A light scalar dark matter extension of the type-II two-Higgs-doublet model

We examine the type-II two-Higgs-doublet model with a light scalar dark matter ($S$) after imposing the constraints from the Higgs searches at the LHC and dark matter experiments. We first assume that both two CP-even Higgses ($h$ and $H$) are portals between the DM and SM sectors, and the CP-odd Higgs ($A$) and $H$ are heavier than 130 GeV. We find that the DM with a mass of $10\sim 50$ GeV is disfavored by the joint constraints of the 125 GeV Higgs signal data, the relic density, XENON1T (2017), PandaX-II (2017) and the Fermi-LAT. Next, we consider a special scenario in which the heavy CP-even Higgs is taken as the 125 GeV Higgs. The light CP-even Higgs is the only portal between the DM and SM sectors, and the DM mass is slightly below Higgs resonance. We find that the signal data of the 125 GeV Higgs restrict $\tan\beta$ to be in the range of $1\sim 1.5$ for $m_h<$ 62 GeV. The $gg\to A\to hZ$ and $b\bar{b}\to h \to \tau^+\tau^-$ channels at the LHC can impose lower limits and upper limits on $\tan\beta$, respectively. For appropriate values of $\tan\beta$, $\lambda_h$ and $m_h$, the DM with a mass of $10\sim 50$ GeV is allowed by the constraints of the Higgs searches at the LHC and dark matter experiments. For example, $\tan\beta$ is restricted to be in the range of $1.0\sim1.5$ for 10 GeV $$ 1.125 is excluded for 30 GeV $


I. INTRODUCTION
The weakly interacting massive particle (WIMP) is one popular candidate of dark matter (DM). The simplest WIMP-DM model is the standard model (SM) plus a real singlet scalar as DM [1]. In the model, the current experiments excluded the DM mass up to 330 GeV, except a small range near 63 GeV [2,3]. Much of the region excluded in this model can be recovered if the Higgs sector is extended to the two-Higgs-doublet model (2HDM) [4] which contains two neutral CP-even Higgs bosons h and H, one neutral pseudoscalar A, and two charged Higgs H ± [3,[5][6][7][8][9][10][11][12][13]. Recently, Ref. [12] took the 125 GeV Higgs with wrong sign Yukawa coupling of the down-type quark as the only portal between the DM and SM sector, and found that the DM mass is allowed to be as low as 50 GeV for appropriate isospin-violating DM interactions with nucleons. The SS → AA annihilation channel can play an important contribution to the relic density, but does not solve the tension between the DM relic density and the signal data of the 125 GeV Higgs, which leads that m S < 50 GeV is excluded. Ref. [3] showed that if the heavy CP-even Higgs boson is the only portal, much of the region below 100 GeV are excluded.
In this paper, the question we want to answer is, which parameter space of the type-II 2HDM with a scalar DM is the DM with a mass below 50 GeV allowed in? We will consider joint constraints from the theory, the precision electroweak data, the flavor observables, the signal data of the 125 GeV Higgs, the searches for the additional Higgs at the LEP and LHC, the relic density, XENON1T (2017), PandaX-II (2017) and the Fermi-LAT searches for DM annihilation from dwarf spheroidal satellite galaxies (dSphs). This paper is organized as follows. In Section II, we introduce some characteristic features of the type-II 2HDM with a scalar DM. In Section III we perform numerical calculations. In Section IV, we examine the allowed parameter space after imposing the relevant theoretical and experimental constraints. Finally, we draw our conclusion in Section V.

