Effective model for heavy meson diffusion

We make use of a model for heavy mesons and their interaction with the quark-gluon plasma, which was introduced in ref. 144. This effective model

describes the interaction with the medium by a dipole approximation. It relies on the large mass of the meson relative to the external momenta of the gauge fields,i. e.the momentum scale given by the temperature of the medium, but does not rely on the smallness of the coupling constant. It was used previously to make a good estimate for the binding ofJ/ψto nuclei[144].

Because the model does not rely on the weakness of the coupling constant, we can make use of it in both the strong and weak coupling regime. At weak coupling we will refer to results from perturbation theory, while the results at strong coupling can be calculated from holographic duals. Since the exact dual to QCD is not known, we once more have to be satisfied with results for N = 4 SYM theory. Therefore, we have to rephrase the model in terms of supersymmetric fields.

Diffusion in largeNQCD

The heavy meson field φdescribes a scalar meson which has a fixed four-velocityu^µ = (γ, γv). Then the effective Lagrangian for this meson field interacting with the gauge fields is[144]

L_eff=−φ^†iu·∂φ+ c_E

N²φ^†O_Eφ+ c_B

N²φ^†O_Bφ, (4.79)

where we refer to the last two terms as theinteraction LagrangianL_int, and O_E =−1

2F^µσaF_σ^νau_µu_ν, (4.80)

O_B=−1

2F^µσaF_σ^νauµuν +1

4F^σβaF_σβ^a. (4.81)

Here,F is the non-Abelian field strength of QCD, with Greek lettersµ, ν, . . . as Lorentz indices and gauge indexa, ThecEandcBare matching coefficients (polarizabilities) to be determined from the QCD dynamics of the heavy quark-antiquark pair. In inserting a factor of^1/_N² into the effective Lagrangian we have anticipated that the couplings of the heavy meson to the field strengths are suppressed byN²in the largeN limit.

In the rest frame of a heavy quark bound state withu= (1,0)the operators O_E andO_Bsimplify to

O_E = 1

2E^a·E^a, (4.82)

O_B= 1

2B^a·B^a, (4.83)

whereE^aandB^aare the color electric and magnetic fields. If the constituents of the dipole are non-relativistic it is expected that the magnetic polarizability c_Bis of orderO(v²)relative to the electric polarizability. For heavy quarks, wherecB is neglected, and largeN these matching coefficients were computed by Peskin[145, 146],

cE = 28π

3Λ³_B, cB = 0. (4.84)

HereΛ_B:= 1/a₀ = (m_q/2)C_Fα_sis the inverse Bohr radius of the mesonic bound state. It is finite at largeN since withCF 'N/2and finiteλwe have ΛB =mqλ/(16π).

The effective Lagrangian can be used to calculate the in-medium mass shift.

We will do so in the subsequent by simply consulting first order perturbation theory which says that

δM =hH_inti=− hL_inti. (4.85)

Translating the model toN = 4Super Yang-Mills theory

Our aim is to calculate the heavy meson diffusion coefficient from gauge/gravity duality. Subsequent to this subsection we explain the Langevin dynamics we use to describe this process, it requires the calculation of the two-point corre-lators as well as of the associated polarizabilitiescE andcB. Because we do not now the gravity dual to QCD we translate the effective meson model to N = 4Super Yang-Mills theory, our standard toy model.

The formalism inN = 4 SU(N)Super Yang-Mills theory is not different from the one we introduced in the preceding section. In general all operators in N = 4SYM which are scalars under under Lorentz transformations andSU(4) R-charge rotations will couple to the meson at some order. The contribution of higher dimensional operators is suppressed by powers of the temperature to the inverse size of the meson. The lowest dimension operator which could couple to the heavy meson field isO_X2 = TrXⁱXⁱ, whereXⁱ denotes the scalar fields of the theory. However, the anomalous dimension of this operator is not protected, and the prediction of the supergravity description ofN = 4 SYM is that these operators decouple in a strong coupling limit[8]. The lowest dimension gauge invariant local operators which are singlets underSU(4)and which have protected anomalous dimension are the stress tensorT_µν which couples to the graviton, and minus the Lagrangian O_F2 = −L_N=4, which couples to the dilaton. (Since we can add a total derivative to the Lagrangian, the operator−Lis ambiguous. The precise form of the operator coupling to the dilaton is given in ref. 147. We neglect this ambiguity here.) There also is the operatorO_F?F = TrF^µν?F_µν+. . ., which couples to the axion. An interaction involvingO_F?F breaksCP-symmetry, which is a symmetry of the Lagrangian of theN = 2hypermultiplet of theN = 4SYM gauge theory.

