Investigation of zero-frequency solutions to the pion dispersion equation

Abstract

The instability of nuclear matter is considered for the case where it is generated by the vanishing of the frequencies of collective excitations belonging to specific types (specifically, excitations that have the pion quantum numbers J π = 0−). The behavior of zero-frequency solutions to the pion dispersion equation is analyzed versus the strength G′ of spin—isospin particle—hole interaction. It is shown that there exists a strength value G′tr (|G′tr| ≪ 1) such that, for G′ < G′tr, zero-frequency solutions are excitations of the ω P type, while, for G′ ≥ G′tr, such solutions are excitations of the ω c type. Excitations of the ω P type for G′ < −1 describe the instability of nuclear matter against small density fluctuations (Pomeranchuk’s instability), while excitations of the ω c type are responsible for the instability associated with pion condensation at G′ ≈ 2. For stable nuclear matter, the solutions ω P(κ) and ω c (κ) lie on unphysical sheets of the complex plane of frequency.