Prescaling and far-from-equilibrium hydrodynamics in the quark-gluon plasma

Prescaling is a far-from-equilibrium phenomenon which describes the rapid establishment of a universal scaling form of distributions much before the universal values of their scaling exponents are realized. We consider the example of the spatio-temporal evolution of the quark-gluon plasma explored in heavy-ion collisions at sufficiently high energies. Solving QCD kinetic theory with elastic and inelastic processes, we demonstrate that the gluon and quark distributions very quickly adapt a self-similar scaling form, which is independent of initial condition details and system parameters. The dynamics in the prescaling regime is then fully encoded in a few time-dependent scaling exponents, whose slow evolution gives rise to far-from-equilibrium hydrodynamic behavior.

Prescaling is a far-from-equilibrium phenomenon which describes the rapid establishment of a universal scaling form of distributions much before the universal values of their scaling exponents are realized. We consider the example of the spatio-temporal evolution of the quark-gluon plasma explored in heavy-ion collisions at sufficiently high energies. Solving QCD kinetic theory with elastic and inelastic processes, we demonstrate that the gluon and quark distributions very quickly adapt a self-similar scaling form, which is independent of initial condition details and system parameters. The dynamics in the prescaling regime is then fully encoded in a few time-dependent scaling exponents, whose slow evolution gives rise to far-from-equilibrium hydrodynamic behavior.
PACS numbers: 11.10.Wx, Introduction. Universal scaling phenomena play an important role in our understanding of the thermalization process in quantum many-body systems. Topical applications range from heavy-ion collisions [1][2][3] to quenches in ultracold quantum gases [4][5][6][7]. Starting far-fromequilibrium, these systems exhibit transiently a nonthermal fixed point regime where the time evolution of characteristic quantities becomes self-similar. As a consequence, details about initial conditions and underlying system parameters become irrelevant in this regime, and the nonequilibrium dynamics is encoded in universal scaling exponents and functions [2,4,[8][9][10][11][12].
In heavy-ion collisions at sufficiently high energies, where the gauge coupling is small due to asymptotic freedom [13,14], the time evolution of gluons (g) and quarks (q) is described by distribution functions f g,q (p ⊥ , p z , τ ). Since the system is longitudinally expanding, the distributions depend on transverse (p ⊥ ) and longitudinal momenta (p z ), and on proper time (τ ) [15,16]. In the scaling regime the gluon distribution obeys with dimensionless τ → τ /τ ref and p ⊥,z → p ⊥,z /Q s in terms of some (arbitrary) time τ ref and characteristic momentum scale Q s . The exponents α, β and γ are universal and the nonthermal fixed-point distribution f S is universal up to normalizations [2], which has been established numerically using classical-statistical lattice simulations [10]. The exponents are expected to be α BMSS = −2/3, β BMSS = 0 and γ BMSS = 1/3 according to the first stage of the 'bottom up' thermalization scenario [1] based on number conserving and small-angle scatterings and neglecting inelastic processes at the same order in the coupling, or α BD = −3/4, β BD = 0 and γ BD = 1/4 in a variant of 'bottom-up' including the effects of plasma instabilities [17].
In this work we compute the evolution of the quarkgluon plasma approaching the nonthermal fixed-point using leading-order QCD kinetic theory [18]. Since this state-of-the-art description involves elastic and inelastic processes, conservation of particle number is not build in, and no small-angle approximation is assumed. In its range of validity it goes beyond classical-statistical approximations by including quantum effects that capture the late-time approach to thermal equilibrium [3,19,20].
We establish that the far-from-equilibrium dynamics according to leading-order QCD kinetic theory exhibits self-similar scaling. Comparing to the dynamics with elastic scatterings only, the softer momentum regions are efficiently populated by collinear radiation processes, which is seen to improve the universal scaling behavior of the distributions.
Most remarkably, we find that much before the scaling (1) with universal exponents is established, the evolution is already governed by the fixed-point distribution f S as with non-universal time-dependent exponents α(τ ), β(τ ) and γ(τ ). This represents a dramatic reduction in complexity already at this early stage: The entire evolution in this prescaling regime is encoded in the time dependence of a few slowly evolving exponents, and we point out the relation to hydrodynamic behavior far from equilibrium. The phenomenon of prescaling describes the rapid establishment of universal nonequilibrium results for certain quantities (f S ), though others still deviate from their universal values (α, β, γ). This has to be distinguished from standard corrections due to finite size/time scaling behavior, from which asymptotic universal values are inferred without taking the infinite volume/time limit. Coined by Wetterich based on the notion of partial fixed points [21,22], prescaling has recently been explored in the context of scaling violations in the short-distance behavior of correlation functions for Bose gases [23].
