Guillaume Idelon-Riton

Quasimodes and a lower bound for the local energy decay of the Dirac equation in Schwarzschild-Anti-de Sitter spacetime

Quasimodes and a lower bound for the local energy decay of the Dirac equation in Schwarzschild-Anti-de Sitter spacetime

Guillaume Idelon-Riton

Introduction

There has been a lot of works concerning the stability of black holes spacetimes in mathematical general relativity. Recently, some significant results have been obtained in this direction. Concerning the Schwarzschild solution, M. Dafermos, G. Holzegel and I. Rodnianski [START_REF] Dafermos | The linear stability of the Schwarzschild solution to gravitational perturbations[END_REF] obtained linear stability to gravitational perturbation by means of vector field methods. F. Finster and J. Smoller [START_REF] Finster | Linear stability of the non-extreme Kerr black hole[END_REF] obtained linear stability of the non-extreme Kerr black hole using an integral representation for the solution of the Teukolsky equation. Concerning the full non-linear stability problem, P. Hintz and A. Vasy [START_REF] Hintz | The global non-linear stability of the Kerr-de Sitter family of black holes[END_REF] obtained the non-linear stability of the Kerr-de Sitter family of black hole by means of a general framework which uses microlocal techniques. In particular, the latest uses a precise understanding of the resonances for their problem. Resonances allow to obtain a precise decay rate for the fields and at which frequency this decay happens. This has been a subject of advanced study in the last decades. In mathematical general relativity, the analysis of resonances, which are also called quasi-normal modes, begins with the work of Bachelot and Motet-Bachelot [START_REF] Bachelot | Motet-Bachelot : Les résonances d'un trou noir de schwarzschild[END_REF] for the Schwarzschild black hole. This study was then pursued by A. Sá Barreto and M. Zworski [START_REF] Barreto | Distribution of resonances for spherical black holes[END_REF] for spherically symmetric black holes where resonances are presented as poles of the meromorphic continuation of the resolvent. J-F Bony and D. Häfner [START_REF] Bony | Decay and Non-Decay of the Local Energy for the Wave Equation[END_REF] gave a full expansion of the solution of the wave equation in terms of resonant states which gives an exponential decay of the local energy for the Schwarzschild-de Sitter solution. R. Melrose, A. Sá Barreto and A. Vasy [START_REF] Melrose | Asymptotics of solutions of the wave equation on de Sitter-Schwarzschild space[END_REF] extended this result to more general manifolds and initial data. This was also extended to the case of rotating black hole with positive cosmological constant by S. Dyatlov [START_REF] Dyatlov | Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole[END_REF], [START_REF] Dyatlov | Asymptotic distribution of quasi-normal modes for Kerr-de Sitter black holes[END_REF] and then by A. Vasy [START_REF] Vasy | Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces[END_REF] who develops more general and flexible techniques. The method developed by S. Dyatlov was also extended to the framework of Dirac operators for the Kerr-Newman-de Sitter black hole by A. Iantchenko [START_REF] Iantchenko | Quasi-normal modes for Dirac fields in Kerr-Newman-de Sitter black holes[END_REF] to define resonances . This result generalizes papers by the same author concerning resonances in the de Sitter-Reissner-Nordström spacetime where expansion of the solution in term of resonant states was also given [START_REF] Iantchenko | Quasi-normal modes for de Sitter-Reissner-Nordström black holes[END_REF], [START_REF] Iantchenko | Resonance expansions of massless Dirac fields propagating in the exterior of a de Sitter-Reissner-Nordström black hole[END_REF]. Concerning resonances in Anti-de Sitter black holes, C. Warnick [START_REF] Warnick | On quasinormal modes of asymptotically anti-de sitter black holes[END_REF] defines resonances in asymptotically Anti-de Sitter black holes. Quasi-normal modes were also defined for Kerr-Anti-de Sitter black holes by O. Gannot [START_REF] Gannot | Existence of quasinormal modes for Kerr-AdS black holes[END_REF]. In the case of Schwarzschild-Anti-de Sitter black hole, the same author proves the existence of resonances using the black box method.

By means of exponentially accurate quasimodes, he obtains the localization of resonances exponentially close to the real axis [START_REF] Gannot | Quasinormal modes for Schwarzschild-ADS black holes: exponential convergence to the real axis[END_REF]. This is linked to a logarithmic local energy decay for the Klein-Gordon fields that was proven by G. Holzegel and J. Smulevici [START_REF] Holzegel | Decay properties of Klein-Gordon fields on Kerr-AdS spacetimes[END_REF], [START_REF] Holzegel | Quasimodes and a lower bound on the uniform energy decay rate for Kerr-AdS spacetimes[END_REF] in the more general setting of the Kerr-Anti-de Sitter family of black holes. This seems to indicate that the Kerr-Anti-de Sitter spacetime may not be stable, at least for the reflecting boundary conditions. For Schwarzschild-Anti-de Sitter, G. Holzegel and J. Smulevici [START_REF] Holzegel | Stability of Schwarzschild-AdS for the spherically symmetric Einstein-Klein-Gordon system[END_REF] proved the stability of this spacetime with respect to spherically symmetric perturbation where the trapping causing the instability is not seen. When looking at dissipative boundary conditions, G. Holzegel, J. Luk, J. Smulevici and C. Warnick [START_REF] Holzegel | Asymptotic properties of linear field equation in anti-de Sitter space[END_REF] prove some estimates that should be useful for proving the non-linear stability of anti-de Sitter space. For higher dimensional Anti-de Sitter spaces, A. Bachelot studied the Cauchy problem for the Klein-Gordon fields in AdS 5 [START_REF] Bachelot | The Klein-Gordon equation in the Anti-de Sitter cosmology[END_REF] and investigate the decay of these fields in several settings. The same author studied the propagation of gravitational waves in a Poincaré patch of AdS 5 [START_REF] Bachelot | New Dynamics in the Anti-de Sitter Universe AdS 5[END_REF] where a new Hilbert functional framework is introduced. Although there's a significant amount of results when considering spacetimes with a negative cosmological constant, most of them are concerned with the Klein-Gordon fields. As far as we know, few things are known concerning Dirac fields for spacetimes with a negative cosmological constant. The first result concerning these fields was obtained by A. Bachelot [START_REF] Bachelot | The Dirac System On The Anti-De Sitter Universe[END_REF] where he studied the well posedness of the Dirac equation on the Anti-de Sitter space and show the existence of a B-F bound. After having reduced the equation to a spectral problem, this relies on a careful analysis of the states near the boundary. Concerning the Schwarzschild-Anti-de Sitter black hole, we obtained asymptotic completeness for massive Dirac fields in [START_REF] Idelon-Riton | Scattering theory for the Dirac equation on the Schwarzschildanti-de Sitter spacetime[END_REF]. This result shows that the particles behave asymptotically like a transport at speed 1 in the direction of the black hole.

In order to have a better understanding of the Dirac fields in the Schwarzschild-Anti-de Sitter spacetime, we are now interested in the local properties of these fields. In particular, this paper is devoted to the proof of a logarithmic lower bound on the local energy decay for the massive Dirac fields in the Schwarzschild-Anti-de Sitter spacetime. More precisely, the massive Dirac equation can be put in the following form:

∂tφ = iHmφ, (1.1)
where:

Hm = iγ 0 γ 1 ∂x + iA (x) D S 2 -mγ 0 B (x) , (1.2)
with the Dirac matrices γ i . When considering this equation for x ∈] -∞, 0[ with reflecting boundary condition, it was proven in a precedent paper [START_REF] Idelon-Riton | Scattering theory for the Dirac equation on the Schwarzschildanti-de Sitter spacetime[END_REF] that this is well-posed. After having given some properties of these fields in the asymptotic regions of our spacetime in [START_REF] Idelon-Riton | Scattering theory for the Dirac equation on the Schwarzschildanti-de Sitter spacetime[END_REF], we are now interested in the decay of the local energy for these fields. In fact, when looking at the potentials A and B, we see that B is confining near 0 whereas A is having the usual non degenerate maximum due to the photon sphere. Even though it cannot be the same picture for Dirac fields due to the matrices, it is good to keep in mind what is happening for the Klein-Gordon fields where this kind of behavior creates a well where the particles are classically trapped. In this picture, this seems to indicate that the local energy will decay slowly. In fact, for the massive Dirac fields, this decay is not faster than logarithmic. This is the subject of the main result of this paper:

Théorème 1.1. For all compact K ⊂] -∞, 0[, there exists a constant C > 0 such that:

lim sup t→+∞ sup ϕ∈H, ϕ =1 ln (t) e itH ϕ L 2 (K)
C.

To prove this result, we will look at the generalized eigenvalue problem for a semi-classical Dirac operator that we introduce now:

H = iγ 0 γ 1 h∂x + γ 0 γ 2 A (x) -hmγ 0 B (x) . (1.3)
This paper is organized as follows. In section 2, we present the Schwarzschild-Anti-de Sitter black hole and the special feature of its geometry. Then we present the Dirac equation in this spacetime. In section 3, we construct the exponentially accurate quasimodes for our operator following the proof of [START_REF] Gannot | Quasinormal modes for Schwarzschild-ADS black holes: exponential convergence to the real axis[END_REF]. More precisely, in subsection 3.1, we present the operators that we will use all along our analysis. In particular, we obtain estimates for elements of the domain of the square of our Dirac operator. We then restrict this operator to a compact set and obtain some formulas already like in [START_REF] Gannot | Quasinormal modes for Schwarzschild-ADS black holes: exponential convergence to the real axis[END_REF]. We prove existence of an eigenvalue close to the local minimum of the potential for an operator P constructed from our Dirac operator by taking a potential having the same asymptotic behavior as our, but being confining. In subsection 3.2, we obtain Agmon estimates. The presence of the mass creates a confining potential at the Anti-de Sitter infinity. Combined with the usual shape of the potential near the photon sphere, this creates a well where the particle is classically trapped. Nevertheless, at a quantum level, and because of the presence of the Schwarzschild black hole, we can see a tunneling effect where the probability of finding the particle in the classically forbidden region is exponentially decreasing. That is exactly what the Agmon estimates say. In subsection 3.3, we gives some estimates on the eigenvalue of P . We then end this section by using Agmon estimates and a cut-off argument to construct exponentially accurate quasimodes for our Dirac operator which gives the theorem presented earlier.

In section 4, we obtain a logarithmic lower bound for the local energy decay of the Dirac fields in the Schwarzschild-Anti-de Sitter spacetime following the ideas of [START_REF] Holzegel | Quasimodes and a lower bound on the uniform energy decay rate for Kerr-AdS spacetimes[END_REF] and obtaining this way a similar result. geodesics and then formulate the Dirac equation as a system of partial differential equations that can be put under a spectral form.

