Effects of a neutral current on the neutrino bremsstrahlung

Abstract

The energy-loss rate due to the neutrino bremsstrahlung from relativistic electrons in a dense matter such as a star is calculated in the framework of Weinberg's unified theory of weak and electromagnetic interactions by using the method of neutrino-energy expansion. The momentum-transfer integration, though complicated, can be carried out analytically, but the subsequent two electron integrals, called the distribution integrals, are helpless and do not allow any further analytic calculation. In order to avoid this difficulty, we set up a set of variables whose integration is simple enough to be expanded in series. It turns out that this series becomes a power series in the Fermi energy inverse after the distribution integrals. This enables us to keep only the leading terms, greatly simplifying the final result. Thus in principle we will be able to calculate arbitrary higher-order terms, even though most of them require prohibitively tedious labor. We carry out the nexthigher-order calculations to find that the correction to the lowest-order contribution is within 5%. This confirms the earlier calculations of Festa and Ruderman, even though the two approaches are quite different: the neutrino-bremsstrahlung energy-loss rate is proportional to ${T}^{6}$ and independent of the density at extremely relativistic degenerate regions. We find that the asymptotic expression for the energy-loss rate is of the form $A(\ensuremath{\beta})+{(\frac{\ensuremath{\rho}}{{\ensuremath{\mu}}_{e}})}^{\ensuremath{-}\frac{1}{3}}kTB(\ensuremath{\beta})$. The most striking effect of the neutral current is the muon-neutrino emission. The ratio of the energy-loss rate in Weinberg's theory to that in the conventional theory is $\frac{1}{2}+4{{sin}^{4}\ensuremath{\theta}}_{W}$ at the extreme relativistic degenerate and nondegenerate nonrelativistic regions.