INTRODUCTION

PRD

PRVDAQ

Physical Review D

Phys. Rev. D

2470-00102470-0029

American Physical Society

10.1103/PhysRevD.103.035012

ARTICLES

Beyond the standard model

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

DARK MATTER, ELECTROWEAK PHASE TRANSITION, AND …

HAN XIAO-FANG, WANG LEI, AND ZHANG YANG

Han

Xiao-Fang

https://orcid.org/0000-0001-8639-4917

Wang

Lei

¹^,*Zhang

Yang

^2,3¹Department of Physics, Yantai University, Yantai 264005, People’s Republic of China²School of Physics and Microelectronics, Zhengzhou University, ZhengZhou 450001, People’s Republic of China³ARC Centre of Excellence for Particle Physics at the Tera-scale, School of Physics and Astronomy, Monash University, Melbourne, Victoria 3800, Australia

Corresponding author.

leiwang@ytu.edu.cn

15February2021

1February2021

1033

035012

24November202020January2021

Published by the American Physical Society

2021

authors

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

In the framework of the type II two-Higgs-doublet model with a singlet scalar dark matter S, we study the dark matter observables, the electroweak phase transition, and the gravitational wave signals by such strongly first order phase transition after imposing the constraints of the LHC Higgs data. We take the heavy CP-even Higgs boson H as the only portal between the dark matter and standard model (SM) sectors, and find that the LHC Higgs data and dark matter observables require mS and mH to be larger than 130 GeV and 360 GeV for mA=600 GeV in the case of the 125 GeV Higgs boson with the SM-like coupling. Next, we carve out some parameter space where a strongly first order electroweak phase transition can be achieved, and find benchmark points for which the amplitudes of gravitational wave spectra reach the sensitivities of the future gravitational wave detectors.

National Natural Science Foundation of China

10.13039/501100001809

11975013

Natural Science Foundation of Shandong Province

10.13039/501100007129

ZR2017MA004

ZR2017JL002

Australian Research Council

10.13039/501100000923

CE110001004

Department of Science and Technology of Shandong Province

10.13039/100012905

2019KJJ007

I.INTRODUCTION

The weakly interacting massive particle is a primary candidate for dark matter (DM) in the present Universe. Many extensions of the SM have been proposed to provide a candidate for DM, and one simple extension is to add a singlet scalar DM to the type II two-Higgs-doublet model (2HDM) [1–3]. The type II 2HDM contains two CP-even states, h and H, one neutral pseudoscalar A, two charged scalars H±, and one CP-even singlet scalar S as the candidate for DM [4–14].

In the type II 2HDM, the Yukawa couplings of the down-type quark and lepton can both be enhanced by a factor of tanβ. Therefore, the flavor observables and the LHC searches for the Higgs boson can impose strong constraints on the type II 2HDM. In the type II 2HDM with a singlet scalar DM (2HDMIID), the two CP-even states h and H may be portals between the DM and SM sectors, and there is plentiful parameter space satisfying the direct and indirect experimental constraints of DM. The scalar potential of 2HDMIID contains the original potential of type II 2HDM and one including DM. For the appropriate Higgs boson mass spectrum and coupling constants, the type II 2HDM can trigger a strong first order electroweak phase transition (SFOEWPT) in the early Universe [15–19], which is required by a successful explanation of the observed baryon asymmetry of the Universe (BAU) [20] and can produce primordial gravitational wave (GW) signals [21].

In this paper, we first examine the parameter space of the 2HDMIID using the recent LHC Higgs data and DM observables. After imposing various theoretical and experimental constraints, we analyze whether a SFOEWPT is achievable in the 2HDMIID, and discuss the resultant GW signals and their detectability at the future GW detectors, such as LISA [22], Taiji [23], TianQin [24], Big Bang Observer (BBO) [25], DECi-hertz Interferometer GW Observatory (DECIGO) [25], and Ultimate-DECIGO [26].

Our work is organized as follows: In Sec. II we give a brief introduction to the 2HDMIID. In Secs. III and IV, we show the allowed parameter space after imposing the limits of the LHC Higgs data and DM observables. In Sec. V, we examine the parameter space leading to a SFOEWPT and the corresponding GW signal. Finally, we give our conclusions in Sec. VI.

II.TYPE II TWO-HIGGS-DOUBLET MODEL WITH A SCALAR DARK MATTER

The scalar potential of 2HDMIID is given as [27] Vtree=m112(Φ1†Φ1)+m222(Φ2†Φ2)-[m122(Φ1†Φ2+H.c.)]+λ12(Φ1†Φ1)2+λ22(Φ2†Φ2)2+λ3(Φ1†Φ1)(Φ2†Φ2)+λ4(Φ1†Φ2)(Φ2†Φ1)+[λ52(Φ1†Φ2)2+H.c.]+12S2(κ1Φ1†Φ1+κ2Φ2†Φ2)+m022S2+λS4!S4.(1)Here we discuss the CP-conserving model in which all λi, κi, and m122 are real. The S is a real singlet scalar field, and Φ1 and Φ2 are complex Higgs doublets with hypercharge Y=1: Φ1=(ϕ1+12(v1+ϕ10+ia1)),Φ2=(ϕ2+12(v2+ϕ20+ia2)).(2)v1 and v2 are the electroweak vacuum expectation values (VEVs) with v2=v12+v22=(246 GeV)2, and the ratio of the two VEVs is defined as tanβ=v2/v1. The linear and cubic terms of the S field are forbidden by a Z2′ symmetry, under which S→-S. The S is a possible DM candidate since it does not acquire a VEV. After spontaneous electroweak symmetry breaking, the remaining physical states are three neutral CP-even states h, H, and S, one neutral pseudoscalar A, and two charged scalars H±.

We can obtain the DM mass and the cubic interactions with the neutral Higgs bosons from Eq. (1), mS2=m02+12κ1v2cos2β+12κ2v2sin2β,-λhvS2h/2≡-(-κ1sinαcosβ+κ2cosαsinβ)vS2h/2,-λHvS2H/2≡-(κ1cosαcosβ+κ2sinαsinβ)vS2H/2,(3)with α being the mixing angle of h and H.

