On the universal identity in second order hydrodynamics

We compute the 't Hooft coupling correction to the infinite coupling expression for the second order transport coefficient $\lambda_2$ in ${\cal N}=4$ $SU(N_c)$ supersymmetric Yang-Mills theory at finite temperature in the limit of infinite $N_c$, which originates from the $R^4$ terms in the low energy effective action of the dual type IIB string theory. Using this result, we show that the identity involving the three second order transport coefficients, $2 \eta \tau_\Pi - 4 \lambda_1 - \lambda_2 =0$, previously shown by Haack and Yarom to hold universally in relativistic conformal field theories with string dual descriptions to leading order in supergravity approximation, holds also at next to leading order in this theory. We also compute corrections to transport coefficients in a (hypothetical) strongly interacting conformal fluid arising from the generic curvature squared terms in the corresponding dual gravity action (in particular, Gauss-Bonnet action), and show that the identity holds to linear order in the higher-derivative couplings. We discuss potential implications of these results for the near-equilibrium entropy production rate at strong coupling.


I. INTRODUCTION
Fluid dynamics is currently understood as an effective theory approximating a given microscopic theory in the long-wavelength, long-time regime via a systematic derivative expansion [1], [2], [3], [4]. The corresponding equations of motion (Navier-Stokes equations, Burnett equations, and their generalizations) follow from combining the equations expressing conservation laws of the microscopic theory with the constitutive relations at a given order in the derivative expansion. 1 In the simplest case of a relativistic neutral conformal fluid in a d-dimensional curved spacetime, the derivative expansion of the stress-energy tensor's expectation value has the form where u a is the velocity of the fluid, ε is the energy density, P is the pressure fixed by the conformal invariance to obey the equation of state ε = (d − 1)P , and the tensor Π ab involving first and second derivatives of velocity is given by where D ≡ u a ∇ a . We use notations and sign conventions of Ref. [2,3], where the definitions of tensor structures such as vorticity Ω ab appearing in Eq. (2) can also be found. The six transport coefficients η, τ Π , κ, λ 1 , λ 2 and λ 3 in Eq. (2) are determined by the underlying microscopic theory. For conformal theories at finite temperature and zero chemical potential, they scale with the appropriate power of temperature (fixed by their scaling dimension) and may depend on coupling constants and other parameters of the theory such as the rank of the gauge group. For some theories, transport coefficients have been computed in the regime of weak coupling using kinetic theory approach and in the regime of strong coupling using gauge-string duality methods [7][8][9][10].
Of particular interest are the properties of transport coefficients universal for all or at least some class of theories. For example, the dimensionless ratio of shear viscosity to entropy density exhibits such a universality: assuming validity of gauge-string duality, one can prove that the ratio is equal to 1 4π for a large class of theories in the limit described by a dual supergravity (usually, in the limit of infinite coupling and infinite rank of the gauge group) [11], [12], [13][14][15]. This result is very robust and holds for any quantum field theory (conformal or not) with a gravity dual description (see [16] for a recent summary and discussion). Coupling constant corrections to the viscosity-entropy ratio are not expected to be universal: in each theory, the ratio is a non-trivial function of the coupling and other parameters. In particular, in the finite-temperature N = 4 SU (N c ) supersymmetric Yang-Mills (SYM) theory in d = 3 + 1 dimensions in the limit of infinite N c and infinite 't Hooft coupling λ = g 2 Y M N c , the shear viscosity to entropy density ratio appears to 1 In this paper, we ignore issues related to non-analytic contributions to correlation functions at small frequency and the breakdown of the derivative expansion [5], [6]. This is justified as long as we work within classical (i.e. not quantum) gravity approximation. For N = 4 SU (Nc) supersymmetric Yang-Mills theory, this means staying in the limit of infinite Nc.
be a monotonic function of the coupling [11], with the correction to the universal infinite coupling result being positive [17,18]: Subsequent calculations revealed that in other (hypothetical) quantum field theories the corrections coming from higher derivative terms in the dual action can have either sign [19,20]. In particular, for a (hypothetical) field theory dual to Einstein gravity with Gauss-Bonnet higher derivative terms 2 where L is the AdS radius, one finds [20] non-perturbatively in the Gauss-Bonnet coupling λ GB .
