Positive definiteness constraints of effective scalar potential in Georgi-Machacek Model

Abstract

The Georgi-Machacek (GM) Model extends the Higgs sector of the Standard Model by introducing additional triplets, preserving custodial symmetry at tree level and allowing large triplet vacuum expectation values (VEVs) of order $\cal{O}$(10) GeV. Theoretical constraints on the model's parameters include bounded-from-below (BFB) conditions for the tree-level scalar potential. This study goes beyond the BFB constraints by examining the positive definiteness of the effective potential in the GM model to ensure the absence of deeper vacua in regions with large field values. Using a one-loop renormalization group-improved (RG-improved) effective potential and new criteria for positive definiteness of homogeneous polynomials with multiple variables (necessary due to custodial symmetry breaking effects from loops), we numerically analyze these constraints. Our results reveal that the parameter ranges allowed by positive definiteness differ significantly from those derived from tree-level BFB conditions. Notably, some regions previously excluded by tree-level BFB constraints remain viable under the one-loop RG-improved scalar potential. Besides, certain parameter spaces that satisfy tree-level BFB constraints with electroweak (EW) scale couplings should be excluded due to violations of positive definiteness in large field-value regions. Our numerical analysis, based on the new criteria for positive definiteness of homogeneous polynomials with multiple variables, not only revises the GM model's viability map but also provides a methodological template for stability studies in other extended Higgs models.