Hydrodynamics of fundamental matter
aa r X i v : . [ h e p - t h ] D ec KUL-TF-09/27
Hydrodynamics of fundamental matter
Francesco Bigazzi a , Aldo L. Cotrone b , Javier Tarr´ıo c . a Physique Th´eorique et Math´ematique and International Solvay Institutes, Universit´eLibre de Bruxelles; CP 231, B-1050 Bruxelles, Belgium. b Institute for theoretical physics, K.U. Leuven; Celestijnenlaan 200D, B-3001 Leuven,Belgium. c Departamento de F´ısica de Part´ıculas, Universidade de Santiago de Compostela andInstituto Galego de F´ısica de Altas Enerx´ıas (IGFAE); E-15782, Santiago de Compostela,Spain. [email protected], [email protected], [email protected]
Abstract
First and second order transport coefficients are calculated for the strongly coupled N = 4 SYMplasma coupled to massless fundamental matter in the Veneziano limit. The results, includingamong others the value of the bulk viscosity and some relaxation times, are presented at next-to-leading order in the flavor contribution. The bulk viscosity is found to saturate Buchel’s bound.This result is also captured by an effective single-scalar five-dimensional holographic dual in theChamblin-Reall class and it is suggested to hold, in the limit of small deformations, for generic plas-mas with gravity duals, whenever the leading conformality breaking effects are driven by marginally(ir)relevant operators. This proposal is then extended to other relations for hydrodynamic coeffi-cients, which are conjectured to be universal for every non-conformal plasma with a dual Chamblin-Reall-like description. Our analysis extends to any strongly coupled gauge theory describing the lowenergy dynamics of N c ≫ ≪ N f ≪ N c homogeneously smeared D7-branes. Introduction
Hydrodynamic models provide a fairly accurate description of the large-time evolution afterthermalization of the QCD plasma produced at RHIC [1]. In this respect, a first principlecomputation of the transport coefficients is both relevant and challenging, due to the stronglycoupled nature of the system. Surprisingly, some features of the known transport coefficientsappear to be common to many plasmas of strongly coupled theories, including QCD. Theprototype example is the shear viscosity, whose ratio with the entropy density has, in theorieswith two-derivative gravity duals, a universal value which is quite close to the QCD one [2].This observation justifies a careful analysis of first and second order transport coefficients inthe class of four dimensional quantum field theories admitting a gravity dual.Hydrodynamic studies by means of the gauge/gravity correspondence have been focusingto-date on theories without fundamental flavors (apart from what concerns the universalshear viscosity [3, 4, 5, 6, 7]). Fundamental flavors are clearly relevant in the RHIC plasma[1, 8]. In this paper, the results for the flavor contribution to the bulk viscosity and to somerelevant second order transport coefficients (relaxation times, κ , κ ∗ ) of the strongly coupled N = 4 SYM plasma are presented.These results are easily extended to N = 1 plasmas describing the thermal low energydynamics of N c ≫ X (the N = 4 SYM case corresponds to X = S ). These areconformal plasmas without fundamental fields. Flavors are added by means of N f ≫ X in the space transverse to the D3-branes [9, 5, 10, 7]. We will just focus onthe case in which all the flavors are massless; the related non-Abelian flavor symmetry groupis explicitly broken into a product of Abelian factors, due to the smearing. The addition ofmassless flavors induces a breaking of conformal invariance at the quantum level.The reason to analyze these theories stems from the fact that, despite being plagued bya Landau pole in the UV, they are the simplest examples where thermal flavor effects atstrong coupling can be reliably studied, providing the first manageable toy models for theQCD plasma in its near conformal regime.Our results are obtained at next-to-leading order in a perturbative expansion in ǫ h ∼ λ h N f /N c where λ h is the ’t Hooft coupling at the energy scale fixed by the plasma tempera-ture. The precise coefficients defining ǫ h depend on the model and hence on the volumes ofthe X and X spaces. For the N = 4 case, for example ǫ h = 18 π λ h N f N c . (1.1)Within this perturbative approach, the flavors can be considered as “deformations” of theconformal plasmas. At zero temperature, the flavor superpotential term, which drives the2reaking of conformality, can be treated as a marginally irrelevant “deformation” [10], ac-cordingly.At second order in ǫ h the gravity solutions derived in [7] provide a completely reliabledescription of the dual field theories in the planar limit and at strong coupling, by accountingfor the backreaction of the branes supporting the flavor degrees of freedom. Thus, they allowfor the calculation of the transport coefficients by means of standard procedures.The main results are collected in the following subsection. A review of the relevant back-grounds can be found in section 2, the main steps of the calculations are reported in section 3and further details are given in appendix A. In section 4 we will provide an alternative simplecalculation for the bulk viscosity, using the arguments presented in [11]. In particular we willshow how an effective single-scalar five-dimensional holographic model in the Chamblin-Reallclass [12] captures the leading conformality breaking effects due to the marginally irrelevantflavor “deformations”. We will also show that the same approach can be successfully appliedto