Prescription for finite oblique parameters $S$ and $U$ in extensions of the SM with $m_W \neq m_Z \cos{\theta_W}$

TL;DR

This paper addresses the calculation issues of oblique parameters S and U in extended Standard Model scenarios with larger scalar multiplets and broken custodial symmetry, proposing modified Feynman rules for consistent one-loop results.

Contribution

It introduces a method to modify Feynman rules to obtain gauge-independent and finite oblique parameters in models with non-standard gauge boson mass relations.

Findings

Modified Feynman rules eliminate gauge dependence and divergences.

Derived expressions for S in models with multiple scalar triplets.

Results recover standard relations when $m_W = m_Z \, \cos\theta_W$.

Abstract

We consider extensions of the Standard Model with neutral scalars in multiplets of $SU(2)$ larger than doublets. When those scalars acquire vacuum expectation values, the resulting masses of the gauge bosons $W^\pm$ and $Z^0$ are not related by $m_W = m_Z \cos{\theta_W}$. In those extensions of the Standard Model the oblique parameters $S$ and $U$, when computed at the one-loop level, turn out to be either gauge-dependent or divergent. We show that one may eliminate this problem by modifying the Feynman rules of the Standard Model for some vertices containing the Higgs boson; the modifying factors are equal to $1$ in the limit $m_W = m_Z \cos{\theta_W}$. We give the result for $S$ in a model with arbitrary numbers of scalar $SU(2)$ triplets with weak hypercharges either $0$ or $1$.