Weak and strong coupling equilibration in nonabelian gauge theories
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This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Weak and strong coupling equilibration in nonabelian gauge theories Keegan, Liam; Kurkela, Aleksi; Romatschke, Paul; van der Schee, Wilke; Zhu, Yan Keegan, L., Kurkela, A., Romatschke, P., van der Schee, W., & Zhu, Y. (2016). Weak and strong coupling equilibration in nonabelian gauge theories. Journal of High Energy Physics, 2016(4), Article 31. https://doi.org/10.1007/JHEP04(2016)031 2016
JHEP04(2016)031 Published for SISSA by Springer Received:December 23, 2015 Accepted:March 11, 2016 Published:April 6, 2016 Weak and strong coupling equilibration in nonabelian gauge theories Liam Keegan,aAleksi Kurkela,a,b Paul Romatschke,c,d Wilke van der Scheee and Yan Zhuf,g aPhysics Department, Theory Unit, CERN, CH-1211 Gen`eve 23, Switzerland bFaculty of Science and Technology, University of Stavanger, 4036 Stavanger, Norway cDepartment of Physics, 390 UCB, University of Colorado at Boulder, Boulder, CO, U.S.A. dCenter for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, U.S.A. eCenter for Theoretical Physics, MIT, Cambridge, MA 02139, U.S.A. fDepartment of Physics, University of Jyv¨askyla, P.O. Box 35, FI-40014 University of Jyv¨askyl¨a, Finland gHelsinki Institute of Physics, P.O. Box 64, 00014 University of Helsinki, Finland E-mail: [email protected],[email protected], [email protected],[email protected],[email protected] Abstract: We present a direct comparison studying equilibration through kinetic theory at weak coupling and through holography at strong coupling in the same set-up. The set-up starts with a homogeneous thermal state, which then smoothly transitions through an outof-equilibrium phase to an expanding system undergoing boost-invariant flow. This first apples-to-apples comparison of equilibration provides a benchmark for similar equilibration processes in heavy-ion collisions, where the equilibration mechanism is still under debate. We find that results at weak and strong coupling can be smoothly connected by simple, empirical power-laws for the viscosity, equilibration time and entropy production of the system. Keywords: AdS-CFT Correspondence, Holography and quark-gluon plasmas, Perturbative QCD, Quark-Gluon Plasma ArXiv ePrint: 1512.05347 Open Access,c The Authors. Article funded by SCOAP3.doi:10.1007/JHEP04(2016)031
JHEP04(2016)031 Contents 1 Introduction 1 2 A simple set-up for studying gauge theory equilibration 3 2.1 Hydrodynamic late time limit 5 3 Gauge theory dynamics from a weak coupling approach 6 4 Gauge theory dynamics from a strong coupling approach 11 5 Results 14 6 Conclusions 21 1 Introduction In the past decade, there has been a wealth of data on nuclear matter at extremely high temperatures from the experimental heavy-ion program at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) [1–7]. To the surprise of many, hydrodynamic models are tremendously successful in describing and often predicting the experimental measurements [8–16]. However, one of the key requirements for the success of these hydrodynamic models is that the matter created after a relativistic ion collision equilibrates quickly, on a time-scale of thydro ∼1−2 fm/c [13,17]. It has been a longstanding theoretical challenge to understand the pre-equilibrium physics that leads to this time scale. As a consequence, the search for a quantitative understanding of equilibration in gauge theories at high temperature has spawned a new subfield of physics. Two branches of this subfield have emerged, based on very different approaches. On the one hand, there is an effort to understand equilibration based on a weakly coupled framework. This branch was pioneered by the early parametric picture of the so-called “Bottom-up” thermalization [18], and since then there has been a continuing effort to elevate the weak-coupling picture from parametric estimates to a quantitative prescription by exploiting the scale separations provided by the weak coupling, admitting different effective theory descriptions [19–28]. While the early parametric estimates were somewhat in tension with early thermalization [29], the modern quantitative calculations are consistent with the fast thermalization [30,31]. The weak coupling approach can be rigorously set up for any gauge theory (such as N= 4 SYM and QCD [19,32]) whenever the coupling is small, but eventually will start to break down as the coupling is increased. – 1 –