QCD kinetic theory. We employ the leading order QCD kinetic theory of Ref. [18] to evolve the gluon and quark distributions by and equivalently for f q with collision integrals C 2↔2 q and C 1↔2 g , using the numerical setup of Refs. [19,20,24]. Here C 2↔2 g,q [f ] represent the collision integrals for leadingorder elastic scatterings in the coupling α s ≡ g 2 /(4π). This involves scatterings gg ↔ gg, qq ↔ qq, gq ↔ gq as well as particle conversion gg ↔ qq processes. To this order we also include number changing processes C 1↔2 g,q [f ] of medium induced collinear gluon radiation g ↔ gg, q ↔ qg and quark pair production g ↔ qq. The effective 1 ↔ 2 splitting rate is calculated by the resummation of multiple interactions with the medium and includes the Landau-Pomeranchuk-Migdal suppression of collinear radiation [25][26][27][28]. The soft momentum exchange is regulated by isotropic screening [29].
From free-streaming to universal scaling. We consider the initial distributions f 0 g,q (p) = A g,q exp{−(p 2 ⊥ + ξ 2 p 2 z )/Q 2 s }, where A q = 0.5 for quarks and A g = σ 0 /g 2 for gluons. This reflects that for energetic collisions the bosonic gluons are expected to be highly occupied ∼ 1/g 2 with a characteristic momentum scale Q s , while fermion occupancies are bounded f q ≤ 1 by Fermi-Dirac statistics [30,31]. Here σ 0 characterizes the initial gluon density, taken to be σ 0 = 0.1, 0.6, and g = 10 −3 in view of the range of validity of kinetic theory. The initial anisotropy is controlled by ξ, and we employ ξ = 2. Starting at τ 0 Q s = 70 and choosing τ ref Q s = 7000, we solve the coupled set of kinetic equations (3) for the gluon and quark distributions numerically [32].
In Fig. 1, we show the evolution of the pressure anisotropy P L /P T (solid curve) as a function of dimensionless time τ → τ /τ ref for σ 0 = 0.1. The longitudinal and transverse pressures are defined from the energymomentum tensor where ν g = 2(N 2 c − 1) = 16 and ν q = 2N c = 6 for N c = 3 Time-dependent scaling exponents from multiple sets of integral moments for gluon density parameter σ0 = 0.1.
colors and N f = 3 quark flavors. One observes how the pressure anisotropy starts to deviate from collisionless expansion (dashed line) relevant at earliest times, and bends over to a milder power-law dependence on time once interactions start to compete with expansion. This is in line with previous results using different approximations [10,19,33] and, for comparison, we also show the result by taking only elastic collisions into account (dotted curve). The difference to the evolution with full collision kernel is comparably small, indicating that the inelastic processes contribute mainly at low momenta, which are phase-space suppressed for bulk quantities such as pressure.
The emergence of scaling can be efficiently analyzed from moments of the distribution functions for gluons and equivalently for quarks. Effectively, different moments probe different momenta and are thus sensitive to   scaling in a particular momentum regime. If a distribution function shows prescaling (2), the precise values of α(τ ), β(τ ) and γ(τ ) in general depend on the history of the evolution from the reference time 1 to τ . Therefore, it can be useful to redefine the exponents in (2) to reflect the instantaneous scaling properties with which for constant α reduces to the power law τ α . Then the rate of change of a particular moment n m,n is given by a linear combination of scaling exponents d log n m,n (τ ) d log τ = α(τ ) − (m + 2) β(τ ) − (n + 1) γ(τ ). (7) By looking at different moments n, m = 0, 1, . . . one obtains a set of algebraic equations from which α(τ ), β(τ ) and γ(τ ) can be determined. Since the choice of moments is not unique, one can probe different momentum regimes and test how well (pre-)scaling is realized. The time-dependent exponents obtained from various combinations of moments n n,m with n, m < 4 for initial conditions with σ 0 = 0.1 are shown in Fig. 2, exhibiting a remarkable overlap of the results from different moment ratios [34]. In this case, one expects free-streaming scaling exponents α → 0, β → 0, γ → 1 at very early times. Accordingly, both α(τ ) and γ(τ ) approach the nonthermal fixed-point limit from above for initial conditions with σ 0 = 0.1 [35]. At later times τ > 1 we fit the powerlaws with constant exponents and obtain α ≈ −0.73, β ≈ −0.01 and γ ≈ 0.29. These values are reasonably close to the analytic estimates given above and consistent with previous lattice results [10]. One clearly observes the prescaling regime, for which different moments can be described by the common set of time-dependent scaling exponents even before the asymptotic scaling is reached. To check the sensitivity of the prescaling regime to initial conditions, we varied the shape of initial gluon distribution, while keeping the initial gluon number and energy densities fixed. Only a moderate spread in exponent values was observed, comparable to the spread from the different sets of moments shown in Fig. 2. However, to emphasize that the time-dependence of the exponents is not universal, we show in Fig. 3 the results for larger initial gluon density σ 0 = 0.6 such that free streaming is suppressed. In this case we also see that at very early times τ < 0.03 there is no unique notion of scaling exponents. But very quickly the results from different sets of moments collapse again to a single curve, much before the exponents attain their universal constant values.
Universal scaling form of the distributions. With the results for exponents, we can now extract the universal scaling form f S . We first consider rescaling with the constant values of exponents obtained from the late time fit. The left panel of Fig. 4 shows the rescaled gluon distribution τ −α g 2 f g as a function of p ⊥ at different times τ for p z = 0 (solid lines) for initial conditions with σ 0 = 0.1. After an initial period, all rescaled curves at different times collapse to a single scaling curve to very good accuracy.
