Prescaling and Far-from-Equilibrium Hydrodynamics in the Quark-Gluon Plasma
Prescaling and Far-from-Equilibrium Hydrodynamics in the Quark-Gluon Plasma
Mazeliauskas, Aleksas;Berges, Jürgen;
2019-03-26 00:00:00
PHYSICAL REVIEW LETTERS 122, 122301 (2019) * † Aleksas Mazeliauskas and Jürgen Berges Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany (Received 8 November 2018; revised manuscript received 5 February 2019; published 26 March 2019) 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 spatiotemporal 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. DOI: 10.1103/PhysRevLett.122.122301 Introduction.—Universal scaling phenomena play an simulations [10]. The exponents are expected to be important role in our understanding of the thermalization α ¼ −2=3, β ¼ 0, and γ ¼ 1=3 according BMSS BMSS BMSS process in quantum many-body systems. Topical applica- to the first stage of the “bottom up” thermalization scenario tions range from heavy-ion collisions [1–3] to quenches in [1] based on number-conserving and small-angle scatter- ultracold quantum gases [4–7]. Starting far from equilib- ings, or α ¼ −3=4, β ¼ 0, and γ ¼ 1=4 in a variant BD BD BD rium, these systems exhibit transiently a nonthermal fixed- of “bottom up” including the effects of plasma instabil- point regime where the time evolution of characteristic ities [17]. quantities becomes self-similar. Consequently, details In this Letter, we compute the evolution of the quark- about initial conditions and underlying system parameters gluon plasma approaching the nonthermal fixed point using become irrelevant in this regime, and the nonequilibrium leading-order QCD kinetic theory [18]. Since this state-of- dynamics is encoded in universal scaling exponents and the-art description involves elastic and inelastic processes, functions [2,4,8–12]. the conservation of particle number is not built in, and no In heavy-ion collisions at sufficiently high energies, where small-angle approximation is assumed [19,20]. the gauge coupling is small due to asymptotic freedom We establish that the far-from-equilibrium dynamics [13,14], the time evolution of gluons (g) and quarks (q)is according to leading-order QCD kinetic theory exhibits described by distribution functions f ðp ;p ; τÞ. Since the self-similar scaling. Comparing to the dynamics with g;q ⊥ z elastic scatterings only, the softer-momentum regions are system is longitudinally expanding, the distributions depend on transverse (p ) and longitudinal momenta (p ), and on efficiently populated by collinear radiation processes, ⊥ z proper time (τ) [15,16]. In the scaling regime, the gluon which is seen to improve the universal scaling behavior distribution obeys of the distributions. Most remarkably, we find that much before the scaling scaling α β γ [Eq. (1)] with universal exponents is established, the f ðp ;p ; τÞ¼ τ f ðτ p ; τ p Þ; ð1Þ g ⊥ z S ⊥ z evolution is already governed by the fixed-point distribu- tion f as with dimensionless τ → τ=τ and p → p =Q in ref ⊥;z ⊥;z s terms of some (arbitrary) time τ and characteristic ref prescaling αðτÞ βðτÞ γðτÞ momentum scale Q . The exponents α, β, and γ are f ðp ;p ; τÞ¼ τ f ðτ p ; τ p Þ; ð2Þ g ⊥ z S ⊥ z universal, and the nonthermal fixed-point distribution f is universal up to normalizations [2], which has been with nonuniversal time-dependent exponents αðτÞ, βðτÞ, established numerically using classical-statistical lattice and γðτÞ. This represents a dramatic reduction in complex- ity already at this early stage: The entire evolution in this prescaling regime is encoded in the time dependence of a Published by the American Physical Society under the terms of few slowly evolving exponents, and we point out the the Creative Commons Attribution 4.0 International license. relation to hydrodynamic behavior far from equilibrium. Further distribution of this work must maintain attribution