R 1 g 0 +µ Sσϕ , Sσϕ + o(c -2 ) = m(α)(1 -2 g 2 1 (g 0 +µ) 2 ϕ , ϕ ) + 2 g 2 1 g 0 +µ ϕ , ϕ + o(c -2 ) = m(α) - g 2 1 2m(α) ϕ , ϕ + g 2 1 m(α) ϕ , ϕ + o(c -2 ) = m(α) + 1 2m(α) g 2 1 ϕ , ϕ + o(c -2 ) = m(α) + 1 2m(α) g 2 1 ψ , ψ + o(c -2 ).
To deal with g0 we use both results |∇| 3 ϕ , ϕ = O(c -3 ) and |g ′ 0 | = O(α) and treat the ((g0 + µ) -1 )'s one after the other. For the last line we use the fact that |∇| 2 χ , χ = O(c -3 ). Writing in terms of ψ:

C 2 0 (E 0 BDF (1)-m(α)) = 1 (g ′ 1 (0)) 2 (2π) 3 c 2 g1 p c 2 | ψ(p)| 2 dp- |ψ(x)| 2 |ψ(y)| 2 |x -y| dxdy+o(1). (85)
We recall (cf Proposition 4, Appendix A) the (g ′ 1 )α,Λ's are uniformly continuous in a neighbourhood of 0; splitting in Fourier space at level |p| = √ c we get

|p|≤ √ c c 2 g1(p/c) 2 | ψ(p)| 2 dp = |p|≤ √ c g ′ 1 (0) 2 |p| 2 | ψ(p)| 2 dp + |p|≤ √ c 1 t=0 (g ′ 1 (tp/c) -g ′ 1 (0))dt 2 |p| 2 | ψ(p)| 2 dp +2g ′ 1 (0) |p|≤ √ c 1 t=0 (g ′ 1 (tp/c) -g ′ 1 (0))dt |p| 2 | ψ(p)| 2 dp = |p|≤ √ c g ′ 1 (0) 2 |p| 2 | ψ(p)| 2 dp + O |∇|ψ 2 sup |q|≤c - 1 2 |g ′ 1 (q) -g ′ 1 (0)| = |p|≤ √ c g ′ 1 (0) 2 |p| 2 | ψ(p)| 2 dp + o c→+∞ (1)
.

Moreover:

|p|≥ √ c c 2 g1(p/c) 2 | ψ(p)| 2 dp |p|≥ √ c |p| 3 |p| | ψ(p)| 2 dp 1 √ c |∇| 3 ψ , ψ c -1/2 → c→+∞ 0.
Thus

1 (g ′ 1 (0) 2 ) c 2 g 2 1 (•/c)ψ , ψ -D(n, n) = |∇| 2 ψ , ψ -D(n, n) + o(1),
By unicity of the asymptotic expansion and by definition of ECP we thus have

E 0 BDF (1) = m(α) + C -2 0 ECP + o((αbΛ(0)) 2 ). (86)
As a consequence, the Choquard-Pekar energy wave function ψ (more specifically ϕ) tends to the minimizer. It is known [START_REF] Lieb | Existence and uniqueness of the minimizing solution to Choquard's nonlinear equation[END_REF] there is but one minimizer in H 1 (R 3 , C) up to translation. The fact that we work with spinors is harmless. By using convexity inequality for gradients [START_REF] Lieb | Loss, Analysis[END_REF] (Theorem 7.8 p.177) and Riesz's rearrangements inequality (sharp version in [START_REF] Lieb | Existence and uniqueness of the minimizing solution to Choquard's nonlinear equation[END_REF]), we have that there is but one minimizer of the Choquard-Pekar energy in H 1 (R 3 , C 4 ) up to translation and overall rotation in C 4 . Keeping track of the mass of ψ with the help of some translation we get that necessarily it tends to a Choquard-Pekar minimizer. Acknowledgment. The author wishes to thank É. Séré and M. Lewin for useful discussions and helpful comments, in particular the latter for suggesting the link with the Choquard-Pekar energy. He is also indebted to the referees and to É. Goujard for useful comments on the paper. This work was partially supported by the Grant ANR-10-BLAN 0101 of the French Ministry of Research. A.1 The functions g 0 and g 1 As established in [START_REF] Hainzl | The Mean-Field Approximation in Quantum Electrodynamics. The no-photon case[END_REF], D 0 is a solution to the following equation in the Fourier space

D 0 = D 0 + α 4π 2 D 0 |D 0 | * 1 | • | 2 in B(B(0, Λ), End(C 4 )) (87)
and by a bootstrap argument D 0 ∈ ∩ m≥1 H m B(0, Λ) . With the notation of 3 (Subsection 2) it shows that g0, g1 are smooth while g1(p) = g1(p) • ωp is a priori in C ∞ (B(0, Λ)\{0}) and we have

g0(|p|) = 1 + α 4π 2 |r|<Λ dr 1 |p -r| 2 g0(|r|) g1(|r|) 2 + g0(|r|) 2 , (88a)
g1(|p|) = |p| + α 4π 2 |r|<Λ dr ωp, ωr |p -r| 2 g1(|r|) g1(|r|) 2 + g0(|r|) 2 . ( 88b
)
Remark 11. We recall here that C1 > 0 is a constant such that g1(r) ≤ C1r and |g0|∞ ≤ C1.

