Dark Matter and Dark Force in the Type-I Inert 2HDM with Local U(1)_H Gauge Symmetry

We discuss dark matter (DM) physics in the Type-I inert two-Higgs-doublet model (2HDM) with local U(1)_H Higgs gauge symmetry. The local U(1)_H gauge symmetry is assigned to the extra Higgs doublet in order to avoid the Higgs-mediated flavor problems, and it spontaneously breaks down to its discrete subgroup. The lightest neutral scalar component H of the U(1)_H-charged Higgs doublet, which does not have Yukawa couplings with the Standard-Model (SM) fermions, is stable because of the remnant discrete symmetry, and it interacts with the SM particles through the U(1)_H gauge boson (Z_H) exchange as well as the SM boson exchange. We first investigate the constraint on the U(1)_H gauge interaction, especially through the kinetic and mass mixing between the SM gauge bosons and the extra gauge boson. Then we discuss dark matter physics in our 2HDM: thermal relic density, and direct/indirect detections of dark matter. The additional U(1)_H gauge interaction plays a crucial role in reducing the DM thermal relic density. The most important result within the inert DM model with local U(1)_H symmetry is that ~ O(10) GeV dark matter scenario, which is strongly disfavored in the usual Inert Doublet Model (IDM) with Z_2 symmetry, is revived in our model because of newly open channels, H H ->Z_H Z_H , Z_H Z. Exotic Higgs decays, h ->Z_H Z_H, Z Z_H, would be distinctive signatures of the inert 2HDM with local U(1)_H symmetry.


I. INTRODUCTION
The discovery of a Standard-Model-like Higgs boson at the LHC [1,2] opens new era in particle physics and cosmology. The precise measurements of its mass and couplings to the Standard Model (SM) particles will reveal the structure of the Higgs sector, which is the least known piece in the SM. Up to now, its couplings to the ordinary particles are consistent with the predictions of the SM within uncertainties and most of results at the LHC can be understood in the framework of the SM [3,4]. On the other hand, there are some clues on new physics beyond the SM: nonbaryonic dark matter (DM), dark energy, neutrino oscillation, baryon asymmetry of the universe, and etc., which cannot be explained by the renormalizable SM and require its extensions beyond the SM.
One simple extension of the SM is to add one extra Higgs doublet. In fact, many highenergy theories predict extra Higgs doublets, and the two-Higgs-doublet models (2HDMs) could be interpreted as the effective theories of those high energy theories after we integrate out heavy particles. Of course, the 2HDMs could be interesting by themselves, because of their rich phenomenology and benchmark models with an extended Higgs sector. 2HDMs predict extra neutral and charged scalar bosons in addition to a SM-like Higgs boson, and the extra scalar bosons may change phenomenology of Higgs boson and SM particles at colliders [5]. We could also find cold dark matter candidates in some 2HDMs: one of the extra scalars [6] or one of the extra fermions, which may be added to the models in 2HDMs with gauged U(1) H symmetry [7]. When the Higgs potential in 2HDMs has a CP-violating source, the baryon asymmetry of the universe may be explained [8]. And small neutrino masses may naturally be generated by one-loop diagram in some 2HDMs [9]. Finally, it is very interesting that in 2HDMs with flavor-dependent U(1) H gauge symmetry, the anomalies in the top forward-backward asymmetry at the Tevatron and B → D ( * ) τ ν decays at BABAR may be reconciled [10].
One important phenomenological issue in models with extra Higgs doublets is the socalled Higgs-mediated flavor changing neutral current (FCNC) problem. If a right-handed (RH) fermion couples with more than two Higgs doublets, FCNCs involving the neutral scalar bosons generally appear after the electro-weak (EW) symmetry breaking. In many cases, this Higgs-mediated flavor problem is resolved by imposing softly-broken Z 2 symmetrý a la Glashow and Weinberg [11]. The discrete Z 2 symmetry could be replaced by other discrete symmetry [12] or continuous local gauge symmetry [7]. In fact, the present authors proposed a new class of 2HDMs where U(1) H Higgs gauge symmetry is introduced instead of softly broken discrete Z 2 symmetry, in order to avoid the flavor problem [7], and discussed the phenomenology of Type-I 2HDM with U(1) H in Ref. [13]. When Higgs doublets are charged under the extra gauge symmetry, the so-called ρ parameter is deviated from the SM prediction at the tree level if the U(1) H -charged Higgs doublet develops a nonzero vacuum expectation value (VEV). In Ref. [13], the authors investigate the constraints not only from the EW precision observables (EWPOs) but also from the recent LHC results on the SM-like Higgs search, especially in Type-I 2HDMs where only one Higgs doublet is charged and the SM particles are neutral under U(1) H . The deviation of the ρ parameter is mainly from the tree-level mass mixing between Z and the U(1) H gauge boson (Z H ), and from the mass differences among the scalar bosons. The bounds on EWPOs require small U(1) H interactions, so that its effects become tiny in physical observables.
In this work, we consider a new scenario where the VEV of the U(1) H -charged Higgs doublet is zero in our Type-I 2HDM and a U(1) H -charged SM-singlet scalar Φ breaks U(1) H spontaneously. In this case, the mass mixing between gauge bosons is also negligible at the tree level. Therefore we can expect that the Z H gauge interaction becomes sizable and then the idea of gauged U(1) H Higgs symmetry might be tested. The bounds on Z H mass and its couplings to the SM particles will come from the loop-level mass and kinetic mixings between the SM gauge bosons and Z H , as well as the tree-level kinetic mixing.
On the other hand, the SM fermions are chiral under new U(1) H gauge symmetry, so that the model may be anomalous unless new chiral fermions are introduced. As discussed in Ref. [7,13], we could consider the anomaly-free U(1) H charge assignment to build the 2HDM with local Higgs symmetry: For example, the SM particles are not charged and only one extra Higgs doublet is charged under U(1) H . In our 2HDM, the U(1) H will be broken by the nonzero VEV of Φ. However, the residual symmetry may still remain and we could find stable cold dark matter (CDM) candidates. In Sec. III, we introduce the conditions for the stability of DM candidates and discuss dark matter physics in the Type-I 2HDMs with local U(1) H gauge symmetry. One well-known dark matter model among 2HDMs is the so-called inert doublet model (IDM) [6,14]. In the ordinary IDM, one extra Z 2 symmetry is imposed and the extra Higgs doublet is the only Z 2 -odd particle. If the Higgs doublet does not develop the nonzero VEV, Z 2 symmetry forbids the decay of the lightest Z 2 -odd scalar boson, and the scalar boson could be a good CDM candidate. The scalar CDM interacts with the SM particle through the scalar exchange and the EW interaction, and one could find the favored regions for the correct thermal relic density around m DM ∼ 60 GeV or m DM > 500 GeV [6], which are safe for the constraints from the collider searches and the DM direct detection searches. Especially, the light CDM scenario faces the strong bound from the invisible search at the LHC, and the only allowed mass region is around the resonance of the SM-Higgs.
In our 2HDM with local U(1) H gauge symmetry, at least one Higgs doublet is charged under the U(1) H symmetry, and the scalar component respects the remnant discrete symmetry of U(1) H after the EW and U(1) H symmetry breaking. This discrete symmetry originates from the local U(1) H and will protect the DM from decay to all orders in perturbation theory even in the presence of higher dimensional operators. Stability of the scalar DM is guaranteed by the local discrete symmetry, and the scalar DM interacts with the SM particles strongly through the U(1) H gauge interaction. The extension of the usual IDM to the U(1) H gauge symmetric one not only suggests the origin of the Z 2 Higgs symmetry, but also may open up a new scenario for dark matter phenomenology, which cannot be achieved in the usual IDM. In fact, the U(1) H gauge interaction can plays an important role in thermalizing the scalar DM, and ∼ O(10) GeV CDM scenario could be revived in our IDM, which is a very interesting aspect of our model. In Sec. IV A and Sec. IV B, we investigate the constraints on not only the relic density and the DM direct detection, but also the DM indirect detection, in the IDM with Z 2 symmetry (IDMwZ 2 ) and in the IDM with local U(1) H symmetry (IDMwU(1) H ). The indirect astrophysical observations would be one of the ways to prove our model so that we calculate the velocity-averaged cross section for dark matter annihilation in the halo, and consider the constraints from the Fermi-LAT by observing the γ-ray flux from the dwarf spheroidal galaxies, which is one of the recent results relevant to the light CDM scenario.
The organization of this paper is as follows. In Sec. II, we introduce the Lagrangian of the Type-I 2HDM with U(1) H Higgs symmetry and discuss the constraints on the U(1) H gauge interactions. In Sec. III, we discuss dark matter physics in the IDMwZ 2 and IDMwU(1) H : the stability of CDM, the thermal relic density, the DM direct and indirect detections. Sec.
V is devoted to our conclusion.

