Holographic transport beyond the supergravity approximation

We set up a unified framework to efficiently compute the shear and bulk viscosities of strongly coupled gauge theories with gravitational holographic duals involving higher derivative corrections. We consider both Weyl$^4$ corrections, encoding the finite 't Hooft coupling corrections of the boundary theory, and Riemann$^2$ corrections, responsible for non-equal central charges $c\ne a$ of the theory at the ultraviolet fixed point. Our expressions for the viscosities in higher derivative holographic models are extracted from a radially conserved current and depend only on the horizon data.

For well over two decades there has been an ongoing program to apply the holographic gauge/gravity duality to gain insights into the dynamics of strongly coupled quantum phases of matter [1].Indeed, the techniques of holography have been adopted to probe the transport properties of a wide spectrum of strongly correlated systems, ranging from the QCD quark gluon plasma (QGP) to high temperature superconductors, strange metals and a variety of electronic materials.Within this program, the most elegant result to date remains the universality [2,3] of the shear viscosity η to entropy s ratio, which holds in strongly coupled gauge theories in the limit of an infinite number of colors, N → ∞, and infinite 't Hooft coupling, λ → ∞.These theories are dual to Einstein gravity coupled to an arbitrary matter sector, with the result (1.1) relying on the additional assumption that rotational invariance is preserved 1 .
The significance of (1.1) can be traced not only to its universal nature, but also to the fact that its value is remarkably close to the experimental range extracted from the QGP data at RHIC and at the LHC.This led to the compelling KSS proposal [5,6] that the shear viscosity might obey a fundamental lower bound in nature, η s ≥ 1 4π (from now on we take = k B = 1).Despite its appeal, it is now well understood that the KSS bound can be violated in a number of ways, either by relaxing symmetries or by introducing higher derivative curvature corrections to the low-energy gravitational action (see i.e. [7] for a review).Indeed, notable early examples of the effects of higher derivatives include Weyl 4 corrections to Einstein gravity [8], which encode finite λ effects in the dual gauge theory, and Riemann2 corrections [9,10], which describe finite N effects in the dual theory 2 .Holographic models involving a more complicated matter sector have been used to study the temperature dependence of η/s [11,12] and have direct applications to the QGP, where the temperature variations of η are expected to play an important role (see e.g. the recent review [13]).A key lesson that has emerged from these studies is that the universality of η/s is generically lost once we move away from the limits λ, N → ∞ (or appropriately break symmetries).Moreover, other transport coefficients have failed to exhibit the simple universal behavior encoded in (1.1).
In addition to η, the bulk viscosity ζ has also attracted considerable attention in holography (see e.g.[14][15][16][17][18][19][20][21][22][23][24]), largely because of its relevance to the physics of the QGP near the deconfinement transition, where ζ is expected to rise dramatically.As non-zero bulk viscosity requires theories with broken conformal symmetry, holographic model building has typically involved adding bulk scalars with non-trivial profiles.Since the latter also yields a non-trivial temperature dependence for η/s, such holographic models have played a prominent role in the attempts to build realistic models of QCD [25,26].
In holography, transport coefficients such as η and ζ can be extracted in a number of complementary ways, e.g.computing correlators of the stress energy tensor and using Kubo formulas, or from linearized quasi-normal modes on black brane backgrounds, which in the hydrodynamic limit correspond to shear and sound modes of the dual field theory.The standard holographic dictionary instructs us to extract correlators from the boundary behavior of fluctuating bulk fields, appropriately supplemented with boundary conditions at the horizon.However, hydrodynamics is an effective description of the system at long wavelengths and small frequencies, and thus one would expect it to be encoded in properties of the geometry and its fluctuations in the IR, i.e. near the horizon.Thus, a natural question is to what extent the horizon of a black brane can fully capture the hydrodynamic behavior of a strongly coupled plasma, and its transport properties.Indeed, the diffusive modes can be understood [5,27] as fluctuations of the black brane horizon using the membrane paradigm [28,29], which identifies the horizon with a fictitious fluid.This approach was made more precise in [30] and led to various formulations for extracting transport coefficients entirely from the horizon geometry.However, these methods have been limited to special cases.
In this paper we revisit some of these questions, and set up a universal -and efficient -framework for extracting the shear and bulk viscosities of strongly coupled gauge theories with holographic duals involving higher derivative corrections.A crucial step in our analysis is the realization that the terms needed to compute both η and ζ can be extracted from radially conserved currents, even in the presence of higher derivatives.In turn, this implies that they can be evaluated at the black brane horizon.As we will see, one clear advantage of our framework is that it avoids having to compute dispersion relations.In our analysis we will consider both Weyl 4 corrections and Riemann 2 corrections, encoding, respectively, finite 't Hooft coupling and finite N effects.Moreover, our gravitational dual has an arbitrary number of scalars, with an arbitrary interaction potential.Having such a framework is especially valuable for theories with higher derivatives, where the computations are intrinsically more cumbersome and a number of subtleties arise, related to the presence of, for instance, additional boundary terms and counterterms.
In closing we should mention that the recent paper [31] has also examined the connection between horizon and boundary data and has put forth an efficient method for computing transport coefficients directly from the horizon3 .However, their analysis is restricted to two-derivative theories, and to matter sectors involving, in addition to Einstein gravity and a U(1) gauge field, only one scalar field.In our analysis, on the other hand, we have included an arbitrary number of scalars and allowed for curvature corrections to the leading gravitational action.

