A guide to functional methods beyond one-loop order Fuentes Martín, Javier Moreno-Sánchez, Adrián Palavric, Ajdin Thomsen, Anders Eller Efective Field Theories Renormalization Group SMEFT 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. 2025-09-26T11:38:07Z 2025-09-26T11:38:07Z 2025-08-13 journal article Fuentes-Martín, J., Moreno-Sánchez, A., Palavrić, A., & Thomsen, A. E. (2025). A guide to functional methods beyond one-loop order. Journal of High Energy Physics, 2025(8). https://doi.org/10.1007/jhep08(2025)099 https://hdl.handle.net/10481/106675 10.1007/jhep08(2025)099 eng http://creativecommons.org/licenses/by/4.0/ open access Atribución 4.0 Internacional Springer