The Yukawa interactions are written as -L=Yu2Q¯LΦ˜2uR+Yd1Q¯LΦ1dR+Yℓ1L¯LΦ1eR+H.c.,(4)where QLT=(uL,dL), LLT=(νL,lL), Φ˜1,2=iτ2Φ1,2*, and Yu2, Yd1, and Yℓ1 are 3×3 matrices in family space.

The Yukawa couplings of the neutral Higgs bosons normalized to the SM are given by yhfi=[sin(β-α)+cos(β-α)κf],yHfi=[cos(β-α)-sin(β-α)κf],yAfi=-iκf (for u),yAfi=iκf (for d, ℓ),withκd=κℓ≡-tanβ,κu≡1/tanβ.(5)

The charged Higgs boson has the following Yukawa interactions, LY=-2vH+{u¯i[κd(VCKM)ijmdjPR-κumui(VCKM)ijPL]dj+κℓν¯mℓPRℓ}+H.c.,(6)where i, j=1, 2, 3.

The neutral Higgs boson couplings with the gauge bosons normalized to the SM are given by yhV=sin(β-α),yHV=cos(β-α),(7)where V denotes Z or W. In the type II 2HDM, the 125 GeV Higgs boson is allowed to have the SM-like coupling and wrong sign Yukawa coupling, where yhfi×yhV>0 for SM-like coupling,yhfi×yhV<0 for wrong sign Yukawa coupling.(8)

III.THE EXPERIMENTAL CONSTRAINTS OF THE HIGGS DATA AT THE LHCA.Numerical calculations

We take the light CP-even Higgs boson h as the SM-like Higgs boson, mh=125 GeV. The measurement of the branching fraction of b→sγ gives the stringent constraints on the charged Higgs mass of the type II 2HDM, mH±>570 GeV [28]. 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. [17] shows the wrong sign Yukawa 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 boson H as the only portal between the DM and SM sectors, and focus on the case of the 125 GeV with the SM-like coupling. The S, T, and U oblique parameters give the stringent constraints on the mass spectrum of Higgs bosons of type II 2HDM [29–31]. One of mA and mH is around 600 GeV, and another is allowed to have a wide mass range, including low mass [30,31]. Therefore, we fix mA=600 GeV to cause the portal H to have a wide mass range.

In our calculations, we consider the following observables and constraints: (1)

The 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. [8,9]. Here we employ the formulas in [8,9] to implement the theoretical constraints. Compared to Refs. [8,9], there are additional factors of 12 in the κ1 term and κ2 terms in this paper. In addition, we require that the potential has a global minimum at the point of (⟨h1⟩=v1, ⟨h2⟩=v2, ⟨S1⟩=0). We do not consider the stability of the scalar potential at a high energy scale using the renormalization group evolution. In the SM there is only one quartic Higgs coupling, while the potential of this model has eight quartic Higgs couplings. In addition, the relations between Yukawa couplings and quark masses are different from the SM, which can be controlled by the two parameters β and α. Therefore, by tunning the parameters, the model should have some different properties from the SM Higgs potential. The detailed study is complicated and beyond the scope of this paper.

(2)

The oblique parameters. The S, T, and U parameters can impose stringent constraints on the mass spectrum of the Higgs bosons of 2hdm. We use 2hdmc [32] to calculate the S, T, and U parameters. Taking the recent fit results of Ref. [33], we use the following values of S, T, and U: S=0.02±0.10,T=0.07±0.12,U=0.00±0.09.(9)The correlation coefficients are ρST=0.89,ρSU=-0.54,ρTU=-0.83.(10)

(3)

The flavor observables and Rb. We employ superiso-3.4 [34] to calculate Br(B→Xsγ), and ΔmBs is calculated following the formulas in [35]. We also include the constraints of bottom quarks produced in Z decays, Rb, which is calculated following the formulas in [36,37].

(4)

The global fit to the 125 GeV Higgs signal data. The version 2.0 of lilith [38] is used to perform the χ2 calculation for the signal strengths of the 125 GeV Higgs boson 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 χ2-χmin2≤6.18, where χmin2 denotes the minimum of χ2. These samples correspond to be within the 2σ range in any two-dimensional 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 [39,40] to implement the exclusion constraints from the neutral and charged Higgs boson searches at LEP at 95% confidence level.

Because the b-quark loop and top quark loop have destructive interference contributions to gg→A production in the type II 2HDM, the cross section decreases with an increase of tanβ, reaches the minimum value for the moderate tanβ, and is dominated by the b-quark loop for enough large tanβ. In addition to tanβ and mH, the cross section of gg→H depends on sin(β-α). We employ sushi to compute the cross sections for H and A in the gluon fusion and bb¯-associated production at next-to-next-to leading order in quantum chromodynamics [41]. The cross sections of H via the vector boson fusion process are deduced from results of the LHC Higgs Cross Section Working Group [42]. We employ 2hdmc to calculate the branching ratios of the various decay modes of H and A. The searches for the additional Higgs bosons considered by us are listed in Tables I and II. The LHC searches for H± can not impose any constraints on the model for mH±>500 GeV and 1≤tanβ≤25 [43]. Therefore, we do not consider the constraints from the searches for the heavily charged Higgs bosons.

10.1103/PhysRevD.103.035012.t1

TABLE I.

The upper limits at 95% C.L. on the production cross section times branching ratio of τ+τ-, μ+μ-, γγ, WW, and ZZ considered in the H and A searches at the LHC.