Proposition 3. We have g1 ∈ C 1 ([0, Λ], R) and g ′ 0 (0) = 0. Writing d 2 g1 ⋆ = sup 0<|p|≤Λ |p|d 2 g1(p) the following holds:

g ′ 0 ∞ = O(α) g ′ 1 ∞ = O(1)
and

g ′′ 0 ∞ = O(α) d 2 g1 ⋆ = O(1)
.

Moreover there exists K > 0 such that ∀ q ∈ B(0, Λ)\B(0, 1),

       |g0(0) -1| ≤ Kα log Λ |q| + 1 |g ′ 1 (q) -1| ≤ Kα log Λ |q| + 1 ,
and we have

g0(0) = 1 + L π + O(L 2 + α), g ′ 1 (0) = 1 + 2L 3π + O(α).
In fact it suffices to differentiate (87) to get g ′ 0 (p) and g ′ 1 (p), we take the norm to obtain the first part; then we differentiate once more to get the second part. The third part is a consequence of those parts. Proposition 3 enables us to prove the following result. Lemma 14. Let p, q ∈ B(0, Λ) and k = p -q. There holds E (p) E (q)g(p), g(q)

E (p) E (q) ( E (p) + E (q)) ≤ min 2, 2K|k| 2
E (p) 2 , 2K|k| 2
E (q) 2 .
where we can choose K ≤ 2 for α log(Λ) sufficiently small.

Proof: In fact we can write for a, b, t

= b -a ∈ R 3 : |a||b| -a, b = a 2 t 2 -t,a 2 |a||b|+ a,b . If a, b > -|a||b| 2 then A = |a||b|-a,b |a||b| ≤ 2a 2 t 2 a 2 b 2 , by symmetry we also have A ≤ 2b 2 t 2 a 2 b 2 . Else -|a||b| ≤ a, b ≤ -|a||b| 2 , then |a||b|(|a||b| + a, b ) -1 ≥ 2(a 2 b 2 ) -1
, so: In particular the same holds for g ′ 1 (t) = dg1(tω), ω .

2 t 2 b 2 ≥ 2 a 2 +b 2 +|a||b| b 2 ≥ 2 2 t 2 a 2 ≥ 2 a 2 +b 2 +|a||b| a 2 ≥ 2.
Proof of Proposition 3 1. We can define dg1(p) for p = 0. First we have

dg0(p)h = α 4π 2 dq |p -q| 2 dg0(q)h E (q)
-g0(q)dg0(q)h + g1(q)dg1(q)h E (q) 2 g0(q) E (q) .

We remark that for p = 0 we have:

dg⋆(p)h = g ′ ⋆ (|p|) ωp , h , ⋆ ∈ {0, 1}, dg1(p) • ωp , ωp = g ′ 1 (|p|).
Then

dg1(p)•h = h+ α 4π 2 dq |p -q| 2 dg1(q) • h E (q) - g0(q)dg0(q)h + g1(q)dg1(q)h E (q) 2 g1(q) E (q) .
So for any ω ∈ S 2 we have

g ′ 1 (x) = 1 + α 4π 2 |q|≤Λ dq |xω -q| 2 g1(q) |q| (1 -ω, ωq 2 ) 1 E (q) + g ′ 1 (q) ωq, ω 2 1 - g 2 1 (q) E(q) 2 1 E (q) - g1(q) E (q) ω, ωq 2 E (q) g0(q)g ′ 0 (q) E (q) . ( 89
)
The regularity of g1 (as a function of R + ) will come from the continuous extension to x = 0 of the formula above.

We have

|g ′ 0 (|p|)| ≤ α 4π 2 dq |p -q| 2 |g ′ 0 |∞ E (q) + |g0|∞ |g ′ 0 |∞ + |g ′ 1 |∞ E (q) 2 (90a) |g ′ 1 (|p|)| ≤ 1 + α 4π 2 dq |p -q| 2 |g ′ 1 |∞ E (q) + |g ′ 0 |∞ + |g ′ 1 |∞ E (q) . ( 90b
) Thus |g ′ 0 |∞ ≤ K1α log(Λ)|g ′ 0 |∞ + K2α|g ′ 1 |∞ |g ′ 1 |∞ ≤ 1 + K3α log(Λ)(|g ′ 0 |∞ + |g ′ 1 |∞). So |g ′ 0 |∞ α and |g ′ 1 |∞ ≤ 1 + Kα log(Λ)
. Since g0 ∈ C ∞ (B(0, Λ), R) and radial, necessarily dg0(0) = 0 and g ′ 0 (0) = dg0(0) ω = 0, ∀ω ∈ S 2 .

2. We treat now the second derivative d 2 D 0 . We write h⋆ = g⋆ E(•)

and

J = E (•) -1 . The coefficient of β in d 2 D 0 (p) h 2 is d 2 g0(p) h 2 = α 4π 2 q dq |p -q| 2 d 2 h0(q) h 2 ,
where

d 2 h0(q) h 2 = d 2 g0(p) • h 2 E (q) - 2
E (q) 3 dg0(q)h [g0(q)dg0(q)h + g1(q)dg1(q)h] -g0(q) E (q) 3 (dg0(q)h) 2 + g0(q)d 2 g0(q)h 2 + (dg1(q)h) 2 + g1(q)d 2 g1(q)h 2 +3 g0(q) E (q) 5 [g0(q)dg0(q)h + g1(q)dg1(q)h] 2 .

