Vacuum Structure of an Extended Standard Model with $U(1)_D$ Symmetry

Technical Summary: Vacuum Structure of an Extended Standard Model with $U(1)_D$ Symmetry

Problem Statement
The Standard Model (SM) faces a significant theoretical challenge regarding vacuum stability. Due to the large top quark Yukawa coupling ($y_t \sim \mathcal{O}(1)$) and the measured Higgs mass ($\sim 125$ GeV), the SM Higgs potential is predicted to become unstable and turn negative at energy scales around $10^{10}$ GeV. This instability suggests the necessity of physics Beyond the Standard Model (BSM). While various BSM theories incorporating new symmetries or scalar fields have been proposed to address this, the specific vacuum structure of extended models with a global $U(1)_D$ symmetry requires detailed investigation to ensure theoretical consistency and phenomenological viability.

Methodology
The authors investigate an extended scalar sector comprising the SM Higgs doublet ($H$), an additional inert $SU(2)$ doublet ($\eta$), a real scalar singlet ($\Phi$), and a complex scalar singlet ($\Phi'$). The model is governed by a global $U(1)_D$ symmetry, where $\eta$ is inert (zero vacuum expectation value, VEV) and preserves a $Z_2$ symmetry, while $\Phi$ and $\Phi'$ acquire VEVs to break the $U(1)_D$ symmetry.

The study employs a multi-stage numerical and analytical approach:

1. Scalar Potential and Mass Spectrum: The authors derive the full scalar potential, including additional quartic terms ($\lambda_{10}, \lambda_{11}, \lambda_{12}$) not present in the original model formulation. They parameterize the fields, determine the minimization conditions for the VEVs, and construct the mass matrices for CP-even, CP-odd, and charged scalars.
2. Vacuum Stability Analysis: To ensure the potential is bounded from below (BfB), the authors apply copositivity criteria to the quartic coupling matrix. They perform a numerical minimization using the Nelder-Mead algorithm to identify the global minimum, specifically searching for configurations that satisfy the required VEV alignment ($\langle \eta \rangle = 0$, $\langle H \rangle \neq 0$, $\langle \Phi \rangle \neq 0$, $\langle \Phi' \rangle \neq 0$).
3. Renormalization Group Evolution (RGE): Using the PyR@TE and RGBeta packages, the authors compute one-loop beta functions for all gauge, Yukawa, and scalar quartic couplings. They numerically evolve these couplings from the electroweak scale ($M_t$) up to the Planck scale ($M_{Pl}$) to test for perturbativity (couplings remaining $< 4\pi$) and stability (Higgs quartic coupling remaining positive).
4. Experimental Constraints: The viable parameter space is constrained by the experimental upper limits on the Higgs invisible decay branching ratio ($BR_{inv} \leq 0.107$ from ATLAS and $\leq 0.15$ from CMS). The authors calculate decay widths for channels involving dark fermions ($\chi$) and various scalar/pseudoscalar pairs ($A_1, H_3, A_3$).

Key Contributions and Results

• Vacuum Structure Characterization: The study establishes that the extended scalar potential can accommodate a stable vacuum. Through statistical analysis of 2 million random samples, the authors demonstrate that while BfB conditions are necessary, they are not sufficient for a "good" vacuum (correct symmetry breaking). By imposing specific constraints on the coupling parameters, the probability of obtaining a valid Type 5 vacuum (satisfying all VEV alignment criteria) increases from $\sim 10\%$ to $\sim 93\%$.
• Perturbativity up to the Planck Scale: The numerical RGE analysis confirms that for specific benchmark points, the scalar quartic couplings remain perturbative and the Higgs-like quartic coupling ($\lambda_4$) remains positive throughout the energy range up to the Planck scale. This indicates the model can be theoretically consistent at very high energies without encountering Landau poles or vacuum instability.
• Parameter Space and Mass Spectra: The authors map the viable parameter space, identifying regions where the charged scalar ($\tilde{\eta}$) and pseudoscalar ($A_3$) masses are consistent with theoretical constraints. They provide five benchmark points (BP1–BP5) that satisfy all conditions, yielding specific mass spectra for the physical scalar states ($H_1, H_2, H_3, \tilde{\eta}, A_3$).
• Invisible Decay Constraints: The analysis reveals that the model is viable under Higgs invisible decay constraints. Specifically, for a Yukawa coupling $\lambda_D < 0.15$, the favored mass region for the lightest scalar $H_3$ is found to be between 45 GeV and 60 GeV. Regions with $m_{H_3} < 45$ GeV are strongly disfavored due to constraints on dark fermion self-interactions and kinematic limits.

Significance and Outlook
The paper claims that the extended Standard Model with a global $U(1)_D$ symmetry offers a viable framework that resolves vacuum instability issues while remaining consistent with current experimental data. The primary significance lies in demonstrating that a stable, perturbative vacuum exists within a small but non-negligible region of the parameter space, satisfying both theoretical bounds (copositivity, global minimum) and experimental limits (Higgs invisible decay).

The authors modestly note that while the tree-level analysis is robust, future work should address loop-corrected effective potentials. Furthermore, they highlight the need for further investigation into the phenomenological constraints of the massless pseudoscalar $A_1$, particularly regarding its potential couplings to gauge fields (which must be anomaly-free) and its cosmological implications (e.g., contributions to $N_{eff}$ and overclosure of the universe). The study concludes that the model provides a testable scenario for dark sector dynamics via Higgs invisible decays and direct dark matter detection.