Derivative expansion for the one-loop effective actions with internal symmetry.

Abstract

A simple systematic method is presented for the evaluation of the derivative expansion of the functional determinant with covariant differential operators, space-time-dependent background fields, and internal symmetry. The results are directly applicable to the one-loop perturbative effective-action expansion for bosons and fermions. Derivative expansions up to four-derivative terms for $\mathrm{Tr}\mathrm{ln}[\ensuremath{-}{\ensuremath{\Pi}}^{2}+U(X)]$ and the effective action of the $\mathrm{SO}(N)$ linear $\ensuremath{\sigma}$ model are calculated.