Hydrodynamic gradient expansion in gauge theory plasmas
HHydrodynamic Gradient Expansion in Gauge Theory Plasmas
Michal P. Heller ∗ Instituut voor Theoretische Fysica, Universiteit van Amsterdam,Science Park 904, 1090 GL Amsterdam, The Netherlands
Romuald A. Janik † and Przemys(cid:32)law Witaszczyk ‡ Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Krak´ow, Poland
We utilize the fluid-gravity duality to investigate the large order behavior of hydrodynamic gradi-ent expansion of the dynamics of a gauge theory plasma system. This corresponds to the inclusionof dissipative terms and transport coefficients of very high order. Using the dual gravity description,we calculate numerically the form of the stress tensor for a boost-invariant flow in a hydrodynamicexpansion up to terms with 240 derivatives. We observe a factorial growth of gradient contributionsat large orders, which indicates a zero radius of convergence of the hydrodynamic series. Further-more, we identify the leading singularity in the Borel transform of the hydrodynamic energy densitywith the lowest nonhydrodynamic excitation corresponding to a ‘nonhydrodynamic’ quasinormalmode on the gravity side.
Introduction.
Hydrodynamics is an effective theory offluids describing long-wavelength evolution of conservedquantities, among which are energy (local temperature)and momentum (local velocity). In the relativistic settingof uncharged hydrodynamics, which we consider here,one starts with the perfect fluid stress tensor and sys-tematically corrects it by including all available gradientterms (dissipative corrections) with their relevance de-creasing with the number of derivatives they carry. Eachgradient term enters with an a priori distinct function oftemperature – transport coefficient, e.g., the first ordersymmetric traceless contribution to the stress tensor isassociated with the shear viscosity.Transport coefficients are, in principle, derivable fromthe microscopic theory underlying the given fluid, beingit kinetic theory for fluids formed out of weakly coupledquantum fields or the gauge-gravity duality for certainstrongly coupled liquids. The exact values of transportcoefficients are, of course, important for the understand-ing of the dynamics of the particular fluid. A famousexample is the shear viscosity of quark-gluon plasma cre-ated in RHIC and LHC experiments, which needs to besufficiently small in order to account for the observedspectra of particles (see [1] for a recent review).Given the ubiquity and universality of the hydrody-namic description of many physical systems, it is inter-esting to investigate the character of the hydrodynamicgradient expansion and its interrelation with nonhydro-dynamic collective excitations or degrees of freedom.On a formal level, it is important to understandthe large-order behavior of any approximate scheme inphysics based on the existence of a small parameter suchas hydrodynamic gradient expansion. Given the com-plexity of typical perturbative calculations at large or-ders, the question we are asking is challenging enough todeserve attention and interest. In fact, no methods wereavailable so far to proceed effectively beyond the first fewterms in the hydrodynamic gradient expansion. The issues of high order hydrodynamics are also in-teresting from a phenomenological point of view, in thecontext of ongoing relativistic heavy ion collision pro-grams [1]. Two examples considered recently in the lit-erature are refining the criterium for the applicability ofhydrodynamics by including terms with more than twoderivatives [2, 3] and trying to resum hydrodynamic ex-pansion in order to account for situations in which indi-vidual gradient terms are not small enough to truncatethe series at lowest orders [4, 5].In this Letter, we provide strong evidence that the ra-dius of convergence of the hydrodynamic gradient expan-sion is zero. Furthermore, using the standard, in the con-text of divergent series, technique of Borel transform, weidentify the leading singularity in Borel-transformed hy-drodynamic stress tensor with the lowest lying nonhydro-dynamic degree of freedom. As a byproduct, we uncoveran intriguing structural similarity of the phenomena aris-ing in perturbative expansions in coupling constant inquantum systems with temporal evolution governed byhydrodynamic and nonhydrodynamic modes.As a way of generating a hydrodynamic stress ten-sor at high orders of gradient expansion, we utilize theAdS/CFT correspondence [6] and the fluid-gravity du-ality [7]. We obtain a gradient-expanded stress ten-sor for a particular solution of the equations of (all-order) relativistic hydrodynamics, generalizing the so-called Bjorken flow [8, 9] - a toy model of expandingquark-gluon plasma. The underlying microscopic theoryis a strongly coupled N = 4 super Yang-Mills theory atlarge number of colors ( N c ) and at strong coupling. Thelowest lying nonhydrodynamic degree of freedom is thenthe metric quasinormal mode sharing the symmetries ofthe problem and having the lowest frequency. Bjorken flow and holography.