Furthermore, we have

d 2 g1(p)h 2 = α 4π 2 dq |p -q| 2 d 2 g1(q)h 2 E (q)
+ 2dg1(q)h dJ (q)h + g1(q)d 2 J (q)h 2 .

Since we have |p|d

2 g1(p)h 2 , ωp = |p|d 2 g p 1 • h 2 + g 1 (p)
|p| ωp, h 2 -|h| 2 , by taking the scalar product with ωp we get

|p||d 2 g1(p)| ≤ C1 + α 4π 2 |p|dq |p -q| 2 |q|E (q) d 2 g1 ⋆ + α 4π 2 |p|dq |p -q| 2 E (q) 2 d 2 g1 ⋆ + α 4π 2 q |p|dq |p -q| 2 1 E (q) 2 (|dg0| 2 + |dg1| 2 ) + g0(q) E (q) 2 |d 2 g0| + 3 E (q) 2 (|dg0| + |dg1|) 2 + 2(|dg1| + C1) |dg0| + |dg1| E (q) 2 + 1 E (q) 2|dg1| + 4C1
|q| .

We also have:

|d 2 g0(p)| ≤ α 4π 2 C1dq E (q) 2 |p -q| 2 d 2 g1 ⋆ q dq |p -q| 2 |d 2 g0| E (q) + 2 |dg0|(|dg0| + |dg1|) E (q) 2 + g0(q) E (q) |dg0| 2 + |dg1| 2 E (q) 2 + g0(q) 2 E (q) 2 |d 2 g0| E (q) + 3 g0(q) E (q) (|dg0| + |dg1|) 2 E (q) 2 .
As

|p| |p-q| 2 |q| ≤ 2 max( 1 |p-q||q| , 1 |p-q| 2 ), we have |q|≤Λ dq|p| |p -q| 2 |q|E (q) ≤ 2 |q|≤Λ dq |p -q||q|E (q) + |q|≤Λ dq |p -q|E (q) .
We recall then that the convolution of radial nonnegative functions is radial nonnegative. So the following holds:

g ′′ 0 ∞ ≤ Kα d 2 g1 ⋆ ≤ C1 + Kα log(Λ)
3. By Ineq (88a) and for p ∈ R 3 , 1 ≤ |p| < Λ we get that:

4π 2 |g0(p) -1| α = |q|<Λ dq |p -q| 2 g0(q) g0(x) 2 + g1(q) 2 ≤ |q|<Λ dq |p -q| 2 1 1 + |q| 2 |g 0 | 2 ∞ ≤ |q|<2Λ dq |q| 2 1 1 + |p+q| 2 |g 0 | 2 ∞ ≤ |q|<2Λ dq |q| 2 |g0|∞ |p + q| = 2π|g0|∞ Λ 0 dr r|p| log r + |p| r -|p| = 2π|g0|∞ Λ 0 r + |p| -r -|p| r|p| 1 + log Λ |p| .
To deal with g ′ 1 we use Eq. ( 89). The integral of the integrand in the second line is O(1): as we multiply by α its contribution is O(α). For 1 ≤ |p| < Λ there holds:

|q|<Λ dq |p -q| 2 | E (q) g1(q) |q| ≤ |q|<2Λ dq |q| 2 |p + q| 1 + log Λ |p| .
For g0(0) we have:

π|g0(0) -1| α = Λ 0 g0(r)dr g0(r) 2 + g1(r) 2 = Λ 1 dr 1 + O(α log Λ r ) √ 1 + r 2 + O(1) = log(Λ) + O(1 + α log(Λ) 2 ).
Let us prove the estimation of g ′ 1 (0). There holds for any 0 < x < Λ and ω ∈ S 2 :

|q|<Λ ω,ωq 2 dq |xω -q| 2 E (q) g1(q) |q| = 2π Λ 0 dr x 2 + r 2 2x 2 x 2 + r 2 2rx log x + r x -r -1 g1(r) r E (r) , = 2π Λ/x 0 dr 1 + r 2 2r 1 + r 2 2r log 1 + r 1 -r -1 g1(xr) E (xr) .
We split at two levels: e -1 and e. The integral over (e -1 , e) is O(1) for log is integrable on (0, e). For x ∈ (0, e -1 ) there holds the following expansion:

1 + r 2 r (log(1 + r) -log(1 -r)) -1 = 4 3 r 2 + O r→0 (r 3 ),
thus the integration over (0, e -1 ) is O(1). For x ∈ (e, Λ/x) there holds: At last we get:

1 + r 2 r (log(1 + r -1 ) -log(1 -r -1 )) -1 = 4 3r 2 + O r→+∞ (r -3
g ′ 1 (0) -1 = α π Λ 0 g1(r)dr r E (r) 1 - 1 3 + O(α) = 2α log(Λ) 3π + O(α).

Proof of Proposition 4

In fact it suffices to use another formulae for d 2 g1 and d 2 g0 consisting in replacing g1(q)dg1(q) by g1(q) , dg1(q) .