ChannelExperimentMass range [GeV]Luminosity

gg/bb¯→H/A→τ+τ-

ATLAS 8 TeV [44]90–1000

19.5–20.3 fb-1

gg/bb¯→H/A→τ+τ-

CMS 8 TeV [45]90–1000

19.7 fb-1

gg/bb¯→H/A→τ+τ-

CMS 13 TeV [46]90–3200

12.9 fb-1

gg/bb¯→H/A→τ+τ-

CMS 13 TeV [47]200–2250

36.1 fb-1

bb¯→H/A→τ+τ-

CMS 8 TeV [48]25–80

19.7 fb-1

gg/bb¯→H/A→τ+τ-

ATLAS 13 TeV [49]200–2500

139 fb-1

bb¯→H/A→μ+μ-

CMS 8 TeV [50]25–60

19.7 fb-1

pp→H/A→γγ

ATLAS 13 TeV [51]200–2400

15.4 fb-1

gg→H/A→γγ

CMS 8+13 TeV [52]500–4000

12.9 fb-1

gg→H/A→γγ+tt¯H/A(H/A→γγ)

CMS 8 TeV [53]80–110

19.7 fb-1

gg→H/A→γγ+tt¯H/A(H/A→γγ)

CMS 13 TeV [53]70–110

35.9 fb-1

VV→H→γγ+VH(H→γγ)

CMS 8 TeV [53]80–110

19.7 fb-1

VV→H→γγ+VH(H→γγ)

CMS 13 TeV [53]70–110

35.9 fb-1

gg/VV→H→W+W-

ATLAS 8 TeV [54]300–1500

20.3 fb-1

gg/VV→H→W+W-(ℓνℓν)

ATLAS 13 TeV [55]300–3000

13.2 fb-1

gg→H→W+W-(ℓνqq)

ATLAS 13 TeV [56]500–3000

13.2 fb-1

gg/VV→H→W+W-(ℓνqq)

ATLAS 13 TeV [57]200–3000

36.1 fb-1

gg/VV→H→W+W-(eνμν)

ATLAS 13 TeV [58]200–3000

36.1 fb-1

gg/VV→H→W+W-

CMS 13 TeV [59]200–3000

35.9 fb-1

gg/VV→H→ZZ

ATLAS 8 TeV [60]160–1000

20.3 fb-1

gg→H→ZZ(ℓℓνν)

ATLAS 13 TeV [61]300–1000

13.3 fb-1

gg→H→ZZ(ννqq)

ATLAS 13 TeV [62]300–3000

13.2 fb-1

gg/VV→H→ZZ(ℓℓqq)

ATLAS 13 TeV [62]300–3000

13.2 fb-1

gg/VV→H→ZZ(ℓℓℓℓ)

ATLAS 13 TeV [63]200–3000

14.8 fb-1

gg/VV→H→ZZ(ℓℓℓℓ+ℓℓνν)

ATLAS 13 TeV [64]200–2000

36.1 fb-1

gg/VV→H→ZZ(ννqq+ℓℓqq)

ATLAS 13 TeV [65]300–5000

36.1 fb-1

10.1103/PhysRevD.103.035012.t2

TABLE II.

The upper limits at 95% C.L. on the production cross section times branching ratio for the channels of Higgs-pair and a Higgs production in association with Z at the LHC.

ChannelExperimentMass range [GeV]Luminosity

gg→H→hh→(γγ)(bb¯)

CMS 8 TeV [66]250–1100

19.7 fb-1

gg→H→hh→(bb¯)(bb¯)

CMS 8 TeV [67]270–1100

17.9 fb-1

gg→H→hh→(bb¯)(τ+τ-)

CMS 8 TeV [68]260–350

19.7 fb-1

gg→H→hh→bb¯bb¯

ATLAS 13 TeV [69]300–3000

13.3 fb-1

gg→H→hh→bb¯bb¯

CMS 13 TeV [70]750–3000

35.9 fb-1

gg→H→hh→(bb¯)(τ+τ-)

CMS 13 TeV [71]250–900

35.9 fb-1

pp→H→hh

CMS 13 TeV [72]250–3000

35.9 fb-1

gg→H→hh→bb¯ZZ

CMS 13 TeV [73]260–1000

35.9 fb-1

gg→H→hh→bb¯τ+τ-

CMS 13 TeV [74]1000–3000

139 fb-1

gg→A→hZ→(τ+τ-)(ℓℓ)

CMS 8 TeV [68]220–350

19.7 fb-1

gg→A→hZ→(bb¯)(ℓℓ)

CMS 8 TeV [75]225–600

19.7 fb-1

gg→A→hZ→(τ+τ-)Z

ATLAS 8 TeV [76]220–1000

20.3 fb-1

gg→A→hZ→(bb¯)Z

ATLAS 8 TeV [76]220–1000

20.3 fb-1

gg/bb¯→A→hZ→(bb¯)Z

ATLAS 13 TeV [77]200–2000

36.1 fb-1

gg/bb¯→A→hZ→(bb¯)Z

CMS 13 TeV [78]225–1000

35.9 fb-1

gg→A→hZ→(τ+τ-)(ℓℓ)

CMS 13 TeV [79]220–400

35.9 fb-1

gg→h→AA→τ+τ-τ+τ-

ATLAS 8 TeV [80]4–50

20.3 fb-1

pp→h→AA→τ+τ-τ+τ-

CMS 8 TeV [81]5–15

19.7 fb-1

pp→h→AA→(μ+μ-)(bb¯)

CMS 8 TeV [81]25–62.5

19.7 fb-1

pp→h→AA→(μ+μ-)(τ+τ-)

CMS 8 TeV [81]15–62.5

19.7 fb-1

pp→h→AA→(bb¯)(τ+τ-)

CMS 13 TeV [82]15–60

35.9 fb-1

pp→h→AA→τ+τ-τ+τ-

CMS 13 TeV [83]4–15

35.9 fb-1

pp→h→AA→μ+μ-τ+τ-

CMS 13 TeV [84]3.6–21

35.9 fb-1

gg→A(H)→H(A)Z→(bb¯)(ℓℓ)

CMS 8 TeV [85]40–1000

19.8 fb-1

gg→A(H)→H(A)Z→(τ+τ-)(ℓℓ)