JHEP04(2016)031 On the other hand, equilibration at strong coupling has been studied using the gauge/ gravity duality or holography [33], in which it is remarkably straightforward to study real time dynamics for certain gauge theories (such as N= 4 SYM, but not QCD). In holography the dynamics of the equilibration of the gauge theory is mapped onto the relaxation of a black hole in an anti-de-Sitter (AdS) space-time with one extra dimension (see [34] for a recent review). Early studies near equilibrium suggested the black hole relaxes fast, with the characteristic time scale being 1/T, with Tthe temperature of the formed plasma [35]. In a non-linear setting the relaxation was pioneered by Chesler and Yaffe, who studied the relaxation of a gauge theory on a non-trivial curved background space-time [36,37]. This was later extended to studies of a gauge theory in flat space-time, prepared with a wide variety of initial states [38–41], which always led to hydrodynamics within t < 1.2/T . Lastly, many studies have been performed in colliding settings, mimicking heavy-ion collisions more closely both in the longitudinal and transverse directions [40,42–46], even allowing for direct comparison with experimental data [47,48]. For QCD at energy scales relevant for heavy-ion collisions the coupling constant is presumably not very small, nor very large. This makes it very interesting to compare the weakly and strongly coupled approaches, as they may bracket what happens in real heavyion collisions. There are several reasons why this comparison is not straightforward. In particular, the initial condition of the pre-equilibrium evolution at weak coupling is usually described in terms of classical fields or distribution functions whereas at strong coupling the initial condition has to be formulated in terms of fields in AdS space-time. In fact, it is not straightforward to characterise a far-from-equilibrium state in terms that are well defined and applicable in the both frameworks, which makes an apples-to-apples comparison of the evolution non-trivial. In this paper, we consider a setup which avoids the question of setting non-equilibrium initial conditions and allows a clean apples-to-apples comparison of the non-equilibrium evolution. We consider a system that is initially in thermal equilibrium but is subsequently pushed out of equilibrium by an external force. In practice we accomplish this by changing the metric rapidly with a pulse of curved space-time from a homogeneous Minkowski space to an expanding space described by Milne coordinates. Finally, we compute the expectation value of the stress-energy tensor and the entropy density in both systems, such that we can follow how the system approaches a hydrodynamical description. We find that at all values of the ’t Hooft coupling λthe system reaches hydrodynamical flow. For small couplings this is preceded by a period of free-streaming type evolution which becomes shorter and shorter as the coupling is increased. As the coupling is increased the evolution starts to resemble the strongly coupled evolution. Indeed, for λ=∞, the system is described by hydrodynamics very quickly after the pulse has ended, but the departure from equilibrium does leave an imprint in non-equilibrium entropy production. In section 2, we explain our setup for driving gauge theories out of equilibrium, including a discussion on the evolution of the stress-energy tensor within hydrodynamics. For this setup, we describe state-of-the-art weak and strong coupling calculations in section 3 and 4, and report our findings in section 5. A summary and the conclusions that we draw from our findings can be found in section 6. – 2 –
JHEP04(2016)031 2 A simple set-up for studying gauge theory equilibration In the following, we consider a gauge theory initially in global equilibrium at the temperature Ti. We then consider this gauge theory to be placed into a space-time with coordinates xa= (t, x, y, L) and line element ds2=−dt2+dx2+dy2+g(t)dL2(2.1) with g(t) a function that smoothly transitions from g(t→ −∞) = 1 to g(t→ ∞)→t2at late times t. This choice of metric tensor implies that for t→ −∞, the gauge theory is in global equilibrium at rest within a flat space-time, as outlined above. By contrast, at late times, the gauge theory experiences stretching of (flat!) space-time in the longitudinal direction z. This late-time behavior corresponds to the familiar Bjorken flow [49], since it is just a coordinate transformation of a gauge theory expanding in the longitudinal direction in Minkowski space. In between, the gauge theory experiences a dynamic, space-time pulse that is driving it (far) from its original equilibrium state. To be concrete, in the following we choose a one-parameter family of metric functions given by the choice g(t) = 1 eα(tTi−1) + 1 +t2T2 i e−α(tTi−1) + 1 ,(2.2) with αa free parameter controlling the rapidity of the change from early to late time behavior. It should be reiterated that for the choice of g(t) in eq. (2.2), the space-time is flat up to exponentially small terms for all texcept the region |tTi−1| ∝ 1 α, where the space-time is curved. It is possible to study the time-evolution of the stress-energy tensor components (such as