The Feynman rules for the SMEFT in the background field gauge

Abstract

We present a package for FeynRules which derives the Feynman rules for the Standard Model Effective Field Theory up to dimension-six using the background field method for gauge fixing. The package includes operators which shift the kinetic and mass terms of the Lagrangian up to dimension-eight and including dimension-six squared effects consistently. To the best of the author’s knowledge this is the first publicly available package to include dimension-six squared effects consistently. The package is validated in a partner publication by analyzing the Ward Identities at dimension-six and one-loop order. We also extend the partner work in this article by including the dimension-six squared effects further demonstrating the consistency of their implementation. In doing so we find that failure to consistently include field shifts to dimension-six squared causes a breakdown in the Ward identities implying concerns about many calculations in the literature which do not properly incorporate these effects.The FeynRules files, as well as Mathematica notebooks performing the relevant calculations, can be downloaded from the FeynRules website and are included as supplementary material to this publication.

One loop verification of SMEFT Ward Identities

“…The results have been determined automatically using a new code package for BFM based SMEFT calculations to one loop order. This code package is reported on in a companion paper [34]. The results have also been directly calculated by hand independently in a cross check and verification of the automated generation of results.…”

The one-loop tadpole in the geoSMEFT

“…In order to define the relevant terms of the Lagrangian for the calculation of the tadpole diagram, we follow the formulation of the geoSMEFT given in [10], as well as the gauge fixing of [9] and [12]. We begin by defining the field content of the geoSMEFT, the Higgs doublet of the SM is rewritten in terms of a four-component real scalar field, φ I , by the following association:…”

“…Where the hatted fields are referred to as the background fields and the unhatted as the quantum fields. The choice of the background field method has various advantages, one of which is the preservation of the naive Ward Identities as discussed in [12,16,17]. This methodology has been adopted in many SMEFT related publications because of its nice properties, see for example [7,18,19].…”

SMEFT calculations for the LHC

“…Significant progress has been made on this front in the past few years, leading for instance to the generalization of R ξ gauges, Ward identities and the Background Field Method [60][61][62]. F R implementations in R ξ gauge have been made available [63][64][65], which partially simplifies the task of computing Feynman diagrams. Full NLO EW calculations in the SMEFT have been completed for µ → eγ [66], 2-body Higgs decays [67][68][69][70][71][72][73][74][75][76], Z, W pole observables [77][78][79] and Drell-Yan [80].…”