Effective equation of state for a spherically expanding pion plasma

Following a relativistic heavy ion collision, the quark-gluon plasma produced eventually undergoes a chiral phase transition. We assume that during this phase transition one can describe the dynamics of the system by the linear $\sigma$ model and that the expansion can be thought of as mostly radial. Because the $\sigma$ model is an effective field theory there is an actual momentum cutoff (Landau pole) in the theory at around 1 GeV. Thus it is necessary to find ways of obtaining a covariantly conserved, renormalized energy-momentum tensor when there is a cutoff present (which breaks covariance), in order to identify the effective equation of state of this time evolving system. We show how to solve this technical problem and then determine the energy density and pressure of the system as a function of the proper time. We consider different initial conditions and search for instabilities which can lead to the formation of disoriented chiral condensates (DCCs). We find that the energy density and pressure both decrease quickly, as is appropriate for a rapidly cooling system, and that the energy is numerically conserved.