Much less is known about bulk viscosity 3 [26], [27], [28]. A proposal for a universal inequality involving bulk viscosity at strong coupling has been made by Buchel [29] but it seems there exist counterexamples to it [30].
Universal behavior is also known to exist for second order transport coefficients. Following the observation made in [31], Haack and Yarom [32] showed that for relativistic conformal fluids with U (1) charges in d > 3 space-time dimensions 4 in the limit described by a dual two-derivative gravity, the following linear combination vanishes: We expect H to be (generically) a non-trivial function of the coupling: perturbative analysis in QED and other theories [33] suggests that H ≠ 0 at weak coupling. In conformal kinetic theory one finds 2ητ Π + λ 2 = 0 [2,33,34]. In the same regime, the ratio λ 1 ητ Π is expected to be relatively close to 1 but the prediction λ 1 = ητ Π [34] is understood to be an artifact of a too restrictive ansatz choice for the collision integral [33]. (If the prediction were true, we would have H = 0 in the kinetic regime.) For the conformal theory dual to Gauss-Bonnet gravity (4), the statement that H is not identically zero would imply that generically one may expect H = O (λ GB ).
Intriguingly, Shaverin and Yarom found that H = 0 still holds in this theory to linear order in λ GB [35]. However, the universal relation does not hold to order λ 2 GB [36] 5 . Indeed, as will be shown elsewhere [36], the full non-perturbative expression for H(λ GB ) implies that the identity (6) holds to linear order in λ GB but is violated at O(λ 2 GB ): 2 As is well known [21][22][23][24][25], such a field theory would be suffering from inconsistencies such as causality violation unless restrictions are imposed on the Gauss-Bonnet coupling or other degrees of freedom are added to the Gauss-Bonnet gravity. Our working assumption is that it is possible to cure the problems in the ultraviolet without affecting the hydrodynamic (infrared) regime. 3 For conformal theories, bulk viscosity is zero. 4 In lower dimension, the coefficient λ1 is undefined.
where γ GB = √ 1 − 4λ GB . Curiously, in that theory H(λ GB ) ≤ 0 for all values of λ GB ∈ (−∞, 1 4]. In this paper, we show that the identity H = 0 holds in N = 4 SYM at next to leading order in the strong coupling expansion (in the limit N c → ∞). In this limit, the shear viscosity and the second order transport coefficients of N = 4 SYM are given by where γ = λ −3 2 ζ(3) 8. To leading order in the strong coupling limit (i.e. at γ → 0), the results (8) - (13) were obtained in [2,39,40] using gauge-gravity and fluid-gravity dualities. Coupling constant corrections to all the coefficients except λ 2 were previously computed from the higher-derivative terms in the low-energy effective action of type IIB string theory [17,18,27,[41][42][43][44]. The O λ −3 2 correction in the expression for λ 2 is the new result obtained in Section II of the present paper. The corrections in formulae (8) - (13) can be trusted so long as they remain (infinitesimally) small relative to the leading order (λ → ∞) result, as they are obtained by treating the higherderivative terms in the equations of motion perturbatively. To leading order in the strong coupling limit, the coefficients (8) - (13) are independent of the coupling, in sharp contrast with their weak coupling behavior [45]. The coefficient λ 3 vanishes at λ → ∞, and was argued to vanish also at λ → 0 (this appears to be a generic property of weakly coupled theories). The full coupling constant dependence of transport coefficients (even at infinite N c ) appears to be beyond reach.