to The phenomenon of prescaling describes the rapid the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP . establishment of universal nonequilibrium results for 0031-9007=19=122(12)=122301(6) 122301-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 122, 122301 (2019) certain quantities (f ), though others still deviate from their universal values (α, β, γ). This has to be distinguished from standard corrections due to finite size or time scaling behavior, from which asymptotic universal values are inferred without taking the infinite volume or 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 FIG. 1. Pressure anisotropy P =P for a longitudinally ex- L T 2↔2 1↔2 panding QCD plasma with overoccupied gluon initial state. ∂ f ðp;τÞ− ∂ f ðp;τÞ¼−C ½f− C ½f; ð3Þ τ g;q p g;q g;q g;q where ν ¼ 2ðN − 1Þ¼ 16 and ν ¼ 2N ¼ 6 for N ¼ 3 g c q c c using the numerical setup developed in Refs. [19,20]. Here colors and N ¼ 3 quark flavors. One observes how the 2↔2 C ½f represents the collision integrals for leading-order g;q pressure anisotropy starts to deviate from collisionless elastic scatterings in the coupling α ≡ g =ð4πÞ. This expansion (dashed line) relevant at earliest times and bends involves scatterings gg ↔ gg, qq ↔ qq, gq ↔ gq as well over to a milder power-law dependence on time once as particle conversion gg ↔ qq¯ processes. To this order we interactions start to compete with expansion, in agreement 1↔2 also include number-changing processes C ½f of g;q with previous results using different approximations medium induced collinear gluon radiation g ↔ gg, [10,31,32]. For comparison, we also show the result by q ↔ qg and quark pair production g ↔ qq¯. The effective taking only elastic collisions into account (dotted curve). 1 ↔ 2 splitting rate is calculated by the resummation of The difference from the evolution with a full collision multiple interactions with the medium and includes the kernel is comparably small, indicating that the inelastic Landau-Pomeranchuk-Migdal suppression of collinear processes contribute mainly at low momenta, which are radiation [24–27]. The soft momentum exchange is regu- phase-space-suppressed for bulk quantities such as lated by isotropic screening [19,20]. pressure. From free streaming to universal scaling.— The emergence of scaling can be efficiently analyzed We consider the initial distributions f ðpÞ¼ g;q from moments of the distribution functions for gluons 2 2 2 2 A expf−ðp þ ξ p Þ=Q g, where A ¼ 0.5 for quarks g;q ⊥ z s q 2 Z and A ¼ σ =g for gluons. For energetic collisions, the g 0 d p m n n ðτÞ ≡ ν p jp j f ðp ;p ; τÞ; ð5Þ bosonic gluons are expected to be highly occupied fðQ Þ ∼ m;n g z g ⊥ z 3 ⊥ ð2πÞ 1=g with a characteristic momentum scale Q , while fermion occupancies are bounded by Fermi-Dirac statistics and equivalently for quarks. Effectively, different moments −3 [28,29]. Here σ is taken to be σ ¼ 0.1, 0.6 and g ¼ 10 0 0 probe different momenta and are thus sensitive to scaling in in view of the range of validity of kinetic theory and in a particular momentum regime. If a distribution function order to make clean comparisons with previous lattice shows prescaling (2), the precise values of αðτÞ, βðτÞ, and simulations [30]. The initial anisotropy is controlled by ξ, γðτÞ in general depend on the history of the evolution from and we employ ξ ¼ 2. Starting at τ Q ¼ 70 and choosing 0 s the reference time, which is normalized to 1, and the final τ Q ¼ 7000, we solve the coupled set of kinetic equa- ref s time τ. Therefore we redefine the exponents in Eq. (2) to tions (3) for the gluon and quark distributions numeri- reflect the instantaneous scaling properties with cally [19,20]. In Fig. 1, we show the evolution of the pressure dτ αðτÞ 0 anisotropy P =P (solid curve) as a function of dimension- τ → exp αðτ Þ ; ð6Þ L T less time τ → τ=τ for σ ¼ 0.1. The longitudinal and ref 0 transverse