II. TYPE-II TWO-HIGGS-DOUBLET MODEL WITH A SCALAR DARK MAT-TER A. Type-II two-Higgs-doublet model
In the type-II 2HDM with a scalar DM, the scalar potential includes two parts, V 2HDM + V S , and they are the potential of type-II 2HDM and the potential of the DM sector, respectively. The V 2HDM with a softly-broken discrete Z 2 symmetry is given by [14] Here we focus on the case of the CP-conserving in which all λ i and m 2 12 are real. The two complex Higgs doublets have hypercharge Y = 1, Where v 1 and v 2 are the electroweak vacuum expectation values (VEVs) with v 2 = v 2 1 + v 2 2 = (246 GeV) 2 and tan β = v 2 /v 1 . After spontaneous electroweak symmetry breaking, the remaining five physical Higgs particles are two neutral CP-even h and H, one neutral pseudoscalar A, and two charged scalars H ± .
In the type II 2HDM, the up-type fermions obtain masses from only Φ 2 field, and the down-type fermions from Φ 1 field [15,16]. The Yukawa interactions are given by where The Yukawa couplings of the neutral Higgs bosons normalized to the SM are given by where α is the mixing angle of the two CP-even Higgs bosons, and V denotes Z or W .

B. A scalar dark matter
Now we add a real singlet scalar S to the type-II 2HDM, and the potential containing the S field is written as The linear and cubic terms of the S field are forbidden by a Z ′ 2 symmetry, under which S → −S. The S is a possible DM candidate provided it does not acquire a VEV. We can obtain the DM mass and the cubic interactions with the neutral Higgses from the Eq. (5),

III. NUMERICAL CALCULATIONS
In this paper, we discuss two different scenarios: In the calculation of the thermal averaged cross section, the kinetic energy of the DM is non negligible in the early universe, and as a result the resonant condition in the DM pair-annihilation can be met for m S slightly smaller than m h /2. The temperature at the present time is much lower compared to the freeze-out temperature, and the velocity of DM is much smaller than that in the early universe. Therefore, the resonant condition for the today DM pair-annihilation is hardly satisfied for m S slightly smaller than m h /2.
In our calculations, to implement the constraints from the Higgs searches at the LHC, we need employ SusHi [17] to compute cross sections of Higgs in the gluon fusion and bbassociated production at NNLO in QCD. Results of SusHi might not be reasonable for a small Higgs mass. Therefore, we take m h > 20 GeV, which determines the DM mass to be larger than 10 GeV in the Case B. The measurement of the branching fraction of b → sγ imposed the strongest lower limit on the charged Higgs mass of type-II 2HDM, m H ± > 580 GeV [18]. The S, T and U oblique parameters give the stringent constraints on the mass spectrum of Higgses of type II 2HDM [19][20][21]. One of m A and m H is around 600 GeV, and another is allowed to have a wide mass range including low mass [19]. Therefore, to allow h to be light enough we fix m A = 600 GeV in the Case B.
In our calculations, we consider the following observables and constraints: (1) Theoretical constraints. The scalar potential of the model contains the potential type-II 2HDM and the potential 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. [3,9]. Here we employ the formulas in [3,9] to implement the theoretical constraints. Compared to Refs. [3,9], there are additional factors of 1 2 in the κ 1 term and the κ 2 term of this paper.
(2) The oblique parameters. The S, T , U parameters can impose strong constraints on the mass spectrum of Higgses of 2HDM. The 2HDMC [22] is employed to implement the constraints from the oblique parameters (S, T , U).
(3) The flavor observables and R b . SuperIso-3.4 [23] is employed to consider the constraint of B → X s γ, and ∆m Bs is calculated following the formulas in [24]. Besides, we perform the constraints of bottom quarks produced in Z decays, R b , which is calculated using the formulas in [25,26].
(4) The global fit to the signal data of the 125 GeV Higgs. Because the 125 GeV Higgs couplings with the SM particles in this model can be modified compared to the SM, the SM-like decay modes will be corrected. In the Case A, h is the 125 GeV Higgs, and the invisible decay h → SS is kinematically allowed, which will be strongly constrained by the experimental data of the 125 GeV Higgs. In the Case B, H is the 125 GeV Higgs, and the invisible decay H → SS is absent since the coupling HSS is taken as zero.
However, the decay H → hh is kinematically allowed for m h < 62.5 GeV. We perform the χ 2 calculation for the signal strengths of the 125 GeV Higgs in the µ ggF +tth (Y ) and µ V BF +V h (Y ) with Y denoting the decay mode γγ, ZZ, W W , τ + τ − and bb, µ ggH+ttH (Y ) and µ V BF +V H (Y ) are the data best-fit values and a Y , b Y and c Y are the parameters of the ellipse, which are given by the combined ATLAS and CMS experiments [27]. We pay particular attention to the surviving samples with χ 2 − χ 2 min ≤ 6.18, where χ 2 min denotes the minimum of χ 2 . These samples correspond to be within the 2σ range in any two-dimension plane of the model parameters when explaining the Higgs data.
In addition, the ATLAS and CMS reported the upper limits on the branching ratio of invisible decay of the 125 GeV Higgs. In our calculation we impose the constraints, Br(h → SS) < 0.34 [27]. for enough large value of tan β. The cross section of the CP-even Higgs in the gluon fusion depends on sin(β−α) in addition to the Higgs mass and tan β. We use SusHi [17] to compute cross sections for Higgs in the gluon fusion and bb-associated production at NNLO in QCD. A complete list of the searches for additional Higgs considered by us is summarized in Table I and Table II where some channels are taken from Ref.

