Quantum corrections to �=s

Robert C. Myers,1,2 Miguel F. Paulos,3 and Aninda Sinha1

1Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y52Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G13Department of Applied Mathematics and Theoretical Physics, Cambridge CB3 0WA, United Kingdom

(Received 25 June 2008; published 4 February 2009)

We consider corrections to the ratio of the shear viscosity to the entropy density in strongly coupled

non-Abelian plasmas using the AdS/CFT correspondence. In particular, higher-derivative terms with the

five-form Ramond-Ramond flux, which have been ignored in all previous calculations, are included. This

provides the first reliable calculation of the leading order correction in the inverse ’t Hooft coupling to the

celebrated result �=s ¼ 1=4�. The leading correction in inverse powers of the number of colors is

computed. Our results hold very generally for quiver gauge theories with an internal manifold Lpqr in the

holographic dual. Our analysis implies that any thermal properties, e.g., all hydrodynamic quantities in

these theories, will not be affected by the five-form flux terms at this order.

DOI: 10.1103/PhysRevD.79.041901 PACS numbers: 11.25.Tq, 12.38.Mh

The gauge/gravity correspondence presents a powerfulnew framework with which to study strongly coupledgauge theories [1]. One of the most striking new insightsis that the ratio of the shear viscosity � to the entropydensity s is universal with �=s ¼ 1=4�, for any gaugetheories with an Einstein gravity dual in the limit of a largenumber of colors and large ’t Hooft coupling, i.e., N, � !1 [2]. In fact, this result has been conjectured to be auniversal lower bound [the Kovtun-Son-Starinets (KSS)bound] in nature [3]. While this bound is far below theratio found for ordinary materials, a great deal of excite-ment has been generated by experimental results from theRelativistic Heavy Ion Collider suggesting that, just abovethe deconfinement phase transition, QCD is very close tosaturating this bound [4]. Certainly to make better contactwith QCD, it is important to understand 1=� and 1=Ncorrections which arise from higher-derivative correctionsto classical Einstein gravity in the holographic framework.For four-dimensional N ¼ 4 super-Yang-Mills (SYM),the leading 1=� correction has been calculated as [5]

�

s¼ 1

4�

�1þ 15�ð3Þ

�3=2

�: (1)

It has often been noted that with this correction added, theSYM theory no longer saturates the KSS bound, but giventhat the correction is positive, the bound is still respected.However, as we explain, the existing calculations produc-ing (1) are incomplete and so the precise coefficient andalso the sign of the correction could not be stated withcertainty.

In the above calculations, in order to introduce a finitetemperature, one considers the background of the AdS-Schwarzschild black hole, AdSBH �M5. In 10D, thefields involved are the metric and the five-form:

ds2 ¼ ð�TLÞ2u

ð�fdt2 þ dx2Þ þ L2du2

4u2fþ L2dS2M5

F5 ¼ � 4

Lð1þ ?ÞvolM5

; x ¼ ðx; y; zÞ:(2)

Here we have f ¼ 1� u2 and T is the temperature of theblack hole. L is the AdS radius of curvature with L4 ¼��02 and volM5

is the volume form on the compact five

manifold M5. Of course, the geometry dual to N ¼ 4SYM has M5 ¼ S5 but our discussion in the followingextends to more general internal manifolds.The coupling constant dependence of thermodynamic

quantities can be obtained by considering higher-derivative�0 corrections to the supergravity action and their effect onthe geometry (2). The first corrections in the IIB theoryarise at �03 relative to the supergravity action, which leads

to thermodynamic corrections suppressed by 1=�3=2. Thebest understood correction is the R4 term [6], which by afield redefinition can be written in terms of the Weyl tensorCabcd:

Sð3ÞR4 ¼ �

16�G

Zd10x

ffiffiffiffiffiffiffi�gp

e�ð3=2Þ ~�WC4

WC4 ¼ CabcdCebcfCagh

e Cdghf �

1

4CabcdC

abefC

ceghC

dfgh:

(3)

where � ¼ 18 �ð3Þ�03. We have written (3) in the Einstein

frame and in the limit of small string coupling. We have

also introduced notation for the dilaton: e� ¼ gse~�, where

gs is the string coupling and ~� is the deviation from theasymptotic value. The corrections to the thermodynamicsofN ¼ 4 SYM arising from this term were first studied in[7,8]. However, this C4 term is only one of a large collec-tion of terms at this �03 order. In particular, there are termsinvolving the five-form Ramond-Ramond (RR) flux whichcannot be ignored if one wishes to make rigorous state-

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ments on the SYM thermodynamics. As such, all existingresults in the literature are incomplete.

