Unified worldline treatment of Yukawa and axial couplings

TL;DR

This paper develops a unified worldline formalism for calculating the one-loop effective action of a Dirac particle with multiple external couplings, addressing ordering issues and providing a consistent regularization scheme.

Contribution

It extends previous worldline methods to include all four couplings (scalar, pseudoscalar, vector, axialvector) and introduces a non-Hermitian Hamiltonian approach with a specific regularization for distribution products.

Findings

Successfully computes heat-kernel coefficients, self energies, and scattering amplitudes.

Achieves perfect agreement with established methods in various tests.

Provides a non-perturbative, unambiguous regularization scheme for worldline path integrals.

Abstract

We provide a worldline representation of the one-loop effective action for a Dirac particle coupled to external scalar, pseudoscalar, vector and axialvector fields. Extending previous work by two of the authors on the pure vector-axialvector case to all four couplings, it allows one to treat the real and the imaginary parts of the effective action in a unified manner, at the price of having a non-Hermitian Hamiltonian. Unlike existing worldline representations, our new worldline action contains terms with an odd number of Grassmann fields, leading to ordering problems that in the worldline formalism are usually encountered only in curved space. Drawing on the highly developed technology for worldline path-integrals in gravity, we employ the Time Slicing regularisation of the path integral which comes about with a specific ``counterterm Lagrangian'', which we calculate once and for all and non-perturbatively, to provide unambiguous rules to treat products of distributions occurring in some diagrams of the one-dimensional worldline theory. We then employ the usual worldline machinery to lay out the rules for the calculation of the effective action itself as well as the corresponding one-loop amplitudes. We test the formalism on the calculation of various heat-kernel coefficients, self energies and scattering amplitudes, including the Higgs decay into two photons or gluons and the PCAC relation. In all cases we find perfect agreement with more established approaches.