Channel Experiment
Mass range (GeV) Luminosity     [67]. Refs. [68,69] show that the LHC searches for the charged Higgs fail to constrain the model for m H ± > 500 GeV. Therefore, the searches channels of the charged Higgs are not included in this paper.
(6) The DM observables. We use micrOMEGAs [70] to calculate the relic density and today DM pair-annihilation. The model file is generated by FeynRules [71]. For 10 GeV < m S < 50 GeV, the DM will annihilate into bb dominantly in this model.
In this model, the elastic scattering of S on a nucleon receives the contributions of the process with t− channel exchange of h and H in the Case A, and only h exchange in the Case B. If both h and H contribute to the DM interactions with nucleons, the spin-independent cross section is given by [72],  limits. This is because the resonant condition for the today DM pair-annihilation is also satisfied for m h 2m S very close to 1.0. The left panel shows that tan β is restricted to be in the range of 1.0 ∼ 1.5 for 10 GeV < m s < 26 GeV. The middle panel shows that | λ h | is allowed to be as low as 10 −5 due to the h resonance contributions to the DM pair-annihilation.
In Fig. 6, we project the surviving samples on the planes of Br(h → SS) versus m h and Br(h → SS) versus m h 2m S . Fig. 6 shows that in the parameter space allowed by the XENON1T (2017), PandaX-II (2017), and the Fermi-LAT, Br(h → SS) is smaller than 3% for m S < 60 GeV and smaller than 0.2% for 60 GeV < m S < 120 GeV. The current searches for the DM at the LHC do not impose the constraints on the parameter space.

V. CONCLUSION
The type-II 2HDM with a scalar DM provides a WIMP-DM candidate economically.
Recent some studies do not find the parameter space of the DM with a mass below 50 GeV in the model. In this paper, we examine the DM with a mass below 50 GeV in the model  1 1.025 1.05 1.075 1.1 1.125 1.15 1.175 1 after imposing the constraints from the Higgs searches at the LHC and the DM experiments.
We first discuss a general scenario in which both two CP-even Higgses (h and H) are portals between the DM and SM sectors, and the CP-odd Higgs (A) and H are heavier than 130 GeV. We find that the DM with a mass of 10 ∼ 50 GeV is disfavored by the joint constraints of the 125 GeV Higgs signal data, the relic density, XENON1T (2017), PandaX-II (2017) and the Fermi-LAT.
Next, we consider a special scenario in which the heavy CP-even Higgs is taken as the 125 GeV Higgs, and the light CP-even Higgs is the only portal between the DM and SM sectors. The DM mass is slightly below Higgs resonance, m h /2 = (1.0 ∼ 1.2) × m S . We find that the signal data of the 125 GeV Higgs restrict tan β to be in the range of 1 ∼ 1.5 for [40] ATLAS Collaboration, "Search for diboson resonance production in the ℓνqq final state using p p collisions at √ s = 13 TeV with the ATLAS detector at the LHC," ATLAS-CONF-2016-062.
[41] ATLAS Collaboration, "Search for WW/WZ resonance production in ℓνqq final states in pp collisions at √ s = 13 TeV with the ATLAS detector," arXiv:1710.07235. [