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Renormalization of general Effective Field Theories: Renormalization of fermionic operators

Renato M. Fonseca, Pablo Olgoso, José Santiago

TL;DR

Problem: achieving a universal, one-loop renormalization framework for general EFTs up to mass-dimension six. Approach: derive the beta functions for all physical fermionic operators within the general EFT, computing off-shell divergences in the Green's basis via the background-field method and projecting onto the physical basis. Contributions: provides explicit, highly correlated beta-function expressions for fermionic dim-6 operators, enabling renormalization of any EFT up to dim-6 at one loop and confirming consistency with SMEFT leptonic sector and related toy models. Significance: accelerates matching and running, reduces model-by-model calculations to a group-theoretical exercise, and paves the way for higher-dimension or higher-loop generalizations.

Abstract

Renormalization group equations play a central role in effective field theories, both maintaining perturbative control and allowing one to determine the correct low-energy phenomenology. In this work, we complete the one-loop renormalization of the recently developed general effective field theory up to mass dimension six by providing the beta functions for physical fermionic operators. Together with Ref. [1], our results allow one to renormalize any effective theory up to dimension six at one loop using only a group-theoretical calculation.

Renormalization of general Effective Field Theories: Renormalization of fermionic operators

TL;DR

Problem: achieving a universal, one-loop renormalization framework for general EFTs up to mass-dimension six. Approach: derive the beta functions for all physical fermionic operators within the general EFT, computing off-shell divergences in the Green's basis via the background-field method and projecting onto the physical basis. Contributions: provides explicit, highly correlated beta-function expressions for fermionic dim-6 operators, enabling renormalization of any EFT up to dim-6 at one loop and confirming consistency with SMEFT leptonic sector and related toy models. Significance: accelerates matching and running, reduces model-by-model calculations to a group-theoretical exercise, and paves the way for higher-dimension or higher-loop generalizations.

Abstract

Renormalization group equations play a central role in effective field theories, both maintaining perturbative control and allowing one to determine the correct low-energy phenomenology. In this work, we complete the one-loop renormalization of the recently developed general effective field theory up to mass dimension six by providing the beta functions for physical fermionic operators. Together with Ref. [1], our results allow one to renormalize any effective theory up to dimension six at one loop using only a group-theoretical calculation.

Paper Structure

This paper contains 7 sections, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The general effective Lagrangian to mass-dimension 6
  3. Results
  4. General formalism
  5. Beta functions of the fermionic operators
  6. Conclusions
  7. UV poles in the Green basis

Figures (1)

  • Figure 1: Renormalization structure of fermionic dimension-six operators, distinguishing contributions proportional to Yukawa $Y_{ija}$ (yellow), gauge $g$ (purple) or scalar quartic $\lambda$ (green) couplings. Double-colored arrows indicate contributions proportional to both couplings simultaneously.