II. 2HDM WITH U (1) H HIGGS SYMMETRY
Based on Ref. [13], we discuss our setup of the Type-I 2HDM with U(1) H Higgs symmetry, where only extra Higgs doublet H 1 is charged under the U(1) H gauge symmetry and all the SM particles including the SM Higgs doublet H 2 are neutral. The Lagrangian for gauge fields and scalar fields is given by where F µν Y , F aµν SU(2) L , and F µν H are the field strengths of U(1) Y , SU(2) L , and U(1) H of the gauge fields, B µ , A a µ , andẐ µ H , and g ′ , g and g H are their gauge couplings. q H 1 is the U(1) H charge of H 1 . κ is the kinetic mixing between U(1) Y and U(1) H field strength tensors, which is allowed by the local gauge symmetry. It is assumed to be a free parameter. Finally V scalar (H) is the potential for the complex scalars: A new scalar Φ is a singlet under the SM gauge group, but is charged under U(1) H gauge symmetry, and thus it breaks the U(1) H by the nonzero VEV ( Φ ≡ v Φ ). c l is the dimensionless coupling of the higher-dimension operator, which is suppressed by one arbitrary scale (Λ Φ ), to make the mass difference between a CP-even scalar and a pseudoscalar bosons. Note that only (H † 1 H 2 ) 2 term can be multiplied by U(1) H -charged operator (Φ/Λ Φ ) l , whereas (Φ † Φ/Λ 2 Φ ) n term can be multiplied to all terms because of U(1) H gauge invariance. If both are included, we would lose predictability because of too many nonrenormalizable interactions. Therefore we will choose U(1) H charge judiciously and l = 1 is allowed by U(1) H gauge symmetry. Then the leading nonrenormalizable operator would be dim-5 operator (the c l=1 term), and other operators with (Φ † Φ/Λ 2 ) n would be at least dim-6 or higher. The single unique dim-5 operator would lift up the mass degeneracy between a CP-even scalar and a pseudoscalar bosons by spontaneous U(1) H breaking. Therefore we do not lose predictability much even if we consider nonrenormalizable operators. We also give comment on how to realize c l in Sec. III. (H † 1 H 2 ) term may be also allowed, multiplying (Φ/Λ Φ ) l following the U(1) H charge assignment. This term may cause the nonzero VEV of H 1 , so that we define the charge assignment to forbid this term, as we see in Sec. III.
After the EW and U(1) H symmetry breaking, the VEVs of H i and Φ give the masses of the gauge bosons. In general, the kinetic mixing between the SM gauge bosons andẐ H may be generated as follows, If H 1 is nonzero, ∆M 2 Z H Z will be generated at the tree level. When H 1 is zero, the tree-level mixing of Z and Z H does not exist, but the mixing may occur at the loop level, as shown in the next subsection.

A. Mass mixing and kinetic mixing at the one-loop level
Before the EW and U(1) H gauge symmetry breaking, the local gauge symmetry forbids the mass mixing and the kinetic mixing among neutral gauge bosons, except for the kinetic mixing κ term in Eq. (1). If we assume that κ is negligible,Ẑ H does not couple with the SM fermions.
After the EW and U(1) H symmetry breaking, the mass mixing and the extra kinetic mixings, κ Z and κ γ , will appear at the loop level, even if κ is negligible at the EW scale. The one-loop contributions of the extra scalars to κ Z and κ γ are, for instance, given in Fig. 1. For nonzero H 1 , they are evaluated as where Z ab is defined as Λ is the cut-off scale, but κ Z,γ are independent of Λ because the Λ dependence is canceled in κ Z,γ . c hm H i and c Am χ i are mixing angles of the CP-even and CP-odd scalars, which are defined by where {h m , A l }({H 0 i , χ i }) are the CP-even and CP-odd scalars in the mass (interaction) bases, respectively, and the formulas for {h m , A l }({H 0 i , χ i }) are referred to Ref. [13]. In the ordinary 2HDM, tan β is defined as tan β = H 0 2 / H 0 1 , and H 1 = 0 corresponds to cos β = 0. In this limit, the physical neutral scalars can be expressed as h = h 2 cos α − h Φ sin α, H = h 1 , andh = h 2 sin α + h Φ cos α, where h 1 , h 2 , and h Φ are the CP-even neutral components of H 1 , H 2 , and Φ after symmetry breaking, respectively, and α is the mixing angle between The mass mixing, ∆M 2 Z H Z , is also induced at the one-loop level through the diagrams in Fig. 1, In the limit cos β → 0, which corresponds to H 0 1 → 0, the mixing parameters converge to Thus the mass and kinetic mixings of the gauge bosons are generated radiatively, even if the tree-level mass mixing is negligible. The U(1) H gauge coupling g H will be constrained by the collider experimental results through the couplings of Z H with the SM fermions induced by the radiative corrections. Even if κ is zero at the cut-off scale (Λ), it may become sizable at the low scale through the RG running. In fact, the RG running correction can easily enhance the mixing in the case with large g H . For example, the RG flow of κ γ is estimated as assuming that the running corrections of e and g H are small. Eventually, g H could be large if the masses of scalars are degenerate, but such large g H coupling would enhance the kinetic mixing easily. In our analyses, we set the cut-off scalar (Λ) at 1 TeV, and include the radiative corrections to κ, assuming κ(Λ) = 0.