Summary of Results
We will work with a five-dimensional theory of gravity in AdS coupled to an arbitrary number of scalars, described by where δL denote terms involving higher derivative corrections to Einstein gravity.In particular, in this paper we consider two classes of models, to leading order4 in β: • four-derivative curvature corrections described by: • eight-derivative curvature corrections described by: where C is the Weyl tensor.
For the shear viscosity to the entropy ratio we find, respectively: where All the quantifies in (1.5) and (1.6) are to be evaluated at the horizon of the dual black brane solution.Since the O(β 0 ) results are universal [3], it is sufficient to evaluate the scalar potential and its derivatives to leading O(β 0 ) order only.
For the bulk viscosity to the entropy ratio we find, respectively: (1.7) (1.8) Once again, all the quantifies in (1.7) and (1.8) are to be evaluated at the horizon of the dual black brane solution.Here z i,0 are the values of the gauge invariant scalar fluctuations, at zero frequency, evaluated at the black brane horizon, see section A.3 and in particular (A.69).While the scalar potential and its derivatives can be evaluated to the leading O(β 0 ) order of the background black brane solution, the horizon values of the scalars z i,0 must be evaluated including O(β) corrections.
The rest of the paper is organized as follows.In section 2 we present the analysis for some specific models, and provide extensive checks on the general formalism.We conclude in section 3 and highlight future directions.The formal proofs of the main results -the final expressions (1.5), (1.6) for the shear viscosity, and (1.7), (1.8) for the bulk viscosity are delegated to appendix A. We discuss the background black brane geometry in A.1, η s is computed in section A.2, and ζ s is computed in section A.3.

