# Feynman Rules for the Standard Model Effective Field Theory in $R_\xi$-gauges

## TL;DR

This paper derives comprehensive Feynman rules for the Standard Model Effective Field Theory in $R_\xi$-gauges, enabling precise calculations of new physics effects up to dimension 6 operators without simplifying assumptions.

## Contribution

It provides the complete set of Feynman rules for SMEFT in $R_\xi$-gauges, including a publicly available Mathematica implementation, facilitating automatic amplitude computations.

## Key findings

- Explicit Feynman rules for SMEFT vertices derived
- Rules include non-renormalizable operator effects in mass basis
- Mathematica code provided for automated calculations

## Abstract

We assume that New Physics effects are parametrized within the Standard Model Effective Field Theory (SMEFT) written in a complete basis of gauge invariant operators up to dimension 6, commonly referred to as "Warsaw basis". We discuss all steps necessary to obtain a consistent transition to the spontaneously broken theory and several other important aspects, including the BRST-invariance of the SMEFT action for linear $R_\xi$-gauges. The final theory is expressed in a basis characterized by SM-like propagators for all physical and unphysical fields. The effect of the non-renormalizable operators appears explicitly in triple or higher multiplicity vertices. In this mass basis we derive the complete set of Feynman rules, without resorting to any simplifying assumptions such as baryon-, lepton-number or CP conservation. As it turns out, for most SMEFT vertices the expressions are reasonably short, with a noticeable exception of those involving 4, 5 and 6 gluons. We have also supplemented our set of Feynman rules, given in an appendix here, with a publicly available Mathematica code working with the FeynRules package and producing output which can be integrated with other symbolic algebra or numerical codes for automatic SMEFT amplitude calculations.