Electroweak baryogenesis, dark matter, and dark CP-symmetry

The baryon asymmetry of the universe (BAU) presents one of the major quests for particle cosmology. By the observation based on the Big-Bang Nucleosynthesis, the BAU is pdg2020

$$Y_{B}\equiv\rho_{B}/s=(8.2-9.2)\times 10^{-11},$$

(1)

where $\rho_{B}$ is the baryon number density and $s$ is the entropy density. Three necessary Sakharov conditions have to be fulfilled for a dynamical generation of BAU: baryon number changing interactions, non-conservation of C and CP, and departure from thermal equilibrium Sakharov . The electroweak baryogenesis (EWBG) ewbg1 ; ewbg2 provides a promising and attractive mechanism of explaining the BAU since it is testable at the energy frontier by the LHC and at the precision frontier by the electric dipole moment (EDM) experiments. To realize the EWBG, one needs extend the SM to produce sufficient large CP-violation and a strongly first-order electroweak phase transition (EWPT), such as the singlet extension of SM (see e.g. bgs-1 ; bgs-3 ; bgs-4 ; bgs-5 ; bgs-6 ; Beniwal:2018hyi ; Huang:2018aja ; Ghorbani:2017jls ; cao ; huang ; Xie:2020wzn ; Ellis:2022lft ; Lewicki:2021pgr ; Idegawa:2023bkh ; Harigaya:2022ptp ) and the two-Higgs-doublet model (2HDM) (see e.g. bg2h-1 ; bg2h-2 ; bg2h-3 ; Kanemura:2004ch ; Basler:2017uxn ; Abe:2013qla ; bg2h-5 ; bg2h-4 ; bg2h-6 ; bg2h-7 ; bg2h-8 ; bg2h-9 ; bg2h-11 ; Basler:2020nrq ; bg2h-13 ; bg2h-12 ; 2111.13079 ; 2207.00060 ; Goncalves:2023svb ).

The negative results in the EDM searches for electrons impose stringent constraints on the explicit CP-violation interactions in the scalar couplings and Yukawa couplings edm-e . There are some cancellation mechanisms of CP-violation effects in the EDM, which can relax the tension between the EWBG and the EDM data 1411.6695 ; Fuyuto:2017ewj ; Modak:2018csw ; Fuyuto:2019svr ; Modak:2020uyq ; 2004.03943 ; 2111.13079 ; 2207.00060 . Even with the cancellation, there are several CP observables of radiative $B$ meson decays that still provide stringent constraints, such as the asymmetry of the CP-asymmetry of inclusive $B\to X_{s}\gamma$ decay Modak:2018csw ; Modak:2020uyq , Alternatively, a finite temperature spontaneous CP-violation mechanism is naturally compatible with the EDM data, where the CP symmetry is spontaneously broken at the high temperature and it is recovered at the present temperature. The novel mechanism was achieved in the singlet scalar extension of the SM cao ; huang in which a high dimension effective operator needs to be added, and the singlet pseudoscalar extension of 2HDM Huber:2022ndk ; Liu:2023sey .

In addition to the BAU, the dark matter (DM) is one of the longstanding questions of particle physics and cosmology. In this paper, we propose a complex singlet scalar extension of the 2HDM respecting a discrete dark CP-symmetry, and simultaneously explain the observed DM relic density and the BAU via the spontaneous CP-violation at high temperature. The dark CP-symmetry allows the imaginary component of singlet scalar to have mixings with the pseudoscalars of scalar doublet fields, which play key roles in generating the CP-violation sources needed by the EWBG at high temperature. On the other hand, the dark CP-symmetry guarantees the real component to be a DM candidate.

Imposing a discrete dark CP-symmetry, we extend the SM by a second Higgs doublet $\Phi_{2}$ and a complex singlet $S$,

$\displaystyle\Phi_{1}=\left(\begin{array}[]{c}\phi_{1}^{+}\\ \frac{(v_{1}+\rho_{1}+i\eta_{1})}{\sqrt{2}}\end{array}\right)\,,\Phi_{2}=\left(\begin{array}[]{c}\phi_{2}^{+}\\ \frac{(v_{2}+\rho_{2}+i\eta_{2})}{\sqrt{2}}\end{array}\right),S=\frac{(\chi+i\eta_{s})}{\sqrt{2}},$

(6)

with $v_{1}$ and $v_{2}$ being the vacuum expectation values (VEVs), $v=\sqrt{v^{2}_{1}+v^{2}_{2}}=(246~{}\rm GeV)^{2}$ and $\tan\beta\equiv v_{2}/v_{1}$. The singlet field $S$ has no VEV. Under the dark CP-symmetry, $S\to-~{}S^{*}$ ($\chi\to-\chi$ ,$\eta_{s}\to\eta_{s}$ in the real parametrization), and while all the other fields are not affected.

