Dark matter, electroweak phase transition and gravitational wave in the type-II two-Higgs-doublet model with a singlet scalar field

We take the light CP-even Higgs boson $h$ as the SM-like Higgs, $m_{h}=125$ GeV. The measurement of the branching fraction of $b\to s\gamma$ gives the stringent constraints on the charged Higgs mass of the type-II 2HDM, $m_{H^{\pm}}>570$ GeV bsr570 . If the 125 GeV Higgs boson is the portal between the DM and SM sectors, it is favored to have wrong sign Yukawa coupling which can realize the isospin-violating DM interactions with nucleons and relax the bounds of direct detection of DM. However, Ref. PT_2HDM2 shows the the wrong sign Yukwa coupling region of type-II 2HDM is strongly restricted by the requirement of SFOEWPT. Therefore, in this paper we take the heavy CP-even Higgs $H$ as the only portal between the DM and SM sectors, and focus on the case of the 125 GeV with the SM-like couping. The $S$, $T$, and $U$ oblique parameters give the stringent constraints on the mass spectrum of Higgses of type-II 2HDM 1604.01406 ; 1701.02678 ; 2003.06170 . One of $m_{A}$ and $m_{H}$ is around 600 GeV, and another is allowed to have a wide mass range including low mass 1701.02678 ; 2003.06170 . Therefore, we fix $m_{A}=600$ GeV to make the portal $H$ to have a wide mass range.

In our calculation, we consider the following observables and constraints:

• (1)

  Theoretical constraints. The scalar potential of the model contains one of the type-II 2HDM and one of the DM sector. The vacuum stability, perturbativity, and tree-level unitarity impose constraints on the relevant parameters, which are discussed in detail in Refs. 2hisos-4 ; 2hisos-6 . Here we employ the formulas in 2hisos-4 ; 2hisos-6 to implement the theoretical constraints. Compared to Refs. 2hisos-4 ; 2hisos-6 , there are additional factors of $\frac{1}{2}$ in the $\kappa_{1}$ term and the $\kappa_{2}$ term of this paper. In addition, we require that the potential has a global minimum at the point of ($<h_{1}>=v_{1}$, $<h_{2}>=v_{2}$, $<S_{1}>=0$).

• (2)

  The oblique parameters. The $S$, $T$, $U$ parameters can impose stringent constraints on the mass spectrum of Higgses of 2HDM. We use 2HDMC 2hc-1 to calculate the $S$, $T$, $U$ parameters. Taking the recent fit results of Ref. pdg2018 , we use the following values of $S$, $T$, $U$,

  $$S=0.02\pm 0.10,~{}~{}T=0.07\pm 0.12,~{}~{}U=0.00\pm 0.09.$$

  (9)

  The correlation coefficients are

  $$\rho_{ST}=0.89,~{}~{}\rho_{SU}=-0.54,~{}~{}\rho_{TU}=-0.83.$$

  (10)

• (3)

  The flavor observables and $R_{b}$. We employ SuperIso-3.4 spriso to calculate $Br(B\to X_{s}\gamma)$, and $\Delta m_{B_{s}}$ is calculated following the formulas in deltmq . Besides, we include the constraints of bottom quarks produced in $Z$ decays, $R_{b}$, which is calculated following the formulas in rb1 ; rb2 .

