Density resummation of perturbation series in a pion gas to leading order in chiral perturbation theory

The mean field (MF) approximation for the pion matter, being equivalent to the leading order chiral perturbation theory, involves no dynamical loops and, if self-consistent, produces finite renormalizations only. The weight factor of the Haar measure of the pion fields, entering the path integral, generates an effective Lagrangian δLH which is generally singular in the continuum limit. There exists only one parametrization of the pion fields, for which the weight factor is equal to unity and δLH=0, respectively. This unique parametrization ensures self-consistency of the MF approximation. We use it to calculate thermal Green functions of the pion gas in the MF approximation as a power series over the density. The Borel transforms of thermal averages of a function J(χαχα) of the pion fields χα with respect to the scalar pion density are found to be (2/√(π))J(4t). The perturbation series over the scalar pion density for basic characteristics of the pion matter, such as the pion propagator, the pion optical potential, the scalar quark condensate , the in-medium pion decay constant F-tilde, and the equation of state of pion matter, appear to be asymptotic ones. These series are summed up using the contour-improved Borel resummation method. The quark scalar condensate decreases smoothly until Tmax≅310 MeV. The temperature Tmax is the maximum temperature admissible for the thermalized nonlinear sigma model at zero pion chemical potentials. The estimate of Tmax is above the chemical freeze-out temperature T≅170 MeV in relativistic heavy-ion collisions and above the phase transition to two-flavor quark matter Tc≅175 MeV, predicted by lattice gauge theories