The full scalar potential is given as

	$\displaystyle\mathrm{V}=m_{11}^{2}(\Phi_{1}^{\dagger}\Phi_{1})+m_{22}^{2}(\Phi_{2}^{\dagger}\Phi_{2})+\frac{\lambda_{1}}{2}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\frac{\lambda_{2}}{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}+\lambda_{3}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})$
	$\displaystyle+\lambda_{4}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})+\left[\frac{\lambda_{5}}{2}(\Phi_{1}^{\dagger}\Phi_{2})^{2}+\lambda_{6}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{1}^{\dagger}\Phi_{2})+\lambda_{7}(\Phi_{2}^{\dagger}\Phi_{2})(\Phi_{1}^{\dagger}\Phi_{2})+\rm h.c.\right]$
	$\displaystyle+m^{2}_{S}SS^{*}+\left[\frac{m^{\prime 2}_{S}}{2}SS+\rm h.c.\right]+\left[\frac{\lambda^{\prime\prime}_{1}}{24}S^{4}+\rm h.c.\right]+\left[\frac{\lambda^{\prime\prime}_{2}}{6}S^{2}SS^{*}+\rm h.c.\right]$
	$\displaystyle+SS^{*}\left[\lambda^{\prime}_{1}\Phi_{1}^{\dagger}\Phi_{1}+\lambda^{\prime}_{2}\Phi_{2}^{\dagger}\Phi_{2}\right]+\frac{\lambda^{\prime\prime}_{3}}{4}(SS^{*})^{2}+\left[S^{2}(\lambda^{\prime}_{4}\Phi_{1}^{\dagger}\Phi_{1}+\lambda^{\prime}_{5}\Phi_{2}^{\dagger}\Phi_{2})+\rm h.c.\right]$
	$\displaystyle+\left[\lambda^{\prime}_{6}SS^{*}\Phi_{2}^{\dagger}\Phi_{1}+\lambda^{\prime}_{7}(SS+S^{*}S^{*})\Phi_{2}^{\dagger}\Phi_{1}+\rm h.c.\right]$
	$\displaystyle+\left[-m_{12}^{2}\Phi_{2}^{\dagger}\Phi_{1}+\frac{\mu}{2}(S-S^{*})\Phi_{1}^{\dagger}\Phi_{2}+\rm h.c.\right],$		(7)

where all the coupling coefficients and mass are real, and thus the scalar potential sector is CP-conserved at zero temperature. The last term leads to the mixings of $\eta_{s}$ and the pseudoscalars of Higgs doublet fields, and $\chi$ is allowed to remain stable. For simplicity, we take $\lambda_{6}=\lambda_{7}=\lambda^{\prime}_{6}=\lambda^{\prime}_{7}=\lambda^{\prime\prime}_{2}=0$ in the following discussions.

The stationary conditions give

	$\displaystyle\quad m_{11}^{2}=m_{12}^{2}t_{\beta}-\frac{1}{2}v^{2}\left(\lambda_{1}c_{\beta}^{2}+\lambda_{345}s_{\beta}^{2}\right)\,,$
	$\displaystyle\quad m_{22}^{2}=m_{12}^{2}/t_{\beta}-\frac{1}{2}v^{2}\left(\lambda_{2}s_{\beta}^{2}+\lambda_{345}c_{\beta}^{2}\right),$		(8)

where $t_{\beta}\equiv\tan\beta$, $s_{\beta}\equiv\sin\beta$, $c_{\beta}\equiv\cos\beta$, and $\lambda_{345}=\lambda_{3}+\lambda_{4}+\lambda_{5}$.

In addition to the 125 GeV Higgs $h$, the physical scalar spectrum contains a CP-even states $H$, a DM candidate $\chi$, two neutral pseudoscalars $A$ and $X$, and a charged scalar $H^{\pm}$. The mass eigenstates $h$, $H$ and $H^{\pm}$ and their masses are the same as those of the pure 2HDM. The $\eta_{1}$, $\eta_{2}$ and $\eta_{s}$ are rotated into the $A$, $X$ and $G$ by the two mixing angles $\theta$ and $\beta$, where $G$ is a Goldstone boson. The parameters $\mu$, $m_{S}^{2}$, and $m_{S}^{\prime 2}$ are given as

	$\displaystyle m_{S}^{2}=\frac{1}{2}\left(m_{\chi}^{2}+m_{A}^{2}s_{\theta}^{2}+m_{X}^{2}c_{\theta}^{2}-\lambda^{\prime}_{1}v^{2}c_{\beta}^{2}-\lambda^{\prime}_{2}v^{2}s_{\beta}^{2}\right),$
	$\displaystyle m^{\prime 2}_{S}=\frac{1}{2}\left(m_{\chi}^{2}-m_{A}^{2}s_{\theta}^{2}-m_{X}^{2}c_{\theta}^{2}-2\lambda^{\prime}_{4}v^{2}c_{\beta}^{2}-2\lambda^{\prime}_{5}v^{2}s_{\beta}^{2}\right),$
	$\displaystyle\mu=\frac{\sqrt{2}(m_{X}^{2}-m_{A}^{2})}{v}s_{\theta}c_{\theta},$		(9)

where $s_{\theta}\equiv\sin\theta$ and $c_{\theta}\equiv\cos\theta$. The couplings $\lambda_{i}$ ($i=1,2,3,4,5$) are determined by