cascading plasmas - where conformality breaking is driven by marginally relevant oper-ators - at leading order in the perturbative expansion introduced in [13]. We will suggest,in turn, that in the limit of small “deformations”, there are certain universal relations fortransport coefficients of gauge theory plasmas (with gravity duals) where the leading con-formality breaking effects are driven by marginally (ir)relevant operators. Our proposal isa natural extension of the results in [14], valid for relevant or exactly marginal deforma-tions. Finally, we will suggest possible universal relations involving the bulk viscosity andthe interaction measure. As we report in the following subsection, the results of section 4allow us to propose a class of universal relations for every non-conformal plasma with a dualChamblin-Reall (effective) description. Up to second order, the hydrodynamic expansion of a non-conformal plasma is known tobe determined by two first order transport coefficients, i.e. the shear and bulk viscosities,and thirteen second order transport coefficients [15], the most important ones being therelaxation times, which are relevant for numerical hydrodynamic simulations. We refer to[15, 16] for the notation of the transport coefficients.The bulk viscosity and a combination of the “shear” and “bulk” relaxation times τ π , τ Π canbe derived from the dispersion relation of the scalar hydrodynamic modes (sound channel)[17, 15] ω = c s q − i Γ q + Γ c s (cid:16) c s τ eff − Γ2 (cid:17) q + O ( q ) where Γ = ηsT (cid:18)
23 + ζ η (cid:19) . (1.2)In this equation, ω is the frequency of the mode and q its momentum; η and ζ are respectivelythe shear and bulk viscosities; s , T and c s represent the entropy density, temperature and3peed of sound of the plasma. Finally, τ eff is an “effective relaxation time” which fornon-conformal plasmas is the combination τ eff = τ π + ζ η τ Π ζ η . (1.3)The second order coefficient κ and a combination of τ π and κ ∗ can be derived from theretarded correlator of the tensorial mode [17, 15] G xy,xyR = p − iηω + (cid:16) ητ π − κ κ ∗ (cid:17) ω − κ q + O ( q , ω ) , (1.4)where p is the pressure.In [7] the following quantities were obtained T = T (cid:16) − ǫ h − ǫ h (cid:17) , p = p (cid:16) − ǫ h (cid:17) , ε − p = p ǫ h ,c s = 1 √ (cid:16) − ǫ h (cid:17) , ηs = 14 π , (1.5)where T and p = π N c T V ol ( X ) are the temperature and pressure of the unflavored conformalplasmas, and ε is the energy density.The new results in this paper, calculated up to O ( ǫ h ), are ζη = 19 ǫ h , (1.6) τ eff T = τ π, T + 16 − π π ǫ h , (1.7) T p κ = T p κ , (1.8) T p ( κ ∗ + ητ π ) = T p η τ π, + T p η (cid:16) τ π, −
16 + π πT (cid:17) ǫ h , (1.9)where τ π, T = 2 − log 22 π , T p κ = 1 π , T η p = 1 π , (1.10)are the corresponding values in the conformal plasmas [17, 18]. Let us comment these results in turn. The bulk viscosity (up to O ( ǫ h )) saturates the boundproposed by Buchel in [19], i.e. ζ /η ≥ / − c s ). In section 4 we obtain again this resultin the framework of [11, 14], showing that it is just the source of the marginally irrelevant This terminology is borrowed from [16]. In the conformal case τ π ≡ τ π, = τ eff , since ζ = 0. ζ /η , of order 10 − for ǫ h ∼ O (10 − ) [7], is quite small as comparedto the expected one in the near-conformal region of QCD, i.e. ζ /η ∼ O (10 − ) [20]. Thisis expected and due to the fact that the models at hand compute just the quantum flavoreffects and do not include the pure YM contribution to the trace anomaly, which is the mainsource to ζ /η in QCD.The result for κ is usually given in the conformal case in the combination T κ /s = T κ / p . In the present case it reads T κs = T κ s (cid:16) − ǫ h (cid:17) . (1.11)As such, all the results (1.6)-(1.9) tell us that the leading corrections to the conformal valuesof dimensionless combinations of transport coefficients consist of a (coefficient dependent)constant times a common function of the temperature, which in the present case is ǫ h ∼ log − T . This is in agreement with an extension of the results presented in [14] to cases withmarginally (ir)relevant deformations. In view of the general behavior discussed in section4, it would not be surprising if the numerical coefficients in (1.6)-(1.9), written in terms of,say, δ s ≡ − c s using eq. (1.5), turned out to be universal in all the marginally (ir)relevantdeformations of 4d conformal theories with two-derivative gravity duals.Unfortunately, τ π , τ Π and κ ∗ cannot be disentangled in the computations reported in thispaper. Nevertheless, the results in (1.7), (1.9) give very interesting indications. Comparing(1.3) with (1.7) and (1.9) using the notation τ π = τ π, + τ π, ǫ h + τ π, ǫ h , τ Π = τ Π , , κ ∗ = κ ∗ + κ ∗ ǫ h + κ ∗ ǫ h , (1.12)where in each of these expressions it is understood that we neglect higher order terms in ǫ h ,it follows that τ π, = 18 τ π, , τ π, + τ Π ,
12 = 17128 τ π, + 16 − π πT , (1.13) κ ∗ = κ ∗ = 0 , τ π, + κ ∗ η = 19384 τ π, −
16 + π πT . (1.14)In section 4 we will note that some theories where conformal invariance is broken bymarginally (ir)relevant operators are effectively described, for what concerns the bulk vis-cosity and speed of sound and at leading order in the deformation, by dual Chamblin-Reall We thank A. Cherman, T. Cohen and A. Nellore for this observation. Since ζ/η ∼ ǫ h , only the zero-th order term in τ Π is relevant in our perturbative analysis. τ π , τ Π which reproduces exactly the result in(1.13). This means that the theories analyzed in this paper, where conformal invariance isbroken by marginally (ir)relevant operators, are effectively described, at least at third orderin the sound channel and at leading order in the deformation, by Chamblin-Reall models.Crucially, the latter include also the non-conformal theory in [22], where the conformaltransport coefficients determine all the others, in particular τ Π = τ π and κ ∗ = − κ c s (1 − c s )[15]. For the D3-D7 plasmas analyzed in this paper, postulating τ Π = τ π (i.e. τ Π , = τ π, )implies from (1.13), (1.14) that κ ∗ = − κ /