Applications
In this section we use simple toy models to validate the general formulas reported in (1.5)-(1.8).First and foremost, note that if the boundary gauge theory is a CFT with we find5 from (1.5) and (1.6) reproducing [9] and [8,36,37] correspondingly.
We discuss the following models: with Note the O(β) modification of the relation between the mass of the bulk scalar and the dimension ∆ of the dual boundary operator 6 .We consider ∆ = {2, 3}.
The bulk viscosity in these models was not discussed in the literature before.
Models A 2,∆ are interesting in that the gravitational holographic bulk is higher derivative and the black brane horizon Wald entropy differs from its Bekenstein entropy, see (A.28).
• (B 2,∆ ): δL 2 model with with The O(β) modification of the relation between the mass of the bulk scalar and the dimension ∆ of the dual boundary operator is precisely as reported in [38].We consider ∆ = {2, 3}.The bulk viscosity in these models was considered in [38], but only to leading order in the non-normalizable coefficient of the scalar φ (albeit to all orders in β).Here we consider leading perturbative in β corrections to transport in these models, but to all orders in the conformal symmetry breaking parameter, i.e., non-perturbatively in the non-normalizable coefficient of the bulk scalar φ.
Models B 2,∆ are interesting in that the gravitational holographic bulk represents a two-derivative model: the coefficients in (2.5) assemble Riemann squared terms into the Gauss-Bonnet combination.Notice that for the black branes in these models the Wald entropy is identical to their Bekenstein entropy, see (A.28).
• (C 2,∆ ): δL 2 model with with The O(β) modification of the relation between the mass of the bulk scalar and the dimension ∆ of the dual boundary operator is identical to the one in models (B 2,∆ ).We consider ∆ = {2, 3}.The bulk viscosity in these models was not discussed in the literature before.
Models C 2,∆ are interesting in that the gravitational holographic bulk is higherderivative, but the horizon physics is effectively two-derivative: as in the case above, in these models there is no difference between the Wald and the Bekenstein entropies of the dual black brane horizon.
• (D 4,∆ ): δL 4 model with (2.9) Notice that here the Weyl 4 higher derivative corrections to the gravitational action (1.4) do not modify the bulk scalar mass/dimension of the dual operator relation.We consider ∆ = {2, 3}.The bulk viscosity in these models was not discussed in the literature before.
Models D 4,∆ are interesting in that here the higher derivative corrections are associated with finite 't Hooft coupling corrections of the UV fixed point CFT, rather than with the difference between the central charges of the UV CFT, encoded by β • α 3 ≡ c−a 8c , as in models {A, B, C} 2,∆ .
In the models just introduced we compute the shear and the bulk viscosities using (1.5)-(1.8),and compare the results with direct computation of these quantities from the dispersion relation of the shear and the sound modes, shear : (2.10) These are the appropriate quasinormal modes of the dual black brane [39].The first non-conformal gauge theory computations of the bulk viscosity from the dispersion relation were performed in [15].In genuinely higher-derivative holographic models the shear and the sound mode dispersion relations where studied only in conformal N = 4 SYM in [35].In this paper, we generalize (and combine) the computation methods of [15] and [35].Such analysis are much more involved and are extremely technical.
We will not provide any details -in fact our motivation of developing the framework explained in sections A.2 and A.3 was precisely to avoid computation of the dispersion relations in the first place.As we already emphasized, here we use such dispersion computations in models A − D as a check on our general framework.
Finally, we mention one additional test we performed.The speed of the sound waves c s in (2.10) is related to the equation of state P = P (E) of the holographic gauge theory plasma via where in computing derivatives of the pressure P with respect to the energy density E one has to keep the non-normalizable coefficient λ ∆ of the bulk scalar (i.e., the coupling constant of the dual operator O ∆ explicitly breaking the conformal invariance) constant.
As we explicitly show in this section, all the validations pass with excellence.

Shear viscosity in models A − D
Given (1.5), we find that in all models A − C the shear viscosity to the entropy density while from (1.6) in models D the shear viscosity to the entropy density is where φ h 0,0 is the leading O(β 0 ) order horizon value of the bulk scalar.It is computed numerically solving the leading order O(β 0 ) background equations of motion (A.7)-(A.10),using the metric parameterization (A.23), subject to the boundary conditions: Without loss of generality we can fix r h = 1, provided we present all results as dimensionless quantities.From (A.14) we find (2.15) Comparisons between the corrections to the shear viscosity (see (2.12) and (2.13)) for the holographic models A − D extracted from the dispersion relation of the shear modes using (2.10) (the solid curves) and using (2.12) and (2.13) (dashed red curves) are presented in figs.1-2.In the left panels we consider models with ∆ = 3 with the non-normalizable gravitational bulk scalar coefficient λ 3 identified as a fermionic mass term λ 3 ≡ m f of the boundary gauge theory.In the right panels we consider models with ∆ = 2 with the non-normalizable gravitational bulk scalar coefficient λ 2 identified as a bosonic mass term λ 2 ≡ m 2 b of the boundary gauge theory 7 .The difference between the solid and the dashed red curves is ∼ 10 −8 • • • 10 −5 % over the ranges of λ ∆ /T 4−∆  reported.When m f = 0 and m 2 b = 0 we recover the conformal gauge theory results (2.2): = 120 . (2.16)

Bulk viscosity in models A − D
Unlike the shear viscosity, the bulk viscosity in models A and B, C differs: using (1.7) we find and where φ h 0,0 is the leading O(β 0 ) order horizon value of the bulk scalar.In (2.17), (2.18) and (2.19) we denoted since our toy models have a single bulk scalar.To proceed further we need to evaluate the gauge invariant bulk scalar fluctuations z 0 at the horizon to order O(β) as emphasized in (2.20).
We present details for the model A 2,∆=3 , and only the final results for the other models.

