Induced Chern-Simons Terms at High Temperatures and Finite Densities

Abstract

The $\mathrm{CP}$-nonconserving portion of the gauge-field effective action for an even number of left-handed SU(2) fermion doublets, with nonzero chemical potentials ${\ensuremath{\mu}}^{i}$, is calculated to leading order in the inverse temperature, ${T}^{\ensuremath{-}1}$. It is shown to be $^{i}\ensuremath{\Sigma}_{i}(\frac{{\ensuremath{\mu}}^{i}}{T})W[A]$, where $W[A]$ is the Chern-Simons topological mass term. This result is also shown to be a consequence of the $\mathrm{U}{(1)}_{L}$ anomaly at zero temperature, but with $\ensuremath{\mu}\ensuremath{\ne}0$. Because of the $i$ in front of the Chern-Simons term, this produces magnetic screening only if $\ensuremath{\mu}$ is complex.