B. Constraints on Z H coupling from the mixings
The kinetic mixings and the mass mixing could be interpreted as the coupling of Z H , after changing the interaction basis to the mass basis. Assuming that the mixing angles are small enough, the couplings could be described as where a mixing angle ξ is given by the mass mixing, The mixings among neutral gauge bosons are strongly constrained by the EWPOs and Z ′ searches at high energy colliders, as discussed in Ref. [13]. If Z H is heavier than the center-of-mass energy of LEP (209 GeV), we can derive the bound on the effective coupling of Z H [15,16], depending on the Z H mass. The lower bound on M Z H /g H would be ∼ O(10) TeV [16]. If Z H is lighter than 209 GeV, the upper bound of Z H coupling would be O(10 −2 ) in order that we avoid conflicts with the data from e + e − → f − f + (f = e, µ) [16,17]. If Z H is lighter than M Z , the upper bound on the kinetic mixing is κ 0.03 [18,19]. In the very light Z H region (100 MeV M Z H 10 GeV), the strong bound comes from the BaBar experiment, κ O(10 −3 ) [19].
The upper bounds on the Z H production at the Tevatron and LHC are investigated in the processes, pp(p) → Z H X → f f X [15,17,20,21], and the stringent bound requires O(10 −3 ) times smaller couplings than the Z-boson couplings around M Z H = 300 GeV [21].
If we require the conservative bound sin ξ 10 −3 , according to Ref.
[21], we could estimate the upper bound in the large tan β case as follows, based on the Eqs. (9), (10), and (11), Including the bounds from the scalar searches, we see the allowed region for g H and M Z H in Fig. 2 in the case with H 0 1 = 0 (cos β = 0).
C. Constraints on the scalar bosons in the 2HDM with cos β = 0 In the limit cos β → 0, the new U(1) H gauge interaction could be large because the constraint from the ρ parameter is drastically relaxed. At the tree level, the ρ parameter constraint in general two Higgs doublet model with U(1) H symmetry (2HDMwU(1) H ) is given by [7] {q Now in the Type-I 2HDMs the SM Higgs doublet and all the SM fields are U(1) H neutral so that q H 2 = 0. In the IDM which we take into account for analysis in this paper, H 1 = 0, namely cos β = 0. Then Eq. (16) implies that the ∆ρ constraint disappears at the tree level, and the U(1) H gauge coupling g H can be large. Also Z H gets its mass only from the VEV of the Φ field and can be relatively light. Furthermore, the scalar component of H 1 may be stable and could be a good cold dark matter candidate, as we discuss in the next section. Below, we introduce the constraints on the extra scalars from the collider experiments and EWPOs.

Constraints on the extra scalar bosons
When H 1 does not develop a nonzero VEV, the scalar components in H 1 , (H, A, H + ), are in the mass eigenstate and do not have the Yukawa couplings with the SM fermions. This corresponds to the setup of the IDM [6,14]. If H is the lightest particle, A and H + decay to H and on-shell or off-shell Z, Z H , W + . The constraints on H, A, and H + from the collider experiments have been widely discussed in the framework of the IDM [22]. The search for multi leptons plus missing energy at LEP gives the lower bounds: m H + 90 GeV and m A 100 GeV with m A − m H 8 GeV. The exotic Z decay may be kinematically forbidden by the condition, m A + m H > M Z in order not to change the decay width of the Z boson.
The mass differences between m H + and m A (m H ) are also strongly constrained by the EWPOs, and it has been studied in the Type-I 2HDMwU(1) H in the case with large scalar masses and generic tan β [13]. When tan β is large, the mass difference between the heavy CP-even scalar boson (H) and the massive CP-odd scalar boson (A) becomes small and the phase of Φ is eaten byẐ H . tan β = 0 makes the masses degenerate in our 2HDM with c l = 0 in Eq. (2), even after the EW and U(1) H symmetry breaking. When H and A are degenerate and become CDM candidates, the Z boson exchange diagram enhances the DM direct detection cross section, for example, through H + N → A + N, where N is a nucleon, and the CDM scenario would be excluded immediately by the XENON100 and LUX experiments. A small mass difference ( O(100) keV) between H and A would be enough to suppress the direct detection cross section. In the 2HDMs with Z 2 symmetry, such a term is generated by λ 5 term, which is clearly invariant under the usual discrete Z 2 symmetry, ( However this terms is not allowed if we implement the discrete Z 2 symmetry to continuous U(1) H gauge symmetry at the renormalizable level. Still the effective λ 5 term may be induced by higher-dimensional operators integrating out heavy particles as we discuss below (see below, Sec. III B.).

Constraints from exotic SM-like Higgs decays
The IDM condition cos β = 0 realizes the situation that H 1 completely decouples with the SM fermions, so that the scalar components could be very light because of the relaxed experimental bounds, and the SM-like Higgs may decay to the extra scalar bosons, as well as where h is the CP-even scalar mainly from Φ.
The extra scalars affect the signal strength µ of the SM-like Higgs boson at the LHC, where µ is defined in Ref. [13]. As discussed in Ref. [  When the Z H boson is light, the SM-like Higgs boson h may decay into Z H Z H or ZZ H . If the mass of the dark matter H is less than the half of the SM-like Higgs mass (m h ), h also can decay invisibly into HH. These exotic h decays are strongly constrained by the search for the invisible and/or nonstandard Higgs decays at the LHC [27]. We set the bound on the exotic Higgs decay to be [28,29] where σ V h SM,2HDM is the cross section for the V h production in the SM and in the 2HDM, respectively, and BR(h → Z (H) Z H ) is the branching ratio for the h → Z (H) Z H decay.
In Fig. 2, the allowed region for M Z H and g H is shown in gray color by taking into account the constraints from the Z-Z H mixing, EWPOs, search for exotic scalars, vacuum stability and unitarity, based on the above arguments and Ref. [13]. For the numerical calculation we choose the following parameter spaces: 20 Since the SM fermions do not have the U(1) H charge, the relatively large g H is allowed as shown in Eq. (15). In the small M Z H region, g H is ∼ O(0.1), but it can be O(1) just below the Z pole and much higher in the large M Z H region. We note that the gray region in Fig. 2 also satisfies the constraints from the thermal relic density of the CDM, DM direct detection searches at the CDMSII, XENON100 and LUX experiments, while the green points satisfy the DM indirect detection search at Fermi-LAT as well as the relic density and the DM direct detection search, which we will discuss in the following section. In the small M Z H region, the allowed values of g H do not change significantly even if we include the constraints from the DM indirect detection search. However, in the large M Z H region, g H should be less than O(1) when the constraints from the DM indirect detection search are taken into account. This is because one of the main channels for the annihilation of dark matters in this region is HH → ZZ H , which would gives a stronger bound on g H .