Model A 2,∆=3
Since we will need the gauge invariant bulk scalar fluctuations z 0 at the horizon to order O(β), we need the background geometry to order O(β).It is convenient to use the metric warp-factor parameterization as in (A.23).Explicitly, at order O(β 0 ), and (2.26) where λ 3 ≡ m f is the non-normalizable coefficient of the bulk scalar, and the coefficients {φ 0;3 , f 4 , φ 1;3 , g 2,1;4 } are related to the thermal expectation values of various boundary gauge theory operators; in the vicinity of the black brane horizon, i.e., as y ≡ (1 − r) → 0, specified by the set of coefficients {φ h 0,0 , g h 0,0 , φ h 1,0 , g h 1,1;0 }.Using (2.29), from (A.14) we compute It is important that s 1 = 0, since as we will show it affects the representation of in the plots.
Conservation of the imaginary part of current (A.77) in particular requires that and provides a stringent test on our numerics.6 × 10 -9 2 × 10 -9 3 × 10 -9 It is important to present the physical results as dimensionless quantities, as we fixed the overall scale on the gravitational side of the computations setting r h = 1.From (2.30), the dimensionless quantity Assume that we have a dimensionless quantity K that is O(β) corrected, and that is extracted from the numerics as a functions of λ 3 , but we need to present it as a function of m f /(2πT ).Then, i.e., the O(β) correction of the quantity K receives an extra derivative in the K 0 term.
This was not an issue in our discussion of the shear viscosity to the entropy density ratio in section A.2, since there the appropriate quantity K 0 ≡ 1 4π is a constant.In fig. 3  In fig. 4 we numerically validate the conservation of the imaginary part of current (A.77).A simple formula to compute the bulk viscosity in two-derivative holographic models was proposed in [23] (EO):

Models
where c s is the speed of the sound waves in the holographic plasma (2.11), and the bulk scalar derivatives are evaluated at the black brane horizon, keeping the nonnormalizable coefficients of these scalars -the mass terms of the boundary gauge theory -fixed [21,24].The EO was extensively tested in many models, and it is verified to leading order O(β 0 ) in the models discussed here.Specifically, in fig. 5   We stress that (2.41) is a simple, novel expression for the bulk viscosity in twoderivative holographic models: unlike [18], there is no restriction to a single bulk scalar, and there is no need to compute scalar derivatives at the horizon -z i,0 are values of the gauge invariant scalar fluctuations 9 at zero frequency evaluated at the horizon.
Furthermore, the first expression in (2.41) is true even in higher-derivative holographic models, provided the dual gravitational physics is effectively two-derivative (as in the Gauss-Bonnet models B (2,∆) ), or, at least, effectively two-derivative at the horizon 10   (as in the class of models C (2,∆) ), see (1.7), where z i,0 has to be evaluated at the horizon including O(β) corrections.
Fig. 6 demonstrates that the naive application of the EO formula (2.40) does not work in higher-derivative models.By "naive" application we mean the evaluation of the speed of the sound waves and the background scalar derivatives at the horizon 9 From the quasinormal mode perspective, the z i 's are spatially SO(3) invariant background scalar fluctuations of the sound channel which decouple from the metric fluctuations as q → 0.
10 Recall that we refer to the physics as being effective two-derivative at the horizon if there is no distinction between the Bekenstein and the Wald entropy densities, see (A.28). in (2.40) to order O(β).Again, the solid curves represent the corrections 11 to the bulk viscosity from (2.17) (the left panel) and (2.18) (the right panel), and the dashed green curves indicate corrections from the EO formula (2.40).Interestingly, there is a disagreement even for models belonging to class B (the right panel), which are effectively two-derivative in the bulk.One might wonder whether the comparison of the ratio ζ s , which is somewhat more universal as it is partly applicable to higherderivative theories (see (2.42)), would fare better.From [23], As we show in fig.7, this is not the case for the effective two-derivative in the bulk model B 2,3 : the solid curve represents the O(β) correction δ ζ s extracted from the quasinormal mode (2.10) analysis, the red dashed curve is obtained from (2.42), and the green dashed curve is the application of the EO formula (2.43).It should probably not come as a surprise that the EO formula for the bulk viscosity fails in higher derivative as well as in Gauss-Bonnet holographic models: the naive application of the EO formalism 11 The quasinormal mode analysis (2.10) validates these results.does not capture the Gauss-Bonnet coupling correction to the shear viscosity either [9].
In the holographic models A − C (1.3) doing a metric field redefinition removes the bulk higher derivative coupling constants α 1 and α 2 [9].However, in the presence of the scalar sector, as in (1.2), such a redefinition generates new higher derivative coupling constants in the scalar sector of the form R • (∂φ) 2 and R µν ∂ µ φ∂ ν φ.Because computing the shear viscosity involves only the metric fluctuations, see (A.30), such a redefinition does not affect it, and the final correction is universal for all these models, see (2.12).On the contrary, computing the bulk viscosity necessitates turning on the bulk scalar fluctuations, see (A.62).As a result, the bulk viscosity is different in models A 2,∆ , B 2,∆ and C 2,∆ even though all these models have the same value of α 3 = 1.This is shown explicitly in fig.8. Black curves represent A 2,∆ models, blue curves represent B 2,∆ models, and magenta curves represent C 2,∆ models.
In fig. 9 we present bulk viscosity corrections in the models D 4,3 (the left panel) and D 4,2 (the right panel).The solid curves show the bulk viscosity corrections extracted from the dispersion relation (2.10), and the dashed red curves represent (2.19).This is an excellent validation of our computational framework in holographic models with Weyl 4 higher derivative corrections.We conclude this section mentioning one of the numerous consistency tests we performed.The speed of the sound waves c s can be extracted from the dispersion relation of the sound channel quasinormal modes of the background black brane (2.10), or from the background black brane equations of state (2.11).We parameterize the speed of the sound waves as