the energy density) using hydrodynamic theory. The stress-energy tensor in hydrodynamic theory is given in terms of a gradient expansion. For a conformal theory, the most general stress tensor complete up to second order gradients in arbitrary d-dimensional space-times is given by [50] Tµν =uµuν+P∆µν +πµν , πµν =−ησµν +ητπhDσµνi+1 d−1σµν (∇·u)+κhRhµνi−(d−2)uαRα<µν>βuβi +λ1σ<µ λσν>λ +λ2σ<µ λΩν>λ +λ3Ω<µ λΩν>λ ≡πµν BRSSS ,(2.3) where , P = d−1are the energy density and pressure for a conformal theory, uµis the fluid four-velocity (normalized to uµuµ=−1), ∆µν =gµν +uµuνwith gµν the metric tensor in the mostly plus sign convention and Rµν, Rαµνβ are the Ricci and Riemann tensors for this spacetime. Furthermore, the definition Ahµνi≡1 2∆µα∆νβ (Aαβ +Aβα)−1 d−1∆µν∆αβAαβ ≡ hAµνi has been used to define e.g. σµν = 2h∇µuνi, where ∇µis the covariant derivative for the metric gµν. Moreover D≡uµ∇µ, and Ωµν =1 2∆µα∆νβ (∇αuβ−∇βuα). The coefficients η, τπ, λ1, λ2, λ3, κ are the first and second-order transport coefficients (material constants – 3 –
JHEP04(2016)031 depending on the specific gauge theory and specific value of the coupling considered). In the limit where all of these transport coefficients are set to zero, one recovers ideal fluid dynamics, for which πµν = 0. In the limit where only the second-order transport coefficients τπ, κ, λ1, λ2, λ3are set to zero one recovers πµν =−ησµν ≡πµν NS ,(2.4) which is the constitutive relation of the relativistic Navier-Stokes equation. We will refer to the equations (2.4) and the expression for Tµν as Navier-Stokes (NS) theory, while the full set of equations (2.3) will be referred to as BRSSS in the following. It turns out that as they stand, both the NS equations (2.4) and the BRSSS equations (2.3) would be acausal, as can be easily seen by working out the group velocity from the dispersion relations [51]. It has proven very useful for practical applications to consider the resummed version of the BRSSS constitutive equations (see e.g. [13,16]). In the following, however, there are no issues with causality, and we can limit our discussion to the case of the un-resummed version of the BRSSS equations. Let us now consider the case of d= 4 in the following. For the line element (2.1), with the initial condition of global equilibrium at temperature Ti, one finds that the fluid dynamic solution maintains the initial condition of vanishing spatial flow velocity, so that uµ= (1,0). This implies =−Tt t, and effective transverse and longitudinal pressures of P⊥≡Tx x=Ty y=P(1 + 2H), PL≡TL L=P(1 −4H), respectively, with H≡−πL L +P. For later convenience, it is useful to define the pressure anisotropy as PL P⊥ =1−4H(t) 1+2H(t).(2.5) The covariant conservation of the stress-energy tensor uµ∇νTµν = 0 leads to ∂tln s=−g0(t) 2g(t)(1 −H(t)) ,(2.6) where s=+P T=4 3Tis the equilibrium entropy density and T(t) is the temperature. It is convenient to express dynamic quantities with respect to their initial (global equilibrium) values, s(t) si≡T3(t) T3 i ,(t) i≡T4(t) T4 i ,etc.(2.7) For ideal hydrodynamics (πµν = 0 and thus H= 0), the conservation of energy can be solved analytically in closed form for an arbitrary metric function g(t): sideal(t) si =g−1/2(t),ideal(t) i =g−2/3(t),PL P⊥ideal = 1 .(2.8) Plots of the ideal hydrodynamic solution for the metric function (2.2) will be shown in section 5. For further reference, it is useful to define the concept of the total equilibrium entropy Seq in the system, defined as Seq Seq,i ≡Rd3xp−detgµνs(t) Rd3xsi =g1/2(t)s(t) si ,(2.9) – 4 –
JHEP04(2016)031 which for ideal hydrodynamics trivially becomes Seq,ideal(t) Seq,i = 1. This just reflects the fact that no entropy is created in ideal (inviscid) hydrodynamics. Note that the equilibrium entropy thus defined will only correspond to the total system entropy if the system is close to equilibrium. Otherwise, non-equilibrium (viscous) corrections to the equilibrium entropy cannot be neglected (see e.g. the discussion in ref. [53]). Including viscous corrections, one finds for Navier-Stokes hydrodynamics HNS(t) = 2 3 η s g0(t) g(t)T(t),(2.10) which has to be solved together with (2.6). Using the fact that g(t) really is a function of t Tionly, the equations of motion for Navier-Stokes become ∂tln s si =−g0 2g 1−2η 3s g0 gTis si−1/3!.(2.11) Going beyond Navier-Stokes, for the BRSSS equations it will be useful to define the common parametrizations τπ=Cτη sT , λ1=Cλη2 sT , κ =Cκs T.(2.12) 2.1 Hydrodynamic late time limit In the late time limit, when g(t)∝t2, the Navier-Stokes equations (2.11) can be solved as a gradient expansion around the ideal hydrodynamic solution. Assuming a constant value of η s, one finds s(t) siNS,tTi1 =χ tTi1−2η s 1 (tTi)2/3χ1/3,PL P⊥NS,tTi1 = 1 −8η s 1 (tTi)2/3χ1/3.