The Schwarzschild Anti-de Sitter space-time

Let Λ < 0. We define l 2 = -3 Λ . We denote by M the black hole mass. In Boyer-Lindquist coordinates, the Schwarzschild-Anti-de Sitter metric is given by:

g ab = 1 - 2M r + r 2 l 2 dt 2 -1 - 2M r + r 2 l 2 -1 dr 2 -r 2 dθ 2 + sin 2 θdϕ 2 (2.1)
We define

F (r) = 1 -2M r + r 2 l 2 .
We can see that F admits two complex conjugate roots and one real root r = r SAdS . We deduce that the singularities of the metric are at r = 0 and r = r SAdS = p++p-where p± = M l 2 ± M 2 l 4 + l 6 . (See [START_REF] Holzegel | Stability of Schwarzschild-AdS for the spherically symmetric Einstein-Klein-Gordon system[END_REF]) The exterior of the black hole will be the region r r SAdS and our spacetime is then seen as Rt×]r SAdS , +∞[×S 2 . It is well-known that the metric can be extended for r r SAdS by a coordinate change which gives the maximally extended Schwarzschild-Anti-de Sitter spacetime. In this paper, we are only interested in the exterior region.

In order to have a better understanding of this geometry, we study the outgoing (respectively ingoing) radial null geodesics. Using the form of the metric we can see that along such geodesics, we have:

dt dr = ±F (r) -1 . (2.2)
We thus introduce a new coordinate x such that tx (respectively t + x) is constant along outgoing (respectively ingoing) radial null geodesics. In other words:

dx dr = F (r) -1 . (2.3)
With this new coordinate, we have:

lim r→r SAdS x (r) = -∞ (2.4) lim r→∞ x (r) = 0. (2.5)
This limit proves that, along radial null geodesic, a particle goes to timelike infinity in finite Boyer-Lindquist time (recall that along these geodesic, tr * and t + r * are constants). This geometric property will lead to several differences compared to other spacetime. In the massless case, we would have to put boundary condition. In fact, even in the massive case, a confining potential appear near the Anti-de Sitter infinity. Combined with the usual bump created by the photon sphere, we expect that the energy will not decay really fast. In fact, for the wave equation, the local energy is decaying only logarithmically as was proven by G. Holzegel and J. Smulevici [START_REF] Holzegel | Decay properties of Klein-Gordon fields on Kerr-AdS spacetimes[END_REF], [START_REF] Holzegel | Quasimodes and a lower bound on the uniform energy decay rate for Kerr-AdS spacetimes[END_REF]. In this work, we obtain a logarithmic lower bound as in [START_REF] Holzegel | Quasimodes and a lower bound on the uniform energy decay rate for Kerr-AdS spacetimes[END_REF].

The Dirac equation on this spacetime

Using the 4-component spinor

ψ =     ψ1 ψ2 ψ3 ψ4     , the Dirac equation in the Schwarzschild-Anti-de
Sitter spacetime takes the form:

∂t + γ 0 γ 1 F (r)∂r + F (r) r + F ′ (r) 4 + F (r) 1 2 r D S 2 + imγ 0 F (r) 1 2 ψ = 0. (2.6)
where m is the mass of the field and D S 2 is the Dirac operator on the sphere. In the coordinate system given by (θ, ϕ) ∈ [0; 2π] × [0; π], we obtain:

D S 2 = γ 0 γ 2 ∂ θ + 1 2 cot θ + γ 0 γ 3 1
sin θ ∂ϕ where singularities appear, but we just have to change our chart in this case. We will now work in these coordinates. For more details about how to obtain this form of the equation, we refer to a previous work [START_REF] Idelon-Riton | Scattering theory for the Dirac equation on the Schwarzschildanti-de Sitter spacetime[END_REF]. Recall that the Dirac matrices γ µ , 0 µ 3, unique up to unitary transform, are given by the following relations:

γ 0 * = γ 0 ; γ j * = -γ j , 1 j 3; γ µ γ ν + γ ν γ µ = 2g µν 1, 0 µ, ν 3. (2.7)
In our representation, the matrices take the form:

γ 0 = i 0 σ 0 -σ 0 0 , γ k = i 0 σ k σ k 0 , k = 1, 2, 3 (2.8)
where the Pauli matrices are given by:

σ 0 = 1 0 0 1 , σ 1 = 1 0 0 -1 , σ 2 = 0 1 1 0 , σ 3 = 0 -i i 0 .
(2.9)

We thus obtain:

γ 0 γ 1 = -σ 1 0 0 σ 1 ; γ 0 γ 2 = -σ 2 0 0 σ 2 ; γ 0 γ 3 = -σ 3 0 0 σ 3 .
(2.10)

We introduce the matrix:

γ 5 = -iγ 0 γ 1 γ 2 γ 3 (2.11)
which satisfies the relations:

γ 5 γ µ + γ µ γ 5 = 0, 0 µ 3. (2.12)
We make the change of spinor φ(t, x, θ, ϕ) = rF (r) 1 4 ψ(t, r, θ, ϕ) and obtain the following equation:

∂tφ = i iγ 0 γ 1 ∂x + i F (r) 1 2 r D S 2 -mγ 0 F (r) 1 2
φ.

(2.13)

We set:

Hm = iγ 0 γ 1 ∂x + i F (r) 1 2 r D S 2 -mγ 0 F (r) 1 2 . (2.14)
We introduce the Hilbert space:

H := L 2 ]-∞, 0[ x × S 2 ω , dxdω 4 (2.15)
Recall that r is now a function of x. Using the spinoidal spherical harmonics (see [START_REF] Idelon-Riton | Scattering theory for the Dirac equation on the Schwarzschildanti-de Sitter spacetime[END_REF] for the details), we are able to diagonalize the Dirac operator on the sphere and we obtain the following operator:

Hm = iγ 0 γ 1 ∂x + γ 0 γ 2 F (r) 1 2 r s + 1 2 -mγ 0 F (r) 1 2 .
The corresponding Hilbert space is now:

Hs,n := L 2 ]-∞, 0[ x 4 ⊗ Ys,n (2.16)
where Ys,n span the corresponding spinoidal spherical harmonic for harmonic s and n fixed.

Next, we introduce a semi-classical parameter h = 1

s+ 1 2
. In the remaining of this article, we will deal with the following operator:

H = hHm = iγ 0 γ 1 h∂x + γ 0 γ 2 F (r) 1 2 r -hmγ 0 F (r) 1 2
(2.17)

where we have omitted to put the indice m for convenience. It was proven in [START_REF] Idelon-Riton | Scattering theory for the Dirac equation on the Schwarzschildanti-de Sitter spacetime[END_REF], that this operator is self-adjoint for all positive masses when equipped with the appropriate domain.

In the sequel, we will be interested in the case where the mass is sufficiently large. The domain of self-adjointness of our operator is then its natural domain, namely:

D (H) = {ϕ ∈ Hs,n|Hϕ ∈ Hs,n}.
3 Quasimodes

In this section, we construct exponentially accurate quasimodes for our Dirac operator. We will follow the same path as the one described in [START_REF] Gannot | Quasinormal modes for Schwarzschild-ADS black holes: exponential convergence to the real axis[END_REF].

Description of the operators

We introduce new comparison operator in the same spirit as in [START_REF] Gannot | Quasinormal modes for Schwarzschild-ADS black holes: exponential convergence to the real axis[END_REF]. First, recall that

H = iγ 0 γ 1 h∂x + γ 0 γ 2 A (x) -hmγ 0 B (x) .
Here, A and B behave at 0 like:

A (x) = 1 l + x 2 2l 3 + o x 2 , B (x) = - l x - x 6l + o (x) .
Thus:

A 2 (x) = 1 l 2 + x 2 l 4 + o x 2 , B 2 (x) = l 2 x 2 + 1 3l + o (1) ,
and:

A ′ (x) = x l 3 + o (x) , B ′ (x) = l x 2 - 1 6l + o (1) .
We will consider the case ml > 1 for which the domain of our operator H is:

D (H) = {ϕ ∈ L 2 |Hϕ ∈ L 2 }.
Furthermore, we have:

γ 0 2 = Id, γ j 2 = -Id, γ µ γ ν = -γ ν γ µ , pour 1 j 3 et 0 µ, ν 3.
The square of H is:

P = H 2 = -h 2 ∂ 2 x + V (x) ,
where:

V (x) = A 2 (x) + h 2 m 2 B 2 (x) -ihγ 1 γ 2 A ′ (x) + ih 2 mγ 1 B ′ (x) . (3.1)
We equip P with the domain:

D (P ) = {ϕ ∈ L 2 |Hϕ ∈ L 2 , P ϕ ∈ L 2 }
on which it is well-defined and self-adjoint. We have γ 0 γ 1 = diag (-1, 1, 1, -1) and:

γ 1 γ 2 =     0 1 0 0 -1 0 0 0 0 0 0 1 0 0 -1 0     ,γ 1 = i     0 0 1 0 0 0 0 -1 1 0 0 0 0 -1 0 0     , γ 0 γ 2 =     0 -1 0 0 -1 0 0 0 0 0 0 1 0 0 1 0     ,γ 0 =     0 0 i 0 0 0 0 i -i 0 0 0 0 -i 0 0     .
First, we study elements of the domain of P . Proposition 3.1. Let ϕ ∈ D (P ). Near 0, we obtain the following behavior:

-If 2ml = 3, ϕ (x) C 4 = O (-x) min( 3 2 ,ml) .
-

If 2ml = 3, ϕ (x) C 4 = O max (-x) 3 2 , (-x) ml (-ln (-x)) .
We also obtain the following estimate for ∂xϕ:

∂xϕ (x) =    O max (-x) 1 2 , (-x) min( 3 2 ,ml)-1 , if 2ml = 3, O max (-x) 1 2 , (-x) ml-1 (-ln (-x)) , if 2ml = 3.
Proof. Let ϕ ∈ D (P ). Since H 2 ϕ ∈ L 2 and using a result from [START_REF] Idelon-Riton | Scattering theory for the Dirac equation on the Schwarzschildanti-de Sitter spacetime[END_REF] saying that, for ψ ∈ D (H) and m > 1 2l , we have

||ψ(x, .)|| L 2 (S 2 ) = O √ -x , x → 0, (3.2)
we deduce that:

Hϕ (x) = O (-x) 1 2
near 0. Thus, we can write:

Hϕ (x) = f (x) with f (x) = O (-x) 1 2
. We obtain the following equations:

         -ih∂xϕ1 -A (x) ϕ2 -ihmB (x) ϕ3 = f1 ih∂xϕ2 -A (x) ϕ1 -ihmB (x) ϕ4 = f2 ih∂xϕ3 + A (x) ϕ4 + ihmB (x) ϕ1 = f3 -ih∂xϕ4 + A (x) ϕ3 + ihmB (x) ϕ2 = f4.
This gives the two following systems:

-ih∂x (ϕ1 + ϕ3) -ihmB (x) (ϕ1 + ϕ3) = g1 -ih∂x (ϕ1 -ϕ3) + ihmB (x) (ϕ1 -ϕ3) = g2 (3.3) ih∂x (ϕ2 + ϕ4) -ihmB (x) (ϕ2 + ϕ4) = g3 ih∂x (ϕ2 -ϕ4) + ihmB (x) (ϕ2 -ϕ4) = g4, (3.4)
where:

g1 (x) = A (x) (ϕ2 + ϕ4) + f1 -f3, g2 (x) = A (x) (ϕ2 -ϕ4) + f1 + f3, g3 (x) = A (x) (ϕ1 + ϕ3) + f2 -f4, g4 (x) = A (x) (ϕ1 -ϕ3) + f2 + f4.
Since A is bounded near 0 and Hϕ ∈ L 2 , we have ϕ (x) = O (-x)

