One-loop renormalization and $\boldsymbol{\rho}$ parameter in the Georgi-Machacek model

We study the one-loop renormalization of the Georgi-Machacek model. At one loop, the renormalization of the model is phenomenologically important when triggered by operators that are absent at the tree level due to the global $SU(2)_R$ symmetry. By computing all the tree-level parameters from the standard input parameters $\alpha_e$, $G_\mu$, and $m_Z$, we show the ultraviolet divergent nature of the electroweak $\rho$ parameter when one-loop corrections are incorporated. In this model, four input parameters are required to completely parametrize the electroweak precision observables at one loop. We study the quantitative impact of the model parameters on the one-loop corrections to the $\rho$ parameter. At one loop, the $\rho$ parameter shows a mild dependence on the mass differences between the custodial fiveplet and the heavy custodial singlet, and mainly depends on the ratio of the doublet and triplet vacuum expectation values, and on the mixing angle between the custodial singlet CP-even scalars.