Induced Chern-Simons Terms at High Temperatures and Finite Densities

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 ${\mu }^i$, is calculated to leading order in the inverse temperature, $T^{-1\mathrm{}}$. It is shown to be ${}_{}{}^i{}_{}{}^{}\Sigma _i^{}{}_{}{}^{}\left(\frac{{\mu }^i}{T}\right)W\left[A\right]\mathrm{}$, where $W\left[A\right]\mathrm{}$ is the Chern-Simons topological mass term. This result is also shown to be a consequence of the $\mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_L$ anomaly at zero temperature, but with $\mu \ne 0$. Because of the $i$ in front of the Chern-Simons term, this produces magnetic screening only if $\mu$ is complex.