Proper time expansions and glasma dynamics

TL;DR

The paper tackles the problem of extending the validity window of proper time expansions for the glasma EMT in the early, highly occupied gluon phase of ultrarelativistic heavy-ion collisions. It assesses three strategies—Li & Kapusta two-scale expansion, Padé approximants, and machine learning—to push the reliable time range beyond the conventional eighth order, with quantified improvements. LK can extend some observables when nuclear-structure effects are small but suppresses derivative terms essential for structure-dependent physics; Padé extrapolants and ML offer more robust extensions, with Padé achieving reliability up to $Q_s\tau \approx 1.5$ (about $0.15$ fm/$c$) for energy density and ML providing access to higher-order coefficients with percent-level uncertainties. These methods collectively improve the utility of glasma-based initial conditions for subsequent hydrodynamic evolution and enhance our understanding of early-time dynamics, although each has limitations depending on the observable of interest.

Abstract

The earliest phase of an ultrarelativistic heavy ion collision can be described as a highly populated system of gluons called glasma. The system's dynamics is governed by the classical Yang-Mills equation. Solutions can be found at early times using a proper time expansion. Since the expansion parameter is the time, this method is necessarily limited to the study of early time dynamics. In addition compute time and memory limitations restrict practical calculations to no more than eighth order in the expansion. The result is that the method produces reliable results only for very early times. In this paper we explore several different methods to increase the maximum time that can be reached. We find that, depending slightly on the quantity being calculated, the latest time for which reliable results are obtained can be extended approximately 1.5 times (from $\sim0.05$~fm/$c$ using previous methods to about $0.08$~fm/$c$).

Figures (9)

• Figure 1: The quantity $A_{TL}$ versus $\Lambda\tau$ in the full theory and the leading order approximation to it, at sixth order in the proper time expansion, with $R=b=0$, using different values of $(\Lambda,Q_s)$ (indicated by the numbers in the square brackets, in GeV). The cyan and light green lines lie on top of each other.
• Figure 2: The quantity $A_{TL}$ versus $\Lambda\tau$ using the leading order LK approximation at different orders in the proper time expansion using $Q_s=1.2$ GeV and two values of $\Lambda$: 8 GeV (solid) and 6 GeV (dotted) with $R=b=0$.
• Figure 3: The quantity $P$ versus $\Lambda\tau$ for different values of $Q_s/\Lambda$ as shown in the legend [in GeV]. The solid lines are the full calculation and the dotted lines are the approximation.
• Figure 4: The quantity $P$ versus $\Lambda\tau$ using the LK approximation at LO (dashed lines)and NLO (solid lines) with $\Lambda=4$ GeV and $Q_s=2.0$ GeV up to ninth order in the proper time expansion. The black line shows the result from the full EMT at fifth order.
• Figure 5: The energy density at eight order (red markers) and the result from the second and fourth order Padé approximates. The blue band shows the difference between the two approximates with maximum value of the data set to $\tau Q_s= 0.45$ and the green band is obtained with a maximum of 0.60.