Two-Loop Rational Terms for Spontaneously Broken Theories

TL;DR

This paper introduces a systematic method to compute two-loop rational counterterms in spontaneously broken theories, simplifying calculations in the Standard Model by leveraging symmetry properties and the unbroken phase.

Contribution

It presents a generalized vev-expansion approach for determining rational counterterms at two loops in broken theories, enabling efficient calculations in the Standard Model.

Findings

Derived the full set of $ ext{O}( ext{α}_s^2)$ rational counterterms for the Standard Model.

Showed that counterterms can be computed once in the symmetric phase and adapted to various schemes.

Demonstrated the method's applicability to all UV-divergent two-loop vertex functions.

Abstract

Rational counterterms are a key ingredient for the automation of loop calculations through numerical methods. Building on the recently established properties of rational terms of UV origin at two loops, in this paper we present a systematic method for the determination of rational counterterms within spontaneously broken theories. In particular we introduce a generalised vev-expansion approach that makes it possible to obtain the rational counterterms of UV origin for a spontaneously broken theory by means of calculations in the unbroken phase. The drastic simplifications that result from the underlying symmetry open the door to the efficient determination of rational counterterms for the full Standard Model at two loops. The renormalisation-scheme dependence is analysed in detail, and we show that rational counterterms need to be determined only once and for all in a generic renormalisation scheme for the symmetric phase and, a posteriori, they can be easily adapted to a wide range of physical renormalisation schemes for the spontaneously broken phase. As a first application we determine the full set of $\mathcal{O}(\alpha_{\mathrm s}^2)$ rational counterterms of UV origin for the full Standard Model, i.e. for all superficially UV-divergent two-loop vertex functions involving combinations of gluons, quarks, electroweak vector bosons and scalar bosons.