(2.13) Unlike solutions to the full evolution equations (2.11), which assume that Navier-Stokes hydrodynamics is accurate through the entire time-evolution, the late-time solution (2.13) is a universal solution to the system evolution for all initial conditions, and as such includes an unknown constant χ. Comparing the total equilibrium entropy resulting from (2.13) to that from ideal hydrodynamics, one can interpret χ=Seq(t→ ∞) Seq,i (2.14) as the total amount of entropy produced. Similarly, one can also go beyond Navier-Stokes and solve the BRSSS equations of motion using a gradient expansion. One finds s(t) siBRSSS,tTi1 =χ tTi1−2η s 1 (tTi)2/3χ1/3+2η2(2 + Cλ−Cτ) 3s2(tTi)4/3χ2/3,(2.15) where it should be noted that the constant Cκdoes not enter the result because the spacetime is flat for g(t)→t2at late times. In a similar fashion one finds PL PTBRSSS,tTi1 = 1 −8η s 1 (tTi)2/3χ1/3+16η2(3 + Cλ−Cτ) 3s2(tTi)4/3χ2/3.(2.16) – 5 –
JHEP04(2016)031 3 Gauge theory dynamics from a weak coupling approach In the non-interacting limit λ→0, the system is described by non-interacting freestreaming particles whose evolution is given by the collisionless transport equation for the on-shell particle distribution f pµ∂µf−Γi αβpαpβ∂f ∂pi= 0,(3.1) where the summation of igoes over the spatial coordinates x, y and L. The transport equation with the initial condition and metric we have chosen, becomes ∂tf−g0 gpL∂f ∂pL= 0 .(3.2) Eq. (3.2) can be solved analytically for arbitrary metric function g(t) using the method of characteristics, and one obtains f=f(px, py, pLg(t)) (cf. ref. [54]). With an initially thermal system with temperature Ti, a full solution to the free-streaming evolution of bosons is thus given by f=1 exp √(px)2+(py)2+g2(t) (pL)2 Ti−1 .(3.3) The solution (3.3) can be brought into the form f=P∞ n=1 exp "− nsp21+ξ(t)pL |p|2 Ti#, where ξ(t) = g(t)−1 (3.4) is the anisotropy parameter defined in ref. [55]. This identification allows direct connection to anisotropic plasma physics literature, and in particular leads to the expressions for the energy density and pressure anisotropy [56]: FS(t) i =1 2 1 1 + ξ(t)+arctanpξ(t) pξ(t)!,PL(t) P⊥(t)FS = 2 (1 + ξ(t))F S (t) i−1 1−(1 −ξ2(t))F S (t) i ,(3.5) for a system of free-streaming (FS) particles experiencing arbitrary metric perturbations of the form (2.1). In the late-time limit, these lead to the following expressions for equilibrium entropy and pressure anisotropy sFS,tTi1(t) si =π 4tTi4/3 ,PL(t) P⊥(t)FS,tTi1 =2 (tTi)2,(3.6) which upon comparison with eqs. (2.13) imply that the system never reaches equilibrium. Thus, free-streaming (non-interacting) evolution is the opposite extreme to ideal hydrodynamic evolution, and one expects these two extreme cases to bound the system evolution for any interaction strength λ∈(0,∞). – 6 –
JHEP04(2016)031 For numerical purposes, it is useful to work with rescaled longitudinal momentum pz=pg(t)pLsuch that f(t, px, py, pL)→ˆ f(t, px, py, pz) = ˆ f(t, px, py,√gpL). There is a Jacobian ∂ f|pL ∂t = ∂ˆ fpL ∂t = ∂ˆ fpz ∂t +∂f ∂pz pL∂pg(t) ∂t associated with this transformation (cf. ref. [57]) which gives rise to the rescaled equation ∂tˆ f−g0 2gpz∂ˆ f ∂pz= 0 .(3.7) (It is easy to check this new form by using the free-streaming solution (3.3), or simply putting ˆ f∝(px)2+ (py)2+g(pz)2). Since it is of no importance, we denote ˆ f→fin the following even though it is a distribution evaluated at constant pz, not constant pL. At small but finite coupling, most of the modes can still be described by a transport equation, which due to collisions among particles will now be of the form (3.2), but with a non-vanishing right-hand side. We will restrict our weak coupling simulations to SU(N) gauge theory. Then assuming that the non-equilibrium distributions are not spin polarized, the transport equation for the spin (and color) independent distribution function of gauge bosons can be written in a form where the right hand side contains two effective collision terms that contribute to leading order in λ[19], one describing elastic (2 ↔2), and the other inelastic (1 ↔2) particle interactions: ∂tf−g0 2gpz∂f ∂pz=−C2↔2−C1↔2 a.(3.8) The collision operators read C2↔2[f](p) = 1 4|p|νZkp0k0|M(p,k;p0,k0)|2(2π)4δ(4)(P+K−P0−K0) ×nf(p)f(k)[1 + f(p0)][1 + f(k0)] −f(p0)f(k0)[1 + f(p)][1 + f(k)]o(3.9) and C1↔2[f](p) = (2π)3 2|p|2νZ∞ 0 dp0dk0δ(|p|−p0−q0)γ(p;p0ˆ p, k0ˆ p) ×nf(p)[1 + f(p0ˆ p)][1 + f(k0ˆ p)] −f(p0ˆ p)f(k0ˆ p)[1 + f(p)]o +(2π)3 |p|2νZ∞ 0 dp0dkδ(|p|+k−p0)γ(p0ˆ p;p, kˆ p) ×nf(p)f(kˆ p)[1 + f(p0ˆ p)] −f(p0ˆ p)[1 + f(p)][1 + f(kˆ p)]o,(3.10) where ν= 2dAis the number of degrees of freedom. Here |M|2and γare effective matrix elements for elastic scattering and collinear splitting, respectively. Both of them have non-trivial structure arising from the soft and collinear divergences of the underlying processes which are regulated dynamically by in-medium physics. The soft tand uchannel divergences present in vacuum are regulated by physics of screening at scale – 7 –
