We consider the O(N ) linear σ model and introduce an auxiliary field to eliminate the scalar self-interaction. Using a suitable limiting process this model can be continuously transformed into the nonlinear version of the O(N ) model. We demonstrate that, up to two-loop order in the CJT formalism, the effective potential of the model with auxiliary field is identical to the one of the standard O(N ) linear σ model, if the auxiliary field is eliminated using the stationary values for the corresponding one-and two-point functions. We numerically compute the chiral condensate and the σ− and π−meson masses at nonzero temperature in the one-loop approximation of the CJT formalism. The order of the chiral phase transition depends sensitively on the choice of the renormalization scheme. In the linear version of the model and for explicitly broken chiral symmetry, it turns from crossover to first order as the mass of the σ particle increases. In the nonlinear case, the order of the phase transition turns out to be of first order. In the region where the parameter space of the model allows for physical solutions, Goldstone's theorem is always fulfilled.
A detailed study of the thermodynamics of the O(N = 3) model in 1+1 dimensions is presented, employing a two-particle-irreducible resummation prescription as well as fully nonperturbative finite-temperature lattice simulations. The analytical results are computed using the Cornwall-Jackiw-Tomboulis (CJT) formalism and the auxiliary field method to one-and to two-loop order. The lattice results are obtained through Monte Carlo simulation for various lattice spacings. The analytical and lattice results for pressure, trace anomaly, and energy density, resembling closely those of four-dimensional Yang-Mills theories, are compared with each other. We find that to one-loop order there is a good correspondence between the CJT formalism and the lattice study for low temperatures. However, at high T the two-loop calculation fares better, correcting for the overestimation from the former approximation.
We study the restoration of spontaneously broken symmetry at nonzero temperature in the framework of the O(2) model using polar coordinates. We apply the CJT formalism to calculate the masses and the condensate in the double-bubble approximation, both with and without a term that explicitly breaks the O(2) symmetry. We find that, in the case with explicitly broken symmetry, the mass of the angular degree of freedom becomes tachyonic above a temperature of about 300 MeV. Taking the term that explicitly breaks the symmetry to be infinitesimally small, we find that the Goldstone theorem is respected below the critical temperature. However, this limit cannot be performed for temperatures above the phase transition. We find that, no matter whether we break the symmetry explicitly or not, there is no region of temperature in which the radial and the angular degree of freedom become degenerate in mass. These results hold also when the mass of the radial mode is sent to infinity.
We use matrix models to characterize deconfinement at a nonzero temperature T for an SU (2) gauge theory in three spacetime dimensions. At one loop order, the potential for a constant vector potential A 0 is ∼ T 3 times a trilogarithm function of A 0 /T . In addition, we add various nonperturbative terms to model deconfinement. The parameters of the model are adjusted by fitting the lattice results for the pressure. The nonperturbative terms are dominated by a constant term ∼ T 2 T d , where T d is the temperature for deconfinement. Besides this constant, we add terms which are nontrivial functions of A 0 /T , both ∼ T 2 T d and ∼ T T 2 d . There is only a mild sensitivity to the details of these nonconstant terms. Overall we find a good agreement with the lattice results.For the pressure, the conformal anomaly, and the Polyakov loop the nonconstant terms are relevant only in a narrow region below ∼ 1.2 T d . We also compute the 't Hooft loop, and find that the details of the nonconstant terms enter in a much wider region, up to ∼ 4 T d .
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