Improved sphaleron decoupling condition and the Higgs coupling constants in the real singlet-extended standard model

We improve the sphaleron decoupling condition in the real singlet-extended standard model (SM). The sphaleron energy is obtained using the finite-temperature one-loop effective potential with daisy resummation. For moderate values of the model parameters, the sphaleron decoupling condition is found to be $v_C/T_C>(1.1-1.2)$, where $T_C$ denotes a critical temperature and $v_C$ is the corresponding vacuum expectation value of the doublet Higgs field at $T_C$. We also investigate the deviation of the triple Higgs boson coupling from its standard model value in the region where the improved sphaleron decoupling condition is satisfied. As a result of the improvement, the deviation of the triple Higgs boson coupling gets more enhanced. In a typical case, if the Higgs couplings to the gauge bosons/fermions deviate from the SM values by about 3 (10)%, the deviation of the triple Higgs boson coupling can be as large as about 16 (50)%, which is about 4 (8)% larger than that based on the conventional criterion $v_C/T_C>1$.

Figure 1

The shape of the tree-level potential $V_0({\phi }_H,{\phi }_S)$ and $V_0^{Z_2}({\phi }_H,{\phi }_S)$ as functions of ${\phi }_H$ and ${\phi }_S$ in left and right figures, respectively. In the left (right) figure, we take $m_{H_1}=125.5\mathrm{GeV}$, $m_{H_2}=150(500)\mathrm{GeV}$, $v_S=100(200)\mathrm{GeV}$, $\alpha =0°(38°)$, ${\mu }_S^{\prime }=-30(0)\mathrm{GeV}$, and ${\mu }_{HS}=-80(0)\mathrm{GeV}$.

Figure 2

The diverse patterns of the EWPT.

Figure 3

The sphaleron energy (left panel) and ${\lambda }_{H,S,HS}$ (right panel) are plotted as a function of $\alpha$. We set $m_{H_1}=125.5\mathrm{GeV}$, $m_{H_2}=500\mathrm{GeV}$, $v_S=200\mathrm{GeV}$, and ${\mu }_S={\mu }_S^{\prime }={\mu }_{HS}=0\mathrm{GeV}$.

Figure 4

The possible region for the strong first-order phase transition in Case (i). In this figure, we set $m_{H_1}=125.5\mathrm{GeV}$, $v_S=200\mathrm{GeV}$, and ${\mu }_S^{\prime }={\mu }_{HS}={\mu }_S=0\mathrm{GeV}$.

Figure 5

The Higgs VEVs at $T_C$ (left panel), and $v_C/T_C$ and ${\zeta }_{\text{sph}}$ (right panel) are presented as functions of $\alpha$ in Case (i).

Figure 6

The possible region for the strong first-order phase transition in Case (ii). In this figure, we set $m_{H_1}=125.5\mathrm{GeV}$, $v_S=90\mathrm{GeV}$, ${\mu }_S^{\prime }=-30\mathrm{GeV}$, ${\mu }_{HS}=-80\mathrm{GeV}$, and ${\mu }_S=0\mathrm{GeV}$.

Figure 7

The Higgs VEVs at $T_C$ (left panel) and ${\lambda }_{H,S,HS}$ (right panel) are shown as functions of $\alpha$ in Case (ii).

Figure 8

(Left panel) $\mathcal{E}(T_C)$ and $\mathcal{E}(0)$ are shown as functions of $\alpha$. (Right panel) The sphaleron profiles are plotted as functions of $\xi$ at $T=0\mathrm{GeV}$ and $T=T_C$, where we take $\alpha =-10°$. The straight (dotted) lines denote $f(\xi )$, $h(\xi )$, and $k(\xi )$ at $T=T_C(0)\mathrm{GeV}$.

Figure 9

(Left panel) ${\zeta }_{\text{sph}}$ at $T_C$ and 0 are plotted as functions of $\alpha$. (Right panel) $v_C/T_C$ vs ${\zeta }_{\text{sph}}$ in the same $\alpha$ range as in the left panel.

Figure 10

$\kappa$ and $\mathrm{\Delta }{\lambda }_{H_1{H}_1{H}_1}$ in the strong first-order EWPT region. The right end point of the red line corresponds to $v_C/T_C\simeq 1.20$ and ${\zeta }_{\text{sph}}(T_C)\simeq 1.19$, while $v_C/T_C\simeq 4.59$ and ${\zeta }_{\text{sph}}(T_C)\simeq 1.19$ at the left end point.

Figure 11

$\mathrm{\Delta }{\lambda }_{H_1{H}_1{H}_1}$ in the ($m_{H_2},\kappa$) plane. The input parameters are the same as in Fig. 6. $v_C/T_C>{\zeta }_{\text{sph}}(T_C)$ is satisfied in the shaded region. For a reference, the $v_C/T_C=1$ contour line is also shown.