# Matching high-energy electroweak fermion loops onto the Fermi theory without higher dimensional operators

## TL;DR

This paper presents a method to match high-energy electroweak fermion loops onto the Fermi theory without introducing higher-dimensional operators, using the FDR approach to maintain accuracy at all loop orders.

## Contribution

It introduces a loop-by-loop matching procedure that avoids adding higher-dimensional operators, enabling exact reproduction of electroweak interactions in the Fermi model.

## Key findings

- FDR approach allows order-by-order matching without extra operators
- Fermi model reproduces electroweak interactions at all loop levels
- Method applicable to massless fermion lines in electroweak theory

## Abstract

We derive the conditions for matching high-energy renormalizable Quantum Field Theories onto low-energy nonrenormalizable ones by means of the FDR approach described in Fortsch.Phys. 63 (2015) 132-141. Our procedure works order-by-order in the loop expansion and avoids the addition of higher dimensional interactions into the nonrenormalizable Lagrangian. To illustrate our strategy, we match the high-energy fermion-loop corrections computed in the complete electroweak theory onto the nonrenormalizable four-fermion Fermi model. As a result, the Fermi Lagrangian can be used without modifications to reproduce, at arbitrary loop orders and energies, the exact electroweak interactions between two massless fermion lines induced by one-fermion-loop resummed gauge boson propagators.