JHEP04(2016)031 Figure 1. For the energy density and transverse pressure we illustrate for α= 8 in eq. (2.2) the renormalization scheme dependence present for the expectation value of stress-energy tensors of QFTs living in an even-dimensional curved spacetime. Clearly, the pressure oscillates wildly during the period where the boundary metric is curved (as also noted in [36]), but these oscillations are mostly due to scheme dependence. We therefore focus on the times after the pulse (t Ti&1.4), where this ambiguity does not arise, and present results for the choice of µ=−2. this particular boundary metric a reasonable choice is µ=−2, which leads to the mildest oscillations possible (though for different αdifferent µwould be preferred in that sense). For the results to be presented in the next section we hence used µ=−2. Lastly, as in the hydrodynamic and weak coupling approaches, we also kept track of a measure of entropy. Here we used the area density (S3) of the apparent horizon, which location is given by ˙ S(t, rAH) = 0. While this measure depends on the time slicing of the AdS metric, it can be determined locally in time (as opposed to the event horizon, which depends on the full future spacetime). Also, this time slicing ambiguity in the definition of the entropy can be compared with similar ambiguities in a field theory farfrom-equilibrium [72]. 5 Results As described in the introduction, the system is prepared in a thermal initial state at time t0<0 and then subjected to the boundary metric pulse of eq. (2.1). For weak coupling λ=O(1), the evolution of the system is solved using the kinetic theory methods described in section 3, while for strong coupling λ→ ∞, gauge gravity methods described in section 4 were used. In figure 2, the time evolution of the energy density is shown for the case of a pulse profile given by eq. (2.2) with α= 8. As expected, the energy density of the system drops at late times consistent with the expansion of the system. For decreasing values of the coupling λ∼1, the kinetic theory simulation approaches the analytic free-streaming result given in eq. (3.5). For strong coupling λ→ ∞, the result is somewhat closer to the analytic ideal hydrodynamics result eq. (2.8) than the kinetic theory result for intermediate coupling λ= 10, but does not coincide with the ideal hydrodynamic result since viscous corrections do not vanish even for λ=∞. – 14 –
JHEP04(2016)031 0.01 0.1 1 0 5 10 15 20 25 30 ε(t)/εinitial t Ti Energy density Ideal hydro λ= ∞ λ=10 λ= 5 λ= 2 Free-streaming Figure 2. Time evolution of the energy density from kinetic theory (λ= 2,5,10) and the gauge/gravity duality (λ=∞). For reference, the analytic results for non-interacting particles (λ= 0, “free-streaming”) and ideal hydrodynamics (“ideal hydro”) are also plotted. 1 1.2 1.4 1.6 1.8 2 0 5 10 15 20 25 30 s(t)/sinitial*f1/2(t) t Ti Total Equilibrium Entropy Ideal hydro late time NS late time BRSSS λ= ∞ λ=10 λ=5 Free-streaming 0 0.2 0.4 0.6 0.8 1 1.2 0 10 20 30 40 50 60 PL/PT t Ti Pressure anisotropy Figure 3. Time evolution of the total system equilibrium entropy (left) and pressure anisotropy (right). Shown are results from kinetic theory (λ= 5,10) and gauge/gravity duality (λ=∞). For reference, the analytic results for non-interacting particles (λ= 0, “free-streaming”), ideal hydrodynamics (“ideal hydro”) as well as the late-time gradient expansion to first-order (NS) and second order (BRSSS) hydrodynamics with transport coefficients from table 1are also shown. This difference to ideal hydrodynamics is highlighted when plotting the total equilibrium entropy and pressure anisotropy, defined in eqs. (2.9), (2.5), as done in figure 3. In this figure, results for λ= 5,10,∞are plotted along with the ideal hydrodynamics and free-streaming results. For the equilibrium entropy one finds that with the exception of the non-interacting case (free-streaming), all curves tend to a constant value for t→ ∞, which quantifies the amount of entropy production during the evolution process. Determining the asymptotic value of entropy corresponds to fixing the parameter χin eqs. (2.13), (2.15). For the pressure anisotropy we clearly see that the systems equilibrate towards isotropy, which allows us to define an isotropization time tiso as the last time when PL/PT= 0.8. – 15 –