1 2 . Thus, g (x) = O (-x) 1
2 . We first study equations (3.3). First equation Here, we study the equation:

∂x (ϕ1 + ϕ3) + mB (x) (ϕ1 + ϕ3) = i h g1.
We write ϕ + 1,3 = ϕ1 + ϕ3. The homogeneous equation has a solution of the form:

ϕ +,EH 1,3 (x) = Ce -x -ǫ mB(y)dy
where ǫ > 0 and C is a constant. With a particular solution of the form:

ϕ +,p 1,3 (x) = C (x) e -x -ǫ mB(y)dy ,
we have:

C (x) = x -ǫ i h g (y) e y -ǫ mB(z)dz dy.
Since, near 0, we have:

B (x) = - l x - x 6l + o (x) ,
so that: e

x -ǫ mB(z)dz = (-x) -ml O (1) , we obtain:

|C (x)| C x -ǫ 1 h (-y) 1 2
(-y) -ml dy

=      C h (-y) 3 2 -ml 3 2 -ml x -ǫ , if ml = 3 2 C h [-ln (-y)] x -ǫ , if ml = 3 2 .
With e -x -ǫ mB(y)dy = (-x) ml O (1), we obtain:

   If 2ml = 3, ϕ +,p 1,3 (x) = O (-x) 3 2
,

If 2ml = 3, ϕ +,p 1,3 (x) = O (-x) ml (-ln (-x)) . Moreover, ϕ +,EH 1,3 (x)
= O (-x) ml , which gives:

-

If 2ml = 3, ϕ + 1,3 (x) = O (-x) min( 3 2 ,ml) .
-

If 2ml = 3, ϕ + 1,3 (x) = O (-x) ml (-ln (-x)) .

Second equation

We write ϕ - 1,3 = ϕ1 -ϕ3. We have the equation:

∂xϕ - 1,3 (x) -mB (x) ϕ - 1,3 (x) = i h g2 (x) .
As before, a solution of the homogeneous equation is: Here, we have e -y -ǫ mB(z)dz = (-y) ml O (1) which gives:

ϕ -,EH 1,3 (x)
ϕ -,p 1,3 (x) C1 (-x) -ml x 0 1 h (-y) 1 2 +ml dy C1 (-x) 3 2
Since ϕ ∈ D (P ), we have Hϕ ∈ L 2 and ϕ (x) = O (-x)

1 2
using a result in [START_REF] Idelon-Riton | Scattering theory for the Dirac equation on the Schwarzschildanti-de Sitter spacetime[END_REF]. It implies that C = 0. We then have:

ϕ - 1,3 (x) = O (-x) 3 2
.

This gives the estimates for ϕ1 and ϕ3:

-

If 2ml = 3, ϕ1 (x) = O (-x) min( 3 2 ,ml)
and ϕ3 (x) = O (-x) min( 3 2 ,ml) .

-

If 2ml = 3, ϕ1 (x) = O max (-x) 3
2 , (-x) ml (-ln (-x)) and the same estimate for ϕ3.

We can do the same with ϕ2 et ϕ4. Using that, if ϕ ∈ D (P ), we have Hϕ ∈ D (H) and

Hϕ (x) = O (-x) 1 2
we deduce that:

iγ 0 γ 1 ∂xϕ (x) = -γ 0 γ 2 A (x) ϕ (x) + hmγ 0 B (x) ϕ (x) + O (-x) 1 2
.

With the previous estimates, we obtain:

-γ 0 γ 2 A (x) ϕ (x) = O (-x) 1 2 hmγ 0 B (x) ϕ (x) =    O (-x) min( 3 2 ,ml)-1 , if 2ml = 3, O max (-x) 1 2 , (-x) ml-1 (-ln (-x)) , if 2ml = 3.
So:

∂xϕ (x) =    O max (-x) 1 2 , (-x) min( 3 2 ,ml)-1 , if 2ml = 3, O max (-x) 1 2 , (-x) ml-1 (-ln (-x)) , if 2ml = 3.

Q.E.D

Remarque. In every case, we have:

∂xϕ (x) , ϕ (x) C 4 → x→0 0.
We now consider P on J =]c, 0[ where c is to be determined. We denote PJ this operator and equip it with the domain:

D (PJ ) = {ϕ ∈ L 2 (J) |Hϕ ∈ L 2 (J) , PJ ϕ ∈ L 2 (J) , ϕ (-c) = 0}.
We have the same lemmas as in [START_REF] Gannot | Quasinormal modes for Schwarzschild-ADS black holes: exponential convergence to the real axis[END_REF]:

Lemme 3.2. Let ϕ ∈ D (PJ ) and Y (x) such that Y (x) = O (x), Y ′ (x) = O (1) at 0. Then: -ih∂xϕ -iF Y ϕ 2 = -h 2 ∂ 2 x ϕ, ϕ + F 2 Y 2 ϕ -hF ∂r (F Y ) ϕ, ϕ
where F (r) = 1 -2M r + r 2 l 2 . Proof. For ϕ ∈ D (PJ ), we have:

-ih∂xϕ -iF Y ϕ 2 = -ih∂xϕ -iF Y ϕ, -ih∂xϕ -iF Y ϕ = -h 2 ∂ 2 x ϕ, ϕ -h ∂x (F Y ) ϕ, ϕ + F 2 Y 2 ϕ, ϕ ,
where the first integration by part is justified by ∂xϕ (x) , ϕ (x)

C 4 → x→0 0. In a neighborhood of 0, we have F (r (x)) = O (-x) -2 , Y (x) = O (-x) et ϕ (x) = o (-x) 1 2
which give F Y ϕ (x) , ϕ (x) C 4 |x=0 = 0 and justify the other integration by part. Since ∂x = ∂r ∂x ∂r = F (r) ∂r we obtain:

-ih∂xϕ -iF Y ϕ 2 = -h 2 ∂ 2 x ϕ, ϕ + F 2 Y 2 ϕ -hF ∂r (F Y ) ϕ, ϕ . Q.E.D Lemme 3.3. Let r+ such that, for r r+, we have F (r) 4l 2 -F ′ (r) 2 16 0 where F (r) = 1 -2M r + r 2 l 2 .
Denote by x+ = x (r+) and P + = P ]x + ,0[ . Let ϕ ∈ D P + , then:

-h 2 ∂ 2 x + h 2 m 2 B 2 (x) + ih 2 mγ 1 B ′ (x) ϕ, ϕ 0.
Proof. We have B 2 (x (r)) = F (r) and:

∂xB (x (r)) = ∂xF (r)

1 2 = F (r) ∂rF (r) 1 2 = 1 2 F ′ (r) F (r) 1 2 .
Moreover:

1 2 F ′ (r) F (r) 1 2 mF (r) + F ′ (r) 2 16m .
We deduce that:

m 2 F (r) + imγ 1 1 2 F ′ (r) F (r) 1 2 m 2 F (r) - m 2 F ′ (r) F (r) 1 2 - F ′ (r) 2 16 = - F ′ (r) 2 16 + F (r) 4l 2 - F (r) 4l 2 ,
in the sense of quadratic forms. Since

F ′ (r) = 2 M r 2 + r l 2 and F (r) 4l 2 = 1 4l 2 -M 2rl 2 + r 2 4l 2 , we obtain: F (r) 4l 2 - F ′ (r) 2 16 = 1 4l 2 - M rl 2 - M 2 4r 4 .
Thus F (r) 4l 2 -F ′ (r) 2 16 0 for r sufficiently large. Let r+ such that, for r r+, we have

F (r) 4l 2 -F ′ (r) 2 16
0. Let x+ = x (r+). On [x+, 0[, we have:

h 2 m 2 B 2 (x) + ih 2 mγ 1 ∂xB (x) -h 2 F (r) 4l 2 . Consider Y (x (r)) = h r-r SAdS 2l 2 F -1 (r). Since r = O (-x) -1 and F (r) = O (-x) -2 , we have Y (x) = O (x). Moreover: ∂xY (x) = F (r) ∂r h r -r SAdS 2l 2 F -1 (r) = h 2l 2 - h 2l 2 (r -r SAdS ) F ′ (r) F (r) . Since F ′ (r (x)) = O (-x) -1 , we deduce that ∂xY (x) = O (1).
Furthermore, as in [START_REF] Gannot | Quasinormal modes for Schwarzschild-ADS black holes: exponential convergence to the real axis[END_REF], we have:

∂r (F Y ) = h 2l 2 and: F 2 Y 2 -hF ∂r (F Y ) = h 2 r 2 -2rr SAdS + r 2 SAdS 4l 4 -h 2 F (r) 2l 2 -h 2 F (r) 4l 2 .
We can now conclude:

-h 2 ∂ 2 x + h 2 m 2 B 2 (x) + ih 2 mγ 1 ∂xB (x) ϕ, ϕ -h 2 ∂ 2 x -h 2 F 4l 2 ϕ, ϕ -h 2 ∂ 2 x + F 2 Y 2 -hF ∂r (F Y ) ϕ, ϕ = -ih∂xϕ -iF Y ϕ 2 0.

Q.E.D

Recall now the asymptotic behavior at 0 of the potentials:

A (x) = 1 l + x 2 2l 3 + o x 2 , B (x) = - l x - x 6l + o (x) ,
and:

A 2 (x) = 1 l 2 + x 2 l 4 + o x 2 , B 2 (x) = l 2 x 2 + 1 3l + o (1) , A ′ (x) = x l 3 + o (x) , B ′ (x) = l x 2 - 1 6l + o (1) .
We introduce the operator:

H = iγ 0 γ 1 h∂x + γ 0 γ 2 1 l 2 + x 2 l 4 1 2 + hmγ 0 l x ,
and its square:

P = H2 = -h 2 ∂ 2 x + 1 l 2 + x 2 l 4 -ihγ 1 γ 2 x l 4 1 l 2 + x 2 l 4 -1 2 + h 2 m 2 l 2 x 2 + iγ 1 ml x 2
on ] -∞, 0[ with domain:

D P = {ϕ ∈ L 2 | Hϕ ∈ L 2 , P ϕ ∈ L 2 }.
The operator H has compact resolvent since its potential is confining. Moreover, we have:

P -E 2 -1 = H + E -1 H -E -1
, which proves that P has compact resolvent and that E 2 is an eigenvalue of P if and only if ±E is an eigenvalue of H.