  Channel	Experiment	Mass range [GeV]	Luminosity
  $gg/b\bar{b}\to H/A\to\tau^{+}\tau^{-}$	ATLAS 8 TeV 47Aad:2014vgg	90-1000	19.5-20.3 fb-1
  $gg/b\bar{b}\to H/A\to\tau^{+}\tau^{-}$	CMS 8 TeV 48CMS:2015mca	90-1000	19.7 fb-1
  $gg/b\bar{b}\to H/A\to\tau^{+}\tau^{-}$	CMS 13 TeV add-hig-16-037	90-3200	12.9 fb-1
  $gg/b\bar{b}\to H/A\to\tau^{+}\tau^{-}$	CMS 13 TeV 1709.07242	200-2250	36.1 fb-1
  $b\bar{b}\to H/A\to\tau^{+}\tau^{-}$	CMS 8 TeV 1511.03610	25-80	19.7 fb-1
  $gg/b\bar{b}\to H/A\to\tau^{+}\tau^{-}$	ATLAS 13 TeV 2002.12223	200-2500	139 fb-1
  $b\bar{b}\to H/A\to\mu^{+}\mu^{-}$	CMS 8 TeV CMS-HIG-15-009	25-60	19.7 fb-1
  $pp\to H/A\to\gamma\gamma$	ATLAS 13 TeV 80lenzi	200-2400	15.4 fb-1
  $gg\to H/A\to\gamma\gamma$	CMS 8+13 TeV 81rovelli	500-4000	12.9 fb-1
  $gg\to H/A\to\gamma\gamma$ + $t\bar{t}H/A~{}(H/A\to\gamma\gamma)$	CMS 8 TeV HIG-17-013-pas	80-110	19.7 fb-1
  $gg\to H/A\to\gamma\gamma$ + $t\bar{t}H/A~{}(H/A\to\gamma\gamma)$	CMS 13 TeV HIG-17-013-pas	70-110	35.9 fb-1
  $VV\to H\to\gamma\gamma$ + $VH~{}(H\to\gamma\gamma)$	CMS 8 TeV HIG-17-013-pas	80-110	19.7 fb-1
  $VV\to H\to\gamma\gamma$ + $VH~{}(H\to\gamma\gamma)$	CMS 13 TeV HIG-17-013-pas	70-110	35.9 fb-1
  $gg/VV\to H\to W^{+}W^{-}$	ATLAS 8 TeV 55Aad:2015agg	300-1500	20.3 fb-1
  $gg/VV\to H\to W^{+}W^{-}~{}(\ell\nu\ell\nu)$	ATLAS 13 TeV 77atlasww13	300-3000	13.2 fb-1
  $gg\to H\to W^{+}W^{-}~{}(\ell\nu qq)$	ATLAS 13 TeV 78atlasww13lvqq	500-3000	13.2 fb-1
  $gg/VV\to H\to W^{+}W^{-}~{}(\ell\nu qq)$	ATLAS 13 TeV 1710.07235	200-3000	36.1 fb-1
  $gg/VV\to H\to W^{+}W^{-}~{}(e\nu\mu\nu)$	ATLAS 13 TeV 1710.01123	200-3000	36.1 fb-1
  $gg/VV\to H\to W^{+}W^{-}$	CMS 13 TeV 1912.01594	200-3000	35.9 fb-1
  $gg/VV\to H\to ZZ$	ATLAS 8 TeV 57Aad:2015kna	160-1000	20.3 fb-1
  $gg\to H\to ZZ(\ell\ell\nu\nu)$	ATLAS 13 TeV 74koeneke4	300-1000	13.3 fb-1
  $gg\to H\to ZZ(\nu\nu qq)$	ATLAS 13 TeV 75koeneke5	300-3000	13.2 fb-1
  $gg/VV\to H\to ZZ(\ell\ell qq)$	ATLAS 13 TeV 75koeneke5	300-3000	13.2 fb-1
  $gg/VV\to H\to ZZ(\ell\ell\ell\ell)$	ATLAS 13 TeV 76koeneke3	200-3000	14.8 fb-1
  $gg/VV\to H\to ZZ(\ell\ell\ell\ell+\ell\ell\nu\nu)$	ATLAS 13 TeV 1712.06386	200-2000	36.1 fb-1
  $gg/VV\to H\to ZZ(\nu\nu qq+\ell\ell qq)$	ATLAS 13 TeV 1708.09638	300-5000	36.1 fb-1