Conclusion
In this paper we developed a novel framework for computing transport coefficients in holographic model with higher derivative corrections.This allowed us to produce compact expressions (1.5)-(1.8)for the shear and bulk viscosities in large classes of nonconformal holographic models with higher derivative corrections.We expect that these formulas would be useful in exploring conditions under which the shear viscosity [6]  or the bulk viscosity [17] bounds are violated.The explicit expressions for the Wald entropy density (A.28) would be useful in searches of stable holographic conformal order [41][42][43][44][45].Moreover, since holographic models with scalar fields have been used, depending on the choice of scalar potential, to generate a wide spectrum of temperature dependence for η, in addition to a non-zero ζ, our analysis is also useful for direct comparison to the physics of the QGP.In particular, our simple expressions for the shear and bulk viscosities in the presence of arbitrary scalars can facilitate holographic model building and guide the efforts to describe the behavior of the QGP near the deconfinement transition.
We demonstrated that there are particularly simple and universal expressions for the ratio ζ s , see (2.42), valid even in models with higher derivatives in the bulk, but effectively two-derivative physics at the horizon, specifically when there is no distinction between the Bekenstein and the Wald entropies of the gauge theory thermal state dual black brane horizon.We also explored the applicability of the Eling-Oz formula for the bulk viscosity [23], and demonstrated that its naive application fails even in effectively two-derivative holographic Gauss-Bonnet models.At this stage it is not clear to us how to extend [23] to capture theories with higher derivatives and whether that construction can be generalized in a simple way.Specific models discussed in section 2 can be of interest to phenomenological applications in heavy ion collisions.To facilitate these applications we recall the relations of some of the parameters used on the gravitational side of the holographic correspondence to the gauge theory observables: • If c and a are the two central charges of a gauge theory UV fixed point, The holographic coupling α 3 appears in models A − C. We are not aware of the simple relation for the other two coupling constants, α 2 and α 3 , in the models B 2,∆ and C 2,∆ .
• Assume for simplicity 12 that the UV fixed point is N = 4 SU(N c ) supersymmetric Yang-Mills theory with a gauge coupling g 2 Y M .Then, in models D 4,∆ , In this paper we focused on the first-order transport coefficients, i.e., η and ζ, of the hydrodynamics theory derivative approximation.Stability and causality of the Landau-frame hydrodynamics can be ensured including higher-order transport coefficients.Holographic computations of the second-order transport coefficients of conformal gauge theories were first done in [47,48] 13 , and finite 't Hooft coupling corrections were discussed in [50][51][52][53].It would be interesting, albeit challenging, to extend the computational framework proposed here to the analysis of these coefficients in holographic models with higher derivative corrections.We expect that such results will be sensitive to the holographic renormalization of the models, as well as to the details of the proper formulation of the variational principle, i.e., the precise expressions for the Gibbons-Hawking terms.
In the future, it would also be interesting to extend the results reported here to holographic models with conserved charges, and to capture the effects of a chemical potential.It is natural to wonder whether, in the presence of generic higher derivative terms, the conductivity can also be extracted from a radially conserved current, and thus entirely from the horizon of the geometry.Finally, it would be useful to extend our framework to magnetohydrodynamics, again in the presence of higher derivatives.