pressures are defined from the energy-momen- which for constant α reduces to the power law τ . Then the tum tensor rate of change of a particular moment n is given by a m;n Z linear combination of scaling exponents 3 μ ν d p p p μν T ðτÞ¼ ðν f ðp; τÞþ 2N ν f ðp; τÞÞ g g f q q 3 0 ð2πÞ p d log n ðτÞ m;n ¼ αðτÞ − ðm þ 2ÞβðτÞ − ðn þ 1ÞγðτÞ: ð7Þ ¼ diagðe; P ;P ;P Þ; ð4Þ d log τ T T L 122301-2 PHYSICAL REVIEW LETTERS 122, 122301 (2019) FIG. 2. Time-dependent scaling exponents from multiple sets FIG. 3. The same as Fig. 2, but for σ ¼ 0.6. of integral moments for gluon density parameter σ ¼ 0.1. We see that with a full collision kernel, the low- By looking at different moments n, m ¼ 0; 1;…, one momentum part of the distribution function develops a obtains a set of algebraic equations from which αðτÞ, ∼1=p behavior. In contrast, only elastic processes are βðτÞ, and γðτÞ can be determined. Since the choice of not efficient in developing these thermallike features of a moments is not unique, one can probe different momentum low-momentum bath (grey dashed curves) [35]. The softer- regimes and test how well (pre)scaling is realized. momentum region is efficiently populated by the collinear The time-dependent exponents obtained from various radiation processes, and we observe excellent scaling combinations of moments n with n, m< 4 for initial n;m properties also in that regime where particle-number- conditions with σ ¼ 0.1 are shown in Fig. 2, exhibiting a changing processes are essential. remarkable overlap of the results from different moment Prescaling states that the very same distribution function ratios [33]. In this case, one expects free-streaming scaling f can be extracted at much earlier times, before the scaling exponents α → 0, β → 0, γ → 1 at very early times. exponents take on their universal values. To verify this, we Accordingly, both αðτÞ and γðτÞ approach the nonthermal rescale the distribution function according to Eq. (2) using fixed-point limit from above for initial conditions with the time-dependent exponents from Fig. 2 and relation (6). σ ¼ 0.1 [34]. At later times τ > 1, we fit the power laws As shown in the right panel of Fig. 4, the rescaled distribution with constant exponents and obtain α≈−0.73, β ≈−0.01, collapses to a single scaling curve even at early times. As can and γ ≈ 0.29. These values are close to both the analytic be seen from Fig. 1, the time-dependent exponents of Fig. 2 values of the BMSS and BD estimates given above and along with the universal scaling form f can be established consistent with previous lattice results within errors [10]. already at a time where the bulk quantity P =P still appears L T One clearly observes the prescaling regime, for which to be deep in the free-streaming regime. different moments can be described by the common set of A corresponding analysis can be done for the longi- time-dependent scaling exponents even before the asymp- tudinal momentum dependence. Figure 5 displays the totic scaling is reached. However, to emphasize that the γ γðτÞ rescaled distribution as a function of τ p and of τ p z z time dependence of the exponents is not universal, we show at different times, and we neglect the nearly vanishing in Fig. 3 the results for larger initial gluon density σ ¼ 0.6 transverse momentum exponent β. Again, a much earlier such that free streaming is suppressed. In this case we see collapse of the curves is observed if time-dependent that at very early times τ < 0.03, there is no unique notion exponents are used. of scaling exponents. But very quickly the results from Like for the case with small-angle scattering approxima- different sets of moments collapse again to a single curve, tion [32], we find that the quarks exhibit similar scaling much before the exponents attain their universal constant behavior as for gluons at late times for the part of the values. distribution function not bounded by the Pauli exclusion Universal scaling form of the distributions.