Charge and isospin fluctuations in a non-ideal pion gas with dynamically fixed particle number

TL;DR

This paper investigates charge and isospin fluctuations in a non-ideal pion gas with fixed particle number, analyzing their behavior near Bose-Einstein condensation and how they depend on isospin composition.

Contribution

It derives general expressions for cross-covariances in a pion gas with arbitrary isospin composition and analyzes fluctuation behavior near the critical temperature for Bose-Einstein condensation.

Findings

Charge variance diverges at the critical temperature for equal isospin species.

Total particle number variance remains finite at the critical point.

Fluctuations depend strongly on isospin composition above the critical temperature.

Abstract

We study the behavior of the non-ideal pion gas with the dynamically fixed number of particles, formed on an intermediate stage in ultra-relativistic heavy-ion collisions. The pion spectrum is calculated within the self-consistent Hartree approximation. General expressions are derived for cross-covariances of the number of various particle species in the pion gas of an arbitrary isospin composition. The behavior of the cross-variances is analyzed for the temperature approaching from above the maximal critical temperature of the Bose-Einstein condensation for the pion species $a=\pm,0$, i.e. for $T>\max T_{\rm cr}^a$. It is shown that in case of the system with equal averaged numbers of isospin species, the variance of the charge, $Q=N_{+}-N_{-}$, diverges at $T\to T_{\rm cr}=T_{\rm cr}^a$, whereas variances of the total particle number, $N=N_{+} + N_{-} + N_{0}$, and of a relative abundance of charged and neutral pions, $G=(N_{+}+N_{-})/2 - N_{0}$, remain finite in the critical point. Then fluctuations are studied in the pion gas with small isospin imbalance $0<|G|\ll N$ and $0<|Q|\ll N$\, and shifts of the effective masses, chemical potentials and values of critical temperatures are calculated for various pion species, and the highest critical temperature, $\mbox{max}T_{\rm cr}^{{a}}$ is found, above which the pion system exists in the non-condensed phase. Various pion cross variances are calculated for $T>\mbox{max}T_{\rm cr}^{{a}}$, which prove to be strongly dependent on the isospin composition of the system, whereas the variances of $N$ and $G$ are found to be independent on the isospin imbalance up to the term linear in $G/N$ and $Q/N$.