By the same method as for dg0, dg1, we get that

d 2 g1 ∞ L. (91)

A.2 The function B Λ

We recall that

BΛ(k) = 1 π 2 |k| 2 |p=l-k 2 |,|q=l+ k 2 |<Λ E (p) E (q) -g(p), g(q) E (p) E (q) ( E (p) + E (q)) dl ≥ 0. (92)
This formula holds only for k = 0: our first purpose is to extend it continuously to 0. Thanks to Lemma 14 we can say that BΛ(k) ≤ K log(Λ). Notation 10. Throughout this part, p = ℓ + k 2 , q = ℓ -k 2 . Proposition 5. Let ω be any in S 2 . For ℓ ∈ B(0, Λ) we write:

g ω ℓ := g ′ 0 (|ℓ|)ω ℓ • ω dg1(ℓ) • ω and E ω ℓ := |g ω ℓ |.
Then we have

BΛ(k) → k→0 1 π 2 |ℓ|≤Λ |g ω ℓ ∧ g ℓ | 2 4 E (ℓ) 5 dℓ =: BΛ(0), (93)
Moreover BΛ(0) = 2 3π log(Λ) + O(L log(Λ) + 1).

Proof: Let us write I = π 2 |k| 2 BΛ(k), its integrand f (ℓ) and x = |k|. Let us consider 0 < ε < 2 3 and s = 1 3 + ε. We assume x < 1 and split the domain in three:

B = {ℓ : |ℓ| ≤ x s }, A = {ℓ : x s < |ℓ| < Λ -x 2 }, C = {ℓ : |ℓ -k 2 |, |ℓ + k 2 | < Λ}\{ℓ : |ℓ| < Λ -x 2 } ⊂ {ℓ : Λ -x 2 < |ℓ| < Λ} = C ′ .
Using Lemma 14 we get the following behaviour independent of α, Λ in the regime (23):

|IB| ≤ Kx 2+3s = Kx 3+3ε = o x→0 (x 3 ), |IC| ≤ Kx 2 log Λ Λ -x 2 ∼ x→0 Kx 3 Λ . (94)
There remains to deal with IA: we rewrite f (ℓ) as follows:

f (ℓ) = |g(p) ∧ g(q)| 2 E (p) E (q) ( E (p) + E (q))( E (p) E (q) + g(p) • g(q)) (95)
where

|g(p) ∧ g(q)| 2 = i |∆0i| 2 + i,j |∆ij | 2 , ∆0i = g0(p) g0(q) (g1(p))i (g1(q))i = δg0 g0(q) (δg1)i (g1(q))i (96a) ∆ij = (g1 (
p))i (g1(q))i (g1(p))j (g1(q))j = (δg1)i (g1(q))i (δg1)j (g1(q))j (96b) δg⋆ = g⋆(p) -g⋆(q). If we take k along a fixed half-line: k = xω we have

1 x δg0(k, ℓ) = 1 t=0 dg0(ℓ + (t -1/2)k) • ωdt → x→0 g ′ 0 (|ℓ|)ω ℓ • ω 1 x δg1(k, ℓ) = 1 t=0 dg1(ℓ + (t -1/2)k) • ωdt → x→0 dg1(ℓ) • ω.
In fact, as A, g0, g1 are radial symmetrics so is IA(k) and for ω ∈ S 2 fixed and p ′ = ℓ + xω 2 , q ′ = ℓ -xω 2 there holds

IA(k = xω k ) = 1 π 2 x 2 x s <|ℓ|<Λ-x 2 E (p ′ ) E (q ′ ) -g(p ′ ), g(q ′ ) E (p ′ ) E (q ′ ) ( E (p ′ ) + E (q ′ )) dl, f0(ℓ) = f (ℓ) x 2 χ ℓ∈A is also symmetric. By Proposition 3 we have |f0(ℓ)| ≤ K 1 (1+|ℓ| 2 ) 3/2 χ |ℓ|≤Λ-x/2
. By dominated convergence we get the integral formula (93). As there holds by symmetry

n∈S 2 n, ω 2 dn = 4 3 π, n∈S 2 |dg1(|ℓ|n)•ω| 2 dn = 4 3 π (g ′ 1 ) 2 (ℓ) + 2 g1(ℓ) 2 |ℓ| 2 (97)
we have

BΛ(0) = 1 3π Λ u=0 u 2 ((g ′ 0 ) 2 (u) + (g ′ 1 ) 2 (u) + 2 g 1 (u) 2 |u| 2 )(g 2 0 (u) + g 2 1 (u)) (g0(u) 2 + g1(u) 2 ) 5/2 du - Λ u=0 u 2 (g0g ′ 0 (u) + g1g ′ 1 (u)) 2 (g0(u) 2 + g1(u) 2 ) 5/2 du , and
BΛ(0) = 1 3π   Λ u=0 u 2 (g ′ 0 ) 2 (u) + (g ′ 1 ) 2 (u) + 2 g 1 (|u|) 2 |u| 2 (g0(u) 2 + g1(u) 2 ) 3/2 du - Λ u=0 u 2 (g0g ′ 0 (u) + g1g ′ 1 (u)) 2 (g0(u) 2 + g1(u) 2 ) 5/2 du   .
Thanks to Proposition 3, we get the estimate of BΛ(0).