CMS 8 TeV [85]20–1000

19.8 fb-1

gg/bb¯→A(H)→H(A)Z→(bb¯)(ℓℓ)

ATLAS 13 TeV [86]130–800

36.1 fb-1

gg→A(H)→H(A)Z→(bb¯)(ℓℓ)

CMS 13 TeV [87]30–1000

35.9 fb-1

B.Results and discussions

In Fig. 1, we show sin(β-α) and tanβ allowed by the 125 GeV Higgs signal data at the LHC. From Fig. 1, we see that tanβ and sin(β-α) 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(β-α)>0, and tanβ is restricted to be in a very narrow range for a given sin(β-α). For the case of the SM-like coupling, sin(β-α) is required to be in two very narrow ranges of -1.0∼-0.99993 and 0.994∼1.0. tanβ is allowed to be as low as 1.0, and its upper bound increases with |sin(β-α)| in the case of the SM-like Higgs coupling.

10.1103/PhysRevD.103.035012.f1

FIG. 1.

Scatter plots of sin(β-α) and tanβ satisfying the constraints of the 125 GeV Higgs boson signal data.

Now, we examine the parameter space of 2HDMIID using the exclusion limits of searches for additional Higgs bosons at the LHC. In the 2HDMIID, we take the heavy CP-even Higgs boson H as the only portal between DM and SM sectors, and the decay H→SS opens for 2mS<mH. 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 2mS>mH, and close the H→SS decay mode. In the next section, the effects of H→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 mH versus tanβ and mH versus |sin(β-α)| after imposing the constraints of pre-LHC (denoting theoretical constraints, electroweak precision data, the flavor observables, Rb, and the exclusion limits from searches for Higgs bosons at the LEP), the 125 GeV Higgs boson signal data, and the searches for additional Higgs bosons at the LHC. Note that in the region of sin(β-α)<0, the signal data of the 125 GeV Higgs boson require sin(β-α) to nearly equal −1.0, as shown in right panel of Fig. 1. For such a case, the couplings of H and A are almost the same as those in the case of sin(β-α)=1.0. Therefore, we do not distinguish the sign of sin(β-α) when discussing the constraints on mH and mA from the LHC direct searches.

10.1103/PhysRevD.103.035012.f2

FIG. 2.

The surviving samples with the SM-like coupling projected on the planes of mH versus tanβ and mH versus |sin(β-α)|. All of the samples are allowed by the constraints of pre-LHC and the 125 GeV Higgs boson signal data. The triangles (sky blue), circles (royal blue), squares (black), inverted triangles (purple), and pluses (red) are respectively excluded by the H/A→τ+τ-, H→WW,ZZ,γγ, H→hh, A→HZ, and A→hZ channels at the LHC. The bullets (green) are allowed by various LHC direct searches.

From Fig. 2, we find the joint constraints of H/A→τ+τ-, A→HZ, H→WW,ZZ,γγ, and H→hh exclude the whole region of mH<360 GeV. The H/A→τ+τ- channels impose an upper bound on tanβ in the whole range of mH, and allow mH to vary from 150 GeV to 800 GeV for appropriate values of tanβ and sin(β-α). The A→HZ channel does not constrain the parameter space of mH>360 GeV since the branching ratio of A→HZ rapidly decreases with an increase of mH. The limits of the A→HZ channel can be relaxed by a small |sin(β-α)| which suppresses the AHZ coupling.

The H→WW,ZZ,γγ,hh, and A→hZ channels impose strong constraints on the regions with small values of |sin(β-α)| and tanβ since the couplings of HWW, HZZ, Hhh, and AhZ increase with the decrease of |sin(β-α)|, and σ(gg→H/A) is enhanced by the top quark loop for a small tanβ. In addition, Fig. 1 shows that the 125 GeV Higgs boson signal data favor a small tanβ for a small |sin(β-α)| in the case of the SM-like coupling. With an increase of mH, the H→tt¯ channel opens and enhances the total width of H sizably, so that the constraints from the H→WW,ZZ,γγ, andhh channels are relaxed. Different from the other channels, the AhZ channel gives the constraints on the region with a large mH. This is because the width of A→HZ decreases with an increase of mH, and thus Br(A→hZ) increases with mH.

It is noted that some allowed samples lie in the region of tanβ<1.5 and the other region is empty for mH>700 GeV in the left panel of Fig. 2. The main reason is from the theoretical constraints. The vacuum stability requires that λ1>0,λ2>0,λ3>-λ1λ2,λ3+λ4-|λ5|>-λ1λ2.(11)Here, we focus on the case of the SM-like coupling, and cos(β-α) is very small. Therefore, we can approximately obtain the following relations [88]: v2λ1=mh2-tβ3(m122-mH2sβcβ)sβ2,v2λ2=mh2-(m122-mH2sβcβ)tβsβ2,v2λ3=mh2+2mH±2-2mH2-tβ(m122-mH2sβcβ)sβ2,v2λ4=mA2-2mH±2+mH2+tβ(m122-mH2sβcβ)sβ2,v2λ5=mH2-mA2+tβ(m122-mH2sβcβ)sβ2,(12)with tβ≡tanβ, sβ≡sinβ, and cβ≡cosβ. If m122-mH2sβcβ→0, the first two requirements in Eq. (11) are simultaneously satisfied, and the last condition will require that mh2+mA2-mH2>0.(13)However, the relation of Eq. (13) is not satisfied for mh=125 GeV, mA=600 GeV, and mH>700 GeV. Therefore, for a large mH, m122-mH2sβcβ is not allowed to approach 0. The first expression of Eq. (12) shows that the term of m122-mH2sβcβ is enhanced by a factor of tβ3. Therefore, the vacuum stability and perturbativity favor a small tanβ for a large mH.