—With the principle. In Fig. 6, we show the fermion distribution along results for exponents, we can now extract the universal the longitudinal momenta and p =Q ¼ 1. Although the ⊥ s scaling form f . We first consider rescaling with the time-dependent exponents capture most of the longitudinal constant values of exponents obtained from the late-time squeeze of the distribution function, the scaling form of the fit. The left panel of Fig. 4 shows the rescaled gluon fermion distribution function is not established as well as for −α 2 distribution τ g f as a function of p at different times τ g ⊥ gluons. Because gluons are highly occupied, the quark for p ¼ 0 (solid lines) for initial conditions with σ ¼ 0.1. z 0 contribution to the total particle number is small at these After an initial period, all rescaled curves at different times times. Therefore, the background evolution of gluons does collapse to a single scaling curve. not change noticeably in the presence of quarks in this regime. 122301-3 PHYSICAL REVIEW LETTERS 122, 122301 (2019) −α FIG. 4. Left: Gluon distribution rescaled with τ versus transverse momentum for the full QCD collision kernels (solid) and with elastic scatterings only (dashed). Right: The same distribution, but rescaled with a time-dependent scaling exponent. Far-from-equilibrium hydrodynamic behavior.—It is For the case of homogeneous boost-invariant expansion remarkable to observe that the dynamics in the prescaling of a conformal system, Eqs. (8) and (9) have only three regime can be explained by the distribution function independent components, namely the particle-number den- 0 0xx 0yy 0zz rescaling [Eq. (2)] with three slowly changing exponents sity n ≡ J and the tensors 2I ð¼ 2I Þ and I , which αðτÞ, βðτÞ, and γðτÞ. This is precisely the situation one together with T evolve according to encounters in hydrodynamics, which is an effective description in terms of a few slowly varying degrees of 00 zz n T þ T ∂ n þ ¼ −C ; ∂ T þ ¼ 0; freedom. To make this link more concrete, we consider the τ J τ τ τ μν energy-momentum tensor T of Eq. (4) along with the 0xx 0zz μ μνσ I 3I 0xx xx 0zz zz particle-number current J and a rank-three tensor I , ∂ I þ ¼ −C ; ∂ I þ ¼ −C ; ð10Þ τ τ I I τ τ 3 μ d p p 3 3 0 J ¼ ν f ; ð8Þ g p 3 0 with the collision integrals C ¼ ν d p=ðð2πÞ p ÞC½f J g ð2πÞ p νσ 3 3 0 ν σ and C ¼ ν d p=ðð2πÞ p Þp p C½f. Noting that I g 3 μ ν σ 0xx 0zz d p p p p n ¼ n , 2I ¼ n , and I ¼ n , we see that the 0;0 2;0 0;2 μνσ I ¼ ν f ; ð9Þ g p 3 0 equations of motion for these moments can be mapped to ð2πÞ p the same number of slowly varying scaling exponents using where we focus on the gluonic part, i.e., ν ¼ 0. Integrating Eq. (7). For a given distribution function f , which is an input from the far-from-equilibrium QCD computation near the kinetic equation (3) with the appropriate powers of p yields the equations of motion for these quantities. FIG. 6. Left: Fermion distribution versus longitudinal momen- FIG. 5. Left: Gluon distribution versus longitudinal momentum tum τ p . Right: Same, but rescaled using the time-dependent τ p . Right: Same, but with time-dependent exponents. gluon scaling exponents employed also in Fig. 2. 122301-4 PHYSICAL REVIEW LETTERS 122, 122301 (2019) the nonthermal attractor, the scaling dependence of the * a.mazeliauskas@thphys.uni-heidelberg.de zz xx collision kernels C , C , and C then closes the system of J I I berges@thphys.uni-heidelberg.de hydrodynamic equation of motions. [1] R. Baier, A. H. Mueller, D. Schiff, and D. T. Son, “Bottom What is special in comparison to more conventional up” thermalization in heavy ion collisions, Phys. Lett. B hydrodynamics descriptions, which describe the evolution 502, 51 (2001). with respect to thermal equilibrium, is that our far-from- [2] J. Berges, K. Boguslavski, S. Schlichting, and R. 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Prescaling and Far-from-Equilibrium Hydrodynamics in the Quark-Gluon Plasma
