Standard model with a complex scalar singlet: Cosmological implications and theoretical considerations

We analyze the theoretical and phenomenological considerations for the electroweak phase transition and dark matter in an extension of the standard model with a complex scalar singlet (cxSM). In contrast with earlier studies, we use a renormalization group improved scalar potential and treat its thermal history in a gauge-invariant manner. We find that the parameter space consistent with a strong first-order electroweak phase transition (SFOEWPT) and present dark matter phenomenological constraints is significantly restricted compared to results of a conventional, gauge-noninvariant analysis. In the simplest variant of the cxSM, recent LUX data and a SFOEWPT require a dark matter mass close to half the mass of the standard model-like Higgs boson. We also comment on various caveats regarding the perturbative treatment of the phase transition dynamics.

Figure 1

Patterns of symmetry breaking at finite temperature for $a_1=0$. For cases (a) and (b), one has ${\delta }_2>0$ and ${\delta }_2<0$, respectively. For case (c), the singlet VEV is nonzero at $T\ne 0$ while 0 at $T=0$. For case (d), the EWPT occurs in one step.

Figure 2

Symmetry breaking at finite temperature for $a_1\ne 0$. In this case the $\mathrm{O}\to \mathrm{A}$ transition is absent, and the initially nonzero ${\overline{v}}_S^{\mathrm{A}}(T)$ smoothly increases until the temperature reaches $T_1$ where EWPT happens.

Figure 3

Evolution of VEVs as a function of $T$ using the high-$T$ effective potential in type-(a) EWPT with $a_1\ne 0$ (left) and type-(c) EWPT with $a_1=0$ (right). For the former, the $A\to B$ transition is first order, with $T_2=T_C=90.4\mathrm{GeV}$ and $\overline{v}(T_C)=158.2\mathrm{GeV}$. For the latter, the $O\to A$ transition is second order while the $A\to B$ transition is first order. It is found that $T_1=224.6\mathrm{GeV}$, $T_2=T_C=99.8\mathrm{GeV}$, and $\overline{v}(T_C)=167.0\mathrm{GeV}$.

Figure 4

Contours of high-$T$ effective potential at $T=250\mathrm{GeV}$ (upper left), $T_C+5\mathrm{GeV}$ (upper right), $T_C$ (lower left), and 0 GeV (Lower Right), where $T_C=99.8\mathrm{GeV}$, $\overline{v}(T_C)=167.0\mathrm{GeV}$, ${\overline{v}}_S^{\mathrm{B}}(T_C)=0\mathrm{GeV}$, ${\overline{v}}_S^{\mathrm{A}}(T_C)=168.4\mathrm{GeV}$ in the case of S2. The black dots denote the minima of the potential.

Figure 5

Renormalization scale dependence of $T_C$ and ${\overline{v}}_C$ in S1. The dashed curves are calculated based on the original PRM scheme to $\mathcal{O}(\hslash )$ while the solid ones are the RG-improved version.

Figure 6

$T_C$ and $\overline{v}(T_C)$ as functions of $\alpha$ in S1. The solid curves are obtained by the PRM scheme with RG improvement while the dashed ones are obtained by the high-$T$ effective potential in Eq. (11).

Figure 7

$S_3(T)/T$ vs $T$ for $\alpha =-20.5°$ in S1. $S_3(T)$ is evaluated by use of the high-$T$ effective potential in Eq. (11). The dotted horizontal line satisfies the condition in Eq. (20). In this case, we have $T_C^{\mathrm{HT}}=90.4\mathrm{GeV}$ and $T_N=84.9\mathrm{GeV}$, with the latter closer to $T_C$ calculated in the PRM scheme.

Figure 8

$E_{\mathrm{sph}}(T)/T$ as a function of $T$, where $E_{\mathrm{sph}}(T)$ is calculated using the high-$T$ effective potential in Eq. (11). From right to left, the dots mark for $E_{\mathrm{sph}}(T_C^{\mathrm{HT}})/T_C^{\mathrm{HT}}=61.31$, $E_{\mathrm{sph}}(T_N)/T_N=74.23$ and $E_{\mathrm{sph}}(T_C)/T_C=78.00$, where $T_C^{\mathrm{HT}}=90.4\mathrm{GeV}$, $T_N=84.9\mathrm{GeV}$ and $T_C=83.1\mathrm{GeV}$.

Figure 9

$\mathcal{E}(T)=g_2{E}_{\mathrm{sph}}(T)/(4\pi \overline{v})$ as a function of $T$. The three dots correspond to $\mathcal{E}(T_C^{\mathrm{HT}})=1.82$, $\mathcal{E}(T_N)=1.86$ and $\mathcal{E}(T_C)=1.87$ from right to left.

Figure 10

$T_C$ and $\overline{v}(T_C)$ as functions of ${\delta }_2$. Below the left end point (${\delta }_2\simeq 0.51$) the EWPT is second order for both HT and PRM cases, while $T_C$ cannot be defined above the right end point (${\delta }_2\simeq 0.68$) in the PRM case. For ${\delta }_2\gtrsim 0.77$, phase A turns into the global minimum at $T=0$ in the HT case.

Figure 11

Scalar dark matter for case S1: (Left) Relic density of $A$ with $\alpha =-22.0°$, $-20.5°$ and $-15.0°$. (Right) Scaled spin-independent DM-nucleon scattering cross section. The allowed region is only at $m_A\simeq m_{H_1}/2\simeq 62.5\mathrm{GeV}$ if a SFOEWPT is required.

Figure 12

Scalar dark matter for case S2: (Left) Relic density of $A$ and $S$ with ${\delta }_2=0.7$, 0.55 and 0.2, where $m_A=m_S$ is assumed. (Right) Scaled spin-independent DM-nucleon scattering cross section.