Let us look at the variations |k| -1 |BΛ(k) -BΛ(0)|. Proposition 6. There exists 0 < rε ∈ R + , independent of α, Λ in the regime (23) such that for |k| < rε:

|k| -1 |BΛ(k) -BΛ(0)| ≤ K(Λ -1 + L 2 |k| + |k| 3ε + |k| 2/3-ε ).
Choosing ε := 6 -1 there holds:

|k| -1 |BΛ(k) -BΛ(0)| ≤ K(Λ -1 + |k| 1/2 ).
Proof: For k ∈ B(0, 1) we write |k| = x. We reconsider the domains A, B and C of the proof of Proposition 5 and write f1 the integrand in (92).

We have

| B f1| ≤ Kx 3s = O x→0 (x 1+3ε ) and | C f1| ≤ K log( Λ Λ-x/2 ) = O x→0 ( x Λ
). There remains the integration over A. For |ℓ| ≥ x s we have x |ℓ| = O(x 2/3-ε ) so we can expand the integrand of IA(x) at order 1. Indeed:

E (p) -1 = E (ℓ) -1 1+ E (p) -E (ℓ) E (ℓ) -1 = E (ℓ) -1 1+ E (ℓ) -E (p) E (ℓ) + O x→0 x 2 E (ℓ) 2 , where the O x→0 (•) is independent of ℓ (because E (ℓ) ≥ 1)
. The same holds for

E (q) -1 and ( E (p) + E (q)) -1 . Writing h(ℓ, k) = E (p) E (q) -g(p) • g(q)
we have:

IA(x) = 1 x 2 A h(ℓ, k) 2 E (ℓ) 3 dℓ + 1 x 2 A h(ℓ, k) 2 E (ℓ) 3 2 E (ℓ) -E (p) -E (q) E (ℓ) + 2 E (ℓ) -E (p) -E (q) 2 E (ℓ) + O x 2 E (ℓ) 2 .
By Taylor formula (at order 2):

|2 E (ℓ) -( E (p) + E (q))| ≤ t u dtduKx 1+2/3-ε = Kx 1+2/3-ε .
By Proposition 4 and by Taylor formula at order 1 we have:

g(p) -g(q) x -g ω l Lx. Thus |k| -1 |BΛ(k) -BΛ(0)| = O k→0 (Λ -1 + L + |k| 3ε ).
B The fixed point method: estimations

B.1 Estimation about the R • operator

Let us generalize Lemma 8. [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF] that states the inequality: RQ R Q Q. Further generalisations are detailed in [START_REF] Sok | Modèle de champ moyen en électrodynamique quantique[END_REF].

Lemma 15. Let f be some function f : B(0, Λ) → R+ and Q ∈ Q f . Then we have:

f (p -q) | RQ(p, q)| 2 |p + q| dpdq f (p -q)|p + q|| Q(p, q)| 2 dpdq. (98)
Proof: The kernel R(p, q) := RQ(p, q) is equal to:

R(p, q) = 1 2π 2 Q(p -ℓ, q -ℓ) |ℓ| 2 dℓ.
We remark the Fourier multiplier:

A(x, y) → F -1 f (p -q) A(p, q) commutes with R• : A → RA. So it suffices to show that: | R(p, q)| 2 |p + q| dpdq |p + q|| Q(p, q)| 2 dpdq.
To this end we follow the proof in [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF], for any θ ∈ (0, 2):

| R(p, q)| 2 |p + q| dpdq = 8 dudv |2u| | R(u + v, u -v)| 2 ≤ 8 (2π 2 ) 2 | Q(ℓ + v, ℓ -v)|| Q(ℓ ′ + v, ℓ ′ -v)| |2u||ℓ -u| 2 |ℓ ′ -u| 2 dudvdℓdℓ ′ ≤ 8 (2π 2 ) 2 1 |2u| | Q(ℓ + v, ℓ -v)| 2 |ℓ -u| 2 |ℓ ′ -u| 2 |2ℓ| 1+θ |2ℓ ′ | 1+θ dudvdℓdℓ ′ ≤ 8 (2π 2 ) 2 |2ℓ|| Q(ℓ + v, ℓ -v)| 2 w θ (ℓ)dvdℓ,
where the weight w θ (ℓ) is:

w θ (ℓ) := |2ℓ| θ dudℓ ′ |2u||2ℓ ′ | 1+θ |ℓ -u| 2 |ℓ ′ -u| 2 .
Then we have:

w θ (ℓ) ≤ u du |2u| 1+θ |u -ℓ| 2 |2u| θ ℓ ′ dℓ ′ |2ℓ ′ | 1+θ |ℓ ′ -u| 2 ≤ 1 2 dx |x| 1+θ |x -e| 2 2
, where e ∈ R 3 is any vector satisfying |e| = 1.

B.2 Estimates for the fixed point method

Let N0 ≥ 0 be in S1(HΛ) and let γ0 be in S In this part f is some function f : R 3 → [1, +∞) satisfying condition (49) and we consider the Fourier multiplier m f : Q(x, y) ∈ L 2 (HΛ × HΛ) → F -1 (f (p -q) Q(p, q)).

