Abstract
In Effective Field Theories (EFTs) with higher-dimensional operators many anomalous dimensions vanish at the one-loop level. With the use of supersymmetry, and a classification of the operators according to their embedding in super-operators, we are able to understand 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 the superfield formalism 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 Lagrangian. 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.
Highlights
Quantum Effective Field Theories (EFTs) provide an excellent framework to describe physical systems, most prominently in particle physics, cosmology and condensed matter
We conclude that the only non-holomorphic anomalous dimension is in the Oyu ↔ Oyd mixing, and its origin can be tracked to the supersymmetry-breaking combination yuyd
5 Conclusions In EFTs with higher-dimensional operators the one-loop anomalous dimension matrix has plenty of vanishing entries apparently not forbidden by the symmetries of the theory
Summary
Quantum Effective Field Theories (EFTs) provide an excellent framework to describe physical systems, most prominently in particle physics, cosmology and condensed matter. It is possible to show this in certain cases by simple inspection of the one-loop diagrams, we present a more compact and systematic approach based on the superfield formalism For this reason we embed the EFT into an effective superfield theory (ESFT), and classify the operators depending on their embedding into super-operators. This implies that cFF is only renormalized by itself at the one-loop level This simple renormalization structure is the starting point from which, by examining more closely the loops involved at the field-component level, we will derive the following non-renormalization results in the non-supersymmetric EFT of Eq (1): Non-renormalization of OF F by Or: The differences between our original EFT in Eq (1) and its supersymmetric version, Eq (6), are the presence of the fermion superpartners for the gauge and scalar: the gaugino, λ, and ”Higgsino”, ψ. We conclude that the loop-operator OFF can only renormalize at the one-loop level the JJ-operators that break supersymmetry, like O6, and not those that can be embedded in a D-term, like Or
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