A Guide to Functional Methods Beyond One-Loop Order

TL;DR

This paper extends functional methods for quantum effective actions to two-loop order and beyond, providing gauge-covariant evaluation techniques and generalizations to mixed spin theories, with practical applications in effective field theory matching.

Contribution

It introduces a gauge-covariant, multi-loop extension of functional methods and proves the hard-region matching formula's validity at all loop orders.

Findings

Extended functional methods to two-loop and higher orders.

Validated the hard-region matching formula for all loops.

Demonstrated practical application with QED Euler-Heisenberg Lagrangian.

Abstract

Functional methods can be applied to the quantum effective action to efficiently determine counterterms and matching conditions for effective field theories. We extend the toolbox to two-loop order and beyond and show how to evaluate the expansion of the path integral in a manifestly gauge-covariant manner. We also generalize the method to theories with mixed spin statistics and prove the validity of the hard-region matching formula to all loop orders. The methods are exemplified with a two-loop matching calculation of the Euler-Heisenberg Lagrangian resulting from decoupling the electron in QED.