Heat kernel of non-minimal second-order operators

TL;DR

This work provides a model-independent recipe for the local part of the trace of the second Seeley-DeWitt coefficient ${\rm tr}\hat{a}_2$ for general non-minimal second-order operators in four dimensions, using a refined Barvinsky–Vilkovisky approach and universal functional traces. It combines a ζ-parameter integration for the principal-part contribution with a systematic V-expansion for the non-minimal rest, delivering explicit, implementable expressions and a Mathematica notebook. The authors illustrate the method through three bosonic applications—vector, Kalb–Ramond, and a torsion toy model—highlighting gauge-invariance features, limiting cases, and the potential implications for quantum corrections and asymptotic-safety analyses in theories with higher spins or non-metricity. The framework offers a practical, computationally efficient tool for evaluating UV divergences in a broad class of non-minimal operators on curved backgrounds.

Abstract

We analyze the spectra of general non-minimal second-order operators. To do this, we derive the local part of the trace of the second Seeley-DeWitt heat kernel coefficient for such operators in a completely model-independent way. Afterwards, we provide three examples to show how our result can be applied in practical scenarios. In particular, we emphasize this discussion when dealing with a toy-model of dynamical torsion, which is viewed as a simple instance of higher-spin fields. All our results are compatible with the literature, and we provide a Mathematica notebook with the model-independent results that are written in the paper.