Bjorken flow is a par-ticular solution of the equations of relativistic hydrody-namics and describes matter expanding in one dimensionwith the additional assumption of boost-invariance in the a r X i v : . [ h e p - t h ] M a y longitudinal direction. This symmetry can be made man-ifest upon passing to curvilinear proper time τ - rapidity y coordinates related to the lab frame time x and posi-tion along the expansion axis x via x = τ cosh y and x = τ sinh y. (1)In the case of (3+1)-dimensional conformal field theoryplasma, the most general stress tensor obeying the sym-metries of the problem in coordinates ( τ, y, x , x ) reads T µν = diag( − (cid:15), p L , p T , p T ) µν , (2)where the energy density (cid:15) is a function of proper timeonly and the longitudinal p L and transverse p T pressuresare fully expressed in terms of the energy density [9] p L = − (cid:15) − τ (cid:15) (cid:48) and p T = (cid:15) + 12 τ (cid:15) (cid:48) . (3)Note that, in the proper time - rapidity coordinates (1),there is no momentum flow in the stress tensor (2) andso the flow velocity is trivial and takes the form u = ∂ τ . Hydrodynamic constituent relations lead, then, togradient expanded energy density of the form (cid:15) = 38 N c π τ / (cid:18) (cid:15) + (cid:15) τ / + (cid:15) τ / + . . . (cid:19) , (4)where the choice of (cid:15) sets an overall energy scale, in par-ticular for the quasinormal frequencies (7) and 9). Theprefactor was chosen to match the N = 4 super Yang-Mills theory at large- N c and strong coupling. In the fol-lowing, we choose the units by setting (cid:15) = π − .Large- τ expansion of the energy density in powers of τ − / , as in (4), is equivalent to the hydrodynamic gra-dient expansion and arises from expressing gradients ofvelocity ( ∇ µ u ν ∼ τ − ) in units of the effective tempera-ture ( T ∼ (cid:15) / ∼ τ − / ). The value of the coefficient (cid:15) is related to the shear viscosity η , whereas (cid:15) is a sumof two transport coefficients: relaxation time τ Π and theso-called λ [10]. Higher order contributions to the en-ergy density are expected to be linear combinations of sofar unidentified transport coefficients. Note also that theexpansion (4) is sensitive to both linear and nonlineargradient terms.As explained in [11, 12] (see also Supplemental ma-terial), higher order contributions to the energy density(4) can be obtained by solving Einstein’s equations witha negative cosmological constant for the metric ansatz ofthe form ds = 2 dτ dr − Adτ +Σ e − B dy +Σ e B ( dx + dx ) , (5)where the warp factors A , Σ and B are functions of r and τ constructed in the gradient expansion as requiredby the fluid-gravity duality. At leading order, the warpfactors are that of a locally boosted black brane and this solution gets systematically corrected in τ − / expansion,as is the case with the energy density in the dual fieldtheory (4).The background expanded in τ − / around a locallyboosted black brane is slowly evolving and captures onlyhydrodynamic degrees of freedom. One can, in ad-dition, consider the incorporation of nonhydrodynamic(fast evolving) degrees of freedom by linearizing Ein-stein’s equations on top of the hydrodynamic solution,i.e. B = B hydro + δB , and similarly for A and Σ, andlooking for δB corresponding to (at very large time) theexponentially decaying contribution to the stress tensordepending only on τ . For the static background analo-gous calculation would lead to the spectrum of nonhydro-dynamic quasinormal modes carrying zero momentum,which is known to be the same as the spectrum of zeromomentum quasinormal modes for the massless scalarfield [13].In the leading order of the gradient expansion, the re-sulting modes, on the gravity side, indeed essentially re-duce to the scalar quasinormal modes but obtain an ad-ditional factor of and are damped exponentially in τ [14]. Upon including viscous correction, the modes ob-tain a further nontrivial powerlike preexponential factor δ(cid:15) ∼ τ α qnm exp ( − i ω qnm τ / ) . (6)Explicit gravity calculation for the lowest mode yield ω qnm = 3 . − . , α qnm = − . . i. (7)The frequency ω qnm agrees with the frequency of thelowest nonhydrodynamic scalar quasinormal mode andwas calculated before in [14], whereas the prediction of α qnm is a new result specific to the dissipative modifi-cations of the expanding black hole geometry (see theSupplemental Material for further details). In the fol-lowing, we will be able to reproduce numerically (7) justfrom the large order behavior of the hydrodynamic series. Large order behavior of hydrodynamic energydensity.
