  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.