For Q0 ∈ Q f , ρ0 ∈ C f we write:

(Q0, ρ0) X f := K (0) (f )( Q0 Q f + ρ0 C f ),
where K (0) (f ) > 0 to be precised later. By Kato's inequality and Sobolev inequality (39) n0 C x 1/2 and n0 L 2 x 3/2 . For the last inequality it suffices to write N0 := ai|fi fi|, ai ≥ 0 and fi L 2 = 1. Then:

n0 L 2 ≤ i ai ∇fi 3/2 L 2 ( i ai ∇fi 2 L 2 ) 3/4 .
The same method enables us to prove that RN 0 S 2 x.

Lemma 16. Let N0 and γ0 be as above. Then we have:

Q0,1[ργ 0 ] Q f log(Λ) ργ 0 C f , Q1,0[γ0] Q f γ0 Q f , ρ1,0[ργ 0 ] C f log(Λ) γ0 Q f . Moreover: Q1,0[N0] E x, ρ1,0[N0] C x.
Lemma 17. Let (Q0, ρ0) be in X f . There exist constants K (1) , K (2) > 0 such that, writing

G f (Q, ρ) := K (1) C(f )( Q Q f + ρ C f )
we have:

∀ℓ ≥ 2 : (Q ℓ , ρ ℓ )[Q0, ρ0] X f ≤ K (2) √ ℓ G f (Q0, ρ0) ℓ . ( 100
)
Assuming these lemmas hold, we follow [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF] to find a ball B(0, R f ) invariant under the function F = FQ × Fρ of the fixed point method ((46) and (47)) and on which F is a contraction. Indeed for some K (4) > 0, we have:

           FQ[Q0, ρ0] Q f ≤ N Q f + K (4) √ Lα( Q0 Q f + ρ0 C f ) + K (2) +∞ ℓ=2 ℓ 1/2 (αG f (Q0, ρ0)) ℓ , Fρ[Q0, ρ0] C f ≤ n C f + K (4) √ Lα( Q0 Q f + ρ0 C f ) + K (2)
+∞ ℓ=2 ℓ 1/2 (αG f (Q0, ρ0)) ℓ , these upper bounds are finite provided αG f (Q0, ρ0) < 1 where G f is defined in Lemma 17. Moreover:

dF [Q0, ρ0] L(X f ) ≤ 2 K (4) √ Lα + αK (3) (f ) +∞ ℓ=2 ℓ 3/2 (αG f (Q0, ρ0)) ℓ-1
where K (3) (f ) = K (1) K (2) C(f )A(f ). The supremum of the above upper bound on BX f (0, R) is written ν = ν(f, R).

We take K (0) (f ) := K (1) C(f ), R f = ε f log(Λ) for some ε f > 0 and assume

( N Q f + n C f ) ≤ εn log(Λ) (with 0 < εn < ε f ).
For any (Q0, ρ0) ∈ BX f (0, R f ) the following holds:

F (Q0, ρ0) X f ≤ ν(f, R f ) (Q0, ρ0) X f + F (0, 0) X f
≤ ν(f, R f )ε f log(Λ) + K (0) (f )εn log(Λ).

We have:

ν(f, ε f log(Λ)) ≤ 2K (4) √ Lα + 2αK (3) (f ) +∞ ℓ=2 ℓ 1/2 αε f log(Λ) C(f ) ℓ-1
To apply the Banach fixed point Theorem it suffices to have: ν(f, ε f log(Λ)) < 1 and ν(f, ε f log(Λ)) + K (1) C(f )εn ε f < 1.

For fj (p -q) = E (p -q) j with j ∈ {0, 1, 2} and provided α log(Λ)ε f is small enough we have:

ν(f1, ε f log(Λ)) √ Lα(1 + α log(Λ)ε f ) = O( √ Lα).
In the case α log(Λ) ≪ 1, it suffices to take εn ε f small enough to apply the fixed point Theorem. Proof of Lemma 16 Let M (•, •) be the function (p, q) ∈ B(0, Λ) 2 → M (p, q) := 1

E (p) + E (q) D 0 (p) E (p) D 0 (q) E (q) -1 .
We write S(p) := D 0 (p) E(p) for short. A direct computation in Fourier space (and Cauchy's formula) gives like in [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF]: Q0,1(ρ; p, q) = 1 2 5/2 π 3/2 ϕρ(p -q)M (p, q), Q1,0(γ; p, q) = -1 2 S(p) Rγ(p, q)S(q) -Rγ (p, q) .

(101) and to deal with E (p + q) we use the following trick:

1

{( E (p) 2 + η 2 )( E (q) 2 + η 2 )} 1/4 1 E (p + q) E (η) . ( 103
)
We consider the integral representation of each term of Q j,ℓ-j [Q0, ρ0]; for convenience we write R0 := R[Q0] and ϕ0 := ϕ[ρ0].

For instance let us treat the term where the j operators R0 are on the left, we take the modulus and get the upper bound: (104) We write p0 := p and p ℓ := q.