Prescaling and Far-from-Equilibrium Hydrodynamics in the Quark-Gluon Plasma
Abstract
PHYSICAL REVIEW LETTERS 122, 122301 (2019) * † Aleksas Mazeliauskas and Jürgen Berges Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany (Received 8 November 2018; revised manuscript received 5 February 2019; published 26 March 2019) 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 spatiotemporal 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. DOI: 10.1103/PhysRevLett.122.122301 Introduction.—Universal scaling phenomena play an simulations [10]. The exponents are expected to be important role in our understanding of the thermalization α ¼ −2=3, β ¼ 0, and γ ¼ 1=3 according BMSS BMSS BMSS process in quantum many-body systems. Topical applica- to the first stage of the “bottom up” thermalization
PHYSICAL REVIEW LETTERS 122, 122301 (2019) * † Aleksas Mazeliauskas and Jürgen Berges Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany (Received 8 November 2018; revised manuscript received 5 February 2019; published 26 March 2019) 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 spatiotemporal 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. DOI: 10.1103/PhysRevLett.122.122301 Introduction.—Universal scaling phenomena play an simulations [10]. The exponents are expected to be important role in our understanding of the thermalization α ¼ −2=3, β ¼ 0, and γ ¼ 1=3 according BMSS BMSS BMSS process in quantum many-body systems. Topical applica- to the first stage of the “bottom up” thermalization scenario tions range from heavy-ion collisions [1–3] to quenches in [1] based on number-conserving and small-angle scatter- ultracold quantum gases [4–7]. Starting far from equilib- ings, or α ¼ −3=4, β ¼ 0, and γ ¼ 1=4 in a variant BD BD BD rium, these systems exhibit transiently a nonthermal fixed- of “bottom up” including the effects of plasma instabil- point regime where the time evolution of characteristic ities [17]. quantities becomes self-similar. Consequently, details In this Letter, we compute the evolution of the quark- about initial conditions and underlying system parameters gluon plasma approaching the nonthermal fixed point using become irrelevant in this regime, and the nonequilibrium leading-order QCD kinetic theory [18]. Since this state-of- dynamics is encoded in universal scaling exponents and the-art description involves elastic and inelastic processes, functions [2,4,8–12]. the conservation of particle number is not built in, and no In heavy-ion collisions at sufficiently high energies, where small-angle approximation is assumed [19,20]. the gauge coupling is small due to asymptotic freedom We establish that the far-from-equilibrium dynamics [13,14], the time evolution of gluons (g) and quarks (q)is according to leading-order QCD kinetic theory exhibits described by distribution functions f ðp ;p ; τÞ. Since the self-similar scaling. Comparing to the dynamics with g;q ⊥ z elastic scatterings only, the softer-momentum regions are system is longitudinally expanding, the distributions depend on transverse (p ) and longitudinal momenta (p ), and on efficiently populated by collinear radiation processes, ⊥ z proper time (τ) [15,16]. In the scaling regime, the gluon which is seen to improve the universal scaling behavior distribution obeys of the distributions. Most remarkably, we find that much before the scaling scaling α β γ [Eq. (1)] with universal exponents is established, the f ðp ;p ; τÞ¼ τ f ðτ p ; τ p Þ; ð1Þ g ⊥ z S ⊥ z evolution is already governed by the fixed-point distribu- tion f as with dimensionless τ → τ=τ and p → p =Q in ref ⊥;z ⊥;z s terms of some (arbitrary) time τ and characteristic ref prescaling αðτÞ βðτÞ γðτÞ momentum scale Q . The