# Effects of custodial symmetry breaking in the Georgi-Machacek model at high energies

## TL;DR

This paper investigates how custodial symmetry breaking affects the Georgi-Machacek model at high energies, deriving the relevant beta functions and analyzing the stability and mass degeneracy of Higgs bosons.

## Contribution

It constructs the most general Higgs potential without custodial symmetry and derives one-loop beta functions, showing custodial breaking remains controlled at high energies.

## Key findings

- Custodial symmetry breaking parameters remain small at high energies.
- Mass degeneracy among Higgs bosons is nearly preserved with tiny corrections.
- Model remains perturbative and vacuum stable up to high energies.

## Abstract

The model proposed by Georgi and Machacek enables the Higgs sector to involve isospin triplet scalar fields while retaining a custodial $SU(2)_V$ symmetry in the potential and thus ensuring the electroweak $\rho$ parameter to be one at tree level. This custodial symmetry, however, is explicitly broken by loop effects of the $U(1)_Y$ hypercharge gauge interaction. In order to make the model consistent at high energies, we construct the most general form of the Higgs potential without the custodial symmetry, and then we derive the one-loop $\beta$-functions for all the model parameters. Assuming the $\delta_i$ quantities describing the custodial symmetry breaking to be zero at low energy, we find that $|\delta_i|$ are typically smaller than the magnitude of the $U(1)_Y$ gauge coupling and the other running parameters in the potential also at high energy without spoiling perturbativity and vacuum stability. We also clarify that the mass degeneracy among the $SU(2)_V$ 5-plet and 3-plet Higgs bosons is smoothly broken by $\sim 0.1\%$ corrections. These results show that the amount of the custodial symmetry breaking is well kept under control up to energies close to the theory cutoff.