  Channel	Experiment	Mass range [GeV]	Luminosity
  $gg\to H\to hh\to(\gamma\gamma)(b\bar{b})$	CMS 8 TeV 64Khachatryan:2016sey	250-1100	19.7 fb-1
  $gg\to H\to hh\to(b\bar{b})(b\bar{b})$	CMS 8 TeV 65Khachatryan:2015yea	270-1100	17.9 fb-1
  $gg\to H\to hh\to(b\bar{b})(\tau^{+}\tau^{-})$	CMS 8 TeV 66Khachatryan:2015tha	260-350	19.7 fb-1
  $gg\to H\to hh\to b\bar{b}b\bar{b}$	ATLAS 13 TeV 84varol	300-3000	13.3 fb-1
  $gg\to H\to hh\to b\bar{b}b\bar{b}$	CMS 13 TeV 1710.04960	750-3000	35.9 fb-1
  $gg\to H\to hh\to(b\bar{b})(\tau^{+}\tau^{-})$	CMS 13 TeV 1707.02909	250-900	35.9 fb-1
  $pp\to H\to hh$	CMS 13 TeV 1811.09689	250-3000	35.9 fb-1
  $gg\to H\to hh\to b\bar{b}ZZ$	CMS 13 TeV 2006.06391	260-1000	35.9 fb-1
  $gg\to H\to hh\to b\bar{b}\tau^{+}\tau^{-}$	CMS 13 TeV 2007.14811	1000-3000	139 fb-1
  $gg\to A\to hZ\to(\tau^{+}\tau^{-})(\ell\ell)$	CMS 8 TeV 66Khachatryan:2015tha	220-350	19.7 fb-1
  $gg\to A\to hZ\to(b\bar{b})(\ell\ell)$	CMS 8 TeV 67Khachatryan:2015lba	225-600	19.7 fb-1
  $gg\to A\to hZ\to(\tau^{+}\tau^{-})Z$	ATLAS 8 TeV 68Aad:2015wra	220-1000	20.3 fb-1
  $gg\to A\to hZ\to(b\bar{b})Z$	ATLAS 8 TeV 68Aad:2015wra	220-1000	20.3 fb-1
  $gg/b\bar{b}\to A\to hZ\to(b\bar{b})Z$	ATLAS 13 TeV 1712.06518	200-2000	36.1 fb-1
  $gg/b\bar{b}\to A\to hZ\to(b\bar{b})Z$	CMS 13 TeV 1903.00941	225-1000	35.9 fb-1
  $gg\to A\to hZ\to(\tau^{+}\tau^{-})(\ell\ell)$	CMS 13 TeV 1910.11634	220-400	35.9 fb-1
  $gg\to h\to AA\to\tau^{+}\tau^{-}\tau^{+}\tau^{-}$	ATLAS 8 TeV 1505.01609	4-50	20.3 fb-1
  $pp\to h\to AA\to\tau^{+}\tau^{-}\tau^{+}\tau^{-}$	CMS 8 TeV 1701.02032	5-15	19.7 fb-1
  $pp\to h\to AA\to(\mu^{+}\mu^{-})(b\bar{b})$	CMS 8 TeV 1701.02032	25-62.5	19.7 fb-1
  $pp\to h\to AA\to(\mu^{+}\mu^{-})(\tau^{+}\tau^{-})$	CMS 8 TeV 1701.02032	15-62.5	19.7 fb-1
  $pp\to h\to AA\to(b\bar{b})(\tau^{+}\tau^{-})$	CMS 13 TeV 1805.10191	15-60	35.9 fb-1
  $pp\to h\to AA\to\tau^{+}\tau^{-}\tau^{+}\tau^{-}$	CMS 13 TeV 1907.07235	4-15	35.9 fb-1
  $pp\to h\to AA\to\mu^{+}\mu^{-}\tau^{+}\tau^{-}$	CMS 13 TeV 2005.08694	3.6-21	35.9 fb-1
  $gg\to A(H)\to H(A)Z\to(b\bar{b})(\ell\ell)$	CMS 8 TeV 160302991	40-1000	19.8 fb-1
  $gg\to A(H)\to H(A)Z\to(\tau^{+}\tau^{-})(\ell\ell)$	CMS 8 TeV 160302991	20-1000	19.8 fb-1
  $gg/b\bar{b}\to A(H)\to H(A)Z\to(b\bar{b})(\ell\ell)$	ATLAS 13 TeV 1804.01126	130-800	36.1 fb-1
  $gg\to A(H)\to H(A)Z\to(b\bar{b})(\ell\ell)$	CMS 13 TeV 1911.03781	30-1000	35.9 fb-1

