Peculiarities of the Schwinger-DeWitt technique: one-loop double poles, total-derivative terms, and determinant anomalies

We discuss peculiarities of the Schwinger-DeWitt technique, associated with the origin of dimensionally regularized double-pole divergences of the 1-loop effective action for massive Proca model in a curved spacetime. These divergences have the form of the total-derivative term generated by integration by parts in the functional trace of the heat kernel for the Proca vector field operator. Because of the nonminimal structure of this operator, its heat kernel has a nontrivial form, and is very different from the universal predictions of Gilkey-Seeley theory. Another hypostasis of total-derivative terms is in the problem of multiplicative determinant anomalies—lack of factorization of the functional determinant of a product of differential operators into the product of their determinants. We demonstrate that this anomaly should have the form of total-derivative terms and verify this statement by direct calculations for several examples.