Generalized Yang-Mills actions from Dirac operator determinants

TL;DR

This paper explicitly computes the quantum effective action of Dirac fermions coupled to generalized Yang-Mills fields in four dimensions, confirming the proportionality of the divergence to the Yang-Mills action and extending it to the chiral case.

Contribution

It introduces an efficient method for computing quantum effective actions using pseudo-differential operators, applicable to both vector and chiral Yang-Mills fields.

Findings

Confirmed the proportionality of the divergent part to the Yang-Mills action in the vector case

Extended the computation to include chiral Yang-Mills fields

Provided an explicit, gauge-invariant expansion of the Dirac operator logarithm

Abstract

We consider the quantum effective action of Dirac fermions on four dimensional flat Euclidean space coupled to external vector- and axial Yang-Mills fields, i.e., the logarithm of the (regularized) determinant of a Dirac operator on flat R^4 twisted by generalized Yang-Mills fields. According to physics folklore, the logarithmic divergent part of this effective action in the pure vector case is proportional to the Yang-Mills action. We present an explicit computation proving this fact, generalized to the chiral case. We use an efficient computation method for quantum effective actions which is based on calculation rules for pseudo-differential operators and which yields an expansion of the logarithm of Dirac operators in local and quasi-gauge invariant polynomials of decreasing scaling dimension.