II. COUPLING CONSTANT CORRECTION TO THE SECOND ORDER TRANSPORT COEFFICIENT λ 2 IN N = 4 SYM THEORY
Coupling constant corrections to transport coefficients in N = 4 SYM can be computed using the following dual five-dimensional gravitational action with the R 4 higher derivative term 6 6 As argued in [46], to compute physical quantities in the hydrodynamic regime of field theories dual to tendimensional type IIB supergravity with five compact dimensions, it is sufficient to consider only the reduced five-dimensional action.
where γ = α ′3 ζ(3) 8 which is related to the value of the 't Hooft coupling λ in N = 4 SYM via We set the AdS radius L = 1 in the following. The effective five-dimensional gravitational constant is connected to the rank of the gauge group by κ 5 = 2π N c . The term W is given in terms of the Weyl tensor C µνρσ by The α ′ -corrected black brane solution corresponding to the action (14) was found in [47]: where f (u) = 1 − u 2 , r 0 is the parameter of non-extremality of the black brane geometry, and the functions Z t and Z u are given by The Hawking temperature corresponding to the metric (16) is T = r 0 (1 + 15γ) π. The standard black three-brane solution is recovered in the limit γ → 0.
To compute the 't Hooft coupling correction to the second order transport coefficient λ 2 , we use the method of three-point functions and the associated Kubo formulae developed by Moore, Sohrabi and Saremi [44,48] and by Arnold, Vaman, Wu and Xiao [49]. The relevant retarded three-point functions of the stress-energy tensor are defined in the Schwinger-Keldysh closed time path formalism [50,51]. We now review the key elements of the method.
Consider a theory described by a microscopic Lagrangian L [φ, h], where φ collectively denotes matter fields and h corresponds to a metric perturbation of a fixed background g (all tensor indices are suppressed). The degrees of freedom of the theory are then doubled, φ → φ ± , g → g ± , h → h ± , where we use the index ± to denote whether the fields are defined on a "+"-time contour running from t 0 towards the final time t f > t 0 , or the "−"-contour with time running from the future t f backwards to t 0 . For the field theory considered at a finite temperature T = 1 β, the two separated real time contours can be joined together by a third, imaginary time part of the contour running between t f and t f − iβ. We use ϕ to denote fields defined in the Euclidean theory on the imaginary time contour. The generating functional of the stress-energy tensor correlation functions can then be written as It is convenient to introduce the Keldysh basis φ R = 1 2 (φ + + φ − ) and φ A = φ + − φ − , and similarly for the metric perturbation and the stress-energy tensor. After variation, the classical expectation values always obey φ + = φ − , hence all fields with an index A will vanish and one can define T ab ≡ T ab R . Explicitly, The expectation value of T R at x = 0 can then be expanded as where G RAA... denote the fully retarded Green's functions [52] obtained by taking the appropriate number of derivatives with respect to h A and h R [48] G ab,cd,...
Denoting the space-time coordinates of the four-dimensional field theory by t, x, y, z and choosing the momentum along the z axis, one can write the following Kubo formula for the coefficient λ 2 [44,48] The three-point functions are calculated by solving the bulk equations of motion to second order in metric perturbations of the background g µν (16), where ǫ serves as a book-keeping parameter indicating the order of the perturbation. We impose the Dirichlet condition h (2) µν = 0 at the boundary [44]. The three-point functions are found by taking functional derivatives of the on-shell action with respect to the boundary value h A simplifying feature of this procedure is that since equations of motion are solved to order ǫ 2 , only the boundary term contributes to the three-point function and hence no bulk-to-bulk propagators appear in the calculation. To compute G xy,ty,xz RAA (p, q), we turn on the following set of metric In the following, we use the notations ω ≡ p 0 , q ≡ q z and T 0 ≡ r 0 π. At first order in ǫ, the metric perturbations can be written as expansions in γ where Z  (14), where the γ-dependent part is treated as a perturbation. The differential equation for Z xy is The functions Z xz and Z ty obey the same differential equation (28) with q and ω, respectively, set to zero. Note that we cannot impose the incoming wave boundary condition at the horizon on Z ty , as it has no time dependence. Instead, the Dirichlet condition Z ty = 0 is used [44].