A few years ago, Green and Stahn [9] proposed that theIIB higher-derivative terms involving the graviton and thefive-form flux can be completely specified. The fields inIIB supergravity can be written compactly as an analyticsuperfield [10] whose first component is � ¼ aþ ie��.Higher-derivative terms at Oð�03Þ can be written as anintegral of functions of this superfield over half the super-space. In particular, [9] proposed a supersymmetric com-pletion of C4 in the case where only the five-form andmetric are present. This proposal was shown [11] to evadepotential problems in defining a supersymmetric measure[12]. Using this proposal, it was found [9] that the D3-brane solution in supergravity does not get renormalized bythe higher-derivative terms. This is a useful check since itis expected that the near-horizon limit AdS5 � S5 is asolution to all orders in �0 [13]. In contrast, the potentialpitfalls faced with incomplete calculations without thefive-form flux terms were demonstrated in [14], where anillustrative analysis with the C4 term alone produced anincorrect modification of the D3-brane solution. Moreconcretely, the correction determined by [9] is

Sð3ÞR4 ¼ �03g3=2s

32�G

Zd10x

Zd16

ffiffiffiffiffiffiffi�gp

fð0;0Þð�; ��Þ� ½ð�mnpÞð�qrsÞRmnpqrs�4 þ c:c:; (4)

where fð0;0Þð�; ��Þ is a modular form [15]. The latter cap-tures the behavior for all values of the string coupling butremarkably contains only string tree-level and one-loopcontributions, as well as an infinite series of instantoncorrections. The six-index tensor R is specified by [9,12]

R mnpqrs ¼ 1

8gpsCmnqr þ i

48DmF

þnpqrs

þ 1

384FþmnptuF

þqrs

tu; (5)

where Fþ ¼ 12 ð1þ �ÞF5. Note that only having this self-

dual combination is essential to the �0-corrected F5 equa-tion being cast as a self-duality constraint [16]:

F5 � 120�WR4

F5

¼ ��F5 � 120�

WR4

F5

�: (6)

Building on methods of [17], the specific tensor structureof the terms contained in (4) was computed in [18]. Theresult is a set of 20 independent terms which can beschematically written as

C4 þ C3T þ C2T 2 þ CT 3 þT 4 þ c:c:; (7)

where the tensor T is defined as

T abcdef � iraFþbcdef þ

1

16ðFþ

abcmnFþdef

mn

� 3FþabfmnF

þdec

mnÞ: (8)

Implicit above are antisymmetry in ½a; b; c� and ½d; e; f�, aswell as symmetry in the exchange of the triples, whicharises from the gamma-matrix structure in (4). The form of(4) also imposes a projection onto the irreducible 1050þrepresentation of rFþ

5 and Fþ25 . The particular combina-

tion of Fþ25 terms in (8) affects this projection, while for F5

satisfying the leading supergravity equation of motionF5 ¼ �F5, this projection reduces to the identity onrFþ

5 . To make (7) more explicit, the C4 term reproduces

(3) and the C3T term is written as [18]

WC3T ¼ 32CabcdC

aefgC

bfhiT

cdeghi þ c:c: (9)

We will now proceed to analyze the effect of these newterms on �=s. Of course, their effect in calculating theshear viscosity � and the entropy density s must be con-sidered separately. The latter requires determining theeffect of the higher order terms on the event horizon inthe background solution [7,19]. � may be calculated byperturbing the black hole background by a metric defor-mation h�� and either applying Kubo formulas to correla-

tors of the dual stress tensor [20] or examining thehydrodynamic form of the resulting stress tensor [21].Again, we want to consider how the �03 terms modifythis ‘‘deformed’’ solution. Hence, in the following, wefocus on how the higher order terms effect the equationsof motion. Our conclusion is that the relevant backgroundsreceive no �03 corrections from five-form terms.We begin with a general supergravity solution of the

form A5 �M5. Our discussion will be general and weonly assume that A5 and M5 are Einstein manifolds withequal and opposite curvatures. However, in our applicationbelow, we intend that A5 is an asymptotically AdS blackhole (with or without the deformation). Beyond the usualchoice of S5 for M5, our general discussion includes theEinstein-Sasaki manifolds Lpqr [22], of which Yp;q and