Proposition 3.4. Let α1 = 1+ √ 1+4ml(ml+1) 2 , E0 (h) := 1 l 2 -h 2 and E2 (h) = 1 l 2 + (2α1 + 1) 1 l 4 + h 2l 6 1 2 h. Then P admit an eigenvalue Ẽ (h) with E0 (h) Ẽ (h) E2 (h) + h 2 .
Proof. Using the fact that ∂xϕ (x) , ϕ (x) C 4 → x→0 0 and that:

m 2 l 2 x 2 + iγ 1 ml x 2 ml (ml -1)
x 2 0, since ml > 1, we have:

-h 2 ∂ 2 x + h 2 m 2 l 2 x 2 + iγ 1 ml x 2 0.
Moreover, we have:

-ihγ 1 γ 2 x l 4 1 l 2 + x 2 l 4 -1 2 ϕ, ϕ h 2 x 2 l 8 1 l 2 + x 2 l 4 -1 ϕ, ϕ + h 2 ϕ 2 h 2 x 2 l 6 ϕ, ϕ + h 2 ϕ 2 . (3.5)
This implies:

1 l 2 + x 2 l 4 -ihγ 1 γ 2 x l 4 1 l 2 + x 2 l 4 -1 2 1 l 2 + 2l 2 x 2 -hx 2 2l 6 - h 2 E0 (h) ,
for h > 0 sufficiently small. Thus:

P ϕ, ϕ E0 (h) ϕ 2 L 2 .
Using (3.5), we have:

P ϕ, ϕ -h 2 ∂ 2 x + 1 l 2 + 1 l 4 + h 2l 6 x 2 + h 2 m 2 l 2 x 2 + iγ 1 ml x 2 ϕ, ϕ + h 2 ϕ 2 .
We then diagonalize γ 1 :

γ 1 = KDK -1
where

K = 1 √ 2     1 0 1 0 0 1 0 1 1 0 -1 0 0 1 0 -1     , D = diag (i, -i, -i, i) , K -1 = K.
We can then write:

-h 2 ∂ 2 x + 1 l 2 + 1 l 4 + h 2l 6 x 2 + h 2 m 2 l 2 x 2 + iγ 1 ml x 2 ϕ, ϕ = -h 2 ∂ 2 x + 1 l 2 + 1 l 4 + h 2l 6 x 2 +h 2     m 2 l 2 x 2 +     -1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -1     ml x 2         K -1 ϕ, K -1 ϕ .
We will be interested in the following equations:

-h 2 ∂ 2 x + 1 l 2 + 1 l 4 + h 2l 6 x 2 + h 2 ml (ml -1) x 2 ψ1 = E1ψ1, (3.6) -h 2 ∂ 2 x + 1 l 2 + 1 l 4 + h 2l 6 x 2 + h 2 ml (ml + 1) x 2 ψ2 = E2ψ2. (3.7)
We look first at the equation (3.6). We choose:

ψ1 (x) = h -1 4 h -1 2 x ml e -1 l 4 + h 2l 6 1 2 x 2 2h ,
where the constants are chosen to normalize ψ1. Then:

-h 2 ∂ 2 x ψ1 (x) = -h 2 ml (ml -1) x 2 ψ1 (x) - ml h 1 l 4 + h 2l 6 1 2 ψ1 (x) - ml + 1 h 1 l 4 + h 2l 6 1 2 ψ1 (x) + 1 h 2 1 l 4 + h 2l 6 x 2 ψ1 (x) .
Consequently:

-h 2 ∂ 2 x + 1 l 2 + 1 l 4 + h 2l 6 x 2 + h 2 ml (ml -1) x 2 ψ1 (x) = E1 (h) ψ1 (x) with E1 (h) = 1 l 2 + (2ml + 1) 1 l 4 + h 2l 6 1 2
h. We now study equation (3.7). We first look for α such that α (α -1) = ml (ml + 1). We

choose α1 = 1+ √ 1+4ml(ml+1) 2
. We then consider:

ψ2 (x) = h -1 4 h -1 2 x α 1 e -1 l 4 + h 2l 6 1 2 x 2 2h ,
where we have also chosen the constant to normalize ψ2. In this case, we obtain:

-h 2 ∂ 2 x + 1 l 2 + 1 l 4 + h 2l 6 x 2 + h 2 ml (ml + 1) x 2 ψ2 = E2 (h) ψ2,
where

E2 (h) = 1 l 2 + (2α1 + 1) 1 l 4 + h 2l 6 1 2
h. Let us remark that:

α1 = 1 + 1 + 4ml (ml + 1) 2 1 + 2ml 2 > ml,
so that E2 (h) > E1 (h). We then see that:

    -h 2 ∂ 2 x + 1 l 2 + 1 l 4 + h 2l 6 x 2 + h 2     m 2 l 2 x 2 +     -1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -1     ml x 2         K -1 ϕ, K -1 ϕ     -h 2 ∂ 2 x + 1 l 2 + 1 l 4 + h 2l 6 x 2 + h 2     m 2 l 2 x 2 +     -1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -1     ml x 2         K -1 ϕ, K -1 ϕ + (E2 -E1) (h)     1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1     K -1 ϕ, K -1 ϕ . With K -1 ϕ (x) =     ψ1 (x) ψ2 (x) ψ2 (x) ψ1 (x)    
, we obtain:

P ϕ, ϕ E2 (h) + h 2 ϕ 2 .
The spectral theorem allows us to conclude the proof. Q.E.D

Agmon Estimates

In this part, we will study the operator P + = P [x + ,0[ . We will prove Agmon estimates for this operator. We begin by a lemma: Then e -φ h f ∈ D P + . Moreover, for all E, we have:

Re e φ h -h 2 ∂ 2 x + V -E e -φ h f, f = -h 2 ∂ 2 x + A 2 + h 2 m 2 B 2 + mγ 1 B ′ -φ ′ 2 -E f, f -2h Im         0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0     A ′ f, f     .
where V is defined in (3.1).

Proof. We have:

V (x) = A 2 (x) + h 2 m 2 B 2 (x) + h 2 mγ 1 B ′ (x) -ihγ 1 γ 2 A ′ (x)
where A, B, A ′ and B ′ are real function and

γ 1 γ 2 =     0 -1 0 0 1 0 0 0 0 0 0 -1 0 0 1 0     . Write f =     f1 f2 f3 f4     . Then: -ihγ 1 γ 2 A ′ (x) f, f = -2h Im A ′ f2, f1 + Im A ′ f4, f3 .
This gives the last term of our formula. For the other term, we have:

-∂ 2 x e -φ h f , e φ h f = -∂ 2 x - (φ ′ ) 2 h 2 f, f + 2i Im φ ′ h ∂xf, f ,
by integration by part and some calculation similar to the ones in [START_REF] Gannot | Quasinormal modes for Schwarzschild-ADS black holes: exponential convergence to the real axis[END_REF]. Taking the real part of this expression gives the result. Q.E.D

We then have:

+ h 2 ϕ, e 2 φ h 2 φ ′ h χχ ′ + χ ′ 2 ϕ
Proof. First, we have:

e φ h P + -E χϕ, e φ h χϕ = e φ h χ P + -E ϕ, e φ h χϕ + -h 2 ∂ 2 x , χ ϕ, e 2 φ h χϕ .
The real part of the last term is:

Re -h 2 ∂ 2 x , χ ϕ, e 2 φ h χϕ = Re -h 2 ∂x χ ′ ϕ , e 2 φ h χϕ + Re -h 2 e φ h χ ′ ϕ ′ , e φ h χϕ = Re h 2 ϕ, e 2 φ h 2 φ ′ h χχ ′ + χ ′ 2 ϕ + Re h 2 e φ h χϕ, e φ h χ ′ ϕ ′ -e φ h χϕ, e φ h χ ′ ϕ ′ = h 2 ϕ, e 2 φ h 2 φ ′ h χχ ′ + χ ′ 2 ϕ ,
since the terms are real. This gives the result.

Q.E.D Let us introduce xA (E) such that A 2 (xA (E)) = E.
Recall that, in a neighborhood of 0, we have:

A 2 (x) = 1 l 2 +
x 2 l 4 + o x 2 , so that:

xA 1 l 2 + T h = -l 2 T 1 2 h 1 2 + o h 1 2
.

We then have:

Lemme 3.7. Let T > 0 and δ > 0. Then, there exist k > 0, h0 > 0 such that, for all ϕ ∈ D P + , we have:

A 2 (x) - 1 l 2 + T h -kx 2 ϕ, ϕ -2h Im         0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0     A ′ ϕ, ϕ     > δh for all x ∈ [x+, xA 1 l 2 + T h [ and h ∈ (0, h0).
Proof. First, we have:

2 Im         0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0     A ′ ϕ, ϕ     A ′ (x) ϕ, ϕ .
Thus:

A 2 (x) - 1 l 2 + T h -kx 2 ϕ, ϕ -2h Im         0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0     A ′ ϕ, ϕ     A 2 (x) -h A ′ (x) - 1 l 2 + T h -kx 2 ϕ, ϕ .
We use the same technique as in [START_REF] Gannot | Quasinormal modes for Schwarzschild-ADS black holes: exponential convergence to the real axis[END_REF] to show that the last term is bounded below by δh.

We introduce the function:

M (x, h) = A 2 (x) + hA ′ (x) - 1 l 2 + T h -kx 2 since A ′ (x) < 0 on [x+, 0[ (by increasing x+ if needed). Choose k = δ 4l 4 (T +2δ
) , then:

M xA 1 l 2 + (T + 2δ) h , h = 2δh - δ 4 h + o (h) > δh,
for h sufficiently small. Note that k 1 8l 4 . We have:

M ′ (x, h) = A 2 ′ (x) + hA ′′ (x) -2kx,
where A 2 ′ (x) = 2x l 4 + o (x) and A ′′ (x) = 1 l 3 + o (1). We obtain:

M ′ xA 1 l 2 + (T + 2δ) h , h 6 4l 4 xA 1 l 2 + (T + 2δ) h + o h 1 2 < - 5 4l 4 l 2 T 1 2 h 1 2 < 0.
Moreover:

M ′′ (x, h) = A 2 ′′ (x) + hA (3) (x) -2k, where A 2 ′′ (x) = 2 l 4 + o (1), A (3) (x) = O (1)
M ′′ (x, h) 0,
for h sufficiently small, on [-A, 0]. Hence, on [-A, xA 1 l 2 + (T + 2δ) h ], M ′ < 0 and:

M (x, h) δh.
On [x+, -A], we have:

A 2 (x) - 1 l 2 + T h -kx 2 A 2 (x) - 1 l 2 + T h -kx 2 + > C1
where we can decrease k if needed. For h sufficiently small, we have:

A 2 (x) + hA ′ (x) - 1 l 2 + T h -kx 2 > C1 -hCA,
where CA = sup

[x + ,-A]
A ′ (x). Thus:

M (x, h) C2 on [x+, -A].
The constant C2 have to satisfy:

C2 δh, that is C1 > (δ + CA) h. For a good choice of k and h0, we obtain the result. Q.E.D

We can now prove the following estimate:

Proposition 3.8. Let T > 0. There exist constants h0 > 0, C > 0, and c > 0 such that:

e x 2 ch ϕ L 2 ([x+,0[) C ϕ L 2 ([x+,0[) + h -1 e x 2 ch P + -E (h) ϕ L 2 ([x+,0[)
for all h ∈ (0, h0), ϕ ∈ D P + and E (h) such that E (h) < 1 + T h.

