Elementary-particle symmetries in relativistic many-body theory at finite temperature and density

The consequences of a phase transition associated with symmetry restoration to SU(2)×SU(2) in nuclear matter are investigated. The changes in the mass spectrum due to the phase transition (a) at zero temperature and high nuclear density, and (b) at high temperature with zero nuclear chemical potential are evaluated in the $\sigma$ model of particle physics. The experimentally observable effects necessitate the measurement of current correlation functions. In this paper we consider the vector-vector-axial-vector ($\mathrm{VVA}$) and the vector-vector-pseudoscalar ($\mathrm{VVP}$) current correlation functions. The relation between the $\mathrm{VVA}$ and $\mathrm{VVP}$ correlation functions is obtained and it is shown that the Adler-Bell-Jackiw anomaly in the divergence of the axial-vector current remains unaltered in the nuclear medium, even though the $\mathrm{VVA}$ and the $\mathrm{VVP}$ correlation functions have additional contributions from processes specific to the many-particle system. The $\mathrm{VVP}$ correlation function is related to the neutral-pion decay amplitude. The changes in the decay rate of ${\pi }^0\to 2\gamma$ in the nuclear medium are evaluated by including the effects of changes in the mass spectrum of particles, and by using the cutting rules of many-body field theory for the real and imaginary parts of the amplitude. The photon in the nuclear medium is dressed by polarization effects and propagates as a plasmon. This effect is taken into account by evaluating the plasmon frequency or the plasmon mass. The changes in the mass spectrum due to symmetry restoration affect the decay rate of ${\pi }^0\to 2\gamma$ by at least 2 orders of magnitude and these results are tabulated. It is suggested that the Primakoff effect might provide the signal for the presence of the "abnormal nuclear matter" phase.