JHEP04(2016)031 Both panels in figure 3also include the curves which follow from late time hydrodynamics, both for first order (“NS”) and second-order (“BRSSS”) hydrodynamics, as given by eqs. (2.13) and (2.16) respectively, whereby we use the transport coefficients from table 1.2While the evolution for λ= 5 and 10 shows that at sufficiently early times the hydrodynamic and kinetic results are clearly different, it is somewhat surprising to see that for λ=∞, there seems to be almost perfect matching after the metric pulse has passed at tTi≃1. This seems to indicate that for λ=∞, the system never actually leaves thermal equilibrium for the type of perturbation studied here (cf. the discussion in ref. [74]). In order to quantitatively study how well the system is described by hydrodynamics we define the first (tNS), and second order (tBRSSS) hydrodynamization times as the time when the first or second order late time hydrodynamic result agrees with the PL/PT within some fiducial range. As the deviations are not monotonic, we also demand that the hydrodynamical expression is within the fiducial range at all later times, whereby we take this range to be 5%. We report the values of all times and transport coefficients mentioned above in table 1. We now first explore the scaling with λof the various quantities extracted, after which we plot a (rescaled) version of figure 3in figure 8, in order to highlight the observed trends. In figure 4the η/s values are plotted as a function of the coupling λ. We find that the analytic leading-log formula of weak coupling SU(3) from ref. [73], η sSU(3),λ1=34.784 λ2log h4.789/√λi accurately captures the kinetic theory result up to λ.5. The kinetic theory results from table 1nicely connect to the N= 4 SYM result for λ→ ∞using the empirical interpolation formula η s≃0.08 + 22λ−1.6.(5.1) However, the result for N= 4 SYM including strong coupling corrections from ref. [76], η sN=4,λ1=1 4π1 + 15ζ(3)λ−3/2≈0.08 + 1.4λ−3/2, significantly underestimates the slope of the kinetic theory η/s values for λ > 10. This behavior has been discussed before in ref. [32], where it was suggested to change the identification of λwhen comparing N= 4 SYM to pure Yang-Mills or QCD. At weak coupling we may estimate the parametric dependence of the equilibration process as a function λor η/s. If the coupling is sufficiently small, the system exhibits large scale separations admitting us to parametrically model the evolution as a three stage process: 2The values of η/s in table 1have been extracted from the late behaviour of stress-energy tensor in our current setup. The values agree with the original calculation of [73] within 10%. The two calculations differ from each other in the way the soft divergence is regulated. Both of these calculations are accurate to leading order but differ at subleading orders in λ, and therefore correspond to different possible definitions of leading order. The small discrepancy between two results can be understood as an estimate of the systematic theory uncertainty introduced in the kinetic theory at finite λ. – 16 –
JHEP04(2016)031 0.1 1 10 0 0.2 0.4 0.6 0.8 1 η/s 1/λ Shear viscosity over entropy ratio η/s 0.08+22 λ-1.6 N=4 strong coupling SU(3) weak coupling Figure 4. Shear viscosity over entropy density for weakly coupled SU(3) [73], for strongly coupled N= 4 SYM [75,76], compared to the values in table 1and the empirical interpolation formula (5.1). See text for details. λ= 1.0 λ= 2.0 λ= 3.0 λ= 4.0 λ= 5.0 λ= 7.5 λ= 10.0 λ=∞ η/s 24.7 7.84 4.09 2.59 1.81 0.966 0.624 0.0796 Cτ5.4 5.3 5.3 5.2 5.2 5.2 5.1 2.6 Cλ4.5 4.3 4.3 4.2 4.2 4.1 4.1 2 tiso(η/s)−4/3176 174 176 178 178 180 181 142 tNSTi6740 387 204 126.5 83 37 21 2.1 tBRSSSTi7596 418 120 73 48 23 13.5 1.7 χ6.29 4.00 3.13 2.66 2.36 1.97 1.71 1.20 Table 1. Summary of transport parameters and derived quantities for SU(N) gauge theory for various values of λas well as for N= 4 SYM for λ=∞. The η/s values are extracted from the late time behaviour of the energy momentum tensor, while the second order parameters are taken from refs. [77–79]. χ=Seq(t→ ∞)/Seq,i is the total (original plus viscously produced) entropy and tiso is the isotropization time. tNS,BRSSS refer to equilibration times from first and second-order hydrodynamics, respectively (see text for details). •at early times tTi<1, the system is in thermal equilibrium, •for 1 < tTi< teqTithe system exhibits free streaming behaviour and is highly anisotropic PLP⊥, •and for t>teq the system has re-equilibrated and follows inviscid hydrodynamics with PL=PT. – 17 –
JHEP04(2016)031 We expect the system to smoothly change its behaviour from free streaming type evolution to hydrodynamical evolution in the time scale determined by the transport mean free time, teq ∼1/λ2T(teq).3Naively the parametric dependence of this equation is ∝λ−2, however the expansion reduces the local energy density of the system and therefore also the target temperature T(teq) to which the system aspires to thermalize. For a freely streaming anisotropic system the energy density evolves as (t)∼i/(Tit), and therefore the target temperature at time tis T(t)∼T3/4 it−1/4. Solving now self-consistently the condition that the system time be of the same order of magnitude as the transport mean free time leads to Titeq ∼λ−8/3∼(η/s)4/3, T(teq)∼λ2/3Ti∼(η/s)−1/3Ti(5.2) and for the total entropy generation during the second stage χ∼Titeq. T(teq)3 T3 i∼1 λ2/3∼(η/s)1/3,(5.3) where η/s ∝λ−2was used. At