Four-fermion operators at dimension 6: dispersion relations and UV completions

TL;DR

This paper derives sum rules linking low-energy 4-fermion operator coefficients in SMEFT to high-energy cross-sections, aiding the construction of UV-complete models and understanding their phenomenological implications.

Contribution

It provides an exhaustive set of sum rules for dimension 6 4-fermion operators, including decomposition under symmetry groups, which was not previously available.

Findings

Sum rules connect Wilson coefficients to UV cross-sections.

Examples show opposite signs of coefficients in different UV completions.

Decomposition under isospin and color tightens IR-UV relations.

Abstract

A major task in phenomenology today is constraining the parameter space of SMEFT and constructing models of fundamental physics that the SM derives from. To this effect, we report an exhaustive list of sum rules for 4-fermion operators of dimension 6, connecting low energy Wilson coefficients to cross-sections in the UV. Unlike their dimension 8 counterparts which are amenable to a positivity bound, the discussion here is more involved due to the weaker convergence and indefinite signs of the dispersion integrals. We illustrate this by providing examples with weakly coupled UV completions leading to opposite signs of the Wilson coefficients for both convergent and non-convergent dispersion integrals. We further decompose dispersion integrals under weak isospin and color groups which lead to a tighter relation between IR measurements and UV models. These sum rules can become an effective tool for constructing consistent UV completions for SMEFT following the prospective measurement of these Wilson coefficients.