T1;1 are special cases. In the following, we denote the A5

and M5 coordinates as ai, mi, respectively. For consis-tency of the expansion in�03, we need only use this leadingorder supergravity solution in evaluating the contributionsof the higher-derivative terms.The key feature of the above background is that the five-

form solution remains as in (2), i.e., it is the sum of thevolume forms on A5 and M5. This five-form solutionimmediately implies that the tensor T vanishes. TherFþ

5 piece vanishes because the volume forms are cova-

riantly constant. The particular combination of Fþ25 terms

appearing in (8) also vanishes in this case. Since T ¼ 0,the only term other than C4 that could affect the equationsof motion is the C3T term, given in (9).To show that this term cannot affect the equations of

motion, we first note that the structure of (9) yields theform PðC3ÞPðFþ2

5 Þ, where P is the 1050þ projection op-

erator, whose action on Fþ25 was described above. On the

C3 factor, the projection operator yields

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PðC3Þabcdef ¼1

2ððC3Þabcdef � 3ðC3Þ½ab½fde�c�Þ: (10)

Now for our purposes, it suffices to consider the tensorWC3T =F5 ¼ PðC3ÞF5 evaluated on the background.First, since A5 and M5 are Einstein, the only nonzerocomponents of the Weyl tensor are Ca1a2a3a4 and

Cm1m2m3m4[23]. Now in WF5

=F5, PðC3Þ contracts intoeither F

a1a2a3a4a55 or F

m1m2m3m4m5

5 and since uncontracted

indices are antisymmetric, the only relevant componentsare PðC3Þa1a2a3a1a2a3 and PðC3Þm1m2m3m1m2m3

. However, us-

ing (10), a simple calculation explicitly shows that in factfor an arbitrary tensor M½abc�½def�, PðMÞ has no such com-

ponents. Therefore WC3T =F5 is always vanishing forthe relevant backgrounds. Given that PðC3ÞF5 ¼ 0 ¼ T , italso follows WC3T =gab vanishes. Hence, we may con-clude that only C4 alters the geometry at order �03. Wewish to emphasize the fact that the five-form did not affectthe equations of motion crucially depended on the tensorstructure of the higher-derivative terms. It is easy to thinkof generic terms C3F2

5 such that the self-duality condition

and the metric equations of motion would get corrected bythe five-form flux. Hence, it is clear that heuristic argu-ments presented in [5,7] for ignoring the five-form fluxterms are unreliable.

With this result, we may conclude that the F5 terms in(4) do not affect the ratio �=s. First of all, to consider �03corrections to the asymptotically AdS black hole in (2), theabove proof indicates that these arise only from theC4 term(3). Hence, we have rigorously established that the entropydensity for the SYM theory takes the form in [7,19]. Notethe observation that only the C4 terms correct the staticblack hole geometry was made in [18] but our discussionextends this result to more general M5.

Corrections to � can be considered with A5 beingasymptotically AdS black hole with an appropriate defor-mation h��. Here again, our proof indicates that only the

C4 term is relevant and so we have rigorously establishedthe finite coupling corrections appearing in (1). In fact, ourresults here have wider significance for general holo-graphic calculations of such transport coefficients. As re-cently explored [21], the hydrodynamics of the conformalplasmas are governed by a variety of higher order coeffi-cients as well as �. Our present discussion indicates that itis sufficient to consider the C4 term to calculate the �03corrections to all of these coefficients e.g., the relaxationtime [24]. We emphasize that the last statement includesthe coefficients for terms nonlinear in the local four veloc-ity, as the above discussion applies to the deformed blackhole fully nonlinear in h��.

One might also consider the universality of the correc-tions to transport coefficients proposed in [25]. There itwas found that the full spectrum of quasinormal modes(and hence implicitly the linear transport coefficients, suchas �) matches for M5 ¼ S5 and T1;1. While it is beyond

the scope of the present article to determine whether or notthis proposal is correct, the present discussion indicates atthe F5 terms at order �03 are irrelevant to determining thespectrum.Implicitly the above conclusions rely on these results not

being modified by new fields which are trivial in theoriginal background. In principle, other terms in the �03action beyond those captured in (4) might source other typeIIB fields. For definiteness, consider the RR axion a whichvanishes at lowest order. There might still be �03 termswhich are linear in a, e.g., C2rFþ

5 r2a. The corrected

solution would then also include an axion of order �03.However, in Einstein’s equations, a will only appear quad-ratically or in terms with an �03 factor. Hence, its effects in,e.g., the quasinormal spectrum will only be felt at order �06and therefore can be neglected here. The same reasoningcan be made for all other fields. In particular, it applies tothe warp factor which is induced at order �03 [5]. This caseis slightly different in that this field is a new correction tothe metric and RR five-form. However, the same reasoningcan be applied in the effective five-dimensional theory onA5, in which, to leading order, the warp factor appears as amassive scalar field with standard quadratic action.Holographic analyses are typically applied with both N,