III. HOW TO STABILIZE DARK MATTER IN THE IDMWU (1) H
In this section, we introduce a dark matter candidate in the IDMwU(1) H and compare predictions in the IDMwU(1) H with those in the IDMwZ 2 [6]. As we discussed in Sec. II, there is a U(1) H gauge boson as well as a U(1) H -charged Higgs doublet in our Type-I 2HDM with U(1) H . If the Higgs doublet H 1 does not develop a VEV, one of the scalar components of the doublet is a CDM candidate and the correct thermal relic density could be achieved. First of all, let us discuss the stability of the dark matter in a generic U(1) H symmetric model in the Sec. III A, and then discuss dark matter physics in the IDMwU(1) H .

A. General conditions for DM stability
In general, we could build a gauge extension of the SM, such as U(1) H . As discussed in Ref. [7], not only additional Higgs doublets but also extra fermions may have to be introduced in order to satisfy the anomaly-free conditions, depending on the charge assignment. The gauge symmetry would be spontaneously broken by extra scalars to avoid an extra massless gauge boson. Still, we could expect that there could be a residual local discrete symmetry after the spontaneous gauge symmetry breaking. Then, if the extra fermions and/or scalars may respect the residual symmetry, their decays may be forbidden by the remaining local discrete symmetry.
Let us discuss the generic U(1) H symmetric models with matters ψ I and Φ whose charges are q I and q Φ . ψ I are U(1) H -charged extra fields. Simply, let us assume that U(1) H is broken only by a nonzero VEV of Φ. Generally, the charges can be described as where n I , n Φ and N are integers, and all the charges are irreducible fractions. Now, we consider the case Φ = 0. In order to have a residual discrete symmetry after U(1) H breaking, the Φ field should be a singlet under the residual one after the U(1) H symmetry breaking. When we assume that U(1) H breaks down to Z m symmetry, the integers (m, m ′ ) can be defined by the relation, This could be satisfied by (m, m ′ ) = (n Φ , N) and we find that the residual symmetry is In our 2HDM with U(1) H , two fields H 1 and Φ are charged under U(1) H . If H 1 also develops a nonzero VEV, H 1 may also break the Z |n Φ | . It depends on q H 1 , and the charge assignment may be fixed by the H † 1 H 2 term, namely by the operator This term should be forbidden because it will correspond to the tadpole term of H 1 which causes nonzero H 1 when H 2 and Φ break the EW and U(1) H symmetries by their nonzero VEVs. Furthermore, if the µ n term is allowed, the relation of the charges, n H 1 = nn Φ , is required, so that H 1 is singlet under Z |n Φ | , as far as n is an integer. In the next section, we discuss the IDMwU(1) H where only Φ breaks U(1) H . q H 1 is defined as q H 1 /q Φ = n H 1 /n Φ is not an integer to realize a stable particle.
B. Toward the IDMwU (1) H One of the motivations for considering Type-I 2HDMs could be CDM, which can be realized in the so-called inert 2HDM (IDMwZ 2 ), where one of the extra neutral scalar bosons could be a good CDM candidate.
In the usual IDMwZ 2 , all the SM particles including the SM Higgs doublet with nonzero VEV are Z 2 -even, whereas the other Higgs doublet without nonzero VEV is Z 2 -odd * . Then the scalar component of the Z 2 -odd doublet could be stable and could be a good CDM candidate, if the Higgs doublet does not get a nonzero VEV. One of the main issue in this type of CDM models is to generate mass differences among the charged, CP-odd, and CP-even components of the Z 2 -odd doublet, in order to avoid strong constraints from the collider experiments and the direct dark matter searches. In the IDMwZ 2 , the λ 4 |H † 1 H 2 | 2 and λ 5 (H † 1 H 2 ) 2 terms in the Higgs potential play an important role in the mass spectrum. Especially the λ 5 term shifts the pseudoscalar mass and thereby suppresses kinematically the interaction of the CP-even component with a nucleus (N) through the Z exchange.
In the IDMwU(1) H , the discrete Z 2 symmetry is gauged to continuous local U(1) H symmetry, so that massless U(1) H gauge boson is predicted if the stability of dark matter is guaranteed by U(1) H Higgs gauge symmetry. The U(1) H symmetry could be spontaneously broken, introducing SM singlet scalar Φ with a nonzero U(1) H charge q Φ . According to the general discussion in the previous subsection, the scalar components of H 1 could be stable if H 1 = 0 and q H 1 /q Φ is not an integer. However, the U(1) H symmetric Higgs potential generates an extra flat direction, so that H and A tend to be degenerate. In fact, the U(1) H symmetry forbids the λ 5 term at the tree level, which was the origin of the mass difference in the ordinary IDMwZ 2 . Without the λ 5 term, too large cross section for the direct detection of dark matter will be predicted, which is in serious conflict with the data. In order to avoid this catastrophe,we generate the effective λ 5 term from higher-dimensional operators, such as the c l term in Eq. (2). Then the λ 5 term depends on Φ in this scenario.
One simple way to realize higher-dimensional operators for an effective λ 5 term is to introduce an extra complex scalar (ϕ) with a nonzero U(1) H charge and φ = 0. Let us consider the following renormalizable potential among the scalar bosons: where λ H and λ ϕ are functions of ϕ and Φ, respectively, and respect the local U(1) H gauge symmetry as well as the SM gauge symmetries. Only Φ breaks U(1) H and only H 2 breaks the EW symmetry, due to nonzero values of V Φ and V H . Now we assume that ϕ does not develop a nonzero VEV, and the direct coupling between H i and Φ is forbidden by a suitable choice of q Φ . If U(1) H is spontaneously broken by the nonzero Φ , the squared mass difference (∆m 2 ) between real and imaginary components of ϕ will be generated by the λ ϕ term. Assuming that ϕ is heavier than the EW scale and λ H = λ 0 H ϕ, λ 5 is induced effectively at low energy when we integrate out the ϕ field: where m ϕ R and m ϕ I are the masses of the real and imaginary parts of the complex scalar ϕ.
After the U(1) H symmetry breaking, the effective scalar potential is the same as the one of the IDMwZ 2 , so that the small mass difference between H and A can be achieved, and we can evade the strong bound from the direct detection search for dark matter.

