Pion-pair formation and the pion dispersion relation in a hot pion gas

The possibility of pion--pair formation in a hot pion gas, based on the bosonic gap equation, is pointed out and discussed in detail. The critical temperature for condensation of pion pairs (Evans--Rashid transition) is determined as a function of the pion density. As for fermions, this phase transition is signaled by the appearance of a pole in the two--particle propagator. In bose systems there exists a second, lower critical temperature, associated with the appearance of the single--particle condensate. Between the two critical temperatures the pion dispersion relation changes from the usual quasiparticle dispersion to a Bogoliubov--like dispersion relation at low momenta. This generalizes the non-relativistic result for an attractive bose gas by Evans et al. Possible consequences for the inclusive pion spectra measured in heavy--ion collisions at ultra--relativistic energies are discussed.