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Higgs phenomenology in Type-I 2HDM with U(1)_H Higgs gauge symmetry

It is well known that generic two-Higgs-doublet models (2HDMs) suffer from potentially large Higgs-mediated flavor-changing neutral current (FCNC) problem, unless additional symmetries are imposed on the Higgs fields thereby respecting the Natural Flavor Conservation Criterion (NFC) by Glashow and Weinberg. A common way to respect the NFC is to impose Z_2 symmetry which is softly broken by a dim-2 operator. Another new way is to introduce local U(1)_H Higgs flavor symmetry that distinguishes one Higgs doublet from the other. In this paper, we consider the Higgs phenomenology in Type-I 2HDMs with the U(1)_H symmetry with the simplest U(1)_H assignments that the SM fermions are all neutral under U(1)_H, and we make detailed comparison with the ordinary Type-I 2HDM. After imposing various constraints such as vacuum stability and perturbativity as well as the electroweak precision observables and collider search bounds on charged Higgs boson, we find that the allowed Higgs signal strengths in our model are much broader than those in the ordinary Type-I 2HDM, because of newly introduced U(1)_H-charged singlet scalar and U(1)_H gauge boson. Still the ATLAS data on gg to h to gamma gamma cannot be accommodated. Our model could be distinguished from the ordinary 2HDM with the Z_2 symmetry in a certain parameter region and some channels. If the couplings of the new boson turn out to be close to those in the SM, it would be essential to search for extra U(1)_H gauge boson and/or one more neutral scalar boson to distinguish two models.

singlet do not couple to them. In the type-I 2HDM with U (1) H symmetry, we can achieve anomaly-free models without extra chiral fermions. Furthermore, constraints from flavor physics and the collider experiments could be relaxed drastically (see Secs. 3

and 4).
This paper is organized as follows. In Sec. 2, we recapitulate the Type-I 2HDM with the spontaneous U (1) H Higgs gauge symmetry breaking including the general Higgs potential, and discuss the vacuum stability condition for the Higgs potential. Then we derive the physical states of the Higgs fields and the masses of the SU (2) gauge bosons in terms of the gauge coupling and Higgs VEVs and discuss the bounds on the physical masses of the charged Higgs and neutral Higgs bosons. Section 3 is devoted to the discussion of the constraints derived from electroweak precision observables (EWPOs), and the comparison of our model with the usual Type-I 2HDM. (The results obtained in Sec. 3 involves only gauge couplings of two Higgs doublets and could be applied to and shared with other types of 2HDM [35].) Then we discuss phenomenology of Higgs bosons in our model at the LHC in Sec. 4. Conclusion of this paper is given in Sec. 5. We present some useful formulas in Appendix.
2 Type-I 2HDM with local U(1) H gauge symmetry

Generalities
In 2HDMs, symmetry to distinguish the two SU (2) L Higgs doublets is required in order to avoid tree-level FCNCs. One usually assign Z 2 parities to two Higgs doublets and the SM fermion fields [6] to achieve the NFC by Glashow and Weinberg. Depending on the charge assignment, one can obtain so-called Type-I 2HDM, Type-II 2HDM, and etc.. Since the Yukawa couplings of the SM fermions are controlled by the Z 2 parities, the models allow the couplings respecting the hypothesis of MFV.
In the usual 2HDMs with the softly broken Z 2 symmetry, there are extra physical scalar bosons: one extra CP-even scalar (H), one pseudoscalar (A), and one charged Higgs pair (H ± ). The scalar masses are given by the Higgs VEVs and dimensionless couplings in the Higgs potential at the renormalizable level. Therefore we can expect that the mass scales of all extra scalar bosons are around the electroweak (EW) scale, like the SM-like Higgs boson observed at the LHC. However, the masses and couplings of the extra scalar bosons are strongly constrained by the collider experiments and the EWPOs as well as the constraints from the flavor physics. One has to introduce the Z 2 symmetry breaking term (soft breaking via dim-2 operators), which generates the pseudo scalar mass (m A ), in order to consider the higher mass scales.
In Ref. [7], the present authors proposed gauged U (1) H symmetry, which may be considered as the origin of the Z 2 symmetry, and constructed a number of well-defined extensions of 2HDMs with only MFV. In this case, the pseudo scalar mass m A is generated by spontaneous symmetry breaking of U (1) H via nonzero VEV of a new U (1) H -charged singlet scalar Φ. The Lagrangian for the two Higgs (H i (i = 1, 2)) and an extra U (1) H -charged scalar (Φ) is is the covariant derivatives for H i under the SM-gauge groups. g H is the U (1) H gauge coupling, and q Hi and q Φ are U (1) H charges of H i 's and Φ, respectively. V scalar is the scalar potential for H i and Φ which breaks U (1) H and the EW symmetry. AndẐ Hµ is the U (1) H gauge boson in the interaction eigenstates. Finally L Yukawa is the Yukawa interaction between the SM fermions and the two Higgs doublets, which would be the same as the Yukawa interactions in Type-I, Type-II, etc.. ‡ This extension might suffer from tree-level deviation of the ρ parameter due to the kinetic and mass mixings between the U (1) H gauge boson and Z boson. Furthermore, this extension would modify relevant collider signatures because of the additional Higgs doublet as well as the extra gauge boson Z H and the complex scalar Φ.