IV. DM PHENOMENOLOGY IN IDMWU (1) H
Base on the above argument and the setup, we discuss the dark matter physics in IDMwU(1) H and compare the results of the IDMwU(1) H with the ones of the IDMwZ 2 .

A. Relic density and direct detection
The most recent measurement for the DM relic density carried out by the PLANCK Collaboration yields its value to be [30] Ω CDM h 2 = 0.1199 ± 0.0027.
We impose that thermal relic densities calculated in the IDMwZ 2 and IDMwU(1) H satisfy this bound within 3σ deviation, assuming that the scalar dark matter in the IDMwZ 2 or IDMwU(1) H is the only CDM of the universe. In case we assume that there exist another dark matter particles which contribute to the dark matter relic density, we impose the bound Ω IDM ≤ 0.1280. On the other hand, dark matter may interact with atomic nuclei in a detector, and it can be detected by underground experiments, such as the DAMA [31], CoGeNT [32], CRESST [33], XENON [34,35], CDMS [36], LUX [37] and etc. The CoGeNT, DAMA, CRESST-II, and CDMS-II experiments show some excesses in the light dark matter region with the dark matter mass of about 10 GeV and the spin-independent cross section for the WIMP-nucleon scattering is predicted to be σ SI ∼ 10 −40 − 10 −42 cm 2 [36]. It is interesting that the four experiments have some excesses in the similar dark matter mass region, but the measured spin-independent cross section is different by about two orders of magnitude from each other. Also these signals are almost ruled out by the XENON100 and LUX experiments, where the upper bound for the spin-independent cross section is ∼ 10 −45 cm 2 at m H ∼ 33 GeV and and ∼ 10 −44 cm 2 at m H ∼ 10 GeV [34,35,37]. The positive signals might be accommodated with each other while being reconciled with the XENON100 and LUX results by introducing isospin-violating dark matter [38] or exothermic inelastic DM scattering.
In our model, we assume that the positive signals at low DM mass regions are excluded by the XENON100 and LUX experiments and impose the bound for the spin-independent scattering cross section for the dark matter and nucleon from the LUX experiment. In Fig. 3, we draw Feynman diagrams which dominantly contribute to the direct detection of dark matter. When the mass difference is negligible between m H and m A , the Z and Z H exchange diagrams become dominant and the spin-independent cross section for the dark matter candidate and nucleon could exceed the bounds from the LUX experiment. This problem is easily cured by the generation of the mass difference between m H and m A with a sizable λ 5 term. Then we can ignore Z and Z H exchanges in Fig. 3, and the scalar (h,h) exchange is dominant in the direct cross section: where f is the form factor, µ is the reduced mass of the DM-nucleon system, m N is the mass of the nucleon and α is the mixing angle between h Φ and H 0 2 . We calculate the elastic cross section of the scattering of H on atomic nuclei by using micrOMEGAs, where the velocity of H near the Earth, v H ≈ 0.001c [39]. In Fig. 5, we show our predictions in this model. All the points in Fig. 5 pass the bound from the LUX experiment, as well as the collider experiments, which we described in Sec. II.
Thermal relic density, direct detections and indirect detection of DM in the IDMwZ 2 have been studied extensively in the literature (see, for example, [6,40]). In the IDMwZ 2 , the extra scalars annihilate into two fermions through the SM-Higgs exchanging and two gauge bosons, as we see in Fig.4. There are two interesting scenarios: light dark matter ( m H M Z ) and heavy dark matter (m H 500GeV). In Fig. 5, we show the relic densities (a) in the light H scenario and (b) in the heavy H scenario, respectively. The pink points correspond to the IDMwZ 2 , whereas the cyan points to the IDMwU(1) H . The horizontal line is the current value of the DM thermal relic density.

The case of IDM with Z 2 symmetry
Let us first explain the results of the IDMwZ 2 . In the light dark matter scenario, the dark matter pair annihilation into the SM fermions (HH → ff ) through the s-channel exchange of the SM Higgs boson h makes a dominant contribution to the relic density in the range of m H 50 GeV. However the direct detection experiments of DM [34,37] and the invisible decay of the SM-like Higgs boson [28,29] strongly constrain the h − H − H coupling. Therefore the annihilation in that light mass region cannot be large enough to thermalize the scalar DM, and the light CDM scenario has been already excluded, as we see in Fig. 5.
In the h resonance region, m H ∼ m h /2 ∼ 60 GeV, the annihilation cross section is enhanced and the thermal relic density could be below the current observation.  region with m H m W , the annihilation channels, HH → W W and HH → ZZ are open so that the IDMwZ 2 predicts the small relic density in Fig. 5. The co-annihilation of H and A(H + ) also becomes relevant in the region with 40 GeV m H 80 GeV, if the mass difference between H and A(H + ) is small. In this case, the pair annihilation of H + H − or AA could also be relevant in the region m H 80 GeV, and their contribution to the relic density could reach 60% of the relic density in a certain parameter region.
In the heavy dark matter scenario, HH → ZZ, W + W − processes are very efficient and many channels like HH → hh, tt are open. They can easily reduce the DM thermal relic density. As shown in Fig. 5 (b), the relic density is below the current data in the region m H 500 GeV, and we have to consider extra DM species in order to account for the DM relic density in Eq. (25). If we wish to accommodate the current data on the DM relic density within uncertainties in the IDMwZ 2 , the DM mass m H should be greater than 500 GeV in the heavy dark matter scenario. The pair annihilations of H + H − and AA also contribute to the DM relic density in the region where the mass difference between H + , A and H is small, but the ratio of their contribution to the relic density is less than 0.5. Typically the HH → W + W − process makes a dominant contribution to the DM relic density. But in some parameter spaces the contribution of the pair annihilation of H + H − and AA to the relic density can be higher than that of the pair annihilation of HH. The process HH → hh can be relevant in some parameter spaces, but the ratio of the contribution is less than 0.3.  Fig. 3. Φ may also work as the mediator if it is light. Fig. 5 (a) shows the relic density in the case m H 126 GeV. We also assume that M Z H is also smaller than 126 GeV because the result in the limit m Z H ≫ m H would not be different from that in the IDMwZ 2 . If M Z + M Z H < 2m H , then the HH → ZZ H channel is also open and it gives another contribution to the CDM relic density. The new channels involving Z H ('s) could be dominant because of the weak bound on g H in Fig. 2, and we could find the annihilations to Z H and SM gauge bosons play a role in decreasing the relic density in Fig. 5 (a). Especially, we can find many allowed points in the region with m H 40 GeV in the IDMwU(1) H (cyan points in Fig.5 (a)), unlike the IDMwZ 2 case where the relic density becomes too large for m H 40 GeV. In this region, the gauge coupling g H of U(1) H prefers a small value, g H 0.5 as shown in Fig. 2.
In the heavy H scenario, the qualitative feature in the IDMwU(1) H is similar to that in the IDMwZ 2 as shown in Fig. 5 (b). For m H 500 GeV, the predicted DM relic density is below the current value and another dark matter is required to make up for the deficit. In the region m H 500 GeV, we can find the parameter regions which satisfy the current relic density. In most parameter spaces, the predicted DM density in the IDMwU(1) H is slightly smaller than that in the IDMwZ 2 . This is because the new channels like HH → Z H Z H and HH → ZZ H are open in the heavy CDM case, too, so that they yield an extra contribution to the dark matter density.