Proof. Fix δ > 0. By the preceding lemma, we can choose a constant c > 0 such that, if φ (x) = x 2 c , we have:

A 2 (x) - 1 l 2 + T h -φ ′ 2 ϕ, ϕ -2h Im         0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0     A ′ ϕ, ϕ     > δh Write X1 = -l 2 (T + 2δ)
Re e φ h -h 2 ∂ 2 x + V -E χϕ, e φ h χϕ = -h 2 ∂ 2 x + A 2 + h 2 m 2 B 2 + h 2 mγ 1 B ′ e φ h χϕ, e φ h χϕ + -E -φ ′ 2 e φ h χϕ, e φ h χϕ -2h Im         0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0     A ′ e φ h χϕ, e φ h χϕ     δh e φ h χϕ 2
Then, by lemma 3.6, we have:

Re e φ h P + -E χϕ, e φ h χϕ L 2 ([x+,0]) = Re e φ h χ P + -E ϕ, e φ h χϕ L 2 ([x+,0]) + h 2 ϕ, e 2 φ h 2 φ ′ h χχ ′ + χ ′ 2 ϕ L 2 ([x+,0]) h 2 ϕ, e 2 φ h 2 φ ′ h χχ ′ + χ ′ 2 ϕ L 2 ([x+,0]) + e φ h χ P + -E ϕ L 2 ([x+,0]) e φ h χϕ L 2 ([x+,0])
.

Thus:

δh e φ h χϕ 2 L 2 ([x+,0]) h 2 ϕ, e 2 φ h 2 φ ′ h χχ ′ + χ ′ 2 ϕ L 2 ([x+,0]) + e φ h χ P + -E ϕ L 2 ([x+,0]) e φ h χϕ L 2 ([x+,0])
.

By the same argument as in [START_REF] Gannot | Quasinormal modes for Schwarzschild-ADS black holes: exponential convergence to the real axis[END_REF] proposition 3.12, we obtain:

e φ h χϕ 2 L 2 ([x+,0]) 2δ -1 h ϕ, e 2 φ h 2 φ ′ h χχ ′ + χ ′ 2 ϕ L 2 ([x+,0]) + (δh) -2 e φ h χ P + -E ϕ 2 L 2 ([x+,0]) . Remark that χ ′ is non zero when x ∈ h 1 2 [X2, X1] so that sup supp(χ ′ ) e φ h = O (1) , sup supp(χ ′ ) φ ′ = O h 1 2
.

We also have:

χ ′ (x) = h -1 2 ζ ′ h -1 2 x; h , which imply sup |χ ′ (x)| = O h -1 2 and sup φ ′ h χχ ′ = O h -1 . Consequently: e φ h ϕ L 2 [x + ,h 1 2 X 2 ] C1 ϕ L 2 ([x+,0]) + C2h -1 e φ h P + -E ϕ L 2 ([x+,0])
.

The result then follow from the fact that:

e φ h ϕ L 2 [h 1 2 X 2 ,0] C3 ϕ L 2 ([x+,0]) . Q.E.D Let S > 0. We choose A1 < A2 < xA 1 l 2 + S and introduce Σi = [x+, Ai[ such that Σ1 ⋐ Σ2 ⋐ [x+, xA 1 l 2 + S ]
. Proposition 3.9. Now, let S > 0 such that 1 l 2 +S < A 2 (x+). There exist constants h0 > 0, C > 0, and ǫ > 0 such that:

ϕ L 2 (Σ 1 ) C e -ǫ h ϕ L 2 (Σ 2 ) + P + -E (h) ϕ L 2 (Σ 2 ) ,
for all h ∈ (0, h0), ϕ ∈ D P + , and for all E (h) such that E (h) < 1 l 2 + S. Proof. For δ sufficiently small, we can suppose that Σ2 ⋐ [x+, xA 1 l 2 + S + 2δ ]. Let χ1 such that χ1 ≡ 1 on Σ1 and supp (χ1) ⊆ Σ2. Let χ2 such that χ2 ≡ 1 on supp (χ1) and supp (χ2) ⊆ Σ2. Then, on Σ2, A 2 -1 l 2 + S > δ. For h and ǫ sufficiently small, we also have:

A 2 -h A ′ - 1 l 2 + S -ǫ 2 χ ′ 1 2 > δ.
Using the same reasoning as in the preceding proof, we obtain:

δ e φ h χ2ϕ 2 L 2 ([x+,0]) h 2 ϕ, e 2 φ h 2 ǫχ ′ 1 h χ2χ ′ 2 + χ ′ 2 2 ϕ L 2 ([x+,0]) + e φ h χ2 P + -E ϕ L 2 ([x+,0]) e φ h χ2ϕ L 2 ([x+,0])
, where we have taken φ (x) = ǫχ1 (x). As before, we obtain:

e φ h χ2ϕ 2 L 2 ([x+,0]) 2δ -1 h 2 ϕ, e 2 φ h 2 ǫχ ′ 1 h χ2χ ′ 2 + χ ′ 2 2 ϕ L 2 ([x+,0]) + (δ) -2 e φ h χ2 P + -E ϕ 2 L 2 ([x+, 0
]) Thanks to the choice of χ1 and χ2, we have:

δ e ǫχ 1 h χ2ϕ L 2 (Σ 1 ) = δ e ǫ h ϕ L 2 (Σ 1 ) , δ e ǫ h ϕ L 2 (Σ 1 ) δ e φ h χ2ϕ L 2 ([x+,0])
.

Since χ1 ≡ 0 on the support of χ ′ 2 and ǫχ1 ǫ, we obtain:

e ǫ h ϕ L 2 (Σ 1 ) 2δ -1 h 2 sup χ ′ 2 2 ϕ L 2 (Σ 2 ) + δ -2 e ǫ h P + -E ϕ L 2 (Σ 2 )
which gives the result when multiplying by e -ǫ h . Q.E.D

Estimates for the eigenvalues of P +

In this section, we prove that it is possible to find an eigenvalue of the operator

P + at distance O h 1 2
from the eigenvalue of P obtained in 3.4. We have:

Proposition 3.10. Let T > 0. There exists h0 > 0 such that for all h ∈ (0, h0), there exists an eigenvalue E + (h) of P + satisfying:

E + (h) -E2 (h) < Ch 1 2 , where E2 (h) = 1 l 2 + (2α1 + 1) 1 l 4 + h 2l 6 1 2 h
Proof. We introduce:

P # = -h 2 ∂ 2 x + 1 l 2 + 1 l 4 + h 2l 6 x 2 + h 2 m 2 l 2 x 2 + iγ 1 ml x 2 + (E2 -E1) (h) K     1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1     K -1
where

E1 (h) = 1 l 2 + (2ml + 1) 1 l 4 + h 2l 6 1 2 h such that (E2 -E1) (h) > 0 and K = 1 √ 2     1 0 1 0 0 1 0 1 1 0 -1 0 0 1 0 -1     .
P + -E2 (h) (χϕ2) = P + -P # (χϕ2)+ P # -E2 (h) (χϕ2) = χR (h) ϕ2+ -h 2 ∂ 2 x , χ ϕ2,
where: . Using the change of variable y = h -1 2 x and the exponential decay of our eigenvector, we see that:

R (h) = A 2 (x) - 1 l 2 + x 2 l 4 -h x 2 l 6 -ihγ 1 γ 2 A ′ (x) + h 2 m 2 B 2 (x) - l 2 x 2 + ih 2 mγ 1 B ′ (x) - l x 2 + (E2 -E1) (h) K     1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1     K -1 = o x 2 + hO x 2 + hO (x) + O h 2 + O (h) . Recall that K -1 ϕ2 (x) =     ψ1 (x) ψ2 (x) ψ2 (x) ψ1 (x)     where ψ1 (x) = h -1 4 h -1 2 x ml e -1 l 4 + h 2l 6 1 2 x 2 2h and ψ2 (x) = h -1 4 h -1 2 x α 1 e -1 l 4 + h
χR (h) ϕ2 L 2 ([x+,0]) = O h 1 2
.

The commutator is equal to:

-h 2 ∂ 2 x , χ = -h 2 χ ′′ -h 2 χ ′ ∂x.
The support of χ ′ and χ ′′ and the formula for ϕ2 give ǫ > 0 such that:

-h 2 ∂ 2 x , χ ϕ2 L 2 ([x+,0]) = O e -ǫ h
.

Consequently, we have:

P + -E2 (h) (χϕ2) L 2 ([x+,0]) = O h 1 2
.

Since χϕ2 L 2 ([x+,0]) = 1 -O e -d h , we can normalize χϕ2. The spectral theorem gives an eigenvalue E + (h) of P + such that:

E + (h) -E2 (h) Ch 1 2
for a certain constant C > 0.

Q.E.D

Construction of quasimodes

We now introduce the operator H + which is H on We have P + = H + 2 and E + 2 is an eigenvalue of P + if and only if ±E + is an eigenvalue of H + . We then have the: Théorème 3.11. Let S > 0 such that 1 l 2 + S < A 2 (x+). There exist constants h0 > 0, D > 0, a real number E + (h) 1 2 > 0 such that E + (h) < 1 l 2 + S and a function ϕ h ∈ D (H) such that ϕ h L 2 (]-∞,0[) = 1 satisfying:

H -E + (h) 1 2 ϕ h L 2 (]-∞,0[) e -D h ,
for all h ∈ (0, h0). such that E + (h) < 1 l 2 + S which exists by the last section. We use ϕ h = χϕ + h that we extend by 0 outside [x+, 0[ so that ϕ h ∈ D (H) and:

H -E + (h) 1 2 ϕ h L 2 (]-∞,0[) = iγ 0 γ 1 h∂x, χ ϕ + h L 2 (]-∞,0[) .
Using Agmon estimates, we obtain:

iγ 0 γ 1 h∂x, χ ϕ + h = iγ 0 γ 1 hχ ′ ϕ + h Ce -ǫ h ϕ + h .

This gives:

H -E + (h)

1 2 ϕ h L 2 (]-∞,0[) e -D h .
Since we can normalize ϕ h , the result is proven. Q.E.D

Lower bound on the local energy

In this section, we will make use of an argument in [START_REF] Holzegel | Quasimodes and a lower bound on the uniform energy decay rate for Kerr-AdS spacetimes[END_REF] to prove the: Proof. Let ϕ h be a quasimode as in the precedent section. We introduce the function ψ h = e iE + (h)t ϕ h . Then: Consequently:

-i∂tψ h -Hψ h = -H -E + (h)
ψ h L 2 (K) -ψh L 2 (K) ψ h -ψh L 2 (K) t 0 e i(t-s)H ϕ h F L 2 (K) ds Ct F Cte -D h .
We deduce that:

e itH ϕ h L 2 (K) ψ h L 2 (K) -Cte -D h = ϕ h L 2 (K) -Cte -D h .
Let λ = ϕ h L 2 (K) . Take t h = λ-h C e D h such that λ -Ct h e -D h = h. We then have:

h = D ln (t h ) + h ln λ-h C ln (t h ) .
For h sufficiently small, we have h ln λ-h C D 2 which gives:

e it h H ϕ h L 2 (K) D 2

ln (t h )

.