The solutions to quadratic order in ω and q are given by We now use these solutions in the full equations of motion to find the differential equations obeyed by Z where the functions G(u) on the right hand side are, respectively, The next step is to use the above solutions to find the second-order perturbation h (2) xy of the metric to linear order in γ and to quadratic order in ω and q. We begin by computing the action (14) with to order ǫ 4 . This gives us the effective action and the equation of motion for the fluctuation h (2) xy that can be solved perturbatively to linear order in γ and quadratic order in ω and q. We can look for a solution in the form At γ = 0, the full fluctuation equation is where only the ω and q-independent parts of Z xz and Z ty are relevant for our purposes.
By further writing Y (γ) xy = 1 − u 2 − iω 4πT 0 y(u), the differential equation for y(u) at quadratic order in ω and q is simply The full solution of the two differential equations is given by We now compute the holographic stress-energy tensor for the induced metric γ µν , which has the same tensorial form as in the Einstein-Hilbert gravity, with no higher-derivative terms contributing [44]. Taking the derivatives of T xy with respect to the boundary values of h (1) xz and h (1) ty , we find the three-point function, and, using the Kubo formula (22), the coefficient λ 2 in Eq. (12).

III. CURVATURE SQUARED CORRECTIONS TO SECOND ORDER TRANSPORT CO-EFFICIENTS
In this Section we determine corrections to the second order transport coefficients in a (hypothetical) four-dimensional CFT dual to a bulk gravity with generic curvature squared terms. The five-dimensional bulk action is where the cosmological constant Λ = −6 L 2 (we set L = 1 in the rest of this Section). For generic values of the coefficients α 1 , α 2 , α 3 , the curvature squared terms are treated perturbatively. The Gauss-Bonnet action is obtained from the action (48) by setting α 1 = α 3 = λ GB 2 and α 2 = −2λ GB .
To compute curvature squared corrections to second order transport coefficients in a dual fourdimensional quantum field theory to linear order in α i , one can use the field redefinition and the already known results for N = 4 SYM and Gauss-Bonnet gravity.
First, we set α 3 = 0 and use the field redefinition discussed in [20], [19] , to rewrite the action (48) in the two-derivative form (to linear order in α 1 and α 2 ): Here, in the notations of [20], K = 2Λ 3 (5α 1 + α 2 ) andΛ = Λ 1+K . The field redefinition (49) implies that the metric satisfies where We can further transform the metricḡ µν to bring the actionS into the standard Einstein-Hilbert form with the cosmological constant Λ dual to N = 4 supersymmetric SU (N c ) Yang-Mills theory in the regime of infinite 't Hooft coupling and infinite N c . Indeed, consider a new metricg µν defined byḡ µν = B 2g µν (the metric determinant and the Ricci scalar transform, correspondingly, as we can now write the action (50) in the standard Einstein-Hilbert form, where the redefined Newton's constant isκ The original metric g µν , which is related tog µν by g µν = A 2 B 2g µν + O(α 2 i ), can be written to linear order in α 1 and α 2 as where e −2ω = 1 A. From Eqs. (54), (56), it is clear that the stress-energy tensorT µν N =4 of N = 4 SYM computed from (54), withκ 2 5 , is related to the stress-energy tensor of a theory dual to S R 2 [α 3 = 0] by a global Weyl transformation and the redefinition of κ 2 5 given by Eq. (55). Then the scaling arguments 7 imply where all the transport coefficients with the overhead tildes depend onκ 2 5 . The N = 4 SYM theory coefficients areη The temperature of the N = 4 SYM theory is given byT = r + π, where r + is the radial position of the black brane horizon. Using Eqs. (52) and (53), we find that the shear viscosity in a (hypothetical) field theory dual to the gravitational background described by the action S R 2 [α 3 = 0] is given by which agrees 8 with the results obtained in [19,53]. Here r + is the location of the event horizon of a black brane solution to the equations of motion following from the action S R 2 [α 3 = 0]. The full result (for arbitrary α 3 to