	$\displaystyle v^{2}\lambda_{1}=\frac{m_{H}^{2}c_{\alpha}^{2}+m_{h}^{2}s_{\alpha}^{2}-m_{12}^{2}t_{\beta}}{c_{\beta}^{2}},\ \ \ v^{2}\lambda_{2}=\frac{m_{H}^{2}s_{\alpha}^{2}+m_{h}^{2}c_{\alpha}^{2}-m_{12}^{2}t_{\beta}^{-1}}{s_{\beta}^{2}},$
	$\displaystyle v^{2}\lambda_{3}=\frac{(m_{H}^{2}-m_{h}^{2})s_{\alpha}c_{\alpha}+2m_{H^{\pm}}^{2}s_{\beta}c_{\beta}-m_{12}^{2}}{s_{\beta}c_{\beta}},\ \ \ v^{2}\lambda_{4}=\frac{(\hat{m}_{A}^{2}-2m_{H^{\pm}}^{2})s_{\beta}c_{\beta}+m_{12}^{2}}{s_{\beta}c_{\beta}},$
	$\displaystyle v^{2}\lambda_{5}=\frac{-\hat{m}_{A}^{2}s_{\beta}c_{\beta}+m_{12}^{2}}{s_{\beta}c_{\beta}}\,,$		(10)

with $\hat{m}_{A}^{2}=m_{A}^{2}c_{\theta}^{2}+m_{X}^{2}s_{\theta}^{2}$.

The general Yukawa interaction is given by

	$\displaystyle-{\cal L}=Y_{u2}\,\overline{Q}_{L}\,\tilde{{\Phi}}_{2}\,u_{R}+\,Y_{d2}\,\overline{Q}_{L}\,{\Phi}_{2}\,d_{R}\,+\,Y_{\ell 2}\,\overline{L}_{L}\,{\Phi}_{2}\,e_{R}\,$
	$\displaystyle+Y_{u1}\,\overline{Q}_{L}\,\tilde{{\Phi}}_{1}\,u_{R}+\,Y_{d1}\,\overline{Q}_{L}\,{\Phi}_{1}\,d_{R}\,+\,Y_{\ell 1}\,\overline{L}_{L}\,{\Phi}_{1}\,e_{R}+\,\mbox{h.c.},$		(11)

where $Q_{L}^{T}=(u_{L}\,,d_{L})$, $L_{L}^{T}=(\nu_{L}\,,l_{L})$, $\widetilde{\Phi}_{1,2}=i\tau_{2}\Phi_{1,2}^{*}$, and $Y_{u1,2}$, $Y_{d1,2}$ and $Y_{\ell 1,2}$ are $3\times 3$ matrices in family space. The Yukawa coupling matrices are taken to be aligned to avoid the tree-level flavour changing neutral current aligned2h ; Wang:2013sha ,

	$\displaystyle(Y_{u1})_{ii}=\frac{\sqrt{2}m_{ui}}{v}\rho_{1u},~{}~{}~{}(Y_{u2})_{ii}=\frac{\sqrt{2}m_{ui}}{v}\rho_{2u},$
	$\displaystyle(Y_{\ell 1})_{ii}=\frac{\sqrt{2}m_{\ell i}}{v}\rho_{1\ell},~{}~{}~{}~{}(Y_{\ell 2})_{ii}=\frac{\sqrt{2}m_{\ell i}}{v}\rho_{2\ell},$
	$\displaystyle(X_{d1})_{ii}=\frac{\sqrt{2}m_{di}}{v}\rho_{1d},~{}~{}~{}(X_{d2})_{ii}=\frac{\sqrt{2}m_{di}}{v}\rho_{2d},$		(12)

where $X_{d1,2}=V_{CKM}^{\dagger}Y_{d1,2}V_{CKM}$, $\rho_{1f}=(c_{\beta}-s_{\beta}\kappa_{f})$ and $\rho_{2f}=(s_{\beta}+c_{\beta}\kappa_{f})$ with $f=u,d,\ell$. All the off-diagonal elements are zero. The couplings of the neutral Higgs bosons normalized to the SM are given by

	$\displaystyle y^{h}_{V}=\sin(\beta-\alpha),~{}y_{f}^{h}=\left[\sin(\beta-\alpha)+\cos(\beta-\alpha)\kappa_{f}\right],$
	$\displaystyle y^{H}_{V}=\cos(\beta-\alpha),~{}y_{f}^{H}=\left[\cos(\beta-\alpha)-\sin(\beta-\alpha)\kappa_{f}\right],$
	$\displaystyle y^{A}_{V}=0,~{}y_{A}^{f}=-i\kappa_{f}~{}{\rm(for}~{}u)c_{\theta},~{}y_{f}^{A}=i\kappa_{f}c_{\theta}~{}{\rm(for}~{}d,~{}\ell),$
	$\displaystyle y^{X}_{V}=0,~{}y_{X}^{f}=-i\kappa_{f}~{}{\rm(for}~{}u)s_{\theta},~{}y_{f}^{X}=i\kappa_{f}s_{\theta}~{}{\rm(for}~{}d,~{}\ell),$		(13)

where $\alpha$ is the mixing angle of the two CP-even Higgs bosons, and $V$ denotes $Z$ or $W$.

