Holographic Gauss-Bonnet transport

We extend the computational framework of \cite{Buchel:2023fst} to analysis of shear and bulk viscosities in generic strongly coupled holographic Gauss-Bonnet gauge theories. The finite Gauss-Bonnet coupling constant encodes holographic plasma with non-equal central charges $c\ne a$ at the ultraviolet fixed point. In a simple model we discuss transport coefficients within the causality window $-\frac12\le \frac{c-a}{c}\le \frac 12$ of the theory.


Introduction and summary
Until the discovery of the gauge theory/gravity correspondence [2,3] it was challenging to compute basic transport coefficients -the shear η and the bulk ζ viscosities -of gauge theory plasmas.In fact, the first reliable computations, relevant for the quarkgluon plasma, were performed holographically in [4] for the shear viscosity, and in [5] for the bulk viscosity.For gravitational duals with multiple scalars, typical in topdown holographic models of strongly coupled gauge theories, the extraction of bulk viscosity involved computation of the dispersion relation of the sound waves 1 .Such computations remained technically challenging until the Eling-Oz (EO) paper [10], where the authors presented simple formulas for the viscosities that involved only the properties of the gravitational background, correspondingly the properties of the holographic thermal equilibrium state.The EO formula was extensively verified in [11,12], and shown to be equivalent to the standard holographic Kubo method in [13].
First holographic computations of the viscosities were performed in two-derivative gravitational duals, correspondingly in gauge theory plasma at infinitely large 't Hooft coupling constant λ = g 2 Y M N c , and for theories with the same central charges c = a at the ultraviolet fixed point.Leading-order corrections due to the finite 't Hooft coupling [14][15][16][17][18] and for theories with c = a [19][20][21][22], until recently, were model specific.
In [1] a unifying framework was presented for the computation of transport coefficients in higher-derivative holographic models.
We summarize now the results of [1] relevant to four-derivative gravitational duals.
Consider a five-dimensional theory of gravity in AdS coupled to an arbitrary number of scalars, described by where δL 2 denotes the four-derivative curvature corrections described by: and the coupling constant β • α 3 is related to the difference of the central charges of the UV fixed point as2 3) The shear viscosity to the entropy ratio is given by: and the bulk viscosity to the entropy ratio is given by: where ∂ i V ≡ ∂V ∂φ i .All the quantifies in (1.4) and (1.5) 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 appendix A. Since δL 2 is generically higher-derivative, its coupling constant β must be treated perturbatively.This is an important exception though: when the combination in (1.2) assembles into a Gauss-Bonnet term, which renders the full gravitational action (1.1) two-derivative.In this case the coupling constant β ≡ λ GB 2 can be finite, and is constraint by the causality as [23,24] In this paper we present extensions of (1.4) and (1.5) to holographic GB models.We find: valid for finite λ GB .The proof of (1.9) and (1.10) is a straightforward generalization of analysis in [1].We will omit the proof and instead refer the reader to a practical guide in applying these formulas in appendix A.
In [25] the authors studied holographic GB transport in models with a single bulk scalar field from entirely different perspective(s).As we report in section 2.1, our results (1.9) and (1.10), restricted to a single scalar models, fully agree with [25].An intriguing claim made in [25] was that the bulk viscosity to the entropy density obeys the unmodified EO formula: where the scalar field derivative with respect to the entropy density is evaluated for its black brane horizon value.This appears to contradict the claim made in [1] that the 'naive' EO formula is not applicable in holographic higher-derivative models.The resolution of this is as follows.The 'naive' EO formula used in [1] was not (1.11), but (1.12) For a two-derivative holographic model, (1.11) and (1.12) are identical because of the universality of the shear viscosity to the entropy density ratio in the supergravity approximation [26] The universality is lost once λ GB is nonzero (1.9), causing the discrepancy.We revisit the δL 2 models of [1] in section (2.2) and show that the bulk viscosity computed from (1.5) agrees with the EO formula (1.11), whenever the gravitational dual is effectively two-derivative at the horizon3 , but could be higher-derivative in the bulk.If the holographic dictionary requires the use of the Wald gravitational entropy [27], the agreement between (1.5) and (1.11) is lost.

