Second order hydrodynamic coefficients from 3-point stress tensor correlators via AdS/CFT

Abstract

We study second order relativistic viscous hydrodynamics in 4-dimensional conformal field theories. We derive Kubo-type relations for second order hydrodynamic coefficients in terms of 3-point stress tensor retarded correlators. For N =4 super Yang-Mills theory at strong coupling and at finite temperature we compute these stress tensor 3-point correlators, using AdS/CFT, by evaluating real-time cubic Witten diagrams in the AdS-Schwarzschild background. The small momentum expansion of the 3-point correlators in terms of first and second order hydrodynamic coefficients is matched with the AdS result. We arrive at the same expressions for the hydrodynamic coefficients which multiply terms quadratic in the shear and vorticity tensors in the hydrodynamic expansion of the stress tensor as did Bhattacharyya, Hubeny, Minwalla and Rangamani [1]. Our method extends the results of Baier et al [2], and allows for a unified treatment of hydrodynamic coefficients, which are extracted from 2-, and now, 3-point retarded stress tensor correlators in the AdS-Schwarzschild background. * E-mail addresses: parnold, dv3h, cw2an, wx2m@virginia.edu 1 1 Readers should be warned that there is an error in the derivation of the Kubo relations for λ 1 , λ 2 and λ 3 in [14]. Specifically, they left out the (ǫ + P )U x U x + P g xx term in deriving their T xx . Also, U i = O(h 2 ) only in the static limit.2 See also [16] for similar causality considerations and [17] for a different take on this subject.

Hydrodynamic transport coefficients for the non-conformal quark-gluon plasma from holography

“…However, the holographic computation of 3-point functions is far more involved than the computation of 2-point functions. In fact, for the strongly coupled SYM plasma λ 3 has been evaluated explicitly by computing the Euclidean 3-point function G xt,yt,xy E (p, q), yielding λ 3 = 0 [93]. Since κ * = 0 in a conformal theory, from (5.15) we obtain that for a strongly coupled SYM lim pz,qz→0…”

“…Among the 13 transport coefficients left in the equations above, results for 8 of them have already been presented in this paper while we have not yet discussed the coefficients λ 1 , λ 2 , ξ 1 , ξ 2 , and τ * π . The coefficients λ 1 and λ 2 have been studied at weak coupling in [122] and at strong coupling in [32,44,93]. The SYM values of these coefficients computed at 30 This theory is qualitatively different than that in the gradient expansion -the dissipative parts of the energy-momentum tensor have their own differential equations and, thus, its initial conditions are not determined by the initial conditions for the hydrodynamic variables ε and u µ .…”

Second order transport coefficients of nonconformal relativistic fluids in various dimensions from Dp-brane

“…This behavior is mostly studied in twoformalisms that are both developed in 2000s. They are the Minkowskian prescription of AdS/CFT duality [1][2][3][4][5][6] and the fluid/gravity correspondence [7][8][9][10][11][12]. Much progress has been made in calculating transport coefficients in both of these formalisms, for various kinds of relativistic fluids.…”

“…The real-time prescription of AdS/CFT correspondence was proposed in [1], with which Policastro et al were able to get the shear viscosity [2] and the sound wave dispersion relation [3] for the 4d conformal fluid. The second order transport coefficients were derived several years later in [4][5][6][7]14]. Then the 1st and 2nd order transport coefficients of conformal relativistic fluid in other dimensions [10,11,[15][16][17][18], and the coupling constant correction to the transport coefficients caused by higher order terms in bulk action [19][20][21][22][23][24][25][26][27][28][29][30] were also studied.…”

Second-order hydrodynamics and universality in non-conformal holographic fluids

“…In order to extract their second-order transport coefficients we need to compute the stress-tensor component T xy to second order O( 2 ) in the perturbations (2.14), (2.16), (2.18) of the field-theory metric and match the result with the effective hydro results (2.15), (2.17), (2.19) [22,65,66]. Perturbations of the external field-theory metric act as boundary sources for perturbations of the dynamical bulk metric g mn in the dual gravity theory.…”