In our calculations, we consider the following theoretical and experimental constraints:

(1) The signal data of the 125 GeV Higgs. We take the light CP even Higgs boson $h$ as the discovered 125 GeV state, and choose $\sin(\beta-\alpha)=1$ to satisfy the bound of the 125 GeV Higgs signal data, for which the $h$ has the same tree-level couplings to the SM particles as the SM.

(2) The direct searches and indirect searches for extra Higgses. From the Eq. (II), one see that the Yukawa couplings of the extra Higgses ($H$, $H^{\pm}$, $A$, $X$) are proportional to $\kappa_{u}$, $\kappa_{d}$ and $\kappa_{\ell}$ for $\sin(\beta-\alpha)=1$. Therefore, we assume $\kappa_{u}$, $\kappa_{d}$ and $\kappa_{\ell}$ to be small enough to suppress the production cross sections of these extra Higgses at the LHC, and satisfy the exclusion limits of searches for additional Higgs bosons at the LHC. Simultaneously, very small $\kappa_{u}$, $\kappa_{d}$ and $\kappa_{\ell}$ can accommodate the indirect searches for these extra Higgses via the $B$-meson decays.

(3) Vaccum stability. We require that the vacuum is stable at tree level, which means that the potential in Eq. (II) has to be bounded from below and the electroweak vacuum is the global minimum of the full scalar potential. To examine bounded from below condition we consider the minimum of quartic part in Eq. (II), $V_{4-min}$, which is written in matrix form in the basis $B=\begin{pmatrix}\Phi_{1}^{\dagger}\Phi_{1},&\Phi_{2}^{\dagger}\Phi_{2},&\chi^{2},&\eta_{S}^{2}\end{pmatrix}^{T}$,

	$\displaystyle V_{4-min}$	$\displaystyle=C^{T}\frac{1}{2}\underbrace{\begin{pmatrix}\lambda_{1}&\lambda_{3}+\Delta&\lambda_{1}^{\prime}+2\lambda_{4}^{\prime}&\lambda_{1}^{\prime}-2\lambda_{4}^{\prime}\\ \lambda_{3}+\Delta&\lambda_{2}&\lambda_{2}^{\prime}+2\lambda_{5}^{\prime}&\lambda_{2}^{\prime}-2\lambda_{5}^{\prime}\\ \lambda_{1}^{\prime}+2\lambda_{4}^{\prime}&\lambda_{2}^{\prime}+2\lambda_{5}^{\prime}&\frac{\lambda_{1}^{\prime\prime}+3\lambda_{3}^{\prime\prime}}{6}&\frac{-\lambda_{1}^{\prime\prime}+\lambda_{3}^{\prime\prime}}{2}\\ \lambda_{1}^{\prime}-2\lambda_{4}^{\prime}&\lambda_{2}^{\prime}-2\lambda_{5}^{\prime}&\frac{-\lambda_{1}^{\prime\prime}+\lambda_{3}^{\prime\prime}}{2}&\frac{\lambda_{1}^{\prime\prime}+3\lambda_{3}^{\prime\prime}}{6}\end{pmatrix}}_{A}C$
		$\displaystyle=\frac{1}{2}C^{T}AC,$		(14)

where $\Delta=0$ for $\lambda_{4}\geq|\lambda_{5}|$ and $\Delta=\lambda_{4}-|\lambda_{5}|$ for $\lambda_{4}<|\lambda_{5}|$.

A copositive matrix $A$ is required to ensure the potential to be bounded from below. Following the approaches described in Kannike:2012pe ; Dutta:2023cig , the matrix $A$ need satisfy $\det(A)\geq 0\lor(\text{adj}A)_{ij}<0$, for some $i,j$. The adjugate of $A$ is the transpose of the cofactor matrix of $A$: $(\text{adj}A)_{ij}=(-1)^{i+j}M_{ji}$, with $M_{ij}$ being the determinant of the submatrix that results from deleting row $i$ and column $j$ of $A$. In addition, one deletes the $i$-th row and column of $A$, $i=1,2,3,4$, and obtains 4 symmetric $3\times 3$ matrices, which are required to be copositive. The copositivity of the symmetric order 3 matrix $B$ with entries $b_{ij}$, $i,j=1,2,3$ requires

	$\displaystyle b_{11}\geq 0,\quad b_{22}\geq 0,\quad b_{33}\geq 0,$
	$\displaystyle\bar{b}_{12}=b_{12}+\sqrt{b_{11}b_{22}}\geq 0,$
	$\displaystyle\bar{b}_{13}=b_{13}+\sqrt{b_{11}b_{33}}\geq 0,$
	$\displaystyle\bar{b}_{23}=b_{23}+\sqrt{b_{22}b_{33}}\geq 0,$
	$\displaystyle\sqrt{b_{11}b_{22}b_{33}}+b_{12}\sqrt{b_{33}}+b_{13}\sqrt{b_{22}}+b_{23}\sqrt{b_{11}}+\sqrt{2\bar{b}_{12}\bar{b}_{13}\bar{b}_{23}}\geq 0.$		(15)

(4) Tree-level perturbative unitarity. We demand that the amplitudes of the scalar quartic interactions leading to $2\to 2$ scattering processes remain below the value of $8\pi$ at tree-level, which is implemented in SPheno-v4.0.5 Porod:2003um using SARAH-SPheno files Staub:2013tta .