IV.THE DARK MATTER OBSERVABLES

We use feynrules [89] to generate the model file, which is called by micromegas [90] to calculate the relic density. In our scenario, the elastic scattering of S on a nucleon receives the contributions of the process with a t-channel exchange of H, and the spin-independent cross section between DM and nucleons is given by [91] σp(n)=μp(n)24πmS2[fp(n)]2,(14)where μp(n)=mSmp(n)mS+mp(n), and fp(n)=∑q=u,d,sfqp(n)CSqmp(n)mq+227fgp(n)∑q=c,b,tCSqmp(n)mq,(15)with CSq=λHmH2mqyHq. The values of the form factors fqp,n and fgp,n are extracted from micromegas [90].

The Planck Collaboration reported the relic density of cold DM in the universe, Ωch2=0.1198±0.0015 [92]. The XENON1T Collaboration reported stringent upper bounds of the spin-independent DM-nucleon cross section [93]. The Fermi-LAT Collaboration searches for the DM annihilation from dwarf spheroidal satellite galaxies gave the upper limits on the averaged cross sections of the DM annihilation to e+e-, μ+μ-, τ+τ-, uu¯, bb¯, and WW [94].

In Fig. 3, we project the surviving samples on the planes of λH versus mS, mH versus mS, and σp versus mS after imposing the constraints of “pre-LHC”, the Higgs data at the LHC, the relic density, XENON1T, and Fermi-LAT. The middle panel shows that the H→SS decay weakens the constraints of the LHC Higgs data compared to Fig. 2. For example, mH is allowed to be as low as 200 GeV for a light DM. However, the upper bounds of the XENON1T and Fermi-LAT Collaborations exclude mS<130 GeV and mH<360 GeV. In order to obtain the correct relic density, |λH| is favored to increase with the decrease of mS. Thus, when mS is small, a large |λH| can enhance the spin-independent DM-nucleon cross section and the averaged cross sections of the DM annihilation to the SM particles, leading that mS<130 GeV and mS<75 GeV are, respectively, excluded by the experimental data of the XENON1T and Fermi-LAT Collaborations. For 180 GeV<mS<340 GeV, |λH| can be allowed to be smaller than 0.01 because of the resonant contribution at 2mS∼mH.

10.1103/PhysRevD.103.035012.f3

FIG. 3.

The surviving samples projected on the planes of mS versus λH, mS versus mH, and mS versus σp. All of the samples are allowed by the constraints of “pre-LHC”, the LHC Higgs data, and the relic density. The circles (royal blue) and pluses (red) are, respectively, excluded by the experimental data of the XENON1T and Fermi-LAT Collaborations, while the bullets (green) are allowed.

V.ELECTROWEAK PHASE TRANSITION AND GRAVITATIONAL WAVE

The phase transition can basically proceed in two different ways. In a first order phase transition, at the critical temperature TC, the two degenerate minima will be at different points in field space, typically with a potential barrier in between. For a second order (crossover) transition, the broken and symmetric minimum are not degenerate until they are at the same point in field space. In this paper we focus on the SFOEWPT, which is required by a successful explanation of the observed BAU and can produce primordial GW signals.

A.The thermal effective potential

In order to examine the electroweak phase transition (EWPT), we first take h1, h2, and S1 as the field configurations, and obtain the field dependent masses of the scalars (h, H, A, H±, S), the Goldstone boson (G, G±), the gauge boson, and fermions. The masses of scalars are given: m^h,H,S2=eigenvalues(MP2^),(16)m^G,A2=eigenvalues(MA2^),(17)m^G±,H±2=eigenvalues(MC2^),(18)MP2^11=3λ12h12+λ3452h22+m122tβ-λ12v2cβ2-λ3452v2sβ2+κ12S12,MP2^22=3λ22h22+λ3452h12+m122tβ-λ22v2sβ2-λ3452v2cβ2+κ22S12,MP2^33=mS2+κ12h12+κ22h22+λS2S12-κ12v2cβ2-κ22v2sβ2,MP2^12=MP2^21=λ345h1h2-m122,MP2^13=MP2^31=κ1h1S1,MP2^23=MP2^32=κ2h2S1,MA2^11=λ12h12+m122tβ-λ12v2cβ2-λ3452v2sβ2+(λ3+λ4-λ5)2h22+κ12S12,MA2^22=λ22h22+m122tβ-λ22v2sβ2-λ3452v2cβ2+(λ3+λ4-λ5)2h12+κ22S12,MA2^12=MA2^21=λ5h1h2-m122,MC2^11=λ12h12+m122tβ-λ12v2cβ2-λ3452v2sβ2+λ32h22+κ12S12,MC2^22=λ22h22+m122tβ-λ22v2sβ2-λ3452v2cβ2+λ32h12+κ22S12,MC2^12=MC2^21=(λ4+λ5)2h1h2-m122,(19)where λ345=λ3+λ4+λ5.

The masses of the gauge boson are given: m^W±2=14g2(h12+h22),m^Z2=14(g2+g′2)(h12+h22),m^γ2=0.(20)

We neglect the contributions of light fermions, and only consider the masses of top quark and bottom quark, m^t2=12yt2h22/sβ2,m^b2=12yb2h12/cβ2,(21)where yt=2mtv and yb=2mbv.

Now, we study the effective potential with thermal correction. The thermal effective potential Veff in terms of the classical fields (h1, h2, S1) is composed of four parts: Veff(h1,h2,S1,T)=V0(h1,h2,S1)+VCW(h1,h2,S1)+VCT(h1,h2,S1)+VT(h1,h2,S1,T)+Vring(h1,h2,S1,T),(22)where V0 is the tree-level potential, VCW is the Coleman-Weinberg potential, VCT is the counter term, VT is the thermal correction, and Vring is the resummed daisy correction. In this paper, we calculate Veff in the Landau gauge.

