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 Trln[-Π²+U(X)] and the effective action of the SO(N) linear σ model are calculated.