(5) The oblique parameters. The oblique parameters ($S$, $T$, $U$) can obtain additional corrections via the self-energy diagrams exchanging $H$, $H^{\pm}$, $A$, and $X$. For $\sin(\beta-\alpha)=1$, the expressions of $S$, $T$ and $U$ in the model are approximately written as stu1 ; stu2

	$\displaystyle S$	$\displaystyle=$	$\displaystyle\frac{1}{\pi m_{Z}^{2}}\left[c_{\theta}^{2}F_{S}(m_{Z}^{2},m_{H}^{2},m_{A}^{2})+s_{\theta}^{2}F_{S}(m_{Z}^{2},m_{H}^{2},m_{X}^{2})-F_{S}(m_{Z}^{2},m_{H^{\pm}}^{2},m_{H^{\pm}}^{2})\right],$
	$\displaystyle T$	$\displaystyle=$	$\displaystyle\frac{1}{16\pi m_{W}^{2}s_{W}^{2}}\left[-c_{\theta}^{2}F_{T}(m_{H}^{2},m_{A}^{2})-s_{\theta}^{2}F_{T}(m_{H}^{2},m_{X}^{2})+F_{T}(m_{H^{\pm}}^{2},m_{H}^{2})\right.$
			$\displaystyle+\left.c_{\theta}^{2}F_{T}(m_{H^{\pm}}^{2},m_{A}^{2})+s_{\theta}^{2}F_{T}(m_{H^{\pm}}^{2},m_{X}^{2})\right],$
	$\displaystyle U$	$\displaystyle=$	$\displaystyle\frac{1}{\pi m_{W}^{2}}\left[F_{S}(m_{W}^{2},m_{H^{\pm}}^{2},m_{H}^{2})-2F_{S}(m_{W}^{2},m_{H^{\pm}}^{2},m_{H^{\pm}}^{2})\right.$		(16)
			$\displaystyle\left.+c_{\theta}^{2}F_{S}(m_{W}^{2},m_{H^{\pm}}^{2},m_{A}^{2})+s_{\theta}^{2}F_{S}(m_{W}^{2},m_{H^{\pm}}^{2},m_{X}^{2})\right]$
			$\displaystyle-\frac{1}{\pi m_{Z}^{2}}\left[c_{\theta}^{2}F_{S}(m_{Z}^{2},m_{H}^{2},m_{A}^{2})+s_{\theta}^{2}F_{S}(m_{Z}^{2},m_{H}^{2},m_{X}^{2})-F_{S}(m_{Z}^{2},m_{H^{\pm}}^{2},m_{H^{\pm}}^{2})\right],$

where

$$F_{T}(a,b)=\frac{1}{2}(a+b)-\frac{ab}{a-b}\log(\frac{a}{b}),~{}~{}F_{S}(a,b,c)=B_{22}(a,b,c)-B_{22}(0,b,c),$$

(17)

with

	$\displaystyle B_{22}(a,b,c)=\frac{1}{4}\left[b+c-\frac{1}{3}a\right]-\frac{1}{2}\int^{1}_{0}dx~{}X\log(X-i\epsilon),$
	$\displaystyle X=bx+c(1-x)-ax(1-x).$		(18)

Ref. pdg2020 gave the fit results of $S$, $T$ and $U$,

$$S=-0.01\pm 0.10,~{}~{}T=0.03\pm 0.12,~{}~{}U=0.02\pm 0.11,$$

(19)

with the correlation coefficients

$$\rho_{ST}=0.92,~{}~{}\rho_{SU}=-0.80,~{}~{}\rho_{TU}=-0.93.$$

(20)

The two neutral CP-even Higgs can mediate the interactions of DM, $\lambda_{h}v\chi^{2}h/2$ and $\lambda_{H}v\chi^{2}H/2$ with

	$\displaystyle\lambda_{h}$	$\displaystyle\equiv$	$\displaystyle(\lambda_{2}^{\prime}+2\lambda_{5}^{\prime})vs_{\beta}c_{\alpha}-(\lambda_{1}^{\prime}+2\lambda_{4}^{\prime})vc_{\beta}s_{\alpha},$
	$\displaystyle\lambda_{H}$	$\displaystyle\equiv$	$\displaystyle(\lambda_{2}^{\prime}+2\lambda_{5}^{\prime})vs_{\beta}s_{\alpha}+(\lambda_{1}^{\prime}+2\lambda_{4}^{\prime})vc_{\beta}c_{\alpha}.$		(21)