4, that is, using the value of the speed of soundin (1.5), precisely the relation above.Thus, ignoring the possibility of a mere coincidence, we are led to conjecture that forall the theories effectively described by a Chamblin-Reall model, the transport coefficientssatisfy the relations reported in [15]: κ ∗ = − κ c s (1 − c s ) , τ ∗ π = − τ π (1 − c s ) , λ = 0 ,ζ = 2 η − c s ) , ζ τ Π = 2 η − c s ) τ π , (1.15) ξ = λ − c s ) , ξ = 2 ητ π c s − c s ) , ξ = λ − c s ) , ξ = 0 . In turn, the relations (1.15) allow to make a prediction for all the second order transportcoefficients, apart from λ , λ , λ , up to O ( ǫ h ) for the D3-D7 plasmas (using the same notationas in (1.12)): κ ∗ = κ ∗ = 0 , κ ∗ = − κ , τ ∗ π, = τ ∗ π, = 0 , τ ∗ π, = − τ π, , λ = 0 ,τ Π , = τ π, , ξ , = ξ , = 0 , ξ , = λ ,
18 = η πT , (1.16) ξ , = ξ , = 0 , ξ , = η τ π, , ξ , = ξ , = 0 , ξ , = λ ,
18 = 0 , ξ = 0 . The results in section 4 suggest that similar predictions can be made for the cascadingplasmas. Note that τ Π = τ π would imply that τ eff = τ π . Moreover, from (1.7) it wouldfollow that τ π T > τ π, T at order O ( ǫ h ). So, these results would support the conjecturein [16] that τ π, T is a lower value for the relaxation times in theories with two-derivative gravity duals. One could also conjecture, analogously, that κ constitues an upper boundfor the same class of theories. We thank T. Springer for pointing out this agreement. Beyond two derivatives, as for the shear viscosity, while finite coupling corrections enhance the relaxationtime [23], finite N c corrections reduce it [24]. Similar bounds on the speed of sound were proposed in [25], [26], and a possible universal relation amongsome second order coefficients were proposed in [27], [28]. τ Π ζ ≥ (cid:16) d − c s (cid:17) τ π η , (1.17)where d is the number of spatial directions of the plasma ( d = 3 for the models consideredin this paper). Again, analogous bounds could be proposed from the relations (1.15). It isnecessary to underline that these proposals are not based on first-principle arguments andthey would clearly require further, more solid, confirmations.In conclusion, the flavor effects are found to enhance both the bulk viscosity (1.6) (simplybecause, breaking conformal invariance, they produce a non-zero ζ ) and the “effective relax-ation time” (1.7), while they reduce κ in the usual combination (1.11) but do not modifyit in the combination (1.8). Finally, breaking conformality, the fundamental flavors shouldgive non-trivial coefficient τ Π , κ ∗ (1.13), (1.14). Unfortunately their values cannot be reli-ably disentangled from τ π with the present gravity analysis, but we believe they satisfy therelations (1.15), which would imply that τ π is enhanced by flavor effects. In this section we summarize the results obtained in [7] for the non-extremal gravity solutiondescribing an intersection of two sets of N c ≫ ≪ N f ≪ N c D7-branes,where the backreaction of both stacks is taken into account. When no D7-branes are presentthe solution is
AdS × X with a black hole. The presence of backreacting D7-branes intro-duces a squashing in the internal X (described as a U (1) fibration over a four-dimensionalK¨ahler-Einstein (KE) base) and the Einstein frame metric reads ds = h − / (cid:2) − b dt + dx i dx i (cid:3) + h / (cid:2) S F b − dr + r (cid:0) S ds KE + F ( dτ + A KE ) (cid:1)(cid:3) . (2.1)Here dA KE / J KE is the Kahler form of the four-dimensional base of X . Moreover h = R r , b = 1 − r h r , R ≡ Q c g s α ′ N c (2 π ) V ol ( X ) , (2.2)where the subscript h stands for “horizon” and F = 1 − ǫ ∗ (cid:18) r − r h r ∗ − r h (cid:19) + ǫ ∗ (cid:18) −
949 2 r − r h r ∗ − r h + 59 (2 r − r h ) (2 r ∗ − r h ) + − r h ( r ∗ − r )(2 r ∗ − r h ) −
48 log rr ∗ (cid:19) , At least one of them is non-zero. With respect to [7]’s notation we have removed the tildes from ˜
F , ˜ S . = 1 + ǫ ∗ (cid:18) − r − r h r ∗ − r h (cid:19) + ǫ ∗ (cid:18) − r − r h r ∗ − r h + 59 (2 r − r h ) (2 r ∗ − r h ) + − r h ( r ∗ − r )(2 r ∗ − r h ) + 48 log rr ∗ (cid:19) , (2.3)up to second order in the perturbative expansion parameter ǫ ∗ ≡ ǫ ( r ∗ ), where ǫ ≡ Q f e Φ ≡ V ol ( X )16 πV ol ( X ) λ N f N c , (2.4)and λ ≡ πg s N c e Φ ; here V ol ( X ) is the volume of the three-dimensional compact mani-folds wrapped by the homogeneously smeared D7-branes. For the N = 4 plasma we have V ol ( X ) = π , V ol ( X ) = 2 π .The D7-brane sources induce a running of the dilaton Φ( r ), which readsΦ = Φ ∗ + ǫ ∗ log rr ∗ + ǫ ∗ (cid:20) − r − r h r ∗ − r h + 12 log rr ∗ + 36 log rr ∗ ++ 92 (cid:18) Li (cid:18) − r h r (cid:19) − Li (cid:18) − r h r ∗ (cid:19)(cid:19)(cid:21) . (2.5)The non-trivial RR field strengths on the background are F (5) = Q c (1 + ∗ ) ε ( X ) , F (1) = Q f ( dτ + A KE ) ⇒ dF (1) = 2 Q f J KE , (2.6)where ε ( X ) is the volume element of the internal space.As mentioned above, at order zero in ǫ ∗ the solution is the standard AdS × X blackhole. The UV cutoff Λ ∗ , corresponding to the radial position r ∗ , is ensured to be well belowthe Landau pole Λ LP (mapped, in turn, to the position r = r LP at which the exact solutionfor Φ( r ) blows up) if ǫ ∗ ≪