• (4)

  The global fit to the 125 GeV Higgs signal data. The version 2.0 of Lilith lilith is used to perform the $\chi^{2}$ calculation for the signal strengths of the 125 GeV Higgs combining the LHC run-I and run-II data (up to datasets of 36 fb^-1). We pay particular attention to the surviving samples with $\chi^{2}-\chi^{2}_{\rm min}\leq 6.18$, where $\chi^{2}_{\rm min}$ denotes the minimum of $\chi^{2}$. These samples correspond to be within the $2\sigma$ range in any two-dimension plane of the model parameters when explaining the Higgs data.

• (5)

  The exclusion limits of searches for additional Higgs bosons. We use HiggsBounds-4.3.1 hb1 ; hb2 to implement the exclusion constraints from the neutral and charged Higgs searches at LEP at 95% confidence level.

  Because the $b$-quark loop and top quark loop have destructive interference contributions to $gg\to A$ production in the type-II 2HDM, the cross section decreases with an increase of $\tan\beta$, reaches the minimum value for the moderate $\tan\beta$, and is dominated by the $b$-quark loop for enough large $\tan\beta$. In addition to $\tan\beta$ and $m_{H}$, the cross section of $gg\to H$ depends on $\sin(\beta-\alpha)$. We employ SusHi to compute the cross sections for $H$ and $A$ in the gluon fusion and $b\bar{b}$-associated production at NNLO in QCD sushi . The cross sections of $H$ via vector boson fusion process are deduced from results of the LHC Higgs Cross Section Working Group higgswg . We employ 2HDMC to calculate the branching ratios of the various decay modes of $H$ and $A$. The searches for the additional Higgs considered by us are listed in Tables 1 and 2. The LHC searches for $H^{\pm}$ can not impose any constraints on the model for $m_{H^{\pm}}>500$ GeV and 1 $\leq\tan\beta\leq 25$ mhp500 . Therefore, we do not consider the constraints from the searches for the heavy charged Higgs.

In Fig. 1, we show $\sin(\beta-\alpha)$ and $\tan\beta$ allowed by the 125 GeV Higgs signal data at the LHC. From Fig. 1, we see that $\tan\beta$ and $\sin(\beta-\alpha)$ have strong correlation due to the constraints of the 125 GeV Higgs data, especially for the case of the wrong sign Yukawa coupling. The wrong sign Yukawa coupling can be achieved only for $\sin(\beta-\alpha)>0$, and $\tan\beta$ is restricted to be in a very narrow range for a given $\sin(\beta-\alpha)$. For the case of the SM-like coupling, $\sin(\beta-\alpha)$ is required to be in two very narrow ranges of $-1.0\sim-0.99993$ and $0.994\sim 1.0$. The $\tan\beta$ is allowed to be as low as 1.0, and its upper bound increases with $\mid\sin(\beta-\alpha)\mid$ in the case of the the SM-like Higgs coupling.

Now we examine the parameter space of 2HDMIID using the exclusion limits of searches for additional Higgses at the LHC. In the 2HDMIID, we take the heavy CP-even Higgs $H$ as only portal between DM and SM sectors, and the decay $H\to SS$ opens for $2m_{S}<m_{H}$. The decay mode possibly affects the allowed parameter space, but the constraints of the DM observables have to be simultaneously considered. Here we temporarily assume $2m_{S}>m_{H}$, and close the $H\to SS$ decay mode. In the next section, the effects of $H\to SS$ will be considered by combining the DM observables.

