Proton-neutron mass difference and the pion mass in a gauge model

Abstract

In a variation of the Weinberg SU(2)×SU(2)×U(1) gauge model of the weak and electromagnetic interactions, we study the proton-neutron mass difference, which is calculable, and investigate the appearance of pions as part of the Higgs system. We find that the proton-neutron mass difference is a function of the way in which the symmetry is broken. We exhibit a possible symmetry breaking which produces the correct sign for the mass difference. In the Higgs sector, we have a mass-degenerate pseudoscalar triplet which interacts with nucleons as pions do in the SU(2)×SU(2) $\sigma$ model. Therefore we identify this triplet with pions. They are massive in zeroth order, but we can calculate the mass difference $\delta m^2$. We find that $\delta m^2$ is of order $\alpha {\mu }^2$ which is too large. If we impose a reflection symmetry on the Lagrangian, the symmetry group of the potential is enlarged and we find that the theory contains three pseudo-Goldstone bosons. These are the pion triplet, which are now massless in zeroth order. When we calculate the pion mass in the one-loop approximation, the ${\Pi }^0$ remains massless while the charged pions pick up mass of order $m^2\sim \alpha {\mu }^2$. This may perhaps be damped numerically to give a suitable estimate of the pion mass, but the mass difference is still too large.