1, and thus represents the scale up to which the solution isunder control. At Λ ∗ a UV completion is needed. We focus here on the IR properties ofthe system, such that we can safely discard contributions coming from the UV completion,which are power-like terms suppressed as (Λ IR / Λ ∗ ) n . Thus, in the computations below wesystematically discard the (subleading) corrections in r h /r ∗ . In this regime the previousexpressions for F , S and Φ get the simpler form F = 1 − ǫ h
24 + 171152 ǫ h − ǫ h
24 log rr h ≡ F h − ǫ h
24 log rr h ,S = 1 + ǫ h
24 + 1128 ǫ h + ǫ h
24 log rr h ≡ S h + ǫ h
24 log rr h , Φ = Φ h + ǫ h log rr h + ǫ h rr h + ǫ h rr h + ǫ h Li (cid:18) − r h r (cid:19) . (2.7)Notice that, being the temperature T proportional to r h at leading order [7], one has ǫ h = ǫ ∗ + ǫ ∗ log r h r ∗ , T dǫ h dT = ǫ h , (2.8)8hich implies that while ǫ runs with the energy scale (and thus with the temperature), ǫ isconstant if we neglect ǫ and higher order terms. The interaction measure ( ε − p ) /T andthe shifted speed of sound c s − / ǫ h as an effect of quantum conformality breaking due to the dynamical masslessflavors [7]. Those observables are thus independent on the temperature at next-to-leadingorder in ǫ h . We will come back to this observation in section 4. In the following we will study hydrodynamic transport properties of the plasmas describedabove. We first rewrite our ten-dimensional model as an effective five-dimensional one byintegrating over the internal manifold X . The five-dimensional reduction of the action reads[10] S = V ol ( X )2 κ Z d x p − det g (cid:20) R [ g ] −
403 ( ∂f ) − ∂w ) −
12 ( ∂ Φ) − V (Φ , f, w ) (cid:21) , (2.9)where g mn is the five-dimensional metric, R [ g ] is its scalar of curvature and the potentialdescribing the interactions of the three scalars reads V (Φ , f, w ) = 4 e f +2 w (cid:0) e w − Q f e Φ (cid:1) + 12 Q f e f − w +2Φ + Q c e f . (2.10)Hereafter we set R = 1 and α ′ = 1 for simplicity. In these units Q c = 4 as can be read from(2.2).To get the above results one starts from a reduction ansatz of the form ds = e f g mn dx m dx n + e − f + w ) ds KE + e w − f ) ( dτ + A KE ) . (2.11)On the background (2.1) we have f = −
15 log (cid:0) S F (cid:1) , w = 15 log ( F/S ) . (2.12)The five-dimensional non-extremal background metric reads ds = r e − f [ − b dt + dx i dx i ] + e − f b − dr r ≡ − c T dt + c X dx i dx i + c R dr . (2.13)From the dimensional reduction we see that we have three scalar fields. In the perturbativeexpansion in ǫ ∗ our models (both in the extremal and in the non-extremal case) can be seenas “deformations” of the unflavored AdS × X (BH) solutions. The AdS background is aminimum of the potential at order zero in ǫ ∗ , in which case f = 0 and w = 0. The field f is dual to an irrelevant operator of dimension ∆ = 8 whose form is Tr F . It drives a9eformation from the AdS × X to the non-near horizon D3-brane background. The field w is dual to a vev for an irrelevant operator of dimension ∆ = 6. It is of the form Tr( W α W α ) and is the responsible of the squashing of the transverse four-dimensional Kahler-Einsteinbase and the fibration. The dilaton Φ is dual to the insertion of a marginally irrelevantoperator, actually the flavor term in the field theory ( T = 0) superpotential [10]. This is thesource term which is responsible for the breaking of conformal invariance at the quantumlevel.Notice that the action (2.9), and so the equations of motion for some of the perturbations,coincides with the one in [29] (but for the definition of the potential). The equations of motionfor the fluctuations not coupling to the scalar fields (that is, insensitive to the potential) canbe read in [30]. Following the standard procedure [31], we will consider fluctuations of the fields present inthe five-dimensional action. It is easily seen from the expansion of the DBI action thatthe perturbations considered in the following do not couple to those of the flavor branesat the linearized level, thus the relevant set of fluctuations we need to consider is Ψ( r ) → Ψ( r ) + δ Ψ( x µ , r ), with Ψ = { g mn , Φ , f, w } .We will assume that the perturbations take a planar wave form in Minkowski space, δ Ψ( x µ , r ) = e − i ( ωt − qz ) Ψ( r ). Thus, these fields can be classified according to their trans-formation under the little group SO (2), which is the remaining symmetry of the system(rotations in the x − y plane). We define δg tt ( x µ , r ) = e − i ( ωt − qz ) c T ( r ) H tt ( r ) , (3.1) δg mn ( x µ , r ) = e − i ( ωt − qz ) c X ( r ) H mn ( r ) , ( m, n ) = ( t, t ) (3.2) δ Φ( x µ , r ) = e − i ( ωt − qz ) ϕ ( r ) , (3.3) δf ( x µ , r ) = e − i ( ωt − qz ) B ( r ) , (3.4) δw ( x µ , r ) = e − i ( ωt − qz ) C ( r ) , (3.5)choosing the gauge H rm ( r ) = 0. The classification of the different channels gives a tensorialmode ( H xy ), vectorial modes ( H tx , H zx , H ty , H zy ) and scalar modes ( H tt , H ⊥⊥ ≡ H xx + H yy , H zz , H tz , ϕ , B , C ). Each kind of perturbation can be expressed in term of gauge invariantquantities under the residual gauge symmetry [31, 29, 30]Tensorial → Z T = H xy , Vectorial → Z V = q H tx + ω H zx , Scalar → Z S = 2 H zz + 4 qω H tz − (cid:20) − q ω c ′ T c T c ′ X c X (cid:21) H ⊥⊥ + 2 q ω c T c X H tt , ϕ = ϕ − Φ ′ log ′ [ c X ] H ⊥⊥ , Z B = B − f ′ log ′ [ c X ] H ⊥⊥ , Z C = C − w ′ log ′ [ c X ] H ⊥⊥ . In all the cases, studying the differential equations