A guide to functional methods beyond one-loop order

Abstract

Functional methods can be applied to the quantum efective action to efciently determine counterterms and matching conditions for efective feld 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 exemplifed with a two-loop matching calculation of the Euler–Heisenberg Lagrangian resulting from decoupling the electron in QED.