We obtain the tree-level potential V0 in terms of the classical fields (h1, h2, S1) V0=[12m122tβ-14λ1v2cβ2-14λ345v2sβ2]h12+[12m1221tβ-14λ2v2sβ2-14λ345v2cβ2]h22+λ18h14+λ28h24-m122h1h2+14λ345h12h22+κ14h12S12+κ24h22S12+12mS2S12+124λsS14-κ14v2cβ2S12-κ24v2sβ2S12.(23)

The Coleman-Weinberg potential in the MS¯ scheme at the 1-loop level has the form [95] VCW(h1,h2,S1)=∑i(-1)2sinim^i4(h1,h2,S1)64π2⁢[lnm^i2(h1,h2,S1)Q2-Ci],(24)where i=h, H, A, H±, S, G, G±, W±, Z, t, b, and si is the spin of particle i. Q is a renormalization scale, and we take Q2=v2. The constants Ci=32 for scalars or fermions and Ci=65 for gauge bosons. ni is the number of degrees of freedom, and nh=nH=nG=nA=1,nH±=nG±=2,nW±=6,nZ=3,nt=nb=12.(25)

With VCW being included in the potential, the minimization conditions of scalar potential in Eq. (22) and the CP-even mass matrix will be shifted slightly. To maintain the minimization conditions at T=0, we add the so-called “counter-terms”, VCT=δm12h12+δm22h22+δλ1h14+δλ12h12h22+δλ2h24+δm02S12+δκ1h12S12+δκ2h22S12,(26)where the relevant coefficients are determined by ∂VCT∂h1=-∂VCW∂h1,∂VCT∂h2=-∂VCW∂h2,∂VCT∂S1=-∂VCW∂S1,(27)∂2VCT∂h1∂h1=-∂2VCW∂h1∂h1,∂2VCT∂h1∂h2=-∂2VCW∂h1∂h2,∂2VCT∂h2∂h2=-∂2VCW∂h2∂h2,∂2VCT∂S1∂S1=-∂2VCW∂S1∂S1,∂2VCT∂h1∂S1=-∂2VCW∂h1∂S1,∂2VCT∂h2∂S1=-∂2VCW∂h2∂S1,(28)which are evaluated at the electroweak (EW) minimum of {h1=vcβ,h2=vsβ,S1=0} on both sides. As a result, the VEVs of h1, h2, S1, and the CP-even mass matrix, will not be shifted.

It is a well-known problem that the second derivative of the Coleman-Weinberg potential at T=0 suffers from logarithmic divergences originating from the vanishing Goldstone masses. To solve the divergence problem, we take a straightforward approach of imposing an IR cutoff at mIR2=mh2 for the masses of Goldstone bosons of the divergent terms, which gives a good approximation to the exact procedure of on-shell renormalization, as argued in [16].

The thermal contributions VT to the potential can be written as [96] Vth(h1,h2,S1,T)=T42π2∑iniJB,F(m^i2(h1,h2,S1)T2),(29)where i=h,H,A,H±,S,G,G±,W±,Z,t,b, and the functions JB,F are JB,F(y)=±∫0∞dxx2ln[1∓exp(-x2+y)].(30)

Finally, the thermal corrections with resummed ring diagrams are given [97,98]: Vring(h1,h2,S1,T)=-T12π∑ini[(M¯i2(h1,h2,S1,T))32-(m^i2(h1,h2,S1,T))32],(31)where i=h,H,A,H±,S,G,G±,WL±,ZL,γL. WL±, ZL, and γL are the longitudinal gauge bosons with nWL±=2, nZL=nγL=1. The thermal Debye masses M¯i2(h1,h2,S1,T) are the eigenvalues of the full mass matrix, M¯i2(h1,h2,T)=eigenvalues[MX2^(h1,h2)+ΠX(T)],(32)where X=P, A, C. ΠX are given by ΠP11=[9g22+3g′22+6yb2cβ2+6λ1+4λ3+2λ4+κ1]T224,ΠP22=[9g22+3g′22+6yt2sβ2+6λ2+4λ3+2λ4+κ2]T224,ΠP33=[4κ1+4κ2+λS]T224,ΠP13=ΠP31=ΠP23=ΠP32=0,ΠA11=ΠC11=ΠP11,ΠA22=ΠC22=ΠP22,ΠA12=ΠA21=ΠC12=ΠC21=0.(33)

The physical mass of the longitudinally polarized W boson is M¯WL±2=14g2(h12+h22)+2g2T2.(34)The physical mass of the longitudinally polarized Z and γ boson is M¯ZL,γL2=18(g2+g′2)(h12+h22)+(g2+g′2)T2±Δ,(35)with Δ2=164(g2+g′2)2(h12+h22+8T2)2-g2g′2T2(h12+h22+4T2).(36)

B.Calculation of electroweak phase transition and gravitational wave

In a first order cosmological phase transition, bubbles nucleate and expand, converting the high temperature phase into the low temperature phase. The bubble nucleation rate per unit volume at finite temperature is given by [99–101] Γ≈A(T)e-SE(T),(37)where A(T)∼T4 is a prefactor and SE is the Euclidean action SE(T)=S3(T)T=∫dx3[12(dϕdr)2+V(ϕ,T)].(38)At the nucleation temperature Tn, the thermal tunneling probability for bubble nucleation, per horizon volume and per horizon time, is of order one, and the conventional condition is S3(T)T≈140. The bubbles nucleated within one Hubble patch proceed to expand and collide, until the entire volume is filled with the true vacuum.

There are two key parameters characterizing the dynamics of the EWPT: β and α. β describes, roughly, the inverse time duration of the strong first order phase transition, βHn=Td(S3(T)/T)dT|T=Tn,(39)where Hn is the Hubble parameter at the bubble nucleation temperature Tn. α is defined as the vacuum energy released from the phase transition normalized by the total radiation energy density ρR at Tn, α=ΔρρR=Δρπ2g*Tn4/30,(40)where g* is the effective number of relativistic degrees of freedom. We use the numerical packages CosmoTransitions [102] and PhaseTracer [103] to analyze the phase transition and compute quantities related to cosmological phase transition.