at the horizon we find that the solutionsbehave as ( rr h − ± i w T T , where w = ω/ (2 r h ) and r h = πT . We choose the index withnegative sign to have incoming boundary conditions at the black hole horizon. Moreover, asin [32], we impose that the fluctuations vanish at the UV cutoff scale related to r ∗ : crucially,our results will turn out to be independent of r ∗ up to suppressed terms in powers of r h /r ∗ .In the following we are going to study the tensorial and scalar perturbations, while thestudy of the vectorial perturbation is relegated to appendix A. We scale w → λ hyd w , q → λ hyd q , where q = q/ (2 r h ) and λ hyd is a parameter keeping trackof the order of the hydrodynamic expansion. Define Z T = C T (cid:18) − r h r (cid:19) − i w T T X j =0 2 X k =0 Z j,kT λ jhyd ǫ k ∗ , (3.6)where higher order terms in ǫ ∗ and λ hyd , which we will not study, are not taken into account.The equation satisfied by the perturbation is reported in appendix A. As expected, Dirichletconditions cannot be imposed, showing the absence of a dispersion relation in this channel[31]. However one can write the hydrodynamic expansion of the retarded correlator. Todo this we have to evaluate the action on-shell. This action is singular when evaluatedat r = r ∗ → ∞ (it goes as r ∗ ). To cure this divergence we have to add the followingcounterterms [33] S bulk → S bulk + V ol ( X )2 κ Z d ξ √− γ K + V ol ( X )2 κ Z d ξ √− γ (cid:16) W [ φ ] − C [ φ ] R [ γ ] (cid:17) , (3.7)where K is the scalar associated to the extrinsic curvature, γ is the four-dimensional metricat the boundary, C [ φ ] is a function of the scalars and W [ φ ] is the superpotential W [ f, w, Φ] = e f (cid:2) Q c e f + Q f e f − w +Φ − e f +6 w − e f − w (cid:3) , (3.8)from which the potential (2.10) can be derived as V = 12 " (cid:18) ∂ W ∂f (cid:19) + 140 (cid:18) ∂ W ∂w (cid:19) + (cid:18) ∂ W ∂ Φ (cid:19) − W . (3.9)11he function C [ φ ] satisfies the differential equation [33]12 − (cid:20) ∂ W ∂f ∂C∂f + 140 ∂ W ∂w ∂C∂w + ∂ W ∂ Φ ∂C∂ Φ (cid:21) + 112 C W = 0 . (3.10)Although we do not know the exact form of C [ φ ], we can extract physical results, sincethe divergence balanced by this function goes in the UV as r ∗ , being the next-to-leadingorder suppressed as r − ∗ , i.e. , it does not affect the finite part from which the hydrodynamictransport coefficients are obtained. For completeness we give its leading behavior, needed tocancel the divergence C [ f, w, Φ] ≈ ǫ ∗ − ǫ ∗ + O (cid:0) r − (cid:1) . (3.11)The Fourier transformed, quadratic-in-fluctuations, on-shell boundary action is S = V ol ( X )2 κ Z d kH − k F ( k, r ∗ ) H k , (3.12)with H k the boundary value of the fluctuation. The retarded correlator of the correspondingcomponents of the energy momentum tensor G xy,xyR ( ω, q ) = − i Z dtd xe i ( ωt − qz ) Θ( t ) h [ T xy ( t, ~x ) , T xy (0 , ~ i , (3.13)is related to the on-shell action by [34] G xy,xyR ( ω, q ) = − F ( k, r ∗ )] . (3.14)Plugging the solution of the equation of motion for Z T , equation (A.1), in the finite action(3.7), it is straightforward to derive the flux F ( k, r ∗ ), and so the correlator G xy,xyR ( ω, q ).Using also (2.8), we get G xy,xyR = π N c T V ol ( X ) (cid:16) [1 − i w − q + 2 w (1 − log 2)] − i w + 2 q − w (1 − log 2)4 ǫ h + −
24 + 19 i w − q + 2 w (2 + 3 π + 22 log 2)192 ǫ h (cid:17) , (3.15)which, compared to (1.4) and using the expression for the temperature in (1.5), confirmsthe values of the pressure p and shear viscosity η from [7] and gives the new results in (1.8),(1.9). 12 .2 Scalar perturbations Focusing now on the scalar fluctuations we write for each perturbabion Z A = S, B , C ,ϕ the ansatz w = X k =0 c s,k ǫ k ∗ q − i X k =0 γ k ǫ k ∗ q + 4 X k =0 t k ǫ k ∗ q , (3.16) Z A = C A (cid:18) − r h r (cid:19) − i w T T X j =0 2 X k =0 Z j,kA q j ǫ k ∗ , (3.17)where γ k , t k are the order ǫ k ∗ coefficients of the adimensional combinations γ ≡ πT Γ , t ≡ ( πT ) Γ c s (cid:16) c s τ eff − Γ2 (cid:17) . (3.18)Higher order terms in ǫ ∗ and q , which we will not study, are not considered.The relevant equations for the perturbations are reported in appendix A. The calculationcan be performed imposing regularity at the horizon. Once this is obtained, one can askfor Dirichlet conditions at the UV cutoff, r ∗ , eventually taking the limit r ∗ → ∞ . Herewe present only the results relevant for the physical observables. The solution is given inappendix A. With the Dirichlet condition at the boundary we find c s, = 1 √ , c s, = 0 , c s, = − √ , (3.19) γ = 16 , γ = 148 , γ = 17 −
16 log[ r ∗ r h ]768 ,t = 3 − √ , t = 3 − √ , t = 57 − π −
22 log 2 − − r ∗ r h ]2304 √ , which confirms the result for the speed of sound found with the thermodynamics in [7] and,using (1.2), (1.5) and (2.8), gives the new results reported in (1.6), (1.7). Let us consider a simple five-dimensional gravity model with a minimally coupled scalar field φ S = 12 κ Z d x p − det g (cid:20) R [ g ] −
12 ( ∂φ ) − V ( φ ) (cid:21) , (4.1)and assume that this model admits an AdS (black hole) vacuum - dual to a four-dimensional(thermal) CFT - when the scalar field is turned off. Considering thermal cases in which φ is13ual to a source for a relevant operator of asymptotic dimension 2 < ∆ <