strong coupling one has η s1, and as a consequence the viscous entropy production can be calculated from the full hydrodynamic evolution equations (2.11) in an arbitrary background g(t). Specifically, when solving eq. (2.11) perturbatively in η s1, one finds χ= 1 + η 3sZ∞ −∞ dt Tig0(t) g2 g1/6(t)≃1+2.0η s,(5.4) where the specific form of g(t) from eq. (2.2) with α= 8 was used to calculate the numerical value of 2.0 in eq. (5.4). Based on the weak and strong coupling results in eqs. (5.3), (5.4) for χ, a model function that obeys both these limits is given by χ≃1+7.0η s1/3,(5.5) where the value 7.0 was adjusted to match the results for χat weak coupling. In figure 5we compare the total entropy generation χfor the weak and strong coupling simulations. We first note that all points follow a monotonous growing curve. The parametric model with (η/s)1/3describes well the scaling of all the kinetic theory points. Extrapolating the model to smaller values eventually predicts isentropic evolution for η/s ≈0.13 when the duration of the second stage goes to zero, thereby clearly signalling the breakdown of the weak coupling picture. It is quite intriguing that the value where the weak coupling theory predicts its own failure happens to be surprisingly close to the strong coupling value of η/s = 0.08. Unlike the parametric model, strong coupling saturates the interactions and the entropy generation remains finite and positive, as born out by the interpolation function (5.5). 3For very small values of λa large scale separation develops between Tiand T(teq) and this estimate should be replaced with the LPM suppressed rate (Ti/T )1/2/λ2Tleading to slightly different power laws [21,31]. Here, in the numerical simulations we do not probe small enough values of λfor this to be numerically relevant. – 18 –
JHEP04(2016)031 0.1 1 10 100 η/s 1 2 4 8 χ = S/Si kinetic AdS (1+7.0η/s)1/3 2.0 (η/s)1/3 Total entropy production Figure 5. Scaling of the total entropy production during the non-equilibrium evolution. The dashed line corresponds to a parametric expectation based on weak coupling picture χ∝(η/s)1/3, while the strong coupling expectation predicts χ−1∝η/s. The empirical result from (5.5) (full line) satisfies both limits. Next, we study the isotropization times by plotting the anisotropy ratio PL/PTin figure 6as function of the rescaled time variable (η/s)−4/3Tit. Upon rescaling, all the kinetic theory simulations approximately collapse onto a single curve, and thus all approach isotropy at same rescaled time. This approach to isotropy can be seen to be governed by viscous hydrodynamics, whereby from (2.16) it is clear that PL PTNS,tTi1 = 1 −8η s 1 (tTi)2/3χ1/3≈1−81 ((η/s)−4/3tTi)2/3η/s 1+7η/s1/9 .(5.6) where we used eq. (5.5). For large viscosity this formula simplifies, which explains why the weak coupling evolutions follow the universal attractor shown in figure 6. Eq. (5.6) can be solved for our fiducial value of 0.8 to give Titiso ≈154 . . . 183 (η/s)4/3,(5.7) for a viscosity between 1/4πand ∞, which compares well with our numerical results as shown in figure 7. Lastly, we focus on hydrodynamization times in figures 8and 9. Figure 8shows the deviation of PLfrom the late time hydro prediction of eqs. (2.13), (2.16) normalized by the transverse pressure. On the one hand, we again observe that the strong coupling simulation is well described by hydrodynamics immediately after the metric pulse has passed. On the other hand we see that the kinetic theory simulations exhibit a breakdown from hydrodynamics roughly at the same time scale of Tit∼40(η/s)4/3. We note that the correspondence with hydrodynamics is slightly improved when the second order coefficients are taken into account, in particular at larger couplings. We note that the exact values of the hydrodynamization times can depend quite strongly on the fiducial range due to – 19 –
JHEP04(2016)031 0 100 200 300 Rescaled time: (η/s)-4/3Tit 0 0.2 0.4 0.6 0.8 1 PL/PT λ=1 λ=2 λ=3 λ=4 λ=5 λ=7.5 λ=10 λ=∞ Pressure anisotropy Figure 6. Pressure anistropy with time rescaled by the weak coupling estimate for the thermalization time. All the simulations, including λ=∞, follow a universal attractor, given by eq. (5.6), towards thermal equilibrium. 0.01 0.1 1 10 100 η/s 1 100 10000 Titiso Kinetic AdS 170 (η/s)4/3 Isotropization time Figure 7. Isotropization time defined by the condition PL/PT= 0.8 as a function of η/s. The parametric model tisoTi∝(η/s)4/3describes the kinetic theory values extremely well and extrapolates to the strong coupling value within 25%. the non-monotonic approach to hydrodynamics. Nevertheless, the overall scaling thyd ∝ (η/s)4/3remains present even when varying the range. We have not yet commented on the generality of our results for different values of α in our metric pulse. We verified that for α= 4 our results change by less than 1%. For significantly faster pulses with α8, however, the calculation starts to differ because of the Hawking radiation generated by the pulse. For α= 16 this leads for instance to χ= 1.28 as compared to χ= 1.20 for α= 8 for λ=∞. For the weak coupling framework we did not include Hawking radiation, which is indeed not needed for the profiles we considered. – 20 –