linear order) is thus where the coefficient C η remains undetermined. Similarly, the second-order transport coefficients to linear order in α 1 and α 2 are given by the corresponding N = 4 SYM results multiplied by AB 5 : Here we added the undetermined terms linear in α 3 . To restore the dependence on α 3 (to linear order), recall that the Gauss-Bonnet expressions for transport coefficients would be restored (to linear order in λ GB ) by substituting α 1 = λ GB 2, α 2 = −2λ GB and α 3 = λ GB 2. For example, according to (63), the coefficient κ in the holographic Gauss-Bonnet liquid to linear order in λ GB should be equal to On the other hand, all transport coefficients of the holographic Gauss-Bonnet liquid are known explicitly (non-perturbatively [36] and to linear order [35]): Comparing Eqs. (68) and (70), and taking into account α 3 = λ GB 2, we read off the coefficient All other coefficients are determined in the same way, and we find the transport coefficients of a (hypothetical) holographic liquid described by the dual gravitational action (48) to linear order in α i : In Eqs. (74)-(79), r + is the location of the event horizon in the full black brane solution involving all three α i corrections. The results for τ Π and κ were previously derived in [53] and are in agreement with our Eqs. (75) and (76). The expressions for λ 1 , λ 2 and λ 3 are new. Finally, by using the expressions (75), (77) and (78), we confirm that the Haack-Yarom relation among the second order coefficients is satisfied to linear order in α i for a (hypothetical) holographic liquid dual to five-dimensional gravity with generic curvature squared terms given by the action (48):

IV. CONCLUSIONS
In this paper, we have made an observation that the universal relation (6) among the second order transport coefficients holds not only to leading order in conformal liquids with dual gravity description, as suggested by Haack-Yarom theorem, but remains valid to next to leading order in N = 4 SYM and in a (hypothetical) fluid dual to five-dimensional gravity with generic curvature squared terms (in particular, Gauss-Bonnet gravity). It is not clear to us whether this result can be generalized to an arbitrary higher-derivative correction to Einstein-Hilbert action (to linear order in the corresponding couplings). Such a generalization would follow from knowing (from e.g. inequalities obeyed by correlation functions or restrictions imposed by the entropy current) that H(λ HD ) ≤ 0, where λ HD is the higher-derivative coupling, since then λ HD = 0 would be a maximum of the function H(λ HD ) with H(0) = 0 by the Haack-Yarom theorem, and the linear term in the expansion of H(λ HD ) for small λ HD would necessarily vanish. This is indeed the case for the Gauss-Bonnet gravity, see Eq. (7). Before looking for a general proof, however, it may be useful to check other examples, including other dimensions and charged backgrounds. Weak coupling calculations seem to suggest that H is a non-trivial function of the coupling, yet it would be desirable to know this explicitly for a conformal theory (e.g. N = 4 SYM) at weak coupling. Also, it may be interesting to generalize the Haack-Yarom theorem to non-conformal holographic liquids 9 .
The physical significance of the function H is not entirely clear but it might be related to one of the parameters regulating dissipation. The normalized rate of the entropy production in a conformal fluid near equilibrium is given by [55] ∇ a s a s = η 2sT σ ab σ ab + κ − 2λ 1 4sT σ ab σ a c σ bc + where the coefficient A 1 remains unknown at present. Dissipationless conformal fluids have η = 0, κ = 2λ 1 and 2ητ Π − 4λ 1 − λ 2 = 0 [56], i.e. for such fluids A 1 = λ 2 2T . Strongly coupled N = 4 SYM is not a dissipationless fluid 10 , but with κ − 2λ 1 = O(λ −3 2 ), low viscosity-entropy ratio and H = 0 it may be not too far from it, comparing especially to the weak coupling limit λ → 0, where η s ∼ 1 λ 2 ln λ −1 , λ 1 ∼ T 2 λ 4 ln 2 λ −1 and κ ∼ T 2 λ 2 [33]. This raises the possibility that the near-equilibrium hydrodynamic entropy production is generically suppressed at strong coupling as discussed recently in [57].