We leave these questions to future work.
with the higher derivative contributions in model (1.3) given by and in model (1.4) by From (A.3) we obtain the following equations of motion14 : We verified that the constraint (A.9) is consistent with the remaining equations to order O(β) inclusive.
On-shell, i.e., evaluated when (A.7)-(A.10)hold, the effective action (A.3) is a total derivative.Specifically, we find with the higher derivative terms δB given by (A. 13) In what follows we will need the entropy density s and the temperature T of the boundary thermal state.The temperature is determined by requiring the vanishing of the conical deficit angle of the analytical continuation of the geometry (A.1), 2πT = lim where to obtain the last equality we used (A.2).The thermal entropy density of the boundary gauge theory is identified with the entropy density of the dual black brane [54].Since our holographic model contains higher-derivative terms, the Bekenstein entropy s B , s B = lim must be replaced with the Wald entropy s W [55], i.e., s = s W .The simplest way to compute the Wald entropy density is instead to use the boundary thermodynamics: According to the holographic correspondence [56,57], the on-shell gravitational action S 1 , properly renormalized [58], has to be identified with the boundary gauge theory free energy density F as follows, where we used (A.11).S GH is a generalized Gibbons-Hawking term [8], necessary to have a well-defined variational principle, and S ct is the counter-term action -we will not need the explicit form of either of these corrections.
Eq.(A.17) can be rearranged to explicitly implement the basic thermodynamic relation −F = sT − E between the free energy density F , the energy density E and the entropy density s [59]: Finally, we identify 15   sT Notice that to leading order O(β 0 ), using (A. 15 Strictly speaking, (A.19) is correct up to an arbitrary constant.But this constant must be set to zero from the comparison with thermal AdS, in which case the black brane geometry is dual to a thermal state of a boundary CFT with vanishing entropy in the limit T → 0.
We proceed to evaluate κ for our two holographic models (1.3) We need to evaluate (A.21) and (A.22) at the horizon, i.e., as r → r h .It is convenient to fix the residual diffeomorphism in (A.1) as16 where {f, g} = {f, g}(r).Given (A.7)-(A.9),to leading order in β, From (A.2), the horizon is located at r h , such that lim Regularity of φ i at the horizon then implies from (A.10) that17 It is convenient to use the idea of the complexified effective action for the fluctuations introduced in [18].This complexified action is a functional of h 12,w (r) and h * 12,w (r) ≡ h 12,−w (r) On-shell, the effective action (A.35) can be re-expressed as a total derivative, with a current (A.37) A crucial observation originally made in [18] was that an analog of J w in two-derivative holographic models (no B i coefficients in (A.37)) has a radially conserved imaginary part, on-shell.It is straightforward to verify that this property holds, even in the presence of higher derivatives of the effective action: for infinitesimal θ-rotations.In [18] this conserved charge was interpreted as the radially conserved number flux of gravitons 18,19 .
Essentially following the discussion of [61], the retarded two point correlation function of the stress-energy tensor (A.29) has to be identified with the boundary limit (i.e., 18 See [18] for further discussion and related earlier work. 19It is an interesting open question as to why the quadratic action for the fluctuations has this peculiar property, (A.36).
Notice that in (A.The equations of motion for Z i completely decouple from the equations for the metric fluctuations, 0 = Z ′′ i + ln (A.66) Furthermore, we have

