Chiral symmetry and pion condensation. I. Model-dependent results

Abstract

The successes of current algebra and partial conservation of axial-vector current can be interpreted as indications that the strong interactions are approximately symmetric under chiral SU(2) × SU(2) transformations. In the absence of matter, the explicit chiral-symmetry breaking, which conserves isospin and gives the pion a small mass, defines a unique vacuum state for the theory. The smallness of the chiral-symmetry breaking, however, implies the existence of many other states, approximately degenerate in energy with, but orthogonal to, the true vaccum. In these states, which can be obtained formally from the vacuum by a chiral rotation, in general the expectation value of the $\pi$ field in the ground state, $〈\pi 〉$, ≠ 0; thus they contain a "pion condensate." At zero baryon density these states are umphysical, but in a macroscopic system—such as a neutron star—at very high baryon density one of these pion-condensed states might become the true ground state. In this first of a series of papers on the implications of chiral symmetry for pion condensation, we study this phenomenon in the linear $\sigma$ model. As a consequence of the simplicity of this model (and of a series of justifiable approximations) we are able to calculate analytically the "phase diagram" describing the ground state of infinite nuclear matter as a function of baryon density. We find that above a critical baryon density ${\rho }_c\simeq O\left({\rho }_{\mathrm{nucl}}\right)$ a phase transition to a pion-condensed ground state occurs. To correct the most obvious phenomenological deficiencies of this simple model we extend the chiral-symmetry approach to include the effects of the $N^{*}\mathrm{}\left(1236\right)\mathrm{}$ and the ill-understood $\pi N$ $s$ waves. Further, we indicate briefly additional affects which must be included in any serious quantitative description of real pion condensation. In addition to analyzing the ground state, we examine briefly the spectrum of excited states, an understanding of which is vital to the explanation of the cooling mechanism of neutron stars. In the condensed phase, the meson excitation spectrum contains a "Goldstone boson" associated with the ground state's not being an eigenstate of the conserved operator $I_3$. However, when electromagnetic interactions are included, this mode disappears via the Higgs phenomenon indicating that the ground state is a superconductor; thus only plasma excitations remain in the meson spectrum. The presence of the pion condensate fosters the $\beta$ decay of the fermion excitations, and the emitted neutrinos provide the cooling mechanism for neutron stars.