In a radiation dominated universe, there are three sources of GW production at an EWPT: bubble collisions (in which the localized energy density generates a quadrupole contribution to the stress-energy tensor, which in turn gives rise to the GW), sound waves in the plasma, and magnetohydrodynamic (MHD) turbulence. The total resultant energy density spectrum can be approximately given as ΩGWh2≃Ωcolh2+Ωswh2+Ωturbh2.(41)Recent studies show that the energy deposited in the bubble walls is negligible, despite the possibility that the bubble walls can run away in some circumstances [104]. Therefore, although a bubble wall can reach relativistic speed, its contribution to the GW can generally be neglected [105,106]. Therefore, in the following discussions we do not include the contribution from bubble collision Ωcol.

The GW spectrum from the sound waves can be obtained by fitting to the result of numerical simulations [107], Ωswh2=2.65×10-6(Hnβ)(κvα1+α)2(100g*)1/3vw×(ffsw)3(74+3(f/fsw)2)7/2,(42)where fsw is the present peak frequency of the spectrum fsw=1.9×10-51vw(βHn)(Tn100 GeV)(g*100)1/6 Hz.(43)vw is the wall velocity and the factor κv is the fraction of latent heat transformed into the kinetic energy of the fluid. κv and vw are difficult to compute, and involve certain assumptions regarding the dynamics of the bubble walls. On the other hand, successful electroweak baryogenesis scenarios favor lower wall velocity vw≤0.15–0.3 [108], which allows the effective diffusion of particle asymmetries near the bubble wall front. In Ref. [109], however, it is pointed out that the relevant velocity for electroweak baryogenesis is not really vw, but the relative velocity between the bubble wall and the plasma in the deflagration front. As a result, the electroweak baryogenesis is not necessarily impossible even in the case with large vw. Therefore, in this paper we take two different cases of vw and κv [110]: (i)

For small wall velocity: vw=0.3 and κv≃vw6/56.9α1.36–0.037α+α.(44)

(ii)

For very large wall velocity: vw=0.9 and κv≃α0.73+0.083α+α.(45)

Considering Kolmogorov-type turbulence as proposed in Ref. [111], the GW spectrum from the MHD turbulence has the form [112,113] Ωturbh2=3.35×10-4(Hnβ)(κturbα1+α)3/2(100g*)1/3vw×(f/fturb)3[1+(f/fturb)]11/3(1+8πf/hn),(46)with the redshifted Hubble rate at GW generation hn=1.65×10-5(Tn100 GeV)(g*100)16 Hz.(47)The peak frequency fturb is given by fturb=2.7×10-51vw(βHn)(Tn100 GeV)(g*100)1/6 Hz.(48)The energy fraction transferred to the MHD turbulence κturb can vary between 5% and 10% of κv [107]. Here we take κturb=0.1κv.

For both sound wave and turbulence contributions as shown in Eqs. (43) and (46), the amplitudes of the GW spectra are proportional to vw and the peak frequencies shift as 1/vw. Therefore, one change in the wall velocity has, approximately, an order one effect on the spectrum and peak frequencies.

C.Results and discussions

The strength of the electroweak phase transition is quantified as ξc=vcTc,(49)with vc=⟨h1⟩2+⟨h2⟩2 at critical temperature Tc. The global minimum of potential has ⟨A⟩=0 because of the CP-conserving case. In order to avoid washing out the baryon number generated during the phase transition, a SFOEWPT is required and the conventional condition is ξc≥1.

After imposing the constraints of “pre-LHC”, the LHC Higgs data, the relic density, XENON1T, and Fermi-LAT, we scan over the parameter space in the previously selected scenario. We find some surviving samples which can achieve a SFOEWPT, and these samples are projected in Figs. 4 and 5. For all of the surviving samples, at Tc the two degenerate minima of potential are at (⟨h1⟩,⟨h2⟩,0) and (0, 0, 0), respectively. In the process of EWPT, ⟨S1⟩ always has no VEV.

10.1103/PhysRevD.103.035012.f4

FIG. 4.

The surviving samples projected on the planes of ⟨h1⟩ versus ⟨h2⟩, Tc versus mH, and Tc versus tanβ. All of the samples achieve a SFOEWPT.

10.1103/PhysRevD.103.035012.f5

FIG. 5.

The surviving samples projected on the planes of |sin(β-α), tanβ, m122 versus mH, m122 versus tanβ, mS versus mH, and λH. All of the samples are allowed by the constraints of “pre-LHC”, the LHC Higgs data, and the DM observables. The squares achieve a SFOEWPT, and bullets fail.

From Fig. 4, we find that ⟨h1⟩ and ⟨h2⟩ can vary in ranges from 20 GeV to 150 GeV and from 125 GeV to 230 GeV with Tc varying from 134 GeV to 240 GeV. Tc tends to increase with mH, and has a relative small value for a large tanβ. It should also be noted that the relic abundance of the DM is achieved by the thermal freeze-out in the early universe when the temperature was about T∼mS/25. In the model, Tc is much larger than mS/25 for 50 GeV<mS<700 GeV. Therefore, the EWPT hardly affects the thermal freeze-out process of DM.

From Fig. 5, we find that a SFOEWPT favors a small mH, namely a large mass splitting between mH and mA, which is consistent with Refs. [17,19]. Most of the samples lie in the region of mH<500 GeV, and there are several samples with mH>500 GeV when |sin(β-α)| is very close to 1.0. Also, a SFOEWPT favors m122 to increase with mH and decrease with an increase of tanβ. There is a relatively strong correlation between m122 and tanβ, and m122 is imposed upper and lower bounds for a given tanβ. With an increase of tanβ, m122 is stringently restricted by the theoretical constraints, leading that it is difficult to achieve a SFOEWPT. Thus, most of the samples lie in the region of the small tanβ. Also, similar results can be expected for type I 2HDM, especially for a large mH. The requirement of a SFOEWPT is not sensitive to mS, and disfavors |λH|>0.3.

