08 3 v 2 2 6 O ct 2 00 6 D ensity resum m ation ofperturbation series in a pion gas to leading order in chiralperturbation theory

The m ean eld (M F)approxim ation forthe pion m atter,being equivalentto the Leading ChPT order, involves no dynam icalloops and, if self-consistent, produces nite renorm alizations only. The weight factor of the Haar m easure of the pion elds, entering the path integral, generates an e ective Lagrangian LH which is generally singular in the continuum lim it. There exists one param eterization ofthe pion eldsonly,forwhich the weightfactorisequalto unity and LH = 0, respectively. This unique param eterization ensures selfconsistency ofthe M F approxim ation. W e useittocalculatetherm alG reen functionsofthepion gasin theM F approxim ation asapowerseries overthetem perature.TheBoreltransform softherm alaveragesofa function J ( )ofthepion elds with respectto the scalarpion density are found to be 2 p J (4t).The perturbation series overthescalarpion density forbasiccharacteristicsofthepion m attersuch asthepion propagator, thepion opticalpotential,the scalarquark condensate < qq> ,thein-m edium pion decay constant ~ F ,and theequation ofstateofpion m atterappearto beasym ptoticones.Theseseriesaresum m ed up using the contour-im proved Borelresum m ation m ethod.The quark scalarcondensate decreases sm oothly untilTm ax ’ 310 M eV.The tem perature Tm ax is the m axim um tem perature adm issible for therm alized non-linearsigm a m odelat zero pion chem icalpotentials. The estim ate ofTm ax is above the chem icalfreeze-outtem perature T ’ 170 M eV atRHIC and above the phase transition to twoavorquark m atterTc ’ 175 M eV,predicted by lattice gauge theories.