Numerical implementation of the methods out-lined in [11, 12] allow for efficient calculation of hydrody-namic series given by (4), up to a very large order, sinceone is effectively solving a set of linear ODE’s (comingfrom Einstein’s equations) at each order. Using spectralmethods we iteratively solved these equations in the largetime expansion reconstructing the energy density up tothe order 240, i.e. up to the term (cid:15) in (4). To the bestof our knowledge this is the first approach allowing us toaccess information about the large order behavior of thehydrodynamic series in any physical system or model.As a way of monitoring the accuracy of our procedureswe compared normalized values of evaluated Einstein’sequations at each order of the τ − / expansion to theratio of coefficients of gradient-expanded energy densityto gradient expanded warp factors. This ensures thatour results for the energy density are reliable. We also FIG. 1. Behavior of the coefficients of hydrodynamic seriesfor the energy density as a function of the order. At largeenough order the coefficients start exhibiting factorial growth.The radius of convergence of the Borel transformed series isestimated to be 6 .
37, in rough agreement with (9): 2 / × .
37 = 4 . verified that we reproduced the known analytic resultsfor the energy density at low ( ≤
3) orders.As anticipated in the introduction, the coefficients ofthe gradient expanded energy density (4) that we ob-tained, indicate that the hydrodynamic gradient expan-sion, as seen in the example of boost-invariant flow, haszero radius of convergence. This is clearly visible onFig. 1, which shows the factorial growth with order ofthe coefficients of the hydrodynamic series for the energydensity (4). Numerical values of coefficients (cid:15) n , as inEq. (4), can be found in the file “eps.m” included in thesubmission (see also Supplemental Material). Singularities of the Borel transform and quasinor-mal modes.
A standard way of dealing with divergentseries, including perturbative quantum field theories [15],is performing a Borel transform of the original power se-ries (here a series in u ≡ τ − ), ˜ (cid:15) n = (cid:15) n / n !, and subse-quently the Borel resummation (cid:15) resum ( τ ) = (cid:90) ∞ ˜ (cid:15) ( ζτ ) e − ζ dζ, (8)where the integral is taken over the real positive axis.A key issue in performing Borel resummation is the iden-tification of the singularities of the Borel transform andits analytical continuation into some neighborhood of thepositive real axis. The singularities of the Borel trans-form are interesting for their own sake as, typically, theyhave a definite physical interpretation (like instantoncontributions in quantum field theories). We will findthat this will also be the case in our context.Since the Borel transform has typically only a finiteradius of convergence, we use the standard technique ofPad´e approximants to provide an analytic continuation.Such an approach was used with success in the context ofresumming perturbative series in quantum field theoriesin [16] and, in principle, can be refined by including the FIG. 2. Real and imaginary parts of poles ζ of the symmet-ric Pad´e approximant of the Borel transform of the energydensity (4). From the plot we removed numerically spuriouspoles. The pole closest to the origin governs the convergenceradius of the Borel transformed series and gives rise to thelowest quasinormal frequency. The poles from the encircledregion (magenta) lead to the powerlike preexponential factorin (6). information about the behavior of the series at large val-ues of the parameter (typically strong coupling behaviorfor perturbative expansion in quantum field theories).In our case, this regime corresponds to small times(high order dissipative terms). Although it was estab-lished that, when expanded around τ = 0, the energydensity contains only even powers of proper time [17],this information turns out to be hard to implement