We multiply (104) by f (p -q) E (p + q) and use tricks (102) and (103). We then use (103) for the terms involving pi and pi+1 (0 ≤ i ≤ j -1) and get:

f (p ′ -q ′ ) 1/2 | R0(p ′ , q ′ )|/ E (p ′ + q ′ ) S 2 Q0 Q f . (105)
Moreover we have by the K.-S.-S. inequality:

( E (p ′ ) 2 + η 2 ) -1/4 | f (p ′ -q ′ ) ϕ0(p ′ -q ′ )|( E (q ′ ) 2 + η 2 ) -1/4 S 6 ρ0 C f E (η) 1/2 , ( E (p ′ ) 2 + η 2 ) -1/4 | f (p ′ -q ′ ) ϕ0(p ′ -q ′ )|( E (q ′ ) 2 + η 2 ) -1/4

S∞

≤ ( E (p ′ ) 2 + η 2 ) -1/4 | f (p ′ -q ′ ) ϕ0(p ′ -q ′ )|( E (q ′ ) 2 + η 2 ) -1/4 S 6 . (106) By using those K.-S.-S. inequalities under the integral sign η in (104) (multiplied by the weight f (p -q) E (p + q)), we get an upper bound of the form:

+∞ -∞ dη E (η) (1+j+ℓ-j)/2 ℓK ℓ C(f ) ℓ Q0 j Q f ρ0 ℓ-j C f .
This upper bound is valid provided (ℓ + 1)/2 > 1 and ℓ ≥ 3 ie if ℓ ≥ 3.

In fact the same method gives:

Q2,0[Q0] Q f C(f ) 2 Q0 2 Q f , Q1,1[Q0, ρ0] Q f C(f ) 2 Q0 Q f ρ0 C f .
Let us now deal with the densities ρ ℓ [Q0, ρ0]. First remark: as recalled in [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF], Furry's Theorem states that for all ℓ = 2ℓ1 ∈ 2N * even, we have ρ 0,2ℓ 1 = 0.

As in [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF], we deal with the other terms by duality: the dual C ′ f of C f is: For the test function defined by ( 25) and (26), it is clear that x = O(Lα) and w(N ) = O((Lα) 3/2 ). 2. The estimate of γ S 2 follows from these estimates. First by computing in Fourier space it is clear that:

C ′ f = ζ ∈ S ′ (R
∀ρ0 ∈ C : Q0,1[ρ0] S 2 ρ0 C .
Then:

γ S 2 ≤ +∞ j=1 α j Qj [γ ′ , ρ ′ γ ] S 2 ≤ α( Q0,1[ρ ′ γ ] S 2 + Q1,0[γ ′ ] S 2 ) + +∞ j=2 α j Qj[γ ′ , ρ ′ γ ] E α ρ ′ γ C + α( RN S 2 + γ Ex ) + O(α 2 (γ ′ , ρ ′ γ ) 2 X ) α √ Lα.
Moreover:

| E (p) -E (q) | 2 | γ(p, q)| 2 dpdq |p -q| 2 | γ(p, q)| 2 dpdq γ 2 E (Lα) 2 .
To conclude this part, there remains to estimate γ|D 0 |ψ λ L 2 and γψ λ L 2 . We have:

γ|D 0 |ψ λ L 2 ≤ γ S 2 |D 0 |ψ λ L 2 α √ Lα, γψ λ L 2 ≤ γ S 2 ψ λ L 2 α √ Lα.
We can get better upper bounds [START_REF] Sok | Modèle de champ moyen en électrodynamique quantique[END_REF] but we do not need them here.

2λ 2 g 4 .

 24 0 (0) |∇| 2 ψ1 , ψ1 + o((Lα) 2 ). Estimation of Tr0(Dγ) Notation 9. Let us write B

1 and

 1 Tr (|D 0 + αB| -|D 0 |)γ 2 = O(α(Lα) 2 ). Supposing those facts are true we get Tr(|D 0 |γ 2 ) = -α 2 Tr0(Bγ) + O(ατ 2

1 and

 1 if Tr( Q(p, p))dp exists then this last integral is equal to Tr0(Q), because P 0 -= f (i∇), in Fourier space we have Tr C 4 ( P 0 -(p) Q(p, p) P 0 + (p)) = Tr C 4 ( P 0 + (p) Q(p, p) P 0 -(p)) = 0. Here, the trace Tr0(ϕ ′ γ γ) is formally equal to

.

  Either x = lim inf k→∞ E 0 BDF (γ) > 0 and E(q) > qE(1) or x = 0. What happens in the second case ? Up to the extraction of a subsequence we can assume that lim inf E 0 BDF (γ) is a limit. Thanks to (20) it implies Tr(|D 0 |γ 2 ) + D(ργ, ργ) → k→∞ 0.

Appendices A The operator D 0 Remark 10 .

 010 In this part the scalar product in R d is written • , • for d = 3, 4.

Proposition 4 .

 4 The functiondg1(p) = id + α 4π 2 |r|<Λ dr |p -r| E (r) dg1(r) -g1(r) g0(r)dg0(r) + g1(r)dg1(r) E (r) 2 is in C 0 (B(0, Λ), L(R 3 , C4)) and |dg1(p) -dg1(q)| ≤ KL|p -q|.

P 0 - 1 (

 01 HΛ).We write n0 := ρN 0 and x(N0) := ∇N0 S 2 . We assume that Tr(N0) 1(99) to simplify. In our problem N0 = |ψ ψ| with ψ L 2 = 1.

|

  ϕ0(p k -p k+1 )| E (p k+1 ) 2 + η 2 .