4, the authors of[11] provided a simple expression relating the bulk viscosity of the dual field theory plasmawith the five-dimensional scalar potential: ζη = | h (0)11 ( φ h ) | (cid:18) V ′ ( φ h ) V ( φ h ) (cid:19) . (4.2)An analogous approximate expression for c s − / φ and φ h is the value of the field at the horizon. The coefficient h (0)11 ( φ h ) is determined by solving the equation of motion for the SO (3) invariant fluctuation H = H = H ≡ e iωt h ( φ ) of the three-dimensional spatial metric components at ω = 0. In the r = φ gauge, in which the five-dimensional radial coordinate is identified withthe background scalar field φ , the equation is given by h ′′ = (cid:20) − A ′ − A ′ + 3 Y ′ − b ′ b (cid:21) h ′ + b ′ b (cid:20) A ′ − Y ′ (cid:21) h , (4.3)where the functions A, Y enter the background metric as ds = e A [ − b dt + dx i dx i ] + e Y dφ b . (4.4)In order to solve eq. (4.3) one imposes regularity at the horizon (selecting only incomingwaves in the ω = 0 case) and the (normalized to fix the residual three-dimensional scalefactor) boundary condition h → AdS boundary.In [14] (see also [36]) it was shown that the coefficient h (0)11 ( φ h ) depends only on ∆ (and noton the details of the five-dimensional potential) at leading order in a perturbative expansionaround the conformal background (i.e. in a large T limit if the deformation related to φ isrelevant). The general results of [14, 36] can be eventually extrapolated to exactly marginaldeformations, in which case they consistently give trivial hydrodynamic coefficients (i.e. ζ = c s − / T ≫ T c , h (0)11 ( φ h ) → h (0)11 ( φ h ) = 1 to satisfy the boundary conditions) for This is the only non trivial first order hydrodynamic coefficient, since η/s = 1 / (4 π ) for any stronglycoupled plasma with a two-derivative gravity dual [3]. In fact the bulk viscosity is related, via Kubo formulas, precisely to the (low frequency limit of the) SO (3) invariant retarded Green’s function of the operator T + T + T , where T µν is the field theorystress-energy tensor. V = V e γφ , with V < γ constants. As we willsee in the following, the 5d duals to the D3-D7 plasmas effectively behave like the Chamblin-Reall models, at leading order in the ǫ -expansion around the conformal fixed point. As wewill show, the same applies to the cascading plasma [13] in an analogous “perturbative”regime. It is worth underlining that other relevant non-conformal plasmas [30, 22] have adual gravity description precisely given by a model in the Chamblin-Reall class [35, 11]. Weargue that this is the reason why the above mentioned systems (at leading “perturbative”order, in the case of the D3-D7 and the cascading plasmas) have so many common features(for example they all saturate the bulk viscosity bound proposed in [19]) despite havingdifferent microscopical content. Reducing the D3-D7 models to five dimensions (see eqs. (2.9), (2.10)) we have seen thatthere are three scalar fields in the action. However, only one of them, namely the dilaton, isdual to the source for the (marginally irrelevant) deformation driving our theories away fromconformality [10]. It is thus conceivable that the bulk viscosity, which is turned on whenconformality is broken, is primarily determined by the dilaton field in an expansion aroundthe
AdS -BH solutions.To better understand the role played by the various scalars, let us consider the quantity V φ ≡ (cid:18) V ′ ( φ h ) V ( φ h ) (cid:19) , (4.5)in three different cases, where we identify φ with one of the three fields entering in thepotential (2.10), taking the other two scalars fixed to their background values. At order ǫ h we get V f,w = 0 , V Φ = ǫ h . (4.6)This indicates that the bulk viscosity (and the speed of sound) can be determined in ourmodels by considering just the dilaton Φ as the “active” field in the game. The other twoscalars, f and w , do not contribute at leading order and they can be fixed to their backgroundvalues. Due to this observation we can immediately apply the recipes of [11], based on asingle-scalar five-dimensional model, to our cases.Let us define, at first order in ǫ h , a new radial variable φ ≡ Φ − Φ h = ǫ h log rr h ⇒ rr h = e φǫh , (4.7)from which we can re-express our five-dimensional metric (2.13), in the form (4.4) with A = φǫ h + ǫ h
24 (1 + φ ) + const , Y = ǫ h φ ) + const . (4.8)15rom these expressions we find that the term in h in (4.3) vanishes (up to O ( ǫ h ) terms).Just as for the Chamblin-Reall models [11], h = constant is thus a solution to (4.3) atleading order. This implies that h (0)11 ( φ h ) = 1 to satisfy the boundary conditions. As a result,Buchel’s bound on the bulk viscosity [19] is saturated. In fact we obtain ζη = V ′ ( φ h ) V ( φ h ) = ǫ h . (4.9)Notice in turn that c s − / − ǫ h /
18 (as computed in [7] and in the previous section) canbe expressed as c s −
13 = − V ′ ( φ h ) V ( φ h ) , (4.10)just as it happens in the Chamblin-Reall cases [35]. Many relevant five-dimensional gravity models, dual to well studied non-conformal fieldtheories, contain more than one scalar. A well known example is given by the cascadingconifold theory [38, 39] whose hydrodynamics has been studied in great detail by Bucheland collaborators (see for example [40]). This theory describes the low energy dynamicsof N regular and M fractional D3-branes on the conifold. When M = 0, the theory hasgauge group SU ( N ) × SU ( N ) and it is conformal with an AdS × T , dual [41]. Theaddition of fractional branes modifies the gauge group to SU ( N + M ) × SU ( N ) and breaksconformal invariance. The combination g − − g − of the gauge couplings, in fact, acquiresa logarithmic running with the scale g − − g − ∼ M log( µ/ Λ IR ). The marginally relevantoperator T rF − T rF is mapped to a massless scalar field in the dual five-dimensionalgravity description (see e.g. [42] for a complete scalar/operator map in the deformed conifoldtheory). This is actually the supergravity modulus arising from the integral of B over thetwo-cycle of the conifold. The other massless field, the dilaton, is dual to T rF + T rF and(differently from the D3-D7 cases examined above) is just a constant on the background at T = 0. The five-dimensional effective action reads S = V ol ( T , )2 κ Z d x p − det g [ R [ g ] − L kin − V ( f, w, Φ , K )] , L kin = 403 ( ∂f ) + 20( ∂w ) + 12 ( ∂ Φ) + 14 P e − Φ+4 f +4 w ( ∂K ) ,V ( f, w, Φ , K ) = 4 e f +2 w (cid:0) e w − (cid:1) + P e Φ+ f − w + 12 K e f . (4.11)In α ′ = 1 units, P ∼ g s M and K is proportional to the effective number of regular D3-branes, which is running with the scale if M = 0 [38]. At P = 0 the previous action has an AdS (BH) vacuum where (setting the