exponents α, β, and γ are f ðp ;p ; τÞ¼ τ f ðτ p ; τ p Þ; ð2Þ g ⊥ z S ⊥ z universal, and the nonthermal fixed-point distribution f is universal up to normalizations [2], which has been with nonuniversal time-dependent exponents αðτÞ, βðτÞ, established numerically using classical-statistical lattice and γðτÞ. This represents a dramatic reduction in complex- ity already at this early stage: The entire evolution in this prescaling regime is encoded in the time dependence of a Published by the American Physical Society under the terms of few slowly evolving exponents, and we point out the the Creative Commons Attribution 4.0 International license. relation to hydrodynamic behavior far from equilibrium. Further distribution of this work must maintain attribution to The phenomenon of prescaling describes the rapid the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP . establishment of universal nonequilibrium results for 0031-9007=19=122(12)=122301(6) 122301-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 122, 122301 (2019) certain quantities (f ), though others still deviate from their universal values (α, β, γ). This has to be distinguished from standard corrections due to finite size or time scaling behavior, from which asymptotic universal values are inferred without taking the infinite volume or 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 FIG. 1. Pressure anisotropy P =P for a longitudinally ex- L T 2↔2 1↔2 panding QCD plasma with overoccupied gluon initial state. ∂ f ðp;τÞ− ∂ f ðp;τÞ¼−C ½f− C ½f; ð3Þ τ g;q p g;q g;q g;q where ν ¼ 2ðN − 1Þ¼ 16 and ν ¼ 2N ¼ 6 for N ¼ 3 g c q c c using the numerical setup developed in Refs. [19,20]. Here colors and N ¼ 3 quark flavors. One observes how the 2↔2 C ½f represents the collision integrals for leading-order g;q pressure anisotropy starts to deviate from collisionless elastic scatterings in the coupling α ≡ g =ð4πÞ. This expansion (dashed line) relevant at earliest times and bends involves scatterings gg ↔ gg, qq ↔ qq, gq ↔ gq as well over to a milder power-law dependence on time once as particle conversion gg ↔ qq¯ processes. To this order we interactions start to compete with expansion, in agreement 1↔2 also include number-changing processes C ½f of g;q with previous results using different approximations medium induced collinear gluon radiation g ↔ gg, [10,31,32]. For comparison, we also show the result by q ↔ qg and quark pair production g ↔ qq¯. The effective taking only elastic collisions into account (dotted curve). 1 ↔ 2 splitting rate is calculated by the resummation of The difference from the evolution with a full collision multiple interactions with the medium and includes the kernel is comparably small, indicating that the inelastic Landau-Pomeranchuk-Migdal suppression of collinear processes contribute mainly at low momenta, which are radiation [24–27]. The soft momentum exchange is regu- phase-space-suppressed for bulk quantities such as lated by isotropic screening [19,20]. pressure. From free streaming to universal scaling.— The emergence of scaling can be efficiently analyzed We consider the initial distributions f ðpÞ¼ g;q from moments of the distribution functions for gluons 2 2 2 2 A expf−ðp þ ξ p Þ=Q g, where A ¼ 0.5 for quarks g;q ⊥ z s q 2 Z and A ¼ σ =g for gluons. For energetic collisions, the g 0 d p m n n ðτÞ ≡ ν p jp j f ðp ;p ; τÞ; ð5Þ bosonic gluons are expected to be highly occupied fðQ Þ ∼ m;n g z g ⊥ z 3 ⊥ ð2πÞ 1=g with a characteristic momentum scale Q , while fermion occupancies are bounded by Fermi-Dirac statistics and equivalently for quarks. Effectively, different moments −3 [28,29]. Here σ is taken to be σ ¼ 0.1, 0.6 and g ¼ 10 0 0 probe different momenta and are thus sensitive to scaling in in view of the range of validity of kinetic theory and in a particular momentum regime. If a distribution function order to make clean comparisons