Applications 2.1 A single bulk scalar model
As an application of holographic GB transport, consider a single scalar field model with a potential where β 2 is given by (A.10). (2. 2) The dashed red lines are the computations of the bulk viscosity using the EO formula (1.11).The red dots represent the holographic GB conformal shear viscosity to the entropy density ratios [19,20]: (2.3) Note that 4πη s ratio (the green curve) is always below the minimal conformal value 16 25 [20], even without the phase transition that decouples the microcausality (the UV property) of the theory from its hydrodynamic transport (the IR property) [28].
There is an excellent numerical agreement between the bulk viscosity results from (the blue curve in the left panel of fig.1).

EO formula in δL 2 models
In this section we revisit δL 2 models of [1] with respect to comparison with an inequivalent EO formula (1.11).We focus on4 A 2,3 , B 2,3 and C 2,3 models discussed there.In all these models the bulk scalar field is dual to a dimension ∆ = 3 boundary gauge theory operator.The sets of coupling constants α i in (1.1) are as follows: Model B is just the model of section 2.1 to the leading order in the GB coupling constant; model C has a peculiar property that while it is higher-derivative in the bulk, the corresponding black brane entropy density is given by the Bekenstein formula -it is effectively two-derivative in the vicinity of the horizon.Finally, model A is genuinely higher-derivative: the correct thermodynamic entropy of the boundary gauge theory thermal state must be identified with the Wald entropy of the dual black brane.As shown in fig.3, there is now an agreement for models B and C for the two methods of computing the bulk viscosity to the entropy density ratio: using (1.5) (which actually reduces to (1.9) for both models) and the EO formula (1.11).Of course, the agreement for model B is expected.The agreement for model C (and similar models of [1]) strongly suggests that the EO formula (1.11) is more widely applicable: it is correct in higher-derivative holographic models that do not distinguish between the Wald and the Bekenstein entropies of the dual black brane.Of course, the holographic dictionary requires to identify s = s W ald .This can be independently verified in model A by extracting the speed of the sound waves c 2 s from the dispersion relation (this computation was performed in [1]) and comparing the result with the sound speed extracted from the thermodynamics: In the right panel of fig. 4 we show δc 2 s from (2.4) (the solid black curve) and the same quantity extracted from (2.5) with the identification s = s Bekenstein (the blue dashed curve) and with the identification s = s W ald (the red dashed curve).
From (1.1) we obtain the following equations of motion: where we defined We verified that the constraint (A.5) is consistent with the remaining equations at finite β.
On shell, i.e., evaluated when (A. where s and T are the entropy density and the temperature of the boundary gauge theory thermal state.
For numerical analysis it is convenient to parameterize the GB coupling constant λ GB as It is also useful to adopt the radial gauge as coefficients of the fluctuations z i,0 , and their n horizon values z h i,0 .From (1.10), the ratio of the bulk viscosity to the entropy density is given by .22)

Figure 3 :
Figure 3: At order O(β) there is a perfect agreement between the ratio of the bulk viscosity to the entropy density evaluated using (1.5) (the solid black curves) and the extension of the EO formula (1.11) to order O(β) (shown in the red dashed curves), in holographic models which are effectively two-derivative at the horizon.

2 sFigure 4 :
Figure 4: Left panel: at order O(β) there is a disagreement between the ratio of the bulk viscosity to the entropy density evaluated using (1.5) (the solid black curve, which is independently verified in [1], using the sound wave attenuation computation), and the extensions of the EO formula (1.11) to order O(β), where the boundary gauge theory thermal state entropy density s is identified with the dual black brane Wald entropy s = s W ald (the red dashed curve), or when the boundary gauge theory thermal state entropy density is identified with the dual black brane Bekenstein entropy s = s Bekenstein (the blue dashed curve).Right panel: O(β) correction to the speed of the sound waves extracted from the sound wave dispersion relation (2.4) is represented by solid black curve; the computation of the same quantity from the thermodynamic relation (2.5) with s = s W ald is shown in the red dashed curve, and with s = s Bekenstein is shown in the blue dashed curve.This validates the holographic dictionary s ≡ s W ald .The agreement between (1.5) and (1.11) is lost for model A, see fig. 4. It is so when one either uses the dual black brane Bekenstein entropy density s = s Bekenstein in (1.11) (the dashed blue curve), or its Wald entropy density s = s W ald (the dashed 3)-(A.6) hold, the effective action (1.1) is a total derivative.Specifically, we find