When h goes to 0, t h goes to +∞ and we have:

  = Ce x -ǫ mB(y)dy with a constant C. This solution behaves like ϕ -,EH 1,3 (x) = C (-x) -ml O (1). A particular solution is: ϕ -,p 1,3 (x) = e x -ǫ mB(y)dy x 0 i h g2 (y) e -y -ǫ mB(z)dz dy.

Lemme 3 . 5 .

 35 Let φ ∈ C ∞ ([x+, 0[) and f ∈ D P + where the domain is given by (3.1).

Lemme 3 . 6 .

 36 Let ϕ ∈ D P + and χ ∈ C ∞ 0 ([x+, 0[). Then: Re e φ h P + -E χϕ, e φ h χϕ = Re e φ h χ P + -E ϕ, e φ h χϕ

1 2 - 1 2

 21 ǫ and X2 = -l 2 (T + 2δ) -2ǫ. Let ζ ∈ C ∞ such that ζ ≡ 0 on [X1, 0] and ζ ≡ 1 on [h -1 2 x+, X2]. Let χ (x, h) = ζ h -1 2 x; h such that χ has support in [x+, xA 1 l 2 + (T + 2δ) h ]. Choose f = e φ h χϕ.By the preceding lemma, we have:

  This operator admit E2 (h) as an eigenvalue. Denote ϕ2 an associated eigenvector. Let S > 0 such that 1 l 2 + S < A 2 (x+) and A < xA 1 l 2 + S . Let χ ∈ C ∞ such that χ ≡ 1 on [A, 0[ and supp (χ) = [x+, 0]. Then χϕ2 ∈ D P + and:

  [x+, 0[ with domain:D H + = {ϕ ∈ L 2 ([x+, 0[) |H + ϕ ∈ L 2 ([x+, 0[) , ϕ (x+) = 0}.

Proof. Let A < xA 1 l 2 +

 2 S and χ ∈ C ∞ 0 such that χ ≡ 1 on [A, 0[ and supp (χ) = [x+, 0[. Denote ϕ + h an eigenvector of H + for the eigenvalue E + (h) 1 2

Théorème 4 . 1 .

 41 For all compact K ⊂] -∞, 0[, there exists a constant C > 0 such that:

  ψ h and H -E + (h) ψ h L 2 (]-∞,0[) e -D h. We consider the preceding equation as non homogeneous equation of the form:-i∂tχ -Hχ = F where F L 2 (]-∞,0[) e -Dh . Let ψh = e itH ϕ h a solution of the homogeneous equation. By Duhamel formula, we have: χ = e itH ϕ h + t 0 e i(t-s)H ϕ h F ds.

The Dirac equation on the Schwarzschild Anti-de Sitter spacetimeIn this section, we present the Schwarzschild Anti-de Sitter space-time and give the coordinate system that we will work with in the rest of the paper. We quickly study the radial null

Acknowledgments

This work was partially supported by the ANR project AARG.


M Ericson

G Chanfray

Interplay between QCD and nuclear responses

Keywords: Pacs: 24, 85, +p 11, 30, Rd 12, 40, Yx 13, 75, Cs 21, 30, -x

Interplay between QCD and nuclear responses

M. Ericson, G.

Introduction

The spectrum of scalar-isoscalar excitations is quite different in the vacuum and in the nuclear medium. In the second case it includes low lying nuclear excitations and also two quasi-pion states i.e. pions dressed by particle-hole excitations. All these lie at lower energies than the vacuum scalar excitations which start at 2 m π . We have shown in previous works [1,2,3,4] that this produces a large increase of the magnitude of the scalar QCD susceptibility over its vacuum value. We have expressed the origin of this increase as arising from the mixing of the nuclear response to a scalar probe coupled to nucleonic scalar density fluctuations into the QCD scalar response.

It is natural to investigate also the reciprocal problem of the influence of the QCD scalar response to a probe which couples to the quark density fluctuations on the ordinary nuclear scalar response of nuclear physics, which is the object of the present work. We will study this influence not only for what concerns the nuclear excitations but also for a single nucleon for which only nucleonic excitations are involved. If this influence indeed exists, does it lead to non-trivial observable consequences ? We will show that this is the case, with two main applications. One concerns the QCD scalar susceptibility of a single nucleon. The second is the possibility to infer medium effects in the propagation of the scalar meson which binds the nucleus from QCD results, such as the lattice ones on the evolution of the nucleon mass with the pion mass.

Our article is organized as follows. In section 2 we remind the mechanisms responsible for the mixing of the nuclear response into the QCD scalar susceptibility. We illustrate it in the framework of a nuclear chiral model with a scalar and vector meson exchange. We show that this mutual influence also exists at the nucleonic level. In section 3 we discuss the influence of the quark structure of the nucleon on the scalar response of nuclear physics in a general framework which is able to incorporate also confinement effects.

2 Mutual influence of the scalar QCD response and nuclear physics response

Study in a nuclear chiral model

We first remind how the usual nuclear physics response to a scalar field enters in the QCD susceptibility. For this, following ref.

[2], we start from the expression of the modification of the quark condensate in the nuclear medium, ∆ qq (ρ) = qq (ρ) -qq vac . We first use, as in [2], its expression for a collection of independent nucleons :

∆ qq (ρ) = Q S ρ S . (1)
where ρ S is the scalar density of nucleons related to the chemical potential µ by :

ρ S = 4 d 3 p (2π) 3 M E p Θ(µ -E p ). ( 2
)
We have introduced the scalar charge of the nucleon, Q S , proportional to the volume integral of the nucleon scalar density of quarks. It is related to the nucleon sigma commutator σ N and the current quark mass, m q , by :

Q S = σ N 2m q = d 3 r N|qq( r) -qq vac |N . (3)
The susceptibility of the nuclear medium, χ A S , is the derivative of the quark scalar density with respect to the quark mass. We define it in such a way that it represents a purely nuclear contribution with the vacuum susceptibility substracted off :

χ A S = ∂∆ qq(ρ) ∂m q µ = ∂(Q S ρ S ) ∂m q µ . (4)
It contains two terms. One arises from the derivative of Q S , which by definition is the free nucleon QCD scalar susceptibility, χ N S = ∂Q S /∂m q . The second one involves the derivative of the nucleon density. This term itself is built of two pieces, one involves antinucleon excitations and is small [2]. The other one, which is larger, involves the nuclear response Π 0 = -2M N p F /π 2 . In this case it is the free Fermi gas one since no interactions between nucleons have been introduced. The result of this derivation is contained in the following equation:

χ A S = ρ S χ S N + 2 Q 2 S Π 0 . (5)
The nuclear susceptibility is thus the sum of a one-body term and of a term due the nuclear excited states, the p-h excitations. This decomposition survives the introduction of the interaction, as will be shown next. In this case the free p-h polarization propagator is replaced by the full RPA one, while the free nucleon susceptibility can undergo medium modifications and become dependent on the density. The previous result has been generalized in ref. [3] to an assembly of nucleons interacting through a scalar and a vector meson exchanges, working at the mean field level as in relativistic mean field theories. In this work the condensate was obtained as the derivative of the grand potential with respect to the quark mass (Feynman-Hellman theorem) and the susceptibility as the derivative of the condensate, both being taken at constant chemical potential. The result is [3] :

χ S = ∂ qq ∂m q µ ≃ -2 qq 2 vac f 2 π ∂ S ∂c µ . (6)
S ≡ f π + s is the expectation value of the chiral invariant scalar field and c = f π m 2 π is the symmetry breaking parameter of the model used in [3]. The quantity ∂ S/∂c µ is related to the in-medium sigma propagator :

∂ S ∂c µ = -D * σ = 1 m * 2 σ - g 2 S m * 2 σ Π S (0) 1 m * 2 σ ( 7
)
where Π S (0) is the full scalar polarization propagator, related to the bare one, Π 0 by :

Π S (0) = M * N E * F Π 0 (0) 1 - g 2 ω m 2 ω E * F M * N - g * 2 S m * 2 σ M * N E * F Π 0 (0) -1 . (8)
In the equations above, m * σ is the in-medium sigma mass, which is obtained from the second derivative of the energy density with respect to the order parameter :

m * 2 σ = ∂ 2 ε ∂s 2 = V ′′ (s) + ∂ (g S ρ S ) ∂s = m 2 σ 1 + 3s f π + 3 2 s f π 2 (9)
where the potential V responsible for the spontaneous symmetry breaking is the standard quartic one of the linear sigma model. In the very last expression of eq. ( 9) we have omitted, as in ref. [3], the small antinucleon contribution embedded in the factor ∂ρ S /∂s. Moreover since for the moment we do not consider, contrary to ref. [3], the scalar response of the nucleon due to confinement, we also ignore the medium renormalization of g S . The mean scalar field s being negative, the term linear in s lowers the sigma mass by an appreciable amount (≃ 30 % at ρ 0 ). This is the chiral dropping associated with chiral restoration [5] and arising from the 3σ interaction as depicted in fig 1 .  Since we are interested only in the medium effects the vacuum value of the quantity ∂ S/∂c µ = 1/m σ 2 has to be subtracted off in eq. ( 7) and the purely nuclear suceptibility, χ A S , writes :

χ A S = 2 qq 2 vac f 2 π 3 s/f π + 3 2 (s/f π ) 2 m * 2 σ + g 2 S m * 2 σ Π S (0) 1 m * 2 σ . ( 10
)
We see that χ A S receives two types of contributions, the second being proportionnal to the full RPA scalar response Π S . The corresponding proportionality factor r between this second contribution and Π S writes, to leading order, i.e., neglecting the medium modification of the sigma mass :

r = 2 g 2 S qq 2 vac f 2 π m 4 σ ≃ 2 (Q s S ) 2 (11)
where we have introduced the nucleon scalar charge Q s S from the scalar field, defined below. In the sigma model the free nucleon sigma commutator is the sum of two contributions, one arising from the pion cloud, which depends on the mean value of the squared pion field, i.e., on the scalar number of pions in the nucleonic cloud. In the mean field approximation where pion loops are ignored this term does not appear. The other one, Q s S , arises from the scalar meson [6,7,8] and is linear in the σ field :

Q s S = σ s N 2m q = - qq vac f π d 3 r N|σ( r)|N = - qq vac f π g S m 2 σ ( 12
)
which establishes relation (11) if we ignore the in-medium modification of Q s S , i.e., the difference beween m * σ and m σ . We now turn to the first part of χ A S which depends on the average scalar field s. In the low density limit, s reduces to s = -g S ρ S /m 2 σ , and we can ignore the term in s2 as well as the difference beween m * σ and m σ . In this limit the first term in the expression (10) of χ A S is linear in the density. If we classify it in the decomposition of eq. ( 5) for χ A S , it obviously belongs to the individual nucleon contribution, ρ S χ N S , to the nuclear susceptibility. Writing the linear term explicitly in eq. ( 10) we deduce the free nucleon scalar susceptibility from the scalar field, (χ N S ) s :

(χ N S ) s = -2 qq 2 vac f 3 π 3 g S m 4 σ , (13)
which is negative (i.e, it favors an increase in magnitude of the field, similar to paramagnetism). It has been obtained here from the low density expression of χ A S . In fact it can also be obtained directly as the derivative with respect to the quark mass of Q s S , the part of the nucleon scalar charge originating in the scalar field written in eq. ( 12) :