  3 ) : | ζ(k)| 2 |k|f (k) dk < +∞ . For any ζ ∈ C ′ f ∩ L 2 and Q ∈ S2(HΛ) we have Qζ = (Q|D 0 | 2 )( 1 |D 0 | 2 ζ) ∈ S1(L 2 (R 3 )).If we consider the norm • C , there holds:α (δ0 -bΛ) * ρ1,0(N ) L 2 √ Lα |p + q||p -q| 2 | N (p, q)| 2 dpdq =: √ Lαw(N )where we have used Lemma 15 with f (p -q) = |p -q| 2 . Provided x = O(Lα) and w(N ) = O(L) the following estimate hold:ργ C , ργ C L √ Lα.

  • L p and • H s for p ∈ [1, +∞) and s ∈ R+ are the usual norms of L p and Sobolev functions. Moreover • Sp is the norm of the space of Schattenclass operators Sp(HΛ) and • B is the usual norm of bounded linear operators in B(HΛ). The norms • C and • Ex are defined in the introduction. A large part of the paper is devoted to estimate Sobolev norms of test functions Q and among them the norm

  ).

	If we multiply by gives:	1 + r 2 2r	we get	2 3r	+ O r→+∞ (r -2 ). Thus the integration over (e, Λ)
	4π 3	e	Λ/x	g1(rx)dr E (rx) r	+ O(1) =	4π 3	Λ ex	g1(r)dr E (r) r	+ O(1).

We will use Lemma 14: it gives an estimation of M (p, q) :

The estimation of Q0,1(ργ 0 ) Q f is then easy. In (101), it suffices to use Lemma 15 to get estimation of Q1,0(γ0) Q f . Then, as RN S 2

x, the estimation of Q1,0(N0) Kin follows from a simple computation of | Q1,0(N0)| 2 . Then the norm ρ1,0[γ0] C f is dealt with in the same way as in [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF]:

By Cauchy-Schwartz inequality we have:

Proof of Lemma 17

We only sketch the proof of Lemma 17 in this paper: we refer the reader to [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF][START_REF] Sok | Modèle de champ moyen en électrodynamique quantique[END_REF] for full details.

The main idea is to use the K.-S.-S. inequality (38) together with the Hölder inequality for Sp(HΛ). For instance, let us take the Hilbert-Schmidt norm of Q0, 3[ρ0]:

By Sobolev inequality we get hρ 0 L 6 ρ0 C and thus there holds:

The term Q2 is dealt with the same way as in [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF]: we refer to this paper for details. The difference between the example above Q0,3

before taking the Hilbert-Schmidt norm. Besides this fact the main idea is the same:

• We take its Hilbert-Schmidt norm and get an upper bound of it using K.-S.-S.

and Hölder inequalities.

To deal with f (p -q) we use condition (49):

Then the following holds:

The idea is to get an upper bound depending only on the C ′ f -norm of ζ and to conclude by density of C ′ f ∩ L 2 in C ′ f . The ingredients are the same but we treat ρ1,1 and ρ0,3 differently (as in [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF]). We use the same K.-S.-S. inequalities and (102).

For instance, for ℓ ≥ 5:

We write

and use (102):

Then it suffices to use 6 times the first inequality of ( 106) and (ℓ + 1 -6) times the second. We refer the reader to [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF][START_REF] Sok | Modèle de champ moyen en électrodynamique quantique[END_REF] for the other terms.

B.3 Estimates of a fixed point

Let (N, n) ∈ X⋆ be given where ⋆ means 0, 1 or no subscript. Let us assume that the norms of N and n are O(1) such that we can apply the fixed point Theorem (cf Lemmas 7 and 8). From now on ν is Lipschitz constant in Lemma 7 that is the one corresponding to F applied on some ball BX (0, R). We write: x = Tr(-∆|N |).

We apply the Banach theorem with initial data (0, 0) ∈ X⋆ : iterations are written (γ ′ (ℓ) , ρ ′ (ℓ) ) and γ (ℓ) , ρ (ℓ) are defined as follows:

The fixed point is written:

Lemma 18. Let N, n, γ, ργ be as above. If (N, n) X⋆ = O(1) then so is (γ, ργ) X⋆ .

Proof: In the regime (23), the Lipschitz constant ν0 in Lemmas 7, 8 is o(1). So:

We want to be more precise and prove Lemma 9. We first have:

Lemma 19. Let N, n, γ, ργ , x(N ) =: x be as above. Let us write:

Then the following estimates hold:

Proof: The first point is devoted to Lemma 19 and the second to the end of Lemma 9. 1. We write n := F -1 ( n(k)/(1 + αBΛ(k))). There holds: F (0, 0) = (N, n); in particular γ (1) = 0 and ρ (1) = n -n = -F -1 (bΛ) * n.

Writing γ = +∞ ℓ=1 (γ (ℓ+1) -γ (ℓ) ) + γ (1) we have:

The first term on the right hand side is equal to

By Lemmas 16 and 17 the following inequalities hold

Using the same method for

There remains to check that x = O(Lα) to get γ E Lα and γ Q α.

For the density we have:

. The first term is O(α 2 ). The last term is the norm of: Here δ0 is the usual Dirac's generalized function.