Analytic and Approximate Solutions to Color Glass Condensate in the Classical Weak-Field Limit

Abstract

We discuss two-point functions and the energy momentum tensor of the classical gluon field after the collision of sheets of color charges on the light cone in the weak-field limit. The classical fields created by such a setup is thought to approximate the behavior of the gluon matter created right after the collision of heavy nuclei at large energies. Our discussion is based on a general expression for the gluon distribution in a nucleus, which contains the McLerran-Venugopalan (MV) Model as a special case. We derive the time-dependence of the energy momentum tensor in this general scenario. We show that the large-time behavior is universal, i.e.\ independent of the specific model for the gluon distribution, e.g.\ for energy density, transverse pressure and longitudinal pressure $\epsilon, P_T \sim 1/\tau$ and $P_L \sim 1/\tau^3$, where $\tau$ is longitudinal proper time. Subsequently, we focus on two special cases, the MV model and a proposed improved Gaussian (iG) model with improved ultraviolet (UV) and infrared (IR) behavior, the latter inspired by earlier work by Lam and Mahlon. We explicitly discuss the time dependence of the energy momentum tensor in both models. In the case of the MV-model, for infinite colliding nuclei, it is possible to give closed-formed analytic solutions for the energy momentum tensor in terms of special functions. Components of the energy momentum tensor take the form $\sim C (m\tau)^{-n} H(m\tau)$, where $n$ is an integer power, $m$ is the infrared cutoff, $H$ is a linear combination of Meijer-G functions with constant asymptotic value, and $C$ is a known constant. For the iG-model, we obtain reliable series expansions for both small and large times and show that the MV-model is recovered qualitatively in the UV limit. We briefly comment on implications for the angular momentum carried by the gluon field.