with previous lattice shows prescaling (2), the precise values of αðτÞ, βðτÞ, and simulations [30]. The initial anisotropy is controlled by ξ, γðτÞ in general depend on the history of the evolution from and we employ ξ ¼ 2. Starting at τ Q ¼ 70 and choosing 0 s the reference time, which is normalized to 1, and the final τ Q ¼ 7000, we solve the coupled set of kinetic equa- ref s time τ. Therefore we redefine the exponents in Eq. (2) to tions (3) for the gluon and quark distributions numeri- reflect the instantaneous scaling properties with cally [19,20]. In Fig. 1, we show the evolution of the pressure dτ αðτÞ 0 anisotropy P =P (solid curve) as a function of dimension- τ → exp αðτ Þ ; ð6Þ L T less time τ → τ=τ for σ ¼ 0.1. The longitudinal and ref 0 transverse pressures are defined from the energy-momen- which for constant α reduces to the power law τ . Then the tum tensor rate of change of a particular moment n is given by a m;n Z linear combination of scaling exponents 3 μ ν d p p p μν T ðτÞ¼ ðν f ðp; τÞþ 2N ν f ðp; τÞÞ g g f q q 3 0 ð2πÞ p d log n ðτÞ m;n ¼ αðτÞ − ðm þ 2ÞβðτÞ − ðn þ 1ÞγðτÞ: ð7Þ ¼ diagðe; P ;P ;P Þ; ð4Þ d log τ T T L 122301-2 PHYSICAL REVIEW LETTERS 122, 122301 (2019) FIG. 2. Time-dependent scaling exponents from multiple sets FIG. 3. The same as Fig. 2, but for σ ¼ 0.6. of integral moments for gluon density parameter σ ¼ 0.1. We see that with a full collision kernel, the low- By looking at different moments n, m ¼ 0; 1;…, one momentum part of the distribution function develops a obtains a set of algebraic equations from which αðτÞ, ∼1=p behavior. In contrast, only elastic processes are βðτÞ, and γðτÞ can be determined. Since the choice of not efficient in developing these thermallike features of a moments is not unique, one can probe different momentum low-momentum bath (grey dashed curves) [35]. The softer- regimes and test how well (pre)scaling is realized. momentum region is efficiently populated by the collinear The time-dependent exponents obtained from various radiation processes, and we observe excellent scaling combinations of moments n with n, m< 4 for initial n;m properties also in that regime where particle-number- conditions with σ ¼ 0.1 are shown in Fig. 2, exhibiting a changing processes are essential. remarkable overlap of the results from different moment Prescaling states that the very same distribution function ratios [33]. In this case, one expects free-streaming scaling f can be extracted at much earlier times, before the scaling exponents α → 0, β → 0, γ → 1 at very early times. exponents take on their universal values. To verify this, we Accordingly, both αðτÞ and γðτÞ approach the nonthermal rescale the distribution function according to Eq. (2) using fixed-point limit from above for initial conditions with the time-dependent exponents from Fig. 2 and relation (6). σ ¼ 0.1 [34]. At later times τ > 1, we fit the power laws As shown in the right panel of Fig. 4, the rescaled distribution with constant exponents and obtain α≈−0.73, β ≈−0.01, collapses to a single scaling curve even at early times. As can and γ ≈ 0.29. These values are close to both the analytic be seen from Fig. 1, the time-dependent exponents of Fig. 2 values of the BMSS and BD estimates given above and along with the universal scaling form f can be established consistent with previous lattice results within errors [10]. already at a time where the bulk quantity P =P still appears L T One clearly observes the prescaling regime, for which to be deep in the free-streaming regime. different moments can be described by the common set of A corresponding analysis can be done for the longi- time-dependent scaling exponents even before the asymp- tudinal momentum dependence. Figure 5 displays the totic scaling is reached. However, to emphasize that the γ γðτÞ rescaled distribution as a function of τ p and of τ p z z time dependence of the