� ! 1. So it is of interest to consider corrections for bothfiniteN and finite �. Higher-derivative corrections in stringtheory at perturbative level in string coupling gs areweighted by �0ng2ms with gs ¼ �=4�N and so one expectsthat �=s can be written as

�

s¼ 1

4�

�1þX

n¼3m¼0

cnm��n=2

���

4�Nc

�2m þ fNP

��; (11)

where the nonperturbative contributions fNP come from D-

instanton effects. Remarkably the modular form fð0;0Þð�; ��Þonly contains two perturbative terms [15]:

fð0;0ÞP ¼ �ð3Þ8

e�3�=2

�1þ �2

3�ð3Þ e2�

�: (12)

Since the dilaton only varies at Oð�03Þ, we simply replace

the term 15�ð3Þ=�3=2 in (1) by 15�ð3Þ=�3=2ð1þ �2=48N2Þas dictated by (12) to obtain

�

s¼ 1

4�

�1þ 15�ð3Þ

�3=2þ 5

16

�1=2

N2þ ~fNP

�: (13)

Examining the nonperturbative contribution in detail

yields ~fNP � 15=ð2�1=2N3=2Þe�ð8�2NÞ=� for small gs [15].Hence, these contributions are subdominant in the fullperturbative expansion (11). However, the second correc-

tion above is enhanced by a factor of �1=2 and so this is theleading order correction in inverse powers ofN forN ¼ 4SYM. The same enhancement appears in the leading 1=N2

correction to the entropy density [7]:

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s ¼ �2N2T3

2

�1þ 15

8

�ð3Þ�3=2

þ 5

128

�1=2

N2

�: (14)

One might be tempted to consider how these first twocorrections might modify �=s in QCD’s strongly coupledquark-gluon plasma. Substituting � ¼ 6� (i.e., �s ¼ 0:5)and Nc ¼ 3, they increase �=s from 1=4� � 0:08 to 0.11.The first and second terms produce roughly 22% and 15%increases, respectively. (Note the nonperturbative piececontributes �2� 10�7 with these parameter values.)

Another interesting case where the five-form termscould have contributed is in the calculation of �=s in thebackground dual to the boost-invariant plasma [26]. In thisbackground the tensor T is no longer vanishing. Indeed,PF2

5 is still zero, but now rF5 is of order �ðu; tÞ, the warpfactor on the five-sphere. Thus, one expects that the termsC2rF2

5 and ðrF5Þ4 will give a nonzero contribution (no

odd powers appear in the real part ofW ). However, in [27]it was shown that the field �ðu; tÞ is zero up to an irrelevantterm for the �=s computation. The equation of motion for� becomes (schematically)

r2�ðu; tÞ ¼ �ðOð1Þ þOð�ðu; tÞÞ þ � � �Þ;and the only relevant piece of the right is the Oð1Þ termscoming from C4. This means that the new terms do notcontribute to the sourcing of �ðu; tÞ, and further they willalso not change Einstein’s equations since �ðu; tÞ shows up

quadratically there and the above equation determines it tobeOð�03Þ. We conclude that the five-form terms containedin W cannot change the �=s computation in this frame-work. Hence, the agreement between this computation andthe equilibrium ones is preserved.Finally, note that the present discussion assumes a sim-

ple product geometry A5 �M5 and so only applies forthermodynamics in the absence of a chemical potential.The latter may be included by considering R-charge blackholes. However, in this case, the equations of motion areknown to be corrected by the F5 terms [18]. In fact, whilethe analysis using the only C4 term seems prohibitivelydifficult in this case, it was found that including the five-form produces a particularly simple result. Hence, it will bevery interesting to compute �=s in this context.

We thank Paolo Benincasa, Alex Buchel, MichaelGreen, Krishna Rajagopal, and Samuel Vazquez for usefuldiscussions. Research at Perimeter Institute is supported bythe Government of Canada through Industry Canada andby the Province of Ontario through the Ministry ofResearch & Innovation. R. C.M. also acknowledges sup-port from an NSERCDiscovery grant and funding from theCanadian Institute for Advanced Research. M. P. is sup-ported by the Portuguese Fundacao para a Ciencia eTecnologia, Grant No. SFRH/BD/23438/2005. M. P. alsogratefully acknowledges Perimeter Institute for its hospi-tality at the beginning of this project.

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