Derivative Expansion for the One-Loop Effective Actions with Internal Symmetry

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 $\mathrm{Tr}\ln [-{\Pi }^2+U(X\left)\right]$ and the effective action of the $\mathrm{SO}\mathrm{}\left(N\right)\mathrm{}$ linear $\sigma$ model are calculated.