In this model, due to the constraints of the DM relic density, generally the couplings κ1 and κ2 of the DM and h1,2 are not large enough to give remarkable effects on the SFOEWPT compared to that of 2HDM without the singlet DM. Certainly, the λi for the SFOEWPT may be slightly changed because of the presence of κ1 and κ2. Via the h1,2-dependent non-SM Higgs squared masses, the thermal correction VT and the resummed daisy correction Vring give the cubic terms of h1,2 proportional to T, which play key roles in generating a potential barrier and achieving the SFOEWPT. Schematically, the h1,2-dependent squared masses of H, A, and H± can be given as [17,114] mϕ2≈yϕm122cβsβ+fϕ(λi,h12,h22),(50)with yϕ=1 for A,H± and yϕ=sin(β-α) for H. When the term of m122 dominates over fϕ(λi,h12,h22), the cubic terms in finite potential are suppressed. As a result, the mass spectrum of H, A, and H± tends to be degenerate, and the phase transition strength tends to be reduced. Conversely, when fϕ(λi,h12,h22) dominates over the term of m122, the cubic terms are not suppressed. Therefore, the phase transition strength tends to be increased and the mass spectrum tends to be split.

There have been many studies of EWPT in the 2HDM [15–19]. Compared with previous studies, we comprehensively examine the current constraints on the parameter space using the latest heavy Higgs boson searches at LHC. Since the SFOEWPT is sensitive to the mass spectrum of the heavy Higgs bosons, the LHC searches for Higgs bosons can be used to reduce the parameter space achieving the SFOEWPT. For example, in this paper we find that the SFOEWPT is not achievable in the region of mH<360 GeV because of the constraints from the heavy Higgs boson searches at LHC.

Now we examine two key parameters, α and β/Hn, which characterize the dynamics of the SFOEWPT and govern the strength of GW spectra. A larger α and a smaller β/Hn can lead to stronger GW signals. In addition to the conditions of the successful bubble nucleations, we require that ξn=vnTn≥1,(51)with vn=⟨h1⟩2+⟨h2⟩2 at the nucleation temperature Tn. In fact, this is a more precise condition of SFOEWPT than ξc≥1. Also note that there generically exists a difficulty for solving bounce solution in a very thin-walled bubble, including in the package CosmoTransitions [115]. Therefore, we will neglect the samples with a very thin-walled bubble. Considering the constraints discussed above, we find some surviving samples and the corresponding α and β/Hn are shown in Fig. 6.

10.1103/PhysRevD.103.035012.f6

FIG. 6.

The parameters α and β/Hn characterizing the dynamics of the SFOEWPT.

The β/Hn may characterize the inverse time duration of the EWPT. A small β/Hn means a long EWPT, and gives strong GW signals. For the GW coming from the sound waves in the plasma, the GW signal will continue being generated and the energy density of the GW is thus proportional to the duration of the EWPT if the mean square fluid velocity of the plasma is non-negligible [107]. In addition, a large β/Hn can enhance the peak frequency of the GW spectra. The parameter α describes the amount of energy released during the EWPT, and therefore a large α leads to strong GW signals.

We pick out two benchmark points (BPs) and examine the corresponding GW spectra. Table III shows the input and output parameters of the BPs. Their phase histories are exhibited in Fig. 7 on field configurations versus the temperature plane. The field configuration S1 is not shown as the minima at any temperatures located at ⟨S1⟩=0. In Fig. 8, we show predicted GW spectra for our BPs along with expected sensitivities of various future interferometer experiments, and find that the amplitudes of the GW spectra reach the sensitivities of LISA, TianQin, BBO, DECIGO, and UDECIGO for BP1 (UDECIGO for BP2).

III

10.1103/PhysRevD.103.035012.t3

TABLE III.

Input and output parameters for two benchmark points for fixed mh=125 GeV, mA=600 GeV, and λh=0.

BP1BP2

sin(β-α)

0.99980.9991

tanβ

1.951.87mH (GeV)369.55387.97mH± (GeV)620.8618.31m122 (GeV)253049.153649.1mS (GeV)479.2501.7

λH

0.133

-0.129

λS

12.310.93Tc (GeV)135.7160.0Tn (GeV)61.095.0

β/Hn

35.6102.8

0.0940.018

10.1103/PhysRevD.103.035012.f7

FIG. 7.

Phase structures for BP1 (left) and BP2 (right). The lines show the field configurations at a particular minimum as a function of temperature. The arrows indicate that at that temperature (TC) the two phases linked by the arrows are degenerate, and can achieve the SFOEWPT.

10.1103/PhysRevD.103.035012.f8

FIG. 8.

Gravitational wave spectra for BP1 and BP2.

VI.CONCLUSION

We examine the status of the 2HDMIID confronted with the recent LHC Higgs data, the DM observables, and SFOEWPT, and discuss the detectability of GW at the future GW detectors. We choose the heavy CP-even Higgs boson H as the only portal between the DM and SM sectors, and focus on the case of the 125 GeV Higgs with the SM-like coupling. We find that for mA=600 GeV, mS<130 GeV and mH<360 GeV are excluded by the joint constraints of the 125 GeV Higgs signal data, the searches for additional Higgs via H/A→τ+τ-, A→HZ, H→WW,ZZ,γγ,hh at the LHC, as well as the relic density XENON1T.

A SFOEWPT can be achieved in the many regions of mH<500 GeV and mA=600 GeV, favors a small tanβ, and is not sensitive to the mass of DM. We find the benchmark points for which the predicted GW spectra can reach the sensitivities of LISA, TianQin, BBO, DECIGO, and UDECIGO.

ACKNOWLEDGMENTS

We thank L. Bian, Wei Chao and Huai-Ke Guo for helpful discussions. This work was supported by the National Natural Science Foundation of China under Grant No. 11975013, by the Natural Science Foundation of Shandong province (Grants No. ZR2017MA004 and ZR2017JL002), and by the ARC Centre of Excellence for Particle Physics at the Tera-scale under Grant No. CE110001004. This work is also supported by the Project of Shandong Province Higher Educational Science and Technology Program under Grant No. 2019KJJ007.

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