Figure 1 :
Figure 1: Corrections to the shear viscosity in models A − C due to the UV fixed point central charges c = a are universal, see (2.12).In the left panel we consider non-conformal gauge theories with a UV fixed point deformed by a dimension ∆ = 3 operator, CF T → CF T + m f O 3 , while in the right panel a UV fixed point is deformed by an operator of dimension ∆ = 2, CF T → CF T + m 2 b O 2 .Solid curves represent the corrections to the shear viscosity extracted from the shear channel quasinormal mode of the background black brane (2.10), while the red dashed curves are obtained applying (2.12).

Figure 2 :
Figure 2: Corrections to the shear viscosity in models D due to finite 't Hooft coupling corrections, see (2.13).In the left panel we consider non-conformal gauge theories with a UV fixed point deformed by a dimension ∆ = 3 operator, CF T → CF T + m f O 3 , while in the right panel a UV fixed point is deformed by an operator of dimension ∆ = 2, CF T → CF T + m 2 b O 2 .Solid curves represent the corrections to the shear viscosity extracted from the shear channel quasinormal mode of the background black brane (2.10), while the red dashed curves are obtained applying (2.13).

Figure 4 :
Figure 4: Numerical test of the conservation of the imaginary part of current (A.77) in model (A 2,∆=3 ) to leading order O(β 0 ) (the left panel), and to subleading order O(β) (the right panel).See (2.37) for more details.
we compare the leading O(β 0 ) (the left panel) and the subleading O(β) correction (the right panel) of the ratio of the bulk viscosity to shear viscosity using the formalism of section A.3 (the red dashed curves), and the same quantities obtained from the computation of the sound channel quasinormal mode of the background black brane (2.10).The difference between the solid and the dashed red curves is ∼ 10 −6 • • • 10 −4 %.

Figure 5 :
Figure 5: To order O(β 0 ) there is a perfect agreement between the ratio of bulk viscosity to the shear viscosity evaluated using our novel formula (2.41), shown in the solid curves, and the Eling-Oz expression (2.40), shown in the dashed green curves.

we compare ζ η 0
for A − D models with ∆ = 3 and ∆ = 2 (the green gashed curves) with the predictions (1.7) of the framework discussed in section A.3 (the solid curves):

Figure 6 :
Figure 6: At order O(β) there is a disagreement between the ratio of bulk viscosity to the shear viscosity evaluated using (2.17) (the left panel) and (2.18) (the right panel), shown in the solid curves, and the extension of the Eling-Oz formula (2.40) to order O(β), shown in the dashed green curves.

Figure 7 :
Figure 7: We compare O(β) correction to ζ s in the holographic model B 2,3 : solid curve is obtained from the quasinormal mode (2.10) analysis, the red dashed curve is obtained from (2.42), and the green dashed curve is the application of the EO formula (2.43).

Figure 8 :
Figure 8: Although models A − C have the same value of the higher derivative coupling α 3 = 1, and the coupling constants α 1 and α 2 can be removed by a metric redefinition, such a redefinition modifies the scalar sector of the model resulting in distinct bulk viscosity corrections.Black curves represent A 2,∆ models, blue curves represent B 2,∆ models, and magenta curves represent C 2,∆ models.

Figure 9 :
Figure 9: Bulk viscosity corrections in the D 4,3 model (the left panel) and in the D 4,2 model (the right panel).Solid curves represent corrections extracted from the sound wave channel quasinormal mode of the background black brane (2.10).The red curves are obtained from (2.19).

. 44 )
In fig.10we compare results for (β 1,0 ) 2 (the left panel) and β 1,1 (the right panel).Solid curves indicate data from the dispersion relation (2.10), and the dashed red curves are the corresponding results obtained from the equation of state (2.11) in the holographic model D 4,3 .

Figure 10 :
Figure 10: We parameterize the speed of the sound waves in the gauge theory plasma as in (2.44).The solid curves represent (β 1,0 ) 2 and β 1,1 in the holographic model D 4,3 obtained from the sound wave channel quasinormal mode (2.10); the dashed red curves represent the same data obtained applying the equation of state (2.11).

∂ 38 )
r (J w − J −w ) The conservation law (A.38) is a direct consequence of the exact U(1) symmetry of L C (A.31) that rotates the phase of fluctuations, namely h 12,w → e iθ h 12,w and h 12,−w → e −iθ h 12,−w .The conserved Noether charge associated with this symmetry is precisely ImJ

r→r h H 1
(r) = finite .(A.46) Using the black brane background equations of motion (A.7)-(A.10),the equation for the fluctuations (A.43), and (A.44), we can evaluate the O(w) part of J w from (A.37)
All the terms in the bracket {} above are O(β), and the equation (A.43) can reduced to a second order equation for h 12,w eliminating h ′′′′ 12,w , h ′′′ 12,w , h ′′ 12,w with B i connection coefficients using the O(β 0 ) equation of motion.We will need to solve the equation (A.43) in the hydrodynamic approximation, specifically to order O(w).