Renormalization of the Standard Model Effective Field Theory from Geometry

TL;DR

This paper introduces a geometric formalism for the Standard Model Effective Field Theory that simplifies the calculation of scattering amplitudes and renormalization group equations by using the curvature of a combined field-space manifold.

Contribution

It develops a unified geometric approach to scalar and gauge fields in SMEFT, enabling streamlined computation of amplitudes and one-loop divergences.

Findings

Derived scattering amplitudes from the Riemann curvature of the combined field space.

Computed RG equations for bosonic operators up to dimension eight.

Simplified renormalization calculations using geometric invariants.

Abstract

$S$-matrix elements are invariant under field redefinitions of the Lagrangian. They are determined by geometric quantities such as the curvature of the field-space manifold of scalar and gauge fields. We present a formalism where scalar and gauge fields are treated together, with a metric on the combined space of both types of fields. Scalar and gauge scattering amplitudes are given by the Riemann curvature $R_{ijkl}$ of this combined space, with indices $i,j,k,l$ chosen to be scalar or gauge indices depending on the type of external particle. One-loop divergences can also be computed in terms of geometric invariants of the combined space, which greatly simplifies the computation of renormalization group equations. We apply our formalism to the Standard Model Effective Field Theory (SMEFT), and compute the renormalization group equations for even-parity bosonic operators to mass dimension eight.