Thus interactions involvingO_F?F can be neglected.

Summarizing the preceding discussion, we find that the effective La-grangian describing the interactions of a heavy meson coupling to the operators in the field theory is

L_eff= −φ^†(t,x)iu·∂φ(t,x) + cT

N² φ^†(t,x)O_T φ(t,x) + cF

N²φ^†(t,x)O_F2φ(t,x), (4.86) which is a linear perturbation ofN = 4Super Yang-Mills theory. The two

composite operators in theinteraction LagrangianL_intare OT =T^µνuµuν

v=0

=↓ T⁰⁰, (4.87)

O_F2 =F^µνF_νµ. (4.88)

They account for the interaction of the mesons with the background. In gauge/gravity duality the modification of the Lagrangian described byO_T is achieved by considering theAdS-Schwarzschild black hole background where hO_F2i = 0. On the other hand, a finitehO_F2i 6= 0is dual to a non-trivial dilaton flow described by Liu and Tseytlin in ref. 143. Details follow below.

The polarization coefficients cT andcF will be determined below from meson mass shifts in gauge/gravity duality. This requires breaking some of the supersymmetry. We work in the linearized limit of small contributions fromO_T andO_F2. This allows to investigate the effects of finite temperature and background gauge fields separately. Additionally, this justifies the use of first order perturbation theory to compute the meson mass shifts in the medium as above by setting δM = − hL_inti. For the contribution of the energy-momentum tensor, this is achieved by switching on the temperature.

Then, the mass shift of the meson is given by expectation value of the stress tensor. Again we consider the rest frame of the mesons,

δM =−cT

N² T⁰⁰

, (4.89)

In contrast, for the meson response tohO_F2ithe mass shift of a heavy meson is given by

δM =−cF

N²hO_F2i. (4.90)

Langevin dynamics

We now turn to the kinetics of the slow moving heavy meson with massM in the medium. The kinetic energyE_kin =pv/2of the meson can be assumed to be of order of the temperatureT of the medium, such thatpv ≈T. With p=M vwe can estimate the velocity and momentum to be

p≈√

M T , v≈ rT

M . (4.91)

For time scales which are long compared to medium correlations, we expect that the kinetics of the meson can be modeled as Brownian motion and can be described by Langevin equations. These are valid for times which are long compared to the inverse temperature but short compared to the lifetime of the quasi-particle state. We model viscous force and random kicks in spatial directionsxiby

dp_i

dt =ξ_i(t)−ηDp_i,

ξ_i(t)ξ_j(t⁰)

=κ δ_ijδ(t−t⁰). (4.92)

Here,ξ_iis a component of the random forceξwith second momentκandηD

is the drag coefficient. The solution forpi(t)is given by p_i(t) =

t

Z

−∞

dt⁰e^ηD(t−t0)ξ_i(t⁰), (4.93) supposed thatηDt1[148]. This allows to relate the drag and fluctuation by

3M T = p²

=

0

Z

−∞

dt₁dt₂ e^ηD(t1+t2)hξ_i(t₂)ξ_i(t₂)i= 3κ 2ηD

. (4.94) This leads to the Einstein relation

ηD = κ

2M T . (4.95)

One of the aims of this section is the calculation of the diffusion coefficients ηDorκ, equivalently. From(4.92)we can obtain these coefficients once we know the microscopical phenomenological force

F_i(t) = dpi

dt (4.96)

acting on the quasiparticle state. We can then compare the response of the Langevin process(4.92)to the microscopic theory(4.96). Over a time interval

∆twhich is long compared to medium correlations but short compared to the time scale of equilibration we can neglect the drag, which is small for the heavy meson withη^D∝^1/_M. Since the considered time interval is long compared to medium correlations we can however equate the stochastic process, the random kicksξ, to the microscopic theory. We average (4.92)

Z

∆t

dt Z

dt⁰

ξ_i(t)ξ_j(t⁰)

= ∆t κ δ_ij

= Z

∆t

dt Z

dt⁰

F_i(t)F_j(t⁰) .