exponents is not universal, we show at different times, and we neglect the nearly vanishing in Fig. 3 the results for larger initial gluon density σ ¼ 0.6 transverse momentum exponent β. Again, a much earlier such that free streaming is suppressed. In this case we see collapse of the curves is observed if time-dependent that at very early times τ < 0.03, there is no unique notion exponents are used. of scaling exponents. But very quickly the results from Like for the case with small-angle scattering approxima- different sets of moments collapse again to a single curve, tion [32], we find that the quarks exhibit similar scaling much before the exponents attain their universal constant behavior as for gluons at late times for the part of the values. distribution function not bounded by the Pauli exclusion Universal scaling form of the distributions.—With the principle. In Fig. 6, we show the fermion distribution along results for exponents, we can now extract the universal the longitudinal momenta and p =Q ¼ 1. Although the ⊥ s scaling form f . We first consider rescaling with the time-dependent exponents capture most of the longitudinal constant values of exponents obtained from the late-time squeeze of the distribution function, the scaling form of the fit. The left panel of Fig. 4 shows the rescaled gluon fermion distribution function is not established as well as for −α 2 distribution τ g f as a function of p at different times τ g ⊥ gluons. Because gluons are highly occupied, the quark for p ¼ 0 (solid lines) for initial conditions with σ ¼ 0.1. z 0 contribution to the total particle number is small at these After an initial period, all rescaled curves at different times times. Therefore, the background evolution of gluons does collapse to a single scaling curve. not change noticeably in the presence of quarks in this regime. 122301-3 PHYSICAL REVIEW LETTERS 122, 122301 (2019) −α FIG. 4. Left: Gluon distribution rescaled with τ versus transverse momentum for the full QCD collision kernels (solid) and with elastic scatterings only (dashed). Right: The same distribution, but rescaled with a time-dependent scaling exponent. Far-from-equilibrium hydrodynamic behavior.—It is For the case of homogeneous boost-invariant expansion remarkable to observe that the dynamics in the prescaling of a conformal system, Eqs. (8) and (9) have only three regime can be explained by the distribution function independent components, namely the particle-number den- 0 0xx 0yy 0zz rescaling [Eq. (2)] with three slowly changing exponents sity n ≡ J and the tensors 2I ð¼ 2I Þ and I , which αðτÞ, βðτÞ, and γðτÞ. This is precisely the situation one together with T evolve according to encounters in hydrodynamics, which is an effective description in terms of a few slowly varying degrees of 00 zz n T þ T ∂ n þ ¼ −C ; ∂ T þ ¼ 0; freedom. To make this link more concrete, we consider the τ J τ τ τ μν energy-momentum tensor T of Eq. (4) along with the 0xx 0zz μ μνσ I 3I 0xx xx 0zz zz particle-number current J and a rank-three tensor I , ∂ I þ ¼ −C ; ∂ I þ ¼ −C ; ð10Þ τ τ I I τ τ 3 μ d p p 3 3 0 J ¼ ν f ; ð8Þ g p 3 0 with the collision integrals C ¼ ν d p=ðð2πÞ p ÞC½f J g ð2πÞ p νσ 3 3 0 ν σ and C ¼ ν d p=ðð2πÞ p Þp p C½f. Noting that I g 3 μ ν σ 0xx 0zz d p p p p n ¼ n , 2I ¼ n , and I ¼ n , we see that the 0;0 2;0 0;2 μνσ I ¼ ν f ; ð9Þ g p 3 0 equations of motion for these moments can be mapped to ð2πÞ p the same number of slowly varying scaling exponents using where we focus on the gluonic part, i.e., ν ¼ 0. Integrating Eq. (7). For a given distribution function f , which is an input from the far-from-equilibrium QCD computation near the kinetic equation (3) with the appropriate powers of p yields the equations of motion for these quantities. FIG. 6. Left: Fermion distribution versus longitudinal momen- FIG. 5. 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