AdS radius to one) K = K ∗ = 4. On this background16 = f = w = 0. The fields f, w are mapped to irrelevant operators with ∆ = 8 , δ = P /K ∗ . This parameter has a non zero betafunction at O ( δ ): its logarithmic running with the temperature (from which it follows that δ ≪ T ≫ Λ IR [13]) can be thus neglected at O ( δ ). At this order and in ourunits, δ = P /
4. If we rewrite K = 4 + √ P χ we can see that the field χ has canonicallynormalized kinetic term and it is massless, around the AdS (BH) vacuum. This is preciselythe field dual to the
T rF − T rF operator mentioned above.In the non-extremal case, the functions f, w, Φ (resp. χ ) receive the first corrections at O ( δ ) (resp. O ( √ δ )). If we now consider the quantity V φ defined in (4.5) we get V f,w, Φ = 0 , V χ = 89 δ , (4.12)at leading order. Precisely as in the D3-D7 case, the only “active” scalar, for what con-cerns the leading perturbative contribution to the bulk viscosity, is the one related to themarginal (in this case marginally relevant) deformation which sources the breaking of con-formal invariance. It is possible to verify, as above, that h (0)11 ( φ h ) = 1 at leading order, suchthat ζη = V ′ ( φ h ) V ( φ h ) = 89 δ , (4.13)nicely reproducing the results found in [40] by means of the alternative analysis of thehydrodynamical pole in the stress-energy tensor. Notice, again, that the speed of sound c s − / − δ/ c s −
13 = − V ′ ( φ h ) V ( φ h ) , (4.14)just as it happens for the Chamblin-Reall models. From the above expressions we see thatBuchel’s bound is saturated at leading order, consistently with the results found in [40]. From the above results it is tempting to propose that ζη = V ′ ( φ h ) V ( φ h ) = 2 (cid:16) − c s (cid:17) , (4.15)in the vicinity of a conformal fixed point - i.e. at leading order in a “small” (resp. “large”) T “perturbative” expansion - for every four-dimensional plasma with a five-dimensional17wo-derivative gravity dual where conformality breaking is driven at the quantum level bya marginally irrelevant (resp. marginally relevant) operator. A common feature of thiskind of plasmas is that the perturbative expansion parameter δ runs as the inverse of thelogarithm of the temperature and hence β T [ δ ] ≡ T ( ∂δ/∂T ) ∼ ± δ . This in turn impliesthat β T [( ε − p ) /T ] is subleading w.r.t. ( ε − p ) /T : the latter (as well as c s − /
3) is thuseffectively constant at leading order.The relations (4.15), easily extended to d = 3 space dimensions (just replacing 1 / /d on the r.h.s), are precisely satisfied by other known non-conformal models [30, 22] whosedual gravity description is in the Chamblin-Reall class. The results of [14, 36], instead, showthat gravity duals of relevant deformations do not belong to this class, as eqns. (4.15) donot hold.A well studied example of a plasma where conformality breaking is driven by relevantdeformations is the N = 2 ∗ one [43]. In this case the conjectured bound in [19] is notsaturated, the perturbative expansion parameters are of the form δ = (cid:16) mT (cid:17) (4 − ∆) , so that β T [ δ ] = (∆ − δ , (4.16)and ( ε − p ) /T ∼ δ . This means that the interaction measure (as well as c s − /
3) is nota constant at leading order: β T [( ε − p ) /T ] ∼ δ is of the same order as ( ε − p ) /T . Thisin turn implies that the model does not have an effective Chamblin-Reall dual description.The above examples suggest that Buchel’s bound could be perturbatively saturated when-ever β T [( ε − p ) /T ] is subleading, i.e. that, more generically ζη − (cid:18) − c s (cid:19) ∼ f ( δ, λ, T ) sT (cid:20) T ∂∂T − (cid:21) ( ε − p ) , (4.17)for some model-dependent dimensionless function f ( δ, λ, T ). This proposal, modulo a slightmodification needed to accommodate the results collected above, is analogous to those pre-sented in [44] by means of exact sum rules and a certain assumption for the spectral densityof the trace of the stress energy tensor. Notice that our formula (4.17) is meant to applyto four-dimensional strongly coupled plasmas with massless flavors having a two-derivativefive-dimensional gravity dual, in a regime where any possible “deconfining temperature” T c can be neglected. Acknowledgments
We are grateful to Aleksey Cherman, Thomas Cohen, Javier Mas, Abhinav Nellore, AngelParedes, Alfonso V. Ramallo and Todd Springer for relevant observations. F. B. is sup-ported by the Belgian Fonds de la Recherche Fondamentale Collective (grant 2.4655.07), Using the results in [36] we find that f → (2 / − / (∆ − − ( π/
9) cot( π ∆ /
4) at large T for strictlyrelevant deformations of planar strongly coupled gauge theories with a gravity dual. In our D3-D7 models T c = 0 exactly, while T c ≪ T in the perturbative regime for the cascading plasma.