We consider a light DM whose freeze-out temperature is much lower than that of EWPT, and thus the effect of EWPT on the current DM relic density can be ignored. We take the new scalars to be much heavier than the DM so that the DM pair-annihilation to these new scalars are kinematically forbidden. In the parameter space chosen previously, the couplings of $H$ to the SM particles can be ignored. Therefore, the DM relic density hardly constrains the $\lambda_{H}$, and $\lambda_{1,2,4,5}^{\prime}$ are allowed to have room enough to produce the pattern of EWPT needed by the EWBG. The annihilation processes with $s$-channel exchange of $h$ are responsible for the relic density. However, for a light $\chi$, the invisible decay $h\to\chi\chi$ is kinematically allowed, and the signal data of the 125 GeV Higgs impose strong upper limits on the $h\chi\chi$ coupling invisible , which is possible to conflict with the requirement of the correct relic density The elastic scattering of $\chi$ on a nucleon receives the contributions of the process with $t$-channel exchange of $h$, which can be strongly constrained by the direct searches experiments of XENON XENON:2018voc . Besides, the indirect searches for DM can impose upper limits on the averaged cross sections of the DM annihilation to $e^{+}e^{-}$, $\mu^{+}\mu^{-}$, $\tau^{+}\tau^{-}$, $u\bar{u}$, $b\bar{b}$, and $WW$ Fermi-LAT:2015att .

After imposing the relevant theoretical and experimental constraints mentioned previously, we show the $\lambda_{h}$ versus $m_{\chi}$ allowed by the invisible decay of the 125 GeV Higgs, the DM relic density, the direct and indirect searches experiments in Fig. 1. From Fig. 1, we find that the DM with a mass of $55$ GeV $\sim 62.5$ GeV is allowed by the joint constraints of the invisible decay of the 125 GeV Higgs, the DM relic density, the direct and indirect searches experiments.

We first consider the effective scalar potential at the finite temperature. The neutral elements of $\Phi_{1}$ and $\Phi_{2}$ are shifted by $\frac{h_{1}}{\sqrt{2}}$ and $\frac{h_{2}+ih_{3}}{\sqrt{2}}$. It is plausible to take the imaginary part of the neutral elements of $\Phi_{1}$ to be zero since the effective potential of Eq. (II) only depends on the relative phase of the neutral elements of $\Phi_{1}$ and $\Phi_{2}$ .

The complete effective potential at finite temperature contains the tree level potential, the Coleman-Weinberg term vcw , the finite temperature corrections vloop and the resummed daisy corrections vring1 ; vring2 , which is gauge-dependent vgauge1 ; vgauge2 . Here we consider the high-temperature approximation of effective potential, which keeps only the thermal mass terms in the high-temperature expansion and the tree-level potential. Therefore, the effective potential is gauge invariant, and it does not depend on the renormalization scheme and the resummation scheme. The high-temperature approximation of effective potential is given by

		$\displaystyle V_{eff}(h_{1},h_{2},h_{3},\chi,\eta_{s})=\frac{1}{2}(m_{11}^{2}+\Pi_{h_{1}})h_{1}^{2}+\frac{1}{2}(m_{22}^{2}+\Pi_{h_{2}})(h_{2}^{2}+h_{3}^{2})+\frac{1}{2}(m_{S}^{2}+{m^{\prime}_{S}}^{2}+\Pi_{\chi})\chi^{2}$
		$\displaystyle+\frac{1}{2}(m_{S}^{2}-{{m^{\prime}_{S}}^{2}}+\Pi_{\eta_{s}})\eta_{s}^{2}+\frac{1}{8}(\lambda_{1}h_{1}^{4}+\lambda_{2}h_{2}^{4}+\lambda_{2}h_{3}^{4})+\frac{1}{4}\lambda_{345}h_{1}^{2}h_{2}^{2}+\frac{1}{4}\bar{\lambda}_{345}h_{1}^{2}h_{3}^{2}$
		$\displaystyle+\frac{\lambda_{2}}{4}h_{3}^{2}h_{2}^{2}-m_{12}^{2}h_{1}h_{2}-\frac{\mu}{\sqrt{2}}h_{3}\eta_{s}h_{1}+\frac{\lambda^{\prime}_{1}}{4}(\chi^{2}+\eta_{s}^{2})h_{1}^{2}+\frac{\lambda^{\prime}_{2}}{4}(\chi^{2}+\eta_{s}^{2})(h_{2}^{2}+h_{3}^{2})$
		$\displaystyle+\frac{\lambda^{\prime}_{4}}{2}(\chi^{2}-\eta_{s}^{2})h_{1}^{2}+\frac{\lambda^{\prime}_{5}}{2}(\chi^{2}-\eta_{s}^{2})(h_{3}^{2}+h_{2}^{2})+(\frac{\lambda^{{}^{\prime\prime}}_{1}}{48}+\frac{\lambda^{\prime\prime}_{3}}{16})(\chi^{4}+\eta_{s}^{4})+\frac{1}{8}(\lambda^{\prime\prime}_{3}-\lambda^{\prime\prime}_{1})\chi^{2}\eta_{s}^{2},$
		$\displaystyle\Pi_{h_{1}}=\left[{9g^{2}\over 2}+{3g^{\prime 2}\over 2}+6\lambda_{1}+4\lambda_{3}+2\lambda_{4}+2\lambda^{\prime}_{1}+6y_{t}^{2}c_{\beta}^{2}\right]{T^{2}\over 24},$
		$\displaystyle\Pi_{h_{2}}=\left[{9g^{2}\over 2}+{3g^{\prime 2}\over 2}+6\lambda_{2}+4\lambda_{3}+2\lambda_{4}+2\lambda^{\prime}_{2}+{6y_{t}^{2}s_{\beta}^{2}}\right]{T^{2}\over 24},$
		$\displaystyle\Pi_{h_{3}}=\Pi_{h_{2}},$
		$\displaystyle\Pi_{\chi}=\left[4\lambda^{\prime}_{1}+4\lambda^{\prime}_{2}+8\lambda^{\prime}_{4}+8\lambda^{\prime}_{5}+2\lambda^{\prime\prime}_{3}\right]{T^{2}\over 24},$
		$\displaystyle\Pi_{\eta_{s}}=\left[4\lambda^{\prime}_{1}+4\lambda^{\prime}_{2}-8\lambda^{\prime}_{4}-8\lambda^{\prime}_{5}+2\lambda^{\prime\prime}_{3}\right]{T^{2}\over 24},$		(22)

