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

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.