18y the Belgian Institut Interuniversitaire des Sciences Nucl´eaires (grant 4.4505.86) and theInteruniversity Attraction Poles Programme (Belgian Science Policy). A. C. is supported bythe FWO -Vlaanderen, project G.0235.05 and by the Federal Office for Scientific, Technicaland Cultural Affairs through the Interuniversity Attraction Poles Programme (Belgian Sci-ence Policy) P6/11-P. J.T. is supported by the MEC and FEDER (grant FPA2008-01838),the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042), Xunta de Galicia(Conselleria de Educacion, grant PGIDIT06PXIB206185PR, project number INCITE09 206121 PR) and by MEC of Spain under a grant of the FPU program. J. T. would like to thankthe Perimeter Institute for hospitality at early stages of this paper.
F. B. and A. L. C. would like to thank the Italian students, parents, teachers and scientistsfor their activity in support of public education and research.
A Details of the calculations
Tensorial perturbations
From the action (2.9) it follows that the equation satisfied by the tensorial perturbation Z T = H xy is Z ′′ T + log ′ (cid:16) c T c X c R (cid:17) Z ′ T + c R c T (cid:16) ω − q c T c X (cid:17) Z T = 0 . (A.1)With the ansatz (3.6) one can check that the only non-zero term in the solution normalizedto one at the horizon up to first order in λ hyd is Z , T = 1. At second order in λ hyd thesolution is too lengthy to be reported here but straightforward to obtain. Scalar perturbations
The equations for the scalar gauge invariant fluctuations are the relevant ones for the soundchannel, giving the dispersion relation in (1.2). They are just combinations of the equationsin [29]: H EOMzz + 4 qω H
EOMtz − (cid:16) − q ω c ′ T c T c ′ X c X (cid:17) H EOMaa + 2 q ω c T c X H EOMtt ++ (cid:16) ωq c X c T Ξ + 8 ω log ′ c X c T (cid:17) H EOMrt + Ξ H EOMrz , (A.2)0 = φ EOM − φ ′ B log ′ c X H EOMaa + ωc X [ c X c ′ X φ ′′ B + φ ′ B ( c ′ X − c X c ′′ X )] c ′ X ( q c T c ′ T c X + 2 q c T c ′ X − ω c X c ′ X ) (cid:16) H EOMrt + qc T ωc X H EOMrz (cid:17) . In these expressions, φ represents each of the scalars f, w, Φ (the form of their equation isthe same) and φ B their background value. Moreover, the notation φ EOM (and H EOMzz and H aa in [29] corresponds to our H ⊥⊥ .
19o on) stands for the corresponding equation for the scalar (and the fluctuation H zz and soon) in section 3 of [29]. For the coefficient Ξ we haveΞ = − qr h (2 q − ω ) rω [ q ( r h − r ) + 3 r ω ] − q r h ( q − ω )( r − r h ) rω [ q ( r h − r ) + 3 r ω ] ǫ ∗ . (A.3)We give here the solution to the non-zero fluctuations entering in the sound channel,satisfying the normalization at the horizon and Dirichlet conditions at the boundary in thecase of Z S Z , S = 1 ρ , Z , ϕ = log ρ − ρ ) , Z , ϕ = i √ − ρ ) h π ( ρ −
1) + 24( ρ −
1) log ρ −
12 log ρ (cid:16) ρ −
1) log (1 + iρ ) +( ρ −
1) log [ i ( i + ρ )( ρ − (cid:17) − ρ − Li ( ρ ) i , where ρ ≡ r/r h . We do not report the expressions for all the q coefficients of the solutionsbecause of their very lengthy form. Vectorial perturbations
The equation in this channel reads Z ′′ V + h log ′ (cid:16) c X c T c R (cid:17) − log ′ (cid:16) c X c T (cid:17)(cid:16) − q ω c T c X (cid:17) − i Z ′ V + c R c T (cid:16) ω − q c T c X (cid:17) Z V = 0 . (A.4)For the vectorial fluctuations we can solve order by order with the scaling w → λ hyd w , q → λ hyd q , imposing regularity at the horizon. The result is Z V = C V (cid:18) − r h r (cid:19) − i w T T (cid:20) r h r + (cid:18) − i q w (cid:19) (cid:18) − r h r (cid:19) (1 + ǫ ∗ + ǫ ∗ ) (cid:21) + O ( w , q ) . (A.5)From Dirichlet conditions at the boundary r ∗ → ∞ we can read off the shear viscosity fromthe dispersion relation ω = − i ηsT q + O ( q ) . (A.6)This calculation is summarized in the membrane paradigm formula given in [45], and it givesthe well-stated ratio η/s = 1 / (4 π ) with corrections in powers of r h /r ∗ → Scaling the frequency also with w → λ hyd w gives the same answer, as is the case in [31]. eferences [1] I. Arsene et al. 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