B. Constraints from indirect detection
In the IDMwU(1) H , the light Z H gauge boson may open the new annihilation channels of the CDM: HH → Z H Z H , ZZ H , γZ H . The Z H contributions could be dominant in the relic density because of the large g H , and then astrophysical observations may give a crucial bound on the Z H -dominant scenario, as we discuss below. In the heavy H scenario, the HH annihilation to the SM particles could be dominant because of gauge interactions. In Fig. 4, we depict representative Feynman diagrams contributing to the HH annihilation.
In the galactic halo, the DM pair annihilation may occur and can be observed through the products of the annihilation : γ-rays, positrons, antiprotons, and etc. † This means that the detection of large signals for the γ-rays, positrons or antiprotons over astrophysical backgrounds would indicate indirect evidence for the dark matter annihilation. There are some tantalizing excesses in the mono-energetic gamma ray signal around 135 GeV observed by the Fermi-LAT [41][42][43] and in the positron excesses at a few GeV ∼ TeV scale observed by PAMELA, Fermi, and AMS02 Collaborations [44][45][46]. However, it is still unclear that the origin of the excesses comes from annihilation or decay of dark matter or astrophysical backgrounds [47]. In this paper, we assume that the excesses are originated in the astrophysical phenomena. Then, indirect detection of dark matter plays a role in constraining models.
Many experiments are devoted to measure the products of the dark matter annihilation typically by observing cosmic rays from dwarf spheroidal satellite galaxies [48][49][50][51] and galactic center [52]. Especially, the dwarf spheroidal satellite galaxies of the Milky Way are excellent targets for the detection of the dark matter annihilation because of its large dark matter content and low astrophysical backgrounds like a hot gas. We calculate the velocityaveraged cross section for the dark matter annihilation in the dwarf spheroidal galaxies by using micrOMEGAs [39].
The integrated γ-ray flux φ, is divided two parts where the second part is related to the dark matter distribution in the galactic halo. Under the Navarro-Frenk-White density profile of dark matter [53] the J factor of each dwarf galaxy is given in Ref. [54]. The quantity Φ PP for the self-conjugate particles is defined by where σv is the velocity-averaged annihilation cross section, B f is the branching ratio for each channel XX → f f ′ , where f and f ′ could be the SM particle or new particle, and dN f γ /dE γ is the photon energy spectrum for each annihilation channel. We note that the direct γ-ray production from the dark matter annihilation which yields the mono-energetic signal is loop-suppressed in both the IDMwZ 2 and the IDMwU(1) H . The photon energy spectrum depends on the decay patterns of intermediate states f and f ′ .
1. The case of the IDM with Z 2 symmetry First, we consider the velocity-averaged annihilation cross section in the IDMwZ 2 by using micrOMEGAs. Figure 6 shows σv in units of cm 3 /s (a) in the light H scenario and (b) in the heavy H scenario. The pink and yellow points correspond to the IDMwZ 2 while the cyan and green points to the IDMwU(1) H , respectively. All the points satisfy thermal relic density and the direct detection constraint from the LUX experiments. And yellow and green points satisfy the additional constraints from indirect detections. The horizontal line around 3 × 10 −26 cm 3 /s is the bound from the relic density when the s-wave annihilation of dark matters is dominant. The upper region of this line would be allowed for the s-wave annihilation dominant case. The solid and dotted curves are the bounds from the Fermi-LAT experiment by assuming that the dominant annihilation channel is HH → bb and HH → W + W − . The lower region of these curves is allowed if the dominant annihilation channel is HH → bb or HH → W + W − . We note that all the points in Fig. 6 satisfy the constraints from the LUX experiment and their relic densities are below the current bound.
In the light H scenario, the HH → bb process can be dominant in the Higgs-resonant region (m H ≃ 60 GeV), whereas the HH → W W process becomes dominant both in the Higgs-resonant region and in the region m H ≃ 80 GeV. In these regions, the bounds from the Fermi-LAT experiment might constrain directly the IDMwU(1) H as well as the IDMwZ 2 . From Fig. 6 (a), we can find the allowed parameter spaces in the Higgs-resonant region, but there is no allowed point at m H 80 GeV. We note that the region for m H 40 GeV is not allowed in the IDMwZ 2 because of overclosing of the Universe. As we will see in the next subsection, this conclusion changes completely when we extend the discrete Z 2 symmetry to local U(1) H symmetry because new channels open, namely HH → Z H Z H , Z H Z for on-shell or off-shell Z.
In the heavy H scenario, the contribution of the HH → bb process is negligible and the HH → W + W − , ZZ processes become effective in the IDMwZ 2 . Depending on the coupling to the SM-like Higgs boson, the HH → hh process could become dominant over the HH → W + W − process. In the HH → W + W − dominant case, we can use the bounds from the Fermi-LAT experiment. We note that in the IDMwZ 2 , σv is less than about 3 × 10 −24 cm 3 /s. In the region m H 200 GeV, there might be the parameter region allowed from the γ-ray observation by the Fermi-LAT when the dominant DM annihilation channel is HH → W + W − .