in anunambiguous way and, in the following, we adopted thesimplest analytic continuation.Figure 2 shows the position of poles of the symmetricPad´e approximant of the Borel transformed energy den-sity. Let us note that some care must be taken as thePad´e approximant exhibits apparent poles on the nega-tive real axis which, however, cancel almost perfectly (upto 10 − accuracy) with zeroes of the numerator. As ourknowledge about the series is limited to a finite numberof terms (first 241), we should only trust the structurein the close vicinity of the origin. Note, however, thatthe lack of poles on the positive real axis seems to in-dicate Borel summability, pointing towards the possibleexistence of Borel-resummed all-order hydrodynamics.The poles approximate some complicated structure ofbranch cuts. The pole nearest the origin, from which amajor branch cut starts, sets the radius of convergenceof the Borel transform of hydrodynamic series. Its nu-merical value (multiplied by the factor 3 / i ) reads ω Borel = 3 . − . i ( | ω Borel | = 4 . / i ) × ω Borel leads to the following contribution to theenergy density at large proper time δ(cid:15) ∼ τ α Borel exp ( − i ω Borel τ / ) , (10)where α Borel = − . . i. (11)The contribution (10) together with (9) and (11)matches, up to numerical accuracy, the gravitational pre-diction for the behavior of the lowest metric quasinormalmode given by (6) and (7), but is derived entirely withinthe hydrodynamic gradient expansion. Summary and conclusions.
We used the fluid-gravityduality to numerically obtain the form of hydrodynamicstress tensor of the Bjorken flow up to 240 th order in gra-dients. This corresponds to the inclusion of dissipativeterms of a very high order. We discovered that the cor-rections grow factorially with the order. This providesstrong indication that hydrodynamic expansion is an ex-ample of asymptotic series; i.e., it has a zero radius ofconvergence.Upon a simple analytic continuation of Borel trans-formed series we found that it is the frequency of thelowest nonhydrodynamic quasinormal mode which setsthe radius of convergence of the Borel transform. Subse-quently, from the hydrodynamic expression alone we ex-tracted the modification of this contribution due to theshear viscosity and matched with the gravity calculation.It is fascinating to observe that nonhydrodynamic col-lective excitations, which are not describable within theframework of hydrodynamics, nevertheless leave their im-print in the behavior of high order transport coefficients.This is reminiscent of the relation of instanton contri-butions with the divergence of the perturbative series inquantum field theory.Note that our observation about the asymptotic char-acter of the hydrodynamic series is made upon evaluatingthe stress tensor of a particular type of hydrodynamicflow. The high degree of symmetry of that flow – boost-invariance – means that at each order in the large propertime expansion, we are indeed studying new dissipativeterms in the hydrodynamic stress tensor with new trans-port coefficients. The dependence on initial conditionsfor this hydrodynamic flow reduces just to a trivial over-all rescaling.We speculate that the most likely source of the facto-rial growth of the coefficients of the hydrodynamic seriesfor the energy density is the fast growth of the number ofgradient terms contributing to the hydrodynamic stress tensor at each order, in analogy with the factorial growthof the number of Feynman graphs at large order of per-turbative calculations.Regarding future directions, a thought-provoking phe-nomenological spin off of our Letter is the possibility ofresumming hydrodynamic description and extending thehydrodynamic stress tensor past the regime in which sub-sequent low order terms give comparable contributions.This might lead to refined criterium for the applicabilityof hydrodynamics used in [2, 3] and goes very much inthe spirit of [4, 5]. Acknowledgments.
