Statistical Properties of the Linear Sigma Model
The statistical equilibrium properties of the linear sigma model are studied, with a view towards characterizing the field configurations employed as initial conditions for numerical simulations of the formation of disoriented chiral condensates in high-energy nuclear collisions. The field is decomposed into its spatial average (the order parameter) and the fluctuations (the quasi- particles) and enclosed in a rectangular box with periodic boundary conditions. The quantized quasi-particle modes are described approximately by Klein-Gordon dispersion relations containing an effective mass that depends on both the temperature and the magnitude of the order parameter. The thermal fluctuations are instrumental in shaping the effective potential governing the order parameter, and the evolution of its statistical distribution with temperature 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 non-equilibrium relaxation of the chiral field. The corresponding correlation function is briefly considered and used to calculate the spectral strength of radiated pions. Finally, by propagating samples of initial configurations by the exact equation of motion, it has been ascertained that the treatment is sufficiently accurate to be of practical utility.