In Fig. 2, we project the surviving samples with the SM-like coupling on the planes of $m_{H}$ versus $\tan\beta$ and $m_{H}$ versus $\mid\sin(\beta-\alpha)\mid$ after imposing the constraints of pre-LHC (denoting theoretical constraints, electroweak precision data, the flavor observables, $R_{b}$, the exclusion limits from searches for Higgs at LEP), the 125 GeV Higgs signal data, and the searches for additional Higgses at the LHC. Note that in the region of $\sin(\beta-\alpha)<0$, the signal data of the 125 GeV Higgs require $\sin(\beta-\alpha)$ to nearly equal to -1.0, as shown in right panel of Fig. 1. For such case, the couplings of $H$ and $A$ are almost the same as those in the case of $\sin(\beta-\alpha)=1.0$. Therefore, we do not distinguish the sign of $\sin(\beta-\alpha)$ when discussing the constraints on $m_{H}$ and $m_{A}$ from the LHC direct searches.

From Fig. 2, we find the joint constraints of $H/A\to\tau^{+}\tau^{-}$, $A\to HZ$, $H\to WW,~{}ZZ,~{}\gamma\gamma$, and $H\to hh$ exclude the whole region of $m_{H}<360$ GeV. The $H/A\to\tau^{+}\tau^{-}$ channels impose upper bound on $\tan\beta$ in the whole range of $m_{H}$, and allow $m_{H}$ to vary from 150 GeV to 800 GeV for appropriate values of $\tan\beta$ and $\sin(\beta-\alpha)$. The $A\to HZ$ channel does not constrain the parameter space of $m_{H}>$ 360 GeV since the branching ratio of $A\to HZ$ rapidly decreases with an increase of $m_{H}$. The limits of $A\to HZ$ channel can be relaxed by a small $\mid\sin(\beta-\alpha)\mid$ which suppresses the $AHZ$ coupling.

The $H\to WW,~{}ZZ,~{}\gamma\gamma,~{}hh$ and $A\to hZ$ channels impose strong constraints on the regions with small values of $\mid\sin(\beta-\alpha)\mid$ and $\tan\beta$ since the couplings of $HWW,~{}HZZ,~{}Hhh$ and $AhZ$ increase with decreasing of $\mid\sin(\beta-\alpha)\mid$, and $\sigma(gg\to H/A)$ is enhanced by the top quark loop for a small $\tan\beta$. In addition, the Fig. 1 shows that the 125 GeV Higgs signal data favor a small $\tan\beta$ for a small $\mid\sin(\beta-\alpha)\mid$ in the case of the SM-like coupling. With an increase of $m_{H}$, the $H\to t\bar{t}$ channel opens and enhances the total width of $H$ sizably, so that the constraints from $H\to WW,~{}ZZ,~{}\gamma\gamma,~{}hh$ channels are relaxed. Different from other channels, the $AhZ$ channel gives the constraints on the region with a large $m_{H}$. This is because the width of $A\to HZ$ decreases with an increase of $m_{H}$, and thus $Br(A\to hZ)$ increases with $m_{H}$.

In order to examine electroweak phase transition (EWPT), we first take $h_{1}$, $h_{2}$, and $S_{1}$ as the field configurations, and obtain the field dependent masses of the scalars ($h,~{}H,~{}A,~{}H^{\pm},~{}S$), the Goldstone boson ($G,~{}G^{\pm}$), the gauge boson, and fermions. The masses of scalars are given

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

The masses of gauge boson are given

We neglect the contributions of light fermions, and only consider the masses of top quark and bottom quark,

where $y_{t}={\sqrt{2}m_{t}\over v}$ and $y_{b}={\sqrt{2}m_{b}\over v}.$

Now we study the effective potential with thermal correction. The thermal effective potential $V_{eff}$ in terms of the classical fields ($h_{1},~{}h_{2},~{}S_{1}$) is composed of four parts:

Where $V_{0}$ is the tree-level potential, $V_{CW}$ is the Coleman-Weinberg potential, $V_{CT}$ is the counter term, $V_{T}$ is the thermal correction, and $V_{ring}$ is the resummed daisy corrections. In this paper, we calculate $V_{eff}$ in the Landau gauge.

We obtain the tree-level potential $V_{0}$ in terms of their classical fields ($h_{1},~{}h_{2},~{}S_{1}$)