The Speed of Sound in Hadronic Matter

TL;DR

This paper calculates the speed of sound in a hadronic resonance gas with a Hagedorn spectrum, revealing critical behavior at the phase transition temperature and analyzing the effects of a resonance mass cutoff.

Contribution

It introduces a model with a Hagedorn spectrum to study the speed of sound and examines how a mass cutoff affects critical behavior near the phase transition.

Findings

Speed of sound vanishes as (T_c - T)^{1/4} near T_c

Finite pressure and energy density at T_c with a diverging specific heat

Mass cutoff diminishes the critical behavior in the resonance gas

Abstract

We calculate the speed of sound $c_s$ in an ideal gas of resonances whose mass spectrum is assumed to have the Hagedorn form $\rho(m) \sim m^{-a}\exp{bm}$, which leads to singular behavior at the critical temperature $T_c = 1/b$. With $a = 4$ the pressure and the energy density remain finite at $T_c$, while the specific heat diverges there. As a function of the temperature the corresponding speed of sound initially increases similarly to that of an ideal pion gas until near $T_c$ where the resonance effects dominate causing $c_s$ to vanish as $(T_c - T)^{1/4}$. In order to compare this result to the physical resonance gas models, we introduce an upper cut-off M in the resonance mass integration. Although the truncated form still decreases somewhat in the region around $T_c$, the actual critical behavior in these models is no longer present.