Type-I 2HDM with local U (1) H symmetry
There are many different ways to assign U (1) H charges to the SM fermions to achieve the NFC in 2HDMs with local U (1) H gauge symmetry. The phenomenology will crucially depend on the U (1) H charge assignments of the SM fermions. In general, the models will be anomalous, even if U (1) H charge assignments are non-chiral, so that one has to achieve anomaly cancellation by adding new chiral fermions to the particle spectrum. For the Type-I case, the present authors noticed that one can achieve an anomalyfree U (1) H assignment even without additional chiral fermions as in Table 1. Only H 2 couples with the SM fermions, and the U (1) H charges of H 1,2 , q H 1 and q H 2 , should be different. Since the U (1) H charges of right-handed up-and down-type quarks (u and d) in Table 1 are arbitrary, one can construct an infinite number of new models from the usual Type-I 2HDM by implementing the softly broken Z 2 symmetry to spontaneously broken local U (1) H gauge symmetry. In the heavy Z H limit, all the models with Type-I models with local U (1) H with arbitrary u and d will get reduced to the conventional Type-I 2HDM with softly broken Z 2 term (see m 2 3 term in Eq. (2) in the next subsection). In Table 1, we present four interesting U (1) H charge assignments: the fermiophobic U (1) H with u = d = 0, U (1) B−L , U (1) R , and U (1) Y cases.

Scalar Potential
The scalar potential of general 2HDMs with U (1) H is completely fixed by local gauge invariance and renormalizability, and given by Φ is a complex singlet scalar with U (1) H charge, q Φ , and contributes to the U (1) H symmetry breaking.m 2 i (|Φ| 2 ) (i = 1, 2) and m 2 3 (Φ) could be functions of Φ: A mass parameter µ can be regarded as real by suitable redefinition of the phase of Φ. Note that the λ 5 in the usual 2HDMs with softly broken Z 2 symmetry does not appear in our models, because we impose the local U (1) H gauge symmetry instead of Z 2 . In our model, the effective λ 5 term would be generated from the scalar exchange, after U (1) H symmetry breaking. The effective λ 5 would contribute to the pseudoscalar mass, the vacuum stability and unitarity conditions like the ordinary 2HDMs. § Expanding the scalar fields around their vacua, one can study the physical spectra in the scalar sector including their masses and couplings. The neutral scalars, h i , χ i , h Φ , and χ Φ , and the charged Higgs, φ + i , in the interaction eigenstates are defined by 3) The scalar VEVs v i and v Φ satisfy the stationary conditions (or vanishing tadpole conditions): The coupling λ5 could also be generated by the dimension six operator λ ′ Then we have to keep all the possible dimension-6 operators in the scalar potential in order to analyze the physical spectra which is a formidable task, and we would lose the predictability. In this paper, we consider only the renormalizable lagrangian and just ignore higher dimensional operators for simplicity and predictability.

Masses and Mixings of Scalar Bosons
In 2HDMs with U (1) H and Φ, there are three CP-even scalars, one pseudoscalar, and one charged Higgs pair after U (1) H and EW symmetry breaking. There is also an additional massless scalar corresponding to U (1) H breaking, which is eaten by the additional gauge boson of U (1) H , called Z H . Without U (1) H -charged Φ, the two CP-odd scalars in H i could be eaten by the gauge bosons, so that we could discuss the effective model with no massive pseudoscalar and U (1) H gauge boson [7,36]. One may consider a model with Z 2 Higgs symmetry instead of U (1) H . In this case, Φ should be a scalar to avoid a massless mode and three CP-even scalars will appear after the symmetry breaking. Both cases will correspond to some limits of the 2HDM with U (1) H and Φ.

