One-loop non-renormalization results in EFTs

TL;DR

This work addresses the puzzling pattern of zeros in the one-loop anomalous-dimension matrix for dimension-six EFT operators. By embedding an EFT into an effective superfield theory (ESFT) and classifying operators as JJ-operators or loop-operators, the authors show that many one-loop mixings are forbidden by Lorentz structure and SUSY-based holomorphy, with superpartner loops often vanishing. A key result is that, in simple U(1) models and their SM extensions, loop-operators such as $O_{FF}$ cannot be renormalized by JJ-operators at one loop, except for a controlled exception involving Yukawa-induced mixing like $y_u y_d$ that renormalizes the dipole operator $O_D$. The approach is then generalized to the SMEFT and to the QCD chiral Lagrangian, providing a unified explanation for which entries of the anomalous-dimension matrix vanish and under what conditions holomorphy constrains mixing. The framework offers a practical, symmetry-driven method to organize operator running and to anticipate rare but important exceptions to the usual non-renormalization rule, with potential broad applicability to other EFTs.

Abstract

In Effective Field Theories (EFTs) with higher-dimensional operators many anomalous dimensions vanish at the one-loop level for no apparent reason. With the use of supersymmetry, and a classification of the operators according to their embedding in super-operators, we are able to show why many of these anomalous dimensions are zero. The key observation is that one-loop contributions from superpartners trivially vanish in many cases under consideration, making supersymmetry a powerful tool even for non-supersymmetric models. We show this in detail in a simple U(1) model with a scalar and fermions, and explain how to extend this to SM EFTs and the QCD Chiral Langrangian. This provides an understanding of why most "current-current" operators do not renormalize "loop" operators at the one-loop level, and allows to find the few exceptions to this ubiquitous rule.

Figures (5)

• Figure 1: A potential contribution from ${\cal O}_{\phi q}$ to ${\cal O}_D$.
• Figure 2: Contributions to $c_{y_uy_d}$ proportional to $y_d y_u$.
• Figure 3: Non-holomorphic mixing between ${\cal O}_{y_u}$ and ${\cal O}_{y_d}$.
• Figure 4: Contributions from ${\cal O}^\dagger_{FF^+}$ to ${\cal O}_{y}$.
• Figure 5: Anomalous-dimension matrix of the dimension-six SM operators showing which entries (red-shaded) vanish following the present analysis. We also show the entries (light blue-shaded) that respect the holomorphic condition Eq. (\ref{['hcond']}). Solid lines separate loop-operators from $JJ$-operators.