Statistical properties of the linearσmodel used in dynamical simulations of DCC formation

The present work develops a simple approximate framework for initializing and interpreting dynamical simulations with the linear $\sigma$ model exploring the formation of disoriented chiral condensates in high-energy collisions. By enclosing the system in a rectangular box with periodic boundary conditions, it is possible to decompose uniquely the chiral field into its spatial average (the order parameter) and its fluctuations (the quasiparticles) which can be treated in the Hartree approximation. The quasiparticle modes are then described approximately by Klein-Gordon dispersion relations containing an effective mass depending on both the temperature and the magnitude of the order parameter; their fluctuations are instrumental in shaping the effective potential governing the order parameter, and the emerging statistical description is thermodynamicially consistent. The temperature dependence of the statistical distribution of the order parameter is discussed, as is the behavior of the associated effective masses; as the system is cooled, the field fluctuations subside, causing a smooth change from the high-temperature phase in which chiral symmetry is approximately restored towards the normal phase. Of practical interest is the fact that the equilibrium field configurations can be sampled in a simple manner, thus providing a convenient means for specifying the initial conditions in dynamical simulations of the nonequilibrium relaxation of the chiral field; in particular, the correlation function is much more realistic than those emerging in previous initialization methods. It is illustrated how such samples remain approximately invariant under propagation by the unapproximated equation of motion over times that are long on the scale of interest, thereby suggesting that the treatment is sufficiently accurate to be of practical utility.