(χ N S ) s = ∂Q s S ∂m q = ∂ ∂m q - qq vac f π g S m 2 σ . (14)
Using the fact that, in the model, qq vac /f π does not depend on m q , only the derivative of the sigma mass with respect to m q enters which, according to the Feynman-Hellmann theorem, is linked to the sigma commutator, σ σ , of the σ. In the linear sigma model the derivative with respect to the quark mass is replaced by the derivative with respect to the symmetry breaking parameter, c = f π m 2 π , keeping the other original parameters of the model, λ and v, constant. The result is :

σ σ = m q ∂m σ ∂m q = 3 2 m 2 π m σ . ( 15
)
When inserted in eq. ( 14), it leads for (χ N S ) s to the expression of eq. ( 13). We now turn to the scattering amplitude for the sigma meson on the nuclear system. In the same framework we will first show that the amplitude for the scattering of the scalar meson on the nucleon has the same relation to the nucleonic susceptibility as the case for the nuclear excitation part. Indeed in the expression (9) of m * 2 σ the term linear in density is obtained from the low density expression :

N σ σ σ

3 s m * 2 σ ≃ -(3 g S /f π ) ρ S .
It represents an optical potential for the sigma propagation. The corresponding σN scattering amplitude, T σ N , which can also be evaluated directly from the graph of fig. 1, is equal to :

T σN = -3 g S /f π . (16)
We are now in a situation to relate the nucleon scalar susceptibility (eq. ( 13)) to the sigma-nucleon amplitude of eq. ( 16), with the result :

(χ N S ) s = 2 (Q s S ) 2 g 2 S T σN . (17)
The proportionality factor, 2 (Q s S ) 2 /g 2 S , is the same as for the purely nuclear excitations. The quantity g S which appears in this factor factor is due to the σNN coupling constant. Adding the two effects from the nucleonic and nuclear excitations the total QCD scalar susceptibility of the nuclear medium (vacuum value substracted) can therefore be related to the total response, T A , to the scalar field through :

χ A S = 2 (Q s S ) 2 g 2 S T A (18)
where the two members include both the individual nucleon contribution and the one arising from the nuclear excitations,with :

T A = ρ S T σN + g 2 S Π SS . (19)
The last term on the r.h.s. represents the (in-medium corrected) Born part of the σN amplitude while the first piece represents the non-Born part linked to nucleonic excitations.

Thus there exists a universal scaling factor between the responses of a nuclear or nucleonic system to probes which couple either to the nucleon scalar density fluctuations or to the quark ones. This relation has allowed us to infer the existence of a contribution to the QCD nucleon scalar susceptibility linked to the scalar meson. To the best of our knowledge this contribution to the nucleon susceptibility has not been discussed previously. It has a link, through the relation ( 17), to the optical potential for the σ propagation, which reduces the sigma mass in the medium.

Effect of the two pion propagator

In order to illustrate the coherence of this approach we will now extend the previous description to incorporate the effect of the two-pion propagator, G, which affects the nucleon susceptibility in the following way. The sigma propagator is renormalized by the σ coupling to two-pion states, as discussed in [4]. At zero four-momentum we have :

-D σ = 1 m 2 σ + 3 λ (m 2 σ -m 2 π ) G 1 -3λG = 1 -3 λ G m 2 σ -3λ m 2 π G ≃ 1 m 2 σ - 3 G 2 f 2 π ( 20
)
where λ = (m 2 σ -m 2 π )/2 f 2 π and both D σ and G are taken at zero four-momentum. In the last term we have restricted to the one pion loop level. We stress that this expression only holds for the sigma, chiral partner of the pion, which is not a chiral invariant field. It does not apply to the scalar field responsible for the nuclear binding which has to be a scalar invariant (that we have denoted s) and which is weakly coupled to two-pion states, while the σ is strongly coupled. Therefore this treatment is done for illustration purpose and not for an application to nuclear physics.

The medium correction to D σ from the coupling of the σ to 2π states is :

∆D σ = 3 ∆G 2 f 2 π , ( 21
)
where ∆G is the in-medium modification of the two-pion propagator. In ∆G, to lowest order, one and only one of the two pions of the two-pion propagator has to be dressed by one p-h bubble. It is again possible to interpret the corresponding modification of the sigma propagator as representing a σN scattering amplitude, T π σN , in which the sigma interacts with the nucleon pion cloud :

T π σN = 3 m 4 σ 2 f 2 π ∆G ρ S . (22)
This is to be compared to the nucleon scalar susceptibility from the pion cloud, which is [4] :

χ N S = 3 ∆G ρ S qq 2 vac f 4 π . ( 23
)
The proportionality factor between the susceptibility (23) and the σN scattering amplitude ( 22) is the same as previously, 2 Q s2 S /g 2 S . We find again that this relation holds not only at the level of p-h excitations but also for a single nucleon, at the level of the nucleonic excitations which in this specific case are of the pionic type. In summary we have seen that the mixing between the quark density fluctuations and the nucleon ones implies that the response of a probe which couples to the nucleonic density fluctuation is proportional to the QCD scalar response. This includes also the nucleonic contribution to these responses. As an example we have shown that the chiral dropping of the sigma mass in the medium has a counterpart in the form of a contribution of the scalar meson to the QCD scalar susceptibility of the nucleon.

Connection with lattice data

It is now interesting to connect our results to the available lattice simulations of the evolution of the nucleon mass with the pion mass, equivalently the quark mass. At present they do not cover the physical region but only the region beyond m π ≃ 400 MeV . The derivative ∂M N /∂m 2 π = σ N /m 2 π provides the nucleon sigma commutator. In turn the derivative of σ N leads to the susceptibility. Both quantities are strongly influenced by the pion cloud contribution which has a non-analytic behavior in the quark mass, preventing a polynomial expansion in this quantity. However the pionic self-energy contribution to the nucleon mass, Σ π , has been separated out in ref. [9] in a model dependent way with different cut-off forms for the pion loops (gaussian, dipole, monopole) with an adjustable parameter Λ. The remaining part is expanded in terms of m 2 π as follows:

M N (m 2 π ) = a 0 + a 2 m 2 π + a 4 m 4 π + Σ π (m π ). ( 24
)
The best fit value of the parameter a 4 which fixes the susceptibility shows little sensitivity to the shape of the form factor, with a value a 4 ≃ -0.5 GeV -3 while a 2 ≃ 1.5 GeV -1 (in a previous work [10] smaller values of a 2 and a 4 were given : a 2 ≃ 1 GeV -1 and a 4 ≃ -0.23 GeV -3 ). We can infer the non-pionic pieces of the sigma commutator and of the susceptibility from the expansion (24) :

σ non-pion N = m 2 π ∂M ∂m 2 π = a 2 m 2 π + 2 a 4 m 4 π ≃ 29 MeV . (25)
It is largely dominated by the a 2 term. The corresponding value for

a 2 ≃ 1 GeV -1 is σ non-pion N = 20
MeV . In turn the nucleon susceptibility is :

χ N,non-pion S = 2 qq 2 vac f 4 π ∂ ∂m 2 π σ non-pion N m 2 π = qq 2 vac f 4 π 4 a 4 ≃ -5.4 GeV -1 (26)
The non-pionic susceptibility is found with a negative sign, as expected from the scalar meson term. In ref. [9] however, the negative sign is interpreted differently. It is attributed to possible deviations from the Gellman-Oakes-Renner (GOR) relation which links quark and pion masses.

It is then interesting to test if the empirical values from the lattice are compatible with a pure scalar meson contribution. We thus tentatively make the following identifications :

Q s S = qq vac f π g S m 2 σ = σ non-pion N (2 m q ) ≃ 2.4, (27)
with 2 m q = 12 MeV (taking a 2 ≃ 1 GeV -1 one would get Q s S = 1.66). It is interesting to translate this number into the value of the mean scalar field in the nuclear medium which, to leading order in density, is :

-s = g s ρ S m 2 σ = Q s S f π ρ S qq vac = σ non-pion N (2 m q ) f π ρ S qq vac = a 2 + a 4 m 2 π f π ρ S . (28)
At normal density the value |s(ρ 0 )| ≃ 21 MeV , quite compatible with nuclear phenomenology. The second identification concerns the susceptibility. If the non pionic susceptibility would arise entirely from the scalar field, we should have :

χ N,non-pion S = - 2 (Q s S ) 2 g 2 S 3 g S f π = qq 2 vac f 4 π 4 a 4 (29)
which would give, using the GOR relation :

-a 4 = 3 2 (σ non-pion N ) 2 g S f π m π 4 = 3.1 GeV -3 , (30)
much larger than the lattice value, -a 4 = 0.5 GeV -3 . Again, taking a 2 ≃ 1 GeV -1 one would get -a 4 ≃ 1.2 GeV -3 , still larger in magnitude than the corresponding lattice value (-0.23 GeV -3 ). Thus the linear σ model which fails to account for the saturation properties, due to the excess attraction produced by the chiral softening of the sigma mass, also leads to too large a susceptibility from the scalar meson. In fact in this work we show that the two problems are linked. Some mechanism suppresses the chiral softening of the σ mass as well as the large nucleonic susceptibility from the scalar meson, incompatible with lattice data. It is indeed likely that the scalar meson is not the only non-pionic contribution. In a previous work [3] we have invoked confinement and the quark meson coupling model (QMC) [11,12] as a source of cancellation for the chiral softening of the sigma mass. It turns out that it has also a cancelling effect in the nucleon scalar susceptibility. Indeed, for 3 valence quarks confined in a bag of radius R, Guichon [13] derived χ N,bag S ≃ + 0.25 R ≃ 1 GeV -1 , for a value R = 0.8 f m. Translated into the parameter a 4 , one has a bag 4 ≃ + 0.1 GeV -3 . Contrary to the other components which are negative (of paramagnetic nature), it has a positive sign (of the diamagnetic type, linked to quark-antiquark excitations). The bag susceptibility indeed produces a mild cancellation effect.

It is then natural to try to extend the linear sigma model description so as to incorporate other effects than the scalar meson ones (or chiral symmetry breaking effects), for instance those arising from confined valence quarks.