where $\bar{\lambda}_{345}=\lambda_{3}+\lambda_{4}-\lambda_{5}$, $y_{t}={\sqrt{2}m_{t}\over v}$, and $\Pi_{i}$ denotes the thermal mass terms of the field $i$.

Because baryogenesis is driven by diffusion processes in front of the bubble wall, one needs to compute the $T_{n}$ at which bubble nucleation actually starts. This can be calculated straightforwardly from the nucleation rate per unit volume by bubble-0 ; bubble-1 ; bubble-2 , $\Gamma\approx A(T)e^{-S_{3}/T}$, where $A(T)\sim T^{4}$ is a prefactor and $S_{3}$ is a three-dimensional Euclidian action. The nucleation temperature $T_{n}$ is obtained by $\Gamma/H^{4}$= 1, where $H$ is the Hubble parameter. It is roughly estimated by $\frac{S_{3}(T)}{T}|_{T=T_{n}}=140$. The bubble wall VEV profiles can be determined by the bounce equations of fields.

The WKB approximation method is used to evaluate the CP-violating source terms and chemical potentials transport equations of particle species in the wall frame with a radial coordinate $z$ bg2h-3 ; 0006119 ; 0604159 . A top quark penetrating the bubble wall acquires a complex mass as a function of $z$,

	$\displaystyle m_{t}(z)=\frac{y_{t}}{\sqrt{2}}\sqrt{(c_{\beta}h_{1}(z)+s_{\beta}h_{2}(z))^{2}+s_{\beta}^{2}h^{2}_{3}(z)}~{}e^{i\Theta_{t}(z)},$
	$\displaystyle{\rm with~{}}\Theta_{t}(z)=\varphi_{Z}(z)+\arctan\frac{s_{\beta}h_{3}(z)}{c_{\beta}h_{1}(z)+s_{\beta}h_{2}(z)},$
	$\displaystyle\partial_{z}\varphi_{Z}(z)=-\frac{h^{2}_{2}(z)+h^{2}_{3}(z)}{h^{2}_{1}(z)+h^{2}_{2}(z)+h^{2}_{3}(z)}\partial_{z}\varphi_{(}z),$
	$\displaystyle\varphi_{(}z)=\arctan\frac{h_{3}(z)}{h_{2}(z)}.$		(23)

In our calculation, the imaginary part of the neutral element of $\Phi_{1}$ is taken to be zero. As a result, the nonvanishing $Z_{\mu}$ field induces an additional CP-violating force acting on the top quark, which is removed a local axial transformation of top quark, reintroducing an additional overall phase $\varphi_{Z}(z)$ into $m_{t}$ bg2h-4 .

The transport equations are derived by the complex mass of the top quark, and contains effects of the strong sphaleron process ($\Gamma_{ss}$) bg2h-3 ; 9311367 , W-scattering ($\Gamma_{W}$) bg2h-3 ; 9506477 , the top Yukawa interaction ($\Gamma_{y}$) bg2h-3 ; 9506477 , the top helicity flips ($\Gamma_{M}$) bg2h-3 ; 9506477 , and the Higgs number violation ($\Gamma_{h}$) bg2h-3 ; 9506477 . The transport equations are written as

