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

TL;DR

This work addresses the instability risk of the Georgi-Machacek model when quantum corrections are included. It uses a one-loop RG-improved effective potential and a new positive definiteness analysis for multi-variable quartics, since loop effects break custodial symmetry and prevent reduction to a quadratic form. The authors find that many regions ruled out by tree-level BFB become viable at high scales, while some EW-scale BFB-allowed regions fail at large field values; the stability, quantified through custodial-breaking measures $\delta_i$, is highly scale-dependent. The results revise the GM model's viability map and offer a general methodological template for stability studies in extended Higgs sectors, with implications for collider phenomenology and future theoretical work on metastability and higher-order corrections.

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.

Figures (3)

• Figure 1: Positive definiteness constraints for relevant quartic couplings. All points can fulfill the perturbative unitarity bounds. The blue points denote the parameter sets excluded by tree-level BFB constraints at the EW scale but validated by positive definiteness of the RG-improved scalar potential above 10 TeV. The cyan points denote the parameter sets satisfying tree-level BFB constraints at the EW scale but violating positive definiteness constraints at high scales (large field values). The red dots denote the parameter sets being consistent with positive definiteness constraints at all scales below the Planck scale or the Landau pole scale. For computational simplicity in the numerical scan, the Landau pole scale is defined as the scale where any relevant coupling exceeds 10 during RG evolution.
• Figure 2: Evolution of quartic coupling $\lambda_2$ values: EW scale vs. high scale after renormalization group evolution. Colors indicate the maximum scale achieved via RG evolution before encountering a Landau pole or reaching the Planck scale. In the panels (a-c), we show the parameter sets satisfying positive definiteness above 10 TeV, only at the EW scale, and at all scales below the Planck/Landau pole scale, respectively. Panel (d) shows the parameter sets satisfying positive definiteness up to the Planck scale (without encountering a Landau pole below it)
• Figure 3: The values of typical physical parameter observables predicted by the GM model, after imposing the new positive definiteness constraints and all other experimental/theoretical constraints (except the tree-level BFB constraints). Panels (a) and (b) show the correlations of triplets VEVs $v_\Delta$ with the masses of the custodial singlets $h,~H_1$, the custodial triplet $H_3$, and the quintuplet $H_5$, respectively. The effective couplings of the SM-like 125 GeV Higgs boson and third generation quarks and the Z vector boson are presented in the panel (c). The panel (d) show the branching ratios of the rare B decays $BR(B\rightarrow X_s \gamma)$ versus $V_\Delta$ and $M_3$.