Generalization and implications for nuclear physics 4.1 General relation

In view of the limitations of the linear sigma model discussed previously, a more general approach is desirable. The aim is to link the response of the nuclear medium to the scalar nuclear field and the QCD responses, in such a way that both quantitites include all the components of the individual nucleon contribution whatever their origin. Of course it is not possible to achieve this goal without some assumptions on the nature of the probe. We keep the basic assumption that the scalar field which couples to the nucleons, couples to the quarks of the nucleon condensate, as is the case in the linear sigma model. Thus its presence can induce a readjustement of the quark structure of the nucleon, that we evaluate in the way described below. Consider a nuclear medium with a scalar nucleon density ρ S . By definition the response of this medium to a scalar field which couples to the nucleon scalar density fluctuations (with a unit coupling constant) is the change in the nucleon scalar density for a small change of the nucleon mass. It is Π S = (∂ρ S /∂M N ) µ , the derivative being taken at constant chemical potential. With a coupling constant g S this result should be multiplied by g 2 S . In the free Fermi gas case this derivative leads the the quantity -2 Mp F /π 2 , the free Fermi gas response. For nucleons interacting via σ and ω exchange, the expression of the scalar nucleon density is :

ρ S = 4 d 3 p (2π) 3 M * N E * p Θ µ -E * p - g 2 ω m 2 ω ρ . ( 31
)
where M * N = M N (1 + s/f π ) is the nucleon effective mass, linked to the mean scalar field s and E * p = p 2 + M * 2 N . The mean field s is obtained from the minimization equation of the energy density ǫ :

∂ε ∂s = g S ρ S + V ′ (s) = 0 . ( 32
)
It is then possible to check that the derivative of the scalar nucleon density with respect to the nucleon mass leads to the full RPA scalar polarization propagator, Π S , as defined in eq. ( 8). In this expression of the response as the derivative of the nucleon density the nucleon structure is not incorporated. It only includes the effect of the nuclear excitations and not that of the nucleonic ones. In order to include them we have to account for the internal nucleon structure, i.e., the quark structure. It is the quark medium, and not only the nucleon one, which responds to the same excitation, i.e., to the modification of the nucleon mass δM N . Accordingly we make the following conjecture, writing the full response R A S as :

R A S = 1 2 Q S ∂ρ q S ∂M N µ (33)
where ρ q S is the quark scalar density and the factor 1/2 Q S in front of the derivative is put for normalization purpose. Each nucleon containing a scalar number of quarks 2 Q S = σ N /m q , the scalar density of quarks is ρ q S = 2 Q S ρ S . The derivative involves two terms :

R A S = 1 Q S ∂ ∂M N (Q S ρ S ) µ = ∂ρ S ∂M N µ + ρ S 2 Q 2 S ∂Q S ∂m q . (34)
In the last term we have replaced the derivative with respect to the nucleon mass by the one with respect to the quark mass, with ∂M N /∂m q = 2 Q S , which introduces the nucleon susceptibility χ N S . The overall result writes :

R A S = χ N S 2 Q 2 S ρ S + Π S . (35)
The interpretation of this equation is clear. This decomposition is obvious and analogous to the one of eq. ( 5). The term linear indensity represents the individual nucleon response from the nucleonic excitations, while the term in Π S embodies nuclear excitations. The new information is that the single nucleon response is proportional to the QCD one, χ N S , with the same proportionality factor 1/(2Q 2 S ), as was found previously for the nuclear excitations. All in all, the eq. ( 35) writes :

R A S = 1 2 Q 2 S χ A S (36)
where χ A S represents the total scalar QCD susceptibility of the nuclear medium (vacuum value substracted) and both members incorporate the individual nucleon contribution. This results holds for a unit coupling constant. For a coupling constant g S (as is the case for the nuclear scalar field) the r.h.s should be multiplied by g 2 S . Accordingly the corresponding σN amplitude is :

T σN = χ N S g 2 S 2 Q 2 S . (37)
We will now comment this result and we then will apply it to the problem of the propagation of the scalar field which mediates the nuclear attraction. Our relation (36) has a close resemblance to the previous one, (18), derived in the linear sigma model but here we do not inquire about the origin of the terms, χ N S and Q S . With the values of the linear sigma model for these quantities we recover the previous result of this model.

Our relation (37) is also very similar to the one of the quark-meson coupling model [11]. In QMC, the bag positive susceptibility manifests itself in the form of a repulsive interaction in the propagation of the scalar nuclear field. The corresponnding scattering amplitude is related to the bag suceptibility by a relation identical to our eq. ( 37), but only bag quantities appear and the scalar charge entering this relation is that of the bag, which is Q bag S ≃ 0.7. As QMC does not incorporate the chiral potential which implies the three-scalar coupling, only the repulsive three-body interaction from the bag structure enters. Our expression (37) thus covers the two extreme situations, when the nucleon mass originates totally from the condensate as is the case in the σ model, or when it is only due to confinement. It is legitimate to believe that it is able to describe a more general situation with a mixed origin.

Illustration in a hybrid model of the nucleon

In the following we will illustrate the relation (37) in a model of the nucleon proposed by Shen and Toki [14], where the nucleon mass originates in part from its coupling to the condensate and in part from confinement. It consists in the following: three constituant quarks, described in the Nambu-Jona-Lasinio model (NJL), are kept together by a central harmonic force so as to mimick confinement. We have chosen for simplicity the form: ((K/4)(1 + γ 0 ) r 2 which leads to analytical results. Although oversimplified the model gives an intuitive picture of the role played by confinement. Denoting M the mass of a free constituant quark and E that of the bound one, the nucleon mass is given :

M N = 3 E = 3 M + 3 2 K E + M . ( 38
)
It is increased as compared to the value, 3M, for three independent constituant quarks.. The nucleon scalar charge, Q S , is :

Q S = 3 2 ∂E ∂m q = 3 2 ∂E ∂M ∂M ∂m q (39)
with :

∂E ∂M = c S = E + 3M 3E + M . (40)
As E > M, c S < 1, the nucleon scalar charge is reduced as compared to a collection of three independent constituant quarks. The nucleon scalar susceptibility, χ N S , given by the next derivative, is composed of two terms arising respectively from the derivative of c S and from that of ∂M/∂m. The second part leads to the susceptibility, χ q S , of a free constituant quark,

χ N S = ∂Q S ∂m q = 3 2 ∂c S ∂M ∂M ∂m q 2 + c S ∂ 2 M ∂ 2 m 2 q (41) = 3 2 ∂c S ∂M ∂M ∂m q 2 + 3 c S χ q S with: ∂c S ∂M = 24 (E 2 -M 2 ) (3E + M) 3 . ( 42
)
Notice that this last derivative is positive since E > M and that it vanishes in the absence of confining force, when E = M. Therefore the first part of the expression of χ N S represents the part of the susceptibility originating in confinement and, as in QMC, it is positive.

The scalar coupling constant g S is linked to the derivative of the nucleon mass with respect to the mean scalar field s:

g S = 3 ∂E ∂s = 3 ∂E ∂M ∂M ∂s = 3 c S g q (43)
where g q is the corresponding coupling constant for a constituant quark.

The nucleon response to the scalar field originating in confinement, κ N S , is linked to the second derivative of the nucleon mass with respect to the scalar field :

κ N S = 3 ∂ 2 E ∂s 2 = 3 ∂c S ∂M ∂M ∂s 2 . ( 44
)
The ratio between the part of the nucleon scalar susceptibility which is due to confinement and κ N S is r = 1 2

( ∂M ∂mq ) 2 ( ∂M ∂s ) 2 = 2 Q 2 S g 2 S , (45)
the same ratio as was previously found.

As for the scalar σN amplitude from the tadpole term, T σN = 3 g S /f π , it should be compared to the other part of the susceptibility. We define r ′ as the corresponding ratio through :

3 2 c S ∂ 2 M ∂m 2 q = r ′ 3 g S f π . (46)
In the semi-bosonized version of the NJL model we have :

∂M ∂m q = -2 g q qq vac f π m 2 σ (47) and ∂ 2 M ∂m 2 q = 2 χ q S = 2 g q qq 2 vac f 3 π m 4 σ (48)
in such a way that the ratio r ′ becomes :

r ′ = 2 qq 2 vac f 2 π m 4 σ = 2 Q 2 S g 2 S ≡ r. (49)
Since the same ratio applies to the two parts, r ′ ≡ r, it can be factorized so as to obtaIn the relation (37), which is thus confirmed in this hybrid model. Numerically a value of the ratio E/M ≃ 2.1, which gives c S ≃ 0.7, leads to a reasonable value for g A . It results in a value of the dimensionless parameter C = (f 2 π /2M) κ N S ≃ 0.1, while the value needed to account for the saturation properties in the framework of chiral models is C ≃ 1 [15]. Even if it fails to account for the numerical value this model has the merit to confirm the validity of the relation between the QCD response and the one to the nuclear scalar field in a situation where confinement enters and to illustrate the role played by confinement, with the introduction of a positive component in the susceptibility which opposes an increase of the nuclear scalar field.

We can now turn to the quantitative applications of the relation (37). Since both the total (non-pionic) nucleonic susceptibility and scalar charge enter in the expression of the (chiral invariant) scalar-nucleon scattering amplitude, it is legitimate to use for these two quantities the phenomenological values obtained from the lattice data. We can therefore infer the medium effects in the propagation of the s field from the lattice results of eq. ( 25) and (26) as :

-D -1 s = m 2 σ + g 2 S 2 Q 2 S χ N S ρ S = m 2 σ + g 2 S 2 a 4 (a 2 + 2 a 4 m 2 π ) 2 ρ S (50)
where in the first equation only non-pionic quantities enter, hence the introduction of the parameters a 2 and a 4 which have been defined in the lattice expansion. Numerically, at normal nuclear density, and for a value of the coupling constant g S = 10, the second term on the rhs of the second equation takes the value 0.06 GeV 2 (a similar value is found for the other set of parameters a 2 and a 4 ). For a sigma mass of m σ = 0.75 GeV , this represents at ρ 0 only a 6% decrease of the mass, much less that the chiral dropping alone and in much better agreement with the nuclear phenomenology [3,15].

Conclusion

In summary we have studied in this work the interplay between the nuclear responses to probes which couple either to nucleon or to quark scalar density fluctuations. We have found that the two responses are closely related, being proportional to each other. The scaling coefficient involves the scalar charge of the nucleon Q S . Our result holds not only at the level of the nuclear excitations but also at the nucleonic ones such that both responses incorporate the individual nucleon contributions to the nuclear response. Thus the scalar response of a nucleon to the nuclear scalar field is proportional to its QCD scalar susceptibility. We have confirmed this relation in the Shen and Toki model of the nucleon where its mass arises in part from the coupling to the condensate and in part from confinement. One application of this relation concerns a free nucleon. The σN amplitude from the tadpole term has a counterpart in the QCD scalar susceptibility in the form of a negative contribution beyond the pionic one. We have tested its existence in the lattice results on the nucleon mass evolution with the pion mass, as analyzed in ref. [9]. The expansion of ref. [9] is indeed compatible with a negative component for the non-pionic susceptibility. However the magnitude does not fit, indicating the existence of other components, with a cancelling effect. In fact a similar cancellation has to occur in the saturation problem of nuclear matter. The 3σ coupling is responsible for a lowering of the σ mass, which produces too much attraction at large densities and destroys saturation. It has to be compensated. In the optics of the present work, the two effects are related. The full σN scattering amplitude being proportional to the susceptibility, a cancelling effect in the amplitude is automatically reflected in the susceptibility. Confinement may be invoked as a natural mechanism for cancellation.

The existence of a link between QCD and nuclear physics quantities allows the derivation of parameters of the σω model from the lattice results on the nucleon mass dependence on the quark mass. This procedure leads to a mean scalar field |s(ρ 0 )| ≃ 20 MeV . In our approach the nucleon response to this field can also be derived from the lattice data. Altogether this method leads to a satisfactory description of the nuclear matter saturation properties.

Figure 1 :

 1 Figure 1: Contribution to the sigma-nucleon scattering amplitude responsible for the lowering of the sigma mass in the medium.