JHEP04(2016)031 −0.2 0 0.2 0.4 0.6 10 100 t Tini (η/s)−4/3 ( PL − PL (NS) ) / PT λ = ∞ λ =10.0 λ = 7.5 λ = 5.0 λ = 4.0 λ = 3.0 λ = 2.0 λ = 1.0 −0.2 0 0.2 0.4 0.6 10 100 t Tini (η/s)−4/3 ( PL − PL (BRSSS) ) / PT λ = ∞ λ =10.0 λ = 7.5 λ = 5.0 λ = 4.0 λ = 3.0 λ = 2.0 λ = 1.0 Figure 8. The deviation of longitudinal pressure from the late time hydrodynamics prediction including the first (left) or the second (right) order terms in the hydrodynamical expansion. 0.01 0.1 1 10 100 η/s 1 100 10000 TitNS and TitBRSSS Kinetic, tNS Kinetic, tBRSSS AdS, tNS AdS, tBRSSS 40(η/s)4/3 Hydrodynamization time tNS tBRSSS Figure 9. Hydrodynamization times as defined by the time when the late time hydrodynamics of eqs. (2.13), (2.16) reproduce the PL/PTratio of the simulation within 5%. The circles correspond to the Navier-Stokes whereas the crosses are second order BRSSS. 6 Conclusions In the present work we have presented a detailed and consistent comparison of the equilibration process of a gauge theory at strong and weak coupling using the same set-up and analysis procedure. Our main conclusion is that while there certainly are differences in the thermalization of these two very different theories, there are also some surprising similarities. We find that the weak coupling thermalization process can be characterized with a simple parametric picture predicting the dependence of of thermalization time and entropy production as a function of the coupling constant λ, or equivalently η/s ∝λ−2. Furthermore, extrapolating the powerlaw model to strong couplings where the parametric picture fails, it is surprising that we still found qualitative agreement even to strong coupling simulations. While at the quantitative level this may be a numerical coincidence, it demonstrates the overall similarities of the thermalization processes both at weak and at strong coupling. – 21 –
JHEP04(2016)031 The present study is probably too simplistic to be directly applicably for heavy-ion phenomenology. However, it provides evidence that treating weak and strong coupling equilibration on the same footing can lead to simple power-law results that smoothly interpolate between weak and strong coupling. Such interpolation functions may be used to effectively estimate viscosity, equilibration time and viscous entropy production (among others) at intermediate values of the coupling where neither the kinetic nor the gauge/gravity approach are applicable. For instance, for our gauge theory with λ≃20, our present study would predict η/s ≃0.3, a hydrodynamization time of τTi≃7 and a viscous entropy production χ−1 of approximately 40 percent. By repeating our methodological approach for a setup applicable to heavy-ion collisions, our goal for future work is to obtain similar quantitative predictions that would then be directly testable when confronted with precision experimental data. Acknowledgments We would like to thank H. Bantilan, P. Chesler, T. Ishii, K. Kajantie, A. Mazeliauskas, K. Rajagopal, K. Skenderis, M. Spalinski, D. Teaney, U. Wiedemann, and C. Wu for fruitful discussions. This work was supported by the Department of Energy, DOE award No. DESC0008132. WS is supported by the U.S. Department of Energy under grant Contract Number DE-SC0011090. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. References [1] PHENIX collaboration, K. Adcox et al., Formation of dense partonic matter in relativistic nucleus-nucleus collisions at RHIC: Experimental evaluation by the PHENIX collaboration, Nucl. Phys. A 757 (2005) 184 [nucl-ex/0410003] [INSPIRE]. [2] BRAHMS collaboration, I. Arsene et al., Quark gluon plasma and color glass condensate at RHIC? The Perspective from the BRAHMS experiment,Nucl. Phys. A 757 (2005) 1 [nucl-ex/0410020] [INSPIRE]. [3] STAR collaboration, J. Adams et al., Experimental and theoretical challenges in the search for the quark gluon plasma: The STAR Collaboration’s critical assessment of the evidence from RHIC collisions,Nucl. Phys. A 757 (2005) 102 [nucl-ex/0501009] [INSPIRE]. [4] B.B. Back et al., The PHOBOS perspective on discoveries at RHIC,Nucl. Phys. A 757 (2005) 28 [nucl-ex/0410022] [INSPIRE]. [5] ALICE collaboration, Elliptic flow of charged particles in Pb-Pb collisions at 2.76 TeV, Phys. Rev. Lett. 105 (2010) 252302 [arXiv:1011.3914] [INSPIRE]. [6] CMS collaboration, Multiplicity and transverse momentum dependence of twoand four-particle correlations in pPb and PbPb collisions,Phys. Lett. B 724 (2013) 213 [arXiv:1305.0609] [INSPIRE]. – 22 –
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