Symmetry-restoring finite counterterms of SMEFT four-fermion operator insertions at one loop

TL;DR

This work addresses the quantum consistency of SMEFT in a chiral setting by computing finite one-loop counterterms needed to restore Slavnov–Taylor identities in the presence of dimension-6 four-fermion operators within the $BMHV$ scheme. Using the Bonneau method and an external-ghost trick, it identifies a complete, non-evanescent set of finite counterterms that cancel BRST-breaking effects without introducing true anomalies. The authors classify these counterterms by operator structure (no scalars, one scalar, two scalars) and provide explicit Lagrangian forms and coefficient relations, including right-handed neutrinos. The results enable reliable higher-order SMEFT calculations, allowing subleading finite effects beyond leading-log accuracy to be consistently incorporated.

Abstract

Some effects induced by SMEFT operators at one loop have attracted a lot of attention in recent years, in particular, the renormalization of divergences by physical operators in single insertions of dimension-6 operators. Important non-logarithmically enhanced contributions must also be calculated. We discuss dimensional regularization in the Breitenlohner-Maison-'t Hooft-Veltman scheme. The goal here consists of determining in this scheme quantum effects in chiral theories at one loop. Namely, the determination of finite counterterms at one loop that reestablish the Slavnov-Taylor identities, which follow from gauge symmetries. These counterterms are necessary due to the presence of evanescent symmetry-breaking terms in the classical Lagrangian needed to regularize fermion propagators. We consider a technique that allows an easier calculation of such finite effects, relying on the identification of $(D-4)/(D-4)$ terms of one-loop amplitudes with an external ghost leg. We focus on dimension-6 four-fermion operators, identifying all finite counterterms in the Breitenlohner-Maison-'t Hooft-Veltman scheme at one loop, and as expected find no obstructions to the Slavnov-Taylor identities that cannot be cured by finite counterterms. This represents one step towards moving to higher order calculations.

Figures (7)

• Figure 1: Examples of 1PI diagrams with single insertions of SMEFT four-fermion operators (represented by filled squares) at one loop. No external scalars are present in these examples. Insertions of the symmetry-breaking operator are indicated by a cross, and ghosts are represented by a dotted line. Gauge bosons are indicated by wiggly lines. The single (double) solid lines represent physical right-handed (left-handed) fermions; analogous diagrams with opposite chiralities are also possible.
• Figure 2: One-loop "wine glass" Feynman diagrams, where insertions of the symmetry breaking operator are indicated by a cross (thus alluding to the chiral flip discussed in the text). Their divergent part is canceled between each other, so they give no contribution to the symmetry breaking. The single (double) lines represent physical right-handed (left-handed) fermions; analogous diagrams with opposite chiralities are also possible, as well as diagrams expressing violation of baryon and lepton numbers. The arrows also indicate the flux of the four-momenta introduced in the text.
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