The case of IDM with local U (1) H symmetry
In Fig. 6, the cyan and green points depict the velocity-averaged annihilation cross section σv versus the dark matter mass m H in the IDMwU(1) H . We note that all points satisfy the bounds from the direct detection search of dark matter at the LUX and the upper bound of the relic density, and the green points satisfy additional constraints from the indirect detection.
In the light H scenario (Fig. 6a), the allowed region in the IDMwU (1)  than that in the IDMwZ 2 because of additional channels like HH → Z H Z H and HH → ZZ H . In the region m H 80 GeV, the overall feature is the same as in the IDMwZ 2 except that in the IDMwU(1) H , σv could be larger by an order of magnitude because of newly open channels such as HH → Z H Z H , Z H Z, etc. In the Higgs-resonant region, m H ∼ 60 GeV, we find that some points are allowed from the indirect detection search at the Fermi-LAT if the dominant annihilation process is HH → bb. As we already discussed in the previous subsection, there are no allowed points below m H ≃ 40 GeV in the IDMwZ 2 , because the model predicts too large relic density. However, in the O(10)GeV CDM scenario of the IDMwU(1) H , we can find the allowed parameter spaces which satisfy the constraints from the relic density and the direct detection search of dark matter. In this region, the dominant annihilation channels are HH → Z H Z H and HH → ZZ H , so that we cannot apply the constraints from the indirect detection search of dark matter directly.
In the heavy H scenario, the allowed parameter region from direct detection search for dark matter and relic density in the IDMwU(1) H is broader than in the IDMwZ 2 . In this scenario, typically the HH → W + W − , ZZ or HH → hh processes could be dominant like the IDMwZ 2 . However, in the IDMwU(1) H , the HH → Z H Z H or HH → ZZ H could be dominant processes over the HH → W + W − , ZZ, hh processes, depending on the values of the parameters in the model. It turns out that the HH → Z H Z H process could become dominant at m H 500 GeV, whereas the HH → ZZ H process could dominate over the other processes at m H 200 GeV.

Φ PP
In the case that the HH → Z H Z H and HH → ZZ H processes are dominant over other processes in the IDMwU(1) H , we cannot apply the bound from the indirect detection search of dark matter at the Fermi-LAT, which assumes that annihilation into the bb or W + W − pair is dominant. The γ-ray spectrum strongly depends on the decay pattern of particles produced from the dark matter annihilation. The Z H boson dominantly decays into a pair of SM fermions through the Z-Z H mixing and its decay pattern is similar to that of the SM Z boson. In this case, the main source of the γ-ray would be the HH → Z H Z H (Z) → bbbb process. The γ-ray spectrum produced from this process would be different from the one from HH → bb. Therefore we should not apply the bound from the Fermi-LAT shown in Fig. 6 directly.
In order to find out if our models can satisfy the constraints from the indirect detection search at the Fermi-LAT, we compute the quantity Φ PP in Eq. (28) by using micrOMEGAs. The range of the photon energy from 500 MeV to 500 GeV is summed. For the comparison, we use the value in Ref. [49], which was obtained by using the joint analysis of seven Milky Way dwarfs and Pass 7 data from the Fermi Gamma-ray Space Telescope. A 95% upper bound is Φ PP = 5.0 +4.3 −4.5 × 10 −30 cm 3 s −1 GeV −2 [49]. In Fig. 6, the green points satisfy this upper limit in the IDMwU(1) H while the yellow points satisfy this limit in the IDMwZ 2 . The pink and cyan points predict more γ-ray than the upper limit.
In the light H scenario, only the Higgs-resonant region is allowed in the IDMwZ 2 , but the region m H 40 GeV is allowed in the IDMwU(1) H too. In the Higgs-resonant region, σv in some parameter spaces is below the reference line for the s-wave annihilation dominant case at the decoupling temperature of dark matter. There are two sources for the smaller σv in the present universe. At the decoupling temperature, the velocity of dark matter is O(0.1), but the velocity at the halo is O(0.001). Another source is that the co-annihilation of HH + or HA and pair annihilation of AA or H + H − also contribute to the relic density, but the co-annihilation does not appear in the halo at the current temperature. In the region m H 40 GeV, only IDMwU(1) H can have a proper dark matter model to explain the bounds from the relic density, the direct and indirect detection searches of dark matter. In this region, most of points predict smaller σv than the reference line for the s-wave dominant annihilation case. We find that the HH → Z (H) Z H process is dominant in this case and m Z H ≈ m H . In this quasi-degenerate case, there is a suppression factor in the phase space, which is proportional to the velocity of dark matter. This can explain the gap of about O(0.01 ∼ 0.001) in σv .
In the heavy H scenario, the allowed region appear at m H 500 GeV in both the IDMwZ 2 and IDMwU(1) H . Most of points predict smaller relic density than the observed one at the Planck experiment, but some of them predict the exact relic density within 3σ uncertainties. We note that the spin-averaged annihilation cross section in the halo at the zero temperature is over the reference line in the whole region. We find that the contribution of HH annihilation to the relic density is at most 90% of the total relic density predicted in the model. Remaining the relic density is accounted for by the co-annihilation of HH + or HA and/or the pair annihilation of AA or H + H − , which do not occur in the present halo. This may explain the small gap between the reference line and the predicted σv in the models.