	$\displaystyle 0=$	$\displaystyle 3v_{W}K_{1,t}\left(\partial_{z}\mu_{t,2}\right)+3v_{W}K_{2,t}\left(\partial_{z}m_{t}^{2}\right)\mu_{t,2}+3\left(\partial_{z}u_{t,2}\right)$
		$\displaystyle-3\Gamma_{y}\left(\mu_{t,2}+\mu_{t^{c},2}+\mu_{h,2}\right)-6\Gamma_{M}\left(\mu_{t,2}+\mu_{t^{c},2}\right)-3\Gamma_{W}\left(\mu_{t,2}-\mu_{b,2}\right)$
		$\displaystyle-3\Gamma_{ss}\left[\left(1+9K_{1,t}\right)\mu_{t,2}+\left(1+9K_{1,b}\right)\mu_{b,2}+\left(1-9K_{1,t}\right)\mu_{t^{c},2}\right]\,,$
	$\displaystyle 0=$	$\displaystyle 3v_{W}K_{1,t}\left(\partial_{z}\mu_{t^{c},2}\right)+3v_{W}K_{2,t}\left(\partial_{z}m_{t}^{2}\right)\mu_{t^{c},2}+3\left(\partial_{z}u_{t^{c},2}\right)$
		$\displaystyle-3\Gamma_{y}\left(\mu_{t,2}+\mu_{b,2}+2\mu_{t^{c},2}+2\mu_{h,2}\right)-6\Gamma_{M}\left(\mu_{t,2}+\mu_{t^{c},2}\right)$
		$\displaystyle-3\Gamma_{ss}\left[\left(1+9K_{1,t}\right)\mu_{t,2}+\left(1+9K_{1,b}\right)\mu_{b,2}+\left(1-9K_{1,t}\right)\mu_{t^{c},2}\right]\,,$
	$\displaystyle 0=$	$\displaystyle 3v_{W}K_{1,b}\left(\partial_{z}\mu_{b,2}\right)+3\left(\partial_{z}u_{b,2}\right)-3\Gamma_{y}\left(\mu_{b,2}+\mu_{t^{c},2}+\mu_{h,2}\right)-3\Gamma_{W}\left(\mu_{b,2}-\mu_{t,2}\right)\,,$
		$\displaystyle-3\Gamma_{ss}\left[\left(1+9K_{1,t}\right)\mu_{t,2}+(1+9K_{1,b})\mu_{b,2}+(1-9K_{1,t})\mu_{t^{c},2}\right]\,,$
	$\displaystyle 0=$	$\displaystyle 4v_{W}K_{1,h}\left(\partial_{z}\mu_{h,2}\right)+4\left(\partial_{z}u_{h,2}\right)-3\Gamma_{y}\left(\mu_{t,2}+\mu_{b,2}+2\mu_{t^{c},2}+2\mu_{h,2}\right)-4\Gamma_{h}\mu_{h,2}\,,$

	$\displaystyle S_{t}=$	$\displaystyle-3K_{4,t}\left(\partial_{z}\mu_{t,2}\right)+3v_{W}\tilde{K}_{5,t}\left(\partial_{z}u_{t,2}\right)+3v_{W}\tilde{K}_{6,t}\left(\partial_{z}m_{t}^{2}\right)u_{t,2}+3\Gamma_{t}^{\mathrm{tot}}u_{t,2}\,,$
	$\displaystyle 0=$	$\displaystyle-3K_{4,b}\left(\partial_{z}\mu_{b,2}\right)+3v_{W}\tilde{K}_{5,b}\left(\partial_{z}u_{b,2}\right)+3\Gamma_{b}^{\mathrm{tot}}u_{b,2}\,,$
	$\displaystyle S_{t}=$	$\displaystyle-3K_{4,t}\left(\partial_{z}\mu_{t^{c},2}\right)+3v_{W}\tilde{K}_{5,t}\left(\partial_{z}u_{t^{c},2}\right)+3v_{W}\tilde{K}_{6,t}\left(\partial_{z}m_{t}^{2}\right)u_{t^{c},2}+3\Gamma_{t}^{\mathrm{tot}}u_{t^{c},2}\,,$
	$\displaystyle 0=$	$\displaystyle-4K_{4,h}\left(\partial_{z}\mu_{h,2}\right)+4v_{W}\tilde{K}_{5,h}\left(\partial_{z}u_{h,2}\right)+4\Gamma_{h}^{\mathrm{tot}}u_{h,2}\,,$		(24)

The $\mu_{i,2}$ and $u_{i,2}$ are the second-order CP-odd chemical potential and the plasma velocity of the particle $i=t,~{}t^{c},~{}b,~{}h$,. The source term $S_{t}$ is

$$S_{t}=-v_{W}K_{8,t}\partial_{z}\left(m_{t}^{2}\partial_{z}\theta_{t}\right)+v_{W}K_{9,t}\left(\partial_{z}\theta_{t}\right)m_{t}^{2}\left(\partial_{z}m_{t}^{2}\right).$$

(25)

The functions $K_{a,i}$ and $\tilde{K}_{a,i}$ ($a=1-9$) are defined in Ref. 0604159 , and the $\Gamma_{i}^{\mathrm{tot}}$ are the total reaction rate of the particle $i$ bg2h-3 ; 0604159 .