MPH acknowledges support fromthe Netherlands Organization for Scientific Research un-der the Veni scheme and would like to thank Norditafor hospitality. This work was supported by NCNgrant 2012/06/A/ST2/00396 (RJ) and IoP JU grantK/DSC/000705 (PW). PW would like to thank theLorentz Center and Nordita for hospitality. We used M.Headrick’s diffgeo.m package for symbolic GR. Finallywe would like to thank M. Baggio, J. de Boer, K. Land-steiner, C. Nunez and D. Teaney for interesting discus-sions and suggestions. ∗ [email protected]; On leave from: National Centre forNuclear Research, Ho˙za 69, 00-681 Warsaw, Poland † [email protected] ‡ [email protected][1] U. W. Heinz and R. Snellings, arXiv:1301.2826 [nucl-th].[2] M. P. Heller, R. A. Janik and P. Witaszczyk, Phys. Rev.Lett. , 201602 (2012) [arXiv:1103.3452 [hep-th]].[3] M. P. Heller, R. A. Janik and P. Witaszczyk, Phys. Rev.D , 126002 (2012) [arXiv:1203.0755 [hep-th]].[4] M. Lublinsky and E. Shuryak, Phys. Rev. C , 021901(2007) [arXiv:0704.1647 [hep-ph]].[5] M. Lublinsky and E. Shuryak, Phys. Rev. D , 065026(2009) [arXiv:0905.4069 [hep-ph]].[6] J. M. Maldacena, Adv. Theor. Math. Phys. , 231(1998) [Int. J. Theor. Phys. , 1113 (1999)] [arXiv:hep-th/9711200].[7] S. Bhattacharyya, V. E. Hubeny, S. Minwalla andM. Rangamani, JHEP , 045 (2008) [arXiv:0712.2456[hep-th]].[8] J. D. Bjorken, Phys. Rev. D , 140 (1983).[9] R. A. Janik and R. B. Peschanski, Phys. Rev. D ,045013 (2006) [arXiv:hep-th/0512162].[10] R. Baier, P. Romatschke, D. T. Son, A. O. Starinetsand M. A. Stephanov, JHEP , 100 (2008)[arXiv:0712.2451 [hep-th]].[11] M. P. Heller, P. Surowka, R. Loganayagam, M. Spalinskiand S. E. Vazquez, Phys. Rev. Lett. , 041601 (2009)[arXiv:0805.3774 [hep-th]].[12] S. Kinoshita, S. Mukohyama, S. Nakamura andK. -y. Oda, Prog. Theor. Phys. , 121 (2009)[arXiv:0807.3797 [hep-th]].[13] P. K. Kovtun and A. O. Starinets, Phys. Rev. D ,086009 (2005) [hep-th/0506184].[14] R. A. Janik and R. B. Peschanski, Phys. Rev. D , , 1350 (2005) [J.Exp. Theor. Phys. , 1188 (2005)] [hep-ph/0510142].[16] J. R. Ellis, E. Gardi, M. Karliner and M. A. Samuel,Phys. Lett. B , 268 (1996) [hep-ph/9509312].[17] G. Beuf, M. P. Heller, R. A. Janik and R. Peschanski,JHEP , 043 (2009) [arXiv:0906.4423 [hep-th]]. SUPPLEMENTAL MATERIAL
Gravity solution and dual energy density.
Themetric ansatz (5) has a residual diffeomorphism freedom r → r + f ( τ ) [12]. The near-boundary expansion of (5)contains information about the expectation of the dualstress tensor. For f ( τ ) = 0 the metric close to the bound-ary take the form A = r + a ( τ ) r − + . . . (12) B = 23 log r rt −
12 [ a ( τ ) + 34 τ a (cid:48) ( τ )] r − + . . . where a is related to the (local) energy density of theplasma (cid:15) ( τ ) = − π N c a ( τ ) . (13)We look for the solution in the form dictated by the fluid-gravity duality, i.e. A = r (cid:88) j =0 τ / j A j ( r τ / ) ,B = 23 log r rt + (cid:88) j =0 τ / j B j ( r τ / ) , Σ = r / (1 + rt ) / (cid:88) j =0 τ / j Σ j ( r τ / ) , (14)where the starting point is the boosted black brane metric A ( v ) = 1 − v , B ( v ) = 0 and Σ ( v ) = 1 . (15) The choice of units and provided data for the en-ergy density.
