Custodial Symmetry (Violation) in SMEFT

TL;DR

This paper develops a new framework within SMEFT to better detect custodial symmetry violation, addressing limitations of the traditional T parameter by introducing a reparameterization-invariant observable called $\\mathscr{T}$.

Contribution

The authors construct a custodial basis of $ u$SMEFT and define a reparameterization-invariant $\\mathscr{T}$ parameter to unambiguously identify custodial symmetry violation at tree level.

Findings

The $\\mathscr{T}$ parameter accurately detects custodial symmetry violation.

The new basis explicitly distinguishes custodial preserving and violating operators.

The approach remains valid despite basis-dependent operator redundancies.

Abstract

We investigate precision observables sensitive to custodial symmetric/violating UV physics beyond the Standard Model. We use the SMEFT framework which in general includes non-oblique corrections that requires a generalization of the Peskin-Takeuchi $T$ parameter to unambiguously detect custodial symmetry/violation. We take a first step towards constructing a SMEFT reparameterization-invariant replacement, that we call $\mathscr{T}$, valid at least for tree-level custodial violating contributions. We utilize a new custodial basis of $\nu$SMEFT (SMEFT augmented by right-handed neutrinos) which explicitly identifies the global $SU(2)_R$ symmetries of the Higgs and fermion sectors, that in turn permits easy identification of higher-dimensional operators that are custodial preserving or violating. We carefully consider equation-of-motion redundancies that cause custodial symmetric operators in one basis to be equivalent to a set of custodial symmetric and/or violating operators in another basis. Utilizing known results about tree/loop operator generation, we demonstrate that the basis-dependent appearance of custodial-violating operators does not invalidate our $\mathscr{T}$ parameter at tree-level. We illustrate our results with several UV theory examples, demonstrating that $\mathscr{T}$ faithfully identifies custodial symmetry violation, while $T$ can fail.