C. Three Benchmark Points for Illustration
Since our model has many parameters, we choose three benchmark points for which we can discuss the underlying physics in a more transparent manner. In Table I, we show three benchmark points, all of which are safe for the astrophysical observations discussed above: the 1st one in the light dark matter region (L1), the 2nd one in the h resonance region (L2), and the 3rd one in the heavy dark matter region (H1).
At the L1 point, the dark matter mass is m H = 38.6 GeV, while the U(1) H gauge boson mass is M Z H = 39.2 GeV, which is almost the same as m H . The coupling λ 5 = −0.174 generates the mass difference between A and H with m A = 110 GeV, where the HN → AN process through the Z or Z H exchange is kinematically forbidden and we can evade the strong bound from the direct detection search of dark matter. λ 4 = −0.957 and the spinindependent cross section for the WIMP-nucleon scattering is σ SI = 2.3 × 10 −46 cm 2 , which is below the LUX bound at m H ∼ 38 GeV. The corresponding relic density of DM is Ωh 2 = 0.113, where HH → Z H Z H is the dominant channel for the correct thermal relic density. The DM annihilation cross section at the present universe is σv 0 = 8.6 × 10 −28 cm 3 /s, where the HH → ZZ H process contributes to the annihilation of HH by 98% and the HH → bb process is about 2%.
At the L2 point, m H = 53.8 GeV and M Z H = 29.6 GeV. λ 5 = −0.144, which leads to m A = 108 GeV. The mass difference between H and A is large enough to avoid the LUX bound and the spin-independent cross section for the H and nucleon is σ SI = 2.9×10 −46 cm 2 , which is below the LUX bound. Here, λ 4 = −0.999. The relic density is Ωh 2 = 0.117, where the contribution of HH → Z H Z H to the relic density is 52% and that of HH → bb is 34%. Finally the DM annihilation cross section at the present universe is σv 0 = 2.20 × 10 −26 cm 3 /s. The contribution of HH → ZZ H to σv 0 is 52% and that of HH → Z H Z H is 45%.
At the H1 point, m H = 821 GeV and M Z H = 985 GeV. λ 5 = −0.164, which leads to m A = 827 GeV. In this case, H + and A are almost degenerate to H, but still m A − m H = 6 GeV, which is large enough that one can evade the strong bound from the direct detection experiments. λ 4 = 0.086 and the spin-independent cross section for the H and nucleon is σ SI = 6.1 × 10 −46 cm 2 , which is below the LUX bound. The relic density is Ωh 2 = 0.119 and a lot of processes contribute to the relic density because dark matter is much heavier than the SM particles (m H ≫ m W , m Z , m f ). The contribution of each process of HH → ZZ H , W + W − , H + H − → W + W − , AZ H is 11%, 10%, 9%, and 9%, respectively. Besides these processes, many processes for HH, H + H + , H + H − , H + H annihilation and so on contribute to the relic density. The DM annihilation cross section at the present universe is σv 0 = 5.89 × 10 −26 cm 3 /s. The contributions of HH → ZZ H , HH → W + W − , and HH → ZZ to σv are 34%, 31%, and 23%, respectively. Those of HH → hh, AW + W − , tt are less than 8%.
In summary, three benchmark points have qualitatively different mass spectra and couplings. Still we find that all these three points produce acceptable phenomenology for the Higgs boson(s) and DM. The inert 2HDM with local U(1) H gauge symmetry has rich phenomenology due to new particles introduced by local U(1) H gauge symmetry. In particular, light DM ( 60 GeV) is still allowed in the gauged inert 2HDM (IDMwU(1) H ), unlike the usual inert 2HDM (IDMwZ 2 ) where DM below ∼ 60 GeV is excluded, because of the strong bound from the direct detection cross section, thermal relic condition and the invisible decay of the SM Higgs. New particles and new interactions in the gauge U(1) H model help to cure these problems in the IDMwZ 2 , when suitable particle spectra and proper couplings are chosen in the IDMwU(1) H .

V. CONCLUSION
In this paper, we constructed inert 2HDM with local U(1) H gauge symmetry instead of the usual discrete Z 2 symmetry. In this IDMwU(1) H , the U(1) H gauge symmetry is spontaneously broken by a nonzero VEV of a U(1) H -charged SM-singlet scalar Φ. The U(1) H -charged new Higgs doublet H 1 neither couples to the SM fermions nor develops a nonzero VEV. The new Higgs doublet H 1 can be decomposed into a CP-even, CP-odd, and charged scalars (H, A, H ± , respectively), and one of the neutral scalar bosons (either H or A) could be a good CDM candidate because of the discrete Z 2 Higgs gauge symmetry, which is a remnant of the original U(1) H gauge symmetry spontaneously broken by nonzero Φ . At the renormalizable level with these particle contents, H and A are degenerate in mass, and the model is immediately excluded by the strong constraints from direct detection experiments on the Z exchange to HN → AN. This problem can be solved by lifting the mass degeneracy between the DM H and the pseudoscalar A by introducing another U(1) Hcharged SM-singlet scalar ϕ which does not develop nonzero VEV. This new singlet scalar ϕ induces an effective λ 5 coupling as described in Eq. (24), and would lift the degeneracy between H and A. Then one can avoid the strong bound from the direct detection cross section from the SM Z exchange in HN → AN.
This scenario is a generalization of the well-known Inert Doublet Model, where the discrete symmetry Z 2 of IDMwZ 2 is replaced by local U(1) H gauge symmetry that is spontaneously broken into its Z 2 subgroup by nonzero VEV of Φ. Like the usual IDMwZ 2 , the new model we presented in this paper has many interesting features, such as the co-annihilation of the scalars and new channels for the DM pair annihilations into Z H Z H , etc.
In the ordinary IDMwZ 2 , there are two interesting CDM mass regions: m H ≃ 60 GeV and m H 500 GeV. In these regions, the correct DM relic density can be achieved without any conflict with the experimental results including the bound from the direct and indirect dark matter detections. Especially, the recent results from the indirect detection experiments may give strong bounds on the DM mass and its interactions. We investigated the allowed region for the recent FERMI-LAT data in Sec. IV B. As we see in Fig. 6, the FERMI-LAT data strongly constrains the DM annihilation cross section, so that the allowed regions are reduced in both of the light and heavy scenarios. In fact, we found that 50 − 60 GeV CDM scenario is only allowed in the light CDM case, because of the low annihilation cross section at the present universe temperature.
In case of the IDMwU(1) H , the nonzero U(1) H charge is assign to one Higgs doublet and one SM singlet, and U(1) H is spontaneously broken by the singlet. The U(1) H breaks down to discrete symmetry, so that we could interpret the continuous symmetry as the origin of the Z 2 Higgs symmetry in the IDMwZ 2 . The additional massive gauge boson (Z H ) interacts with the extra Higgs doublet and weakly interacts with the SM particles through the mass and kinetic mixing with the SM gauge bosons in the IDMwU(1) H . If we assume the kinetic mixing between U(1) H and U(1) Y is negligibly small before the EW and U(1) H symmetry breaking, we could expect that the Z H interaction could be sizable as we see in Fig. 2. In the dark matter physics, the Z H contribution might be dominant, and it may make the IDMwU(1) H distinguishable from the IDMwZ 2 . In fact, we could find many allowed points for the bounds from the collider and dark matter experiments below m H = 60GeV in the IDMwU(1) H , where the IDMwZ 2 is totally excluded. We also investigated the consistency with the recent FERMI-LAT data, and we concluded that many points are still allowed in the light and heavy CDM mass regions. In particular a new possibility opens up that the inert 2HDM can accommodate the γ-ray excess from the galactic center, if we promote the discrete Z 2 symmetry to local U(1) H symmetry. This case will be discussed in detail elsewhere [55].
Our light CDM scenario predicts the exotic SM-Higgs decays: h → Z H Z H and Z H Z. Z H could decay to the SM fermions like the Z boson, so that it may be possible to observe Z H at LHC, as studied by the CMS collaboration [56]. Developing the analysis of the SM-Higgs branching ratio, we may be able to draw the stronger bound on the exotic SM-Higgs decay according to the global fitting [28,29].