Momentum dependence of the spectral functions in theO(4)linear sigma model at finite temperature

Abstract

The spatial momentum dependence of the spectral function for $\ensuremath{\pi}$ and $\ensuremath{\sigma}$ at finite temperature is studied by employing the $O(4)$ linear sigma model and adopting a resummation technique called optimized perturbation theory (OPT). The poles of the propagators are also searched for. We analyze the spatial momentum dependence of the imaginary part of the self-energy and find that its temperature-dependent part vanishes in the large momentum limit. This is because the energy of the particles in the heat bath which participate in the process becomes large and therefore the Bose distribution function vanishes. We then calculate the spectral functions and search for the poles of the propagators. First, we discuss the temperature dependence for zero spatial momentum. We reproduce the spectral functions in both \ensuremath{\pi} and $\ensuremath{\sigma}$ channels in the previous work and find that the pole, which is responsible for the threshold enhancement of the spectral functions in both \ensuremath{\pi} and \ensuremath{\sigma} channels, is different from the one for the peak at zero temperature. Second, we discuss the spatial-momentum dependence. When the temperature is low the spatial momentum dependence is also small. As the temperature becomes higher the spatial momentum dependence also becomes large. When the spatial momentum is smaller than the temperature, the spectral functions do not deviate much from those at zero spatial momentum. Once the spatial momentum becomes comparable with the temperature, the deviation becomes considerable. As the momentum becomes much larger than the temperature, thermal effects effectively decrease.