Charged Higgs (H ± )
After the EW symmetry breaking, one Goldstone pair (G ± ) and one massive charged Higgs pair (H ± ) appear. The directions of Goldstone bosons are fixed by the Higgs VEVs: The squared mass of the charged Higgs boson H + is given by In the 2HDM without Φ, m 2 3 is zero and m 2 H + is determined only by the second term with negative λ 4 . In the 2HDM with Φ, λ 4 could be either negative or positive.

Pseudoscalar boson (A)
In 2HDMs with discrete Z 2 symmetry, one CP-odd mode is eaten by the Z boson and the other becomes massive. In the 2HDM with a complex scalar, Φ, there is an additional CPodd mode and two Goldstone bosons (G 1,2 ) appear after the EW and U (1) H symmetry breaking. m 2 3 (Φ) plays a crucial role in the mass of A, m A . m 2 3 (Φ) is m 2 3 (Φ) = µΦ or µΦ 2 in the renormalizable potential depending on the definition of The directions of G 1,2 and A are defined as The squared pseudoscalar mass m 2 A is given by where n = 1 or 2 depending on m 2 3 (Φ). G 1 corresponds to the Goldstone boson in the ordinary 2HDMs and could be eaten by the Z boson. In the limit, v Φ → ∞, χ Φ is G 2 and eaten by Z H . Also the direction of A and m 2 A become the same as in the ordinary 2HDMs. In the 2HDM with local U (1) H symmetry but without Φ, A does not exist, so that it could corresponds to the limit, m A → ∞ and v Φ → 0. In the following section, we discuss our 2HDMs assuming m 2 3 (Φ) = µΦ and q Φ = (q H 1 − q H 2 ).

CP-even scalar bosons (h, H,h)
After the EW and U (1) H symmetry breaking, three massive CP-even scalars appear and they generally mix with each other as follows: where α corresponds to the mixing angle between two neutral scalars in the ordinary 2HDM and α 1,2 are additional mixing angles that newly appear in our model with local U (1) H and a singlet scalar Φ. The mixing is given by the mass matrix which is introduced in Appendix A. In the limit of α 1,2 → 0 one can interpret h Φ as the field in the mass basis and h Φ does not mix with h 1,2 . Throughout this paper, we assume that h is the SM-like scalar boson with its mass (m h ) being fixed around 126 GeV.

Gauge bosons
In 2HDMs with local U (1) H Higgs symmetry, at least one of the Higgs doublets H i=1,2 should be charged under U (1) H . Therefore tree-level mass mixing between Z and Z H would appear after spontaneous breaking of the EW and U (1) H symmetries. Let us describe the mass matrix of Z and Z H as 13) and the mass mixing term between Z and Z H is (2.14) Here g, g ′ and g H are the gauge couplings of U (1) Y , SU (2) L , and U (1) H gauge interactions, respectively. And q H i and q Φ are the U (1) H charges of the Higgs doublet H i 's and the singlet scalar Φ, respectively. Some examples of the charge assignments within Type-I 2HDM are shown in Table 1. U (1) H charge assignments for other types of 2HDMs can be found in Ref. [7]. The tree-level masses in the mass eigenstates are given by Then the mixing between Z and Z H is described by the mixing angle ξ, which is defined as (2.17) Note that we omit the symbol "0" for the physical (renormalized) masses for the gauge bosons. The extra gauge boson couples with the SM fermions through the mixing even if the SM fermions are not charged under U (1) H . Furthermore, this mixing modifies the coupling of the Z boson with the fermions, which has been well-investigated at the LEP experiments. The Z boson mass is also deviated from the SM prediction according to Eq. (2.15) and the allowed size of the deviation is evaluated by the ρ parameter. Our 2HDMs are strongly constrained not only by the Z H search in the experiments but also by the EWPOs, as we will see in the next section.

Vacuum stability condition and perturbative unitarity bounds
There are many theoretical and experimental constraints on our model. First we consider theoretical bounds on Higgs self couplings from vacuum stability condition and perturbative unitarity.
In order to break the U (1) H and EW symmetry, the potential (2.2) should have a stable vacuum with nonzero VEVs, namely the scalar potential is bounded from below. We impose the vacuum stability bounds, which require that the dimensionless couplings λ 1,2,3