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

Abstract

In this research, we investigate the vacuum structure of an extended standard model with a $U(1)_D$ global symmetry. The scalar sector consists of two $SU(2)$ doublets as well as one complex singlet and one real singlet, resulting in a more complicated vacuum structure compared to that of the Standard Model. We analyze various theoretical constraints, including the conditions for being bounded from below, the existence of a global minimum, and perturbativity up to the Planck scale. Additionally, we consider experimental constraints from the Higgs invisible decay. Through a detailed statistical analysis using numerical methods, we show that the extended scalar potential can accommodate a stable vacuum while satisfying both theoretical and experimental constraints for a small region of the parameter space.

Figures (6)

• Figure 1: Percentages of the five types of vacuum found in the randomly generated samples. Type 1 to 5 cover around 33.4%, 35.7%, 0%, 20.5%, and 10.4%, respectively. The definition of each type of the obtained vacuum is given in Table \ref{['table:1']}.
• Figure 2: Percentages of the five types of vacuum when the potential parameters are generated with additional constraints given in Eq.\ref{['eq41']} (left panel) and Eq.\ref{['eq42']} (right panel). In the left panel, the percentages are 7.8%, 4.8%, 0%, 26.4%, and 61.0%. In the right panel, the percentages are 3.5%, 0.2%, 0%, 3.2%, and 93.1%. All for types 1 to 5, respectively.
• Figure 3: The numerical minimization of the scalar potential in Eq.\ref{['eq.1']} is plotted for different pairs of quartic couplings $\lambda\l_i\,(i=5,6,8,9)$. Other parameters, besides the ones shown as axes, are fixed at the benchmark value in Eq.\ref{['eq43']}. The gray region violates the BfB condition. The blue region satisfies the BfB condition but does not include a type 5 vacuum. The green region represents a BfB potential and fulfills the type 5 vacuum. The red region indicates all data points exceeding the perturbative limit $4\pi$.
• Figure 4: RG running of the chosen quartic couplings $\lambda_i\,(i=1,4,5,6)$ of the scalar potential (Eq.\ref{['eq.1']}) with initial values given in Table \ref{['table:2']} and BP1 in Table \ref{['table:3']}. The other couplings, besides $\lambda_i$, overlap within the region below $0.05$ and are not shown for clarity.
• Figure 5: Distribution of parameter points in the $m_{\tilde{\eta}}-m_{A_3}$ plane satisfying all theoretical constraints. The color shading indicates the density of viable solutions, with yellow corresponding to higher concentrations of allowed points and blue to lower concentrations. The overlaid contours highlight levels of constant density, while the darkest region outside the contours is excluded.