(4.97)

In a rotationally invariant medium we have fori=j κ= 1

3 Z

dt hF_j(t)F_j(0)i. (4.98)

We now identify the force with the negative of the gradient of the potential V that we read off from the Lagrangian or our theory, i. e. the interaction LagrangianV =−L_int. For the case of QCD with onlyO_E switched on we get

F(t) = Z

d³x φ^†(t,x) cE

N²∇O_E(t,x)φ(t,x), (4.99)

which is the usual form of a dipole force averaged over the wave function of the meson.

In our caseκis a constant in space and time,i. e.we consider situations with constant diffusion parameters in a homogeneous medium, for instance slight deviations from equilibrium. From the point of view of a more general description in Fourier space withκ(ω)we therefore are only interested in the hydrodynamic limit ofω→0. The fluctuation dissipation theorem relates the spectrum of(4.98)(with the specified time order of operators) to the imaginary part of the retarded force-force correlation functionG^R ∝ hF_j(t)F_j(0)ion the right hand side. In the hydrodynamic limit we get

κ=−1 3 lim

ω→0

2T

ω ImG^R(ω), (4.100)

where the full form of the retarded correlator is G^R=−i

Z

dt e^+iωtθ(t)h[F_j(t),F_j(0)]i. (4.101) Integrating out the heavy meson field as discussed in detail in ref. 99, which treated the heavy quark case, we obtain a formula for the momentum diffusion coefficient

κ= 1 3

c²_E N⁴

Z d³q (2π)³ q²

−2T

ω ImG^R(ω,q)

, (4.102)

with the retardedO_EO_E correlator given by G^R(ω,q) =−i

Z

d⁴x e^{+iωt−iq·x}θ(t)h[O_E(t,0),O_E(0,0)]i . (4.103) We can understand this result with simple kinetic theory. Examining the Langevin dynamics we see that3κis the mean squared momentum transfer to the meson per unit time. The factor of three arises from the number of spatial dimensions. In perturbation theory this momentum transfer is easily computed by weighting the square of the transferred momentum of each scattering with the transition rate for any gluon in the bath to scatter with the heavy quark,

3κ=

Z d³p (2π)³2Ep

d³p⁰

(2π)³2E_p⁰ |M|² np(1 +np⁰)q²(2π)³δ³(q−p+p⁰). (4.104) Here,pis the spatial momentum of the incoming gluon,p⁰is the momentum of the outgoing gluon andqis the momentum transferq=p−p⁰, and|M|²is the gluon meson scattering amplitude computed with the effective Lagrangian in(4.79)and weighted by the appropriate momentum distributionsnof the incoming and outgoing gluons,

|M|²= c²_E

N²ω⁴ 1 + cos²(θ_pp⁰)

. (4.105)

Alternatively (as detailed in appendix A of ref. 2), we can simply evaluate the imaginary part of the retarded amplitude written in(4.102)to obtain the same result.

InN = 4theory the generalized force is given by F(t) =−

Z

d³x φ^†(t,x)∇cT

N²O_T(t,x) + cF

N² O_F2(t,x)+

φ(t,x) (4.106) which results in a momentum broadening

κ=−1 3 lim

ω→0

Z d³q (2π)³q²2T

ω c²_T

N⁴ImG^R_T(ω,q) + c²_F

N⁴ImG^R_F(ω,q)

, (4.107) where the retarded correlators at vanishing velocity are

G^R_{T T} =−i Z

d⁴x e^{+iωt−iq·x}θ(t)

T⁰⁰(t,x),T⁰⁰(0,0)

, (4.108) G^R_{F F} =−i

Z

d⁴x e^{+iωt−iq·x}θ(t)h[O_F2(t,x),O_F2(0,0)]i. (4.109) In writing (4.107)we have implicitly assumed that there is no cross term betweenO_F2 andOT. In the gauge/gravity duality this is reflected in the fact that at tree level in supergravity _δg00^δ2(x)^SsugraδΦ(y) = 0.

Im Dokument In-medium effects in the holographic quark-gluon plasma (Seite 120-126)