We see that with full collision kernel, the lowmomentum part of the distribution function develops a ∼ 1/p T behavior. In contrast, only elastic processes are not efficient in developing these thermal-like features of a low-momentum bath (grey dashed curves) [36]. The softer momentum region is efficiently populated by the collinear radiation processes, and we observe excellent scaling properties also in that regime where particle number changing processes are essential.
Prescaling states that the very same distribution function f S can be extracted at much earlier times, before the scaling exponents take on their universal values. To verify this, we rescale the distribution function according to  (2) using the time dependent exponents from Fig. 2 and relation (6). As shown in the right panel of Fig. 4 the rescaled distribution collapses to a single scaling curve even at early times. To judge how early this collapse is happening, it is instructive to consider again the evolution of the pressure anisotropy shown in Fig. 1. The time-dependent exponents of Fig. 2 along with the universal scaling form f S can be established already at a time, where the bulk quantity P L /P T still appears to be deep in the free-streaming regime. A corresponding analysis can be done for the longitudinal momentum dependence. Fig. 5 displays the rescaled distribution as a function of τ γ p z and τ γ(τ ) p z , respectively, at different times and we neglect the nearly vanishing transverse momentum exponent β. Again a much earlier collapse of the curves is observed if time-dependent exponents are used.
Like for the case with small-angle scattering approximation [33], we find that the quarks exhibit similar scaling behavior as for gluons at late times for the part of the distribution function not bounded by the Pauli exclusion principle. In Fig. 5 we show the fermion distribution along the longitudinal momenta and p ⊥ /Q s = 1. Although the time-dependent exponents capture most of the longitudinal squeeze of the distribution function, the scaling form of the fermion distribution function is not established as well as for gluons. Because the typical gluon momenta are highly occupied, the quark contribution to the total particle number is small at these times. Therefore, the background evolution of gluons does not change noticeably in the presence of quarks in this regime.
Far-from-equilibrium hydrodynamic behavior. It is remarkable to observe that the dynamics in the prescaling regime can be explained by the distribution function rescaling (2) with three slowly changing exponents α(τ ), β(τ ) and γ(τ ). This is precisely the situation one  encounters in hydrodynamics, which is an effective description in terms of a few slowly varying degrees of freedom. To make this link more concrete, we consider the energy-momentum tensor T µν of (4) along with the particle number current J µ and a rank-three tensor I µνσ , where we focus on the gluonic part, i.e. ν q = 0. Integrating the kinetic equation (3) with the appropriate powers of p µ yields the equations of motion for these quantities. For the case of homogeneous boost-invariant expansion of a conformal system, (8) and (9) have only three independent components, namely the particle number density n ≡ J 0 and the tensors 2I 0xx (= 2I 0yy ) and I 0zz , which together with T 00 evolve according to [37]. Noting that n = n 0,0 , 2I 0xx = n 2,0 and I 0zz = n 0,2 , we see that the equations of motion for these moments can be mapped to the scaling exponents using (7). Specifying the scaling dependence of the collision kernels C J , C xx I and C zz I based on the prescaling property (2) of the distributions then closes the system of hydrodynamic equation of motions. This generalized hydrodynamic description around the nonthermal fixed-point distribution f S resembles anisotropic hydrodynamic formulations, which are based on expansion around the deformed equilibrium distribution [38][39][40]. A major difference is that in our formulation the anisotropic distribution is a scaling solution which is a result of the far-from-equilibrium dynamics. We leave the solution of the generalized hydrodynamic equations to a subsequent study. Conclusion.
The scaling regime of far-fromequilibrium QGP near the nonthermal fixed point reflects an enormous simplification of the plasma's temporal evolution, which is governed only by the rescaling of the arguments and the overall normalization of the universal scaling distribution function f S . We point out that this loss of detail does not rely on the scaling exponents being constant, but imposes relations between different moments of the distribution function. Using QCD kinetic theory we demonstrate that longitudinally expanding weakly coupled QGP evolves transiently towards a nonthermal fixed-point distribution with universal scaling exponents, which we confirm in the presence of both elastic and particle number changing processes. The approach to the asymptotics in the prescaling regime is characterised by slowly changing scaling exponents, which give rise to the hydrodynamic description of the far-from-equilibrium system, which is not based on an approach to local thermal equilibrium. The subsequent evolution of QGP towards thermal equilibrium with QCD kinetic theory is the subject of a separate work [20].
Though we have focussed on prescaling in the quarkgluon plasma, the generalized prescaling relation (2) should be relevant to other far-from-equilibrium systems exhibiting scaling. For instance, earlier theoretical [4,5,12] and very recently also experimental [6,7] studies on quenches in nonequilibrium Bose gases may be interpreted along these lines, and it would be very interesting to revisit the results in view of our findings. In particular, our method of moments for the extraction of time-dependent exponents may also be efficiently applied in the analysis for the experiments.