In (15) we chose the units by settingits event horizon to lie at r = 1 and this choice leadsto (cid:15) = π − in the main body of the paper. The file“eps.m”, included in the submission, contains coefficientsof the energy density in the late time expansion writtenin Mathematica format as { j, (cid:15) j } .In order to restore general units, i.e. bring the energydensity to the form (cid:15) ( τ ) = 38 π N c Λ (Λ τ ) / (cid:26) c + c τ ) / + . . . (cid:27) , (16)where Λ is an overall scale and c = 1, one needs tomultiply each (cid:15) j> that we provide by π − j (cid:15) j = π j − c j . (17) Numerical implementation.
At each order order of τ / expansion we are solving a set of coupled linear or-dinary differential equations for 3 functions. There are5 different equations and we explicitly solve 3 of themand use the remaining 2 to provide boundary conditionsfor the functions at v = 1. In the numerical implemen-tation we fixed the residual diffeomorphism freedom bydemanding that A j (1) = 0 for all j . The bulk regular-ity requires that the mode diverging logarithmically inthe vicinity of the horizon vanishes. The flatness of theboundary metric provides the remaining boundary con-ditions for the metric at each order. See [12] for detaileddiscussion of the late time gravity solution to the Bjorkenflow in the Eddington-Finkelstein coordinates.At each order we generated the equations symbolically.The kernels for all equations take particularly simpleform and the whole complication arises from source termswhich significantly grow in length with order. We solvedthese equations in Mathematica via matrix inversion us-ing spectral discretization in v direction [using z = 1 /v running from 0 (the boundary) to 1 (the horizon)]. Theenergy density at each order is obtained by taking thefourth derivative of A j (1 /z ) at z = 0.Going to 240 th order required a grid of 250 points withextended precision (keeping the first 450 digits) and tookabout 4 weeks on a desktop computer with 3.4 GHz pro-cessor and 16 GB of memory. QNMs on top of the expanding plasma.
Small per-turbations on top of the expanding plasma are capturedby the linearized Einstein’s equations B = B hydro + δB, (18)with analogous expressions for A and Σ. The fluid-gravity duality dictates that at late time the perturba-tions coincide with corresponding perturbations of thestatic black brane with the temperature replaced by thelocal temperature of the flow [14]. Within the gradientexpansion one thus expects that δB =exp [ − i (cid:90) dτ ( ω τ − / + ω τ − + . . . ] × (cid:26) δB ( r τ / ) + 1 τ / δB ( r τ / ) + . . . (cid:27) (19)and similar expressions for δA and δ Σ.Upon linearizing equations on top of (14) and keepingterms up to the first order in gradients one discovers that δB given by (19) obeys the same equation as the mass-less scalar field. Subleading correction takes into accountviscous effects present in the expanding plasma describedby hydrodynamics.The overall normalization does not matter, which al-lows to set δB (1) = 1. This choice fixes both the nor-malization and chooses the ingoing boundary conditionat the horizon. Imposing Dirichlet boundary conditionat v = ∞ leads to the discrete spectrum of frequenciesthe lowest one giving ω = ω qnm = 3 . − . i, (20)which is the value quoted in the main body of the article.Repeating the analysis in the first order allows to ob-tain ω = − . . i. (21)It is perhaps worth stressing that the value of δB ( v ) at v = 1 turns out not to matter for obtaining ω , as theassociated solution vanishes at the boundary.Finally, in order to obtain the correction to the energydensity δ(cid:15) ( τ ), one needs to take into account that theasymptotic behavior of δB is indirectly related to theenergy density δ(cid:15) ( τ ) through δB (cid:12)(cid:12)(cid:12) r →∞ ∼ (cid:26) δ(cid:15) ( τ ) + 34 τ δ(cid:15) (cid:48) ( τ ) (cid:27) r − . (22) Solving this relation in the late time expansion isstraightforward and leads to α qnm = − iω − −
23 = − . . i, (23)as quoted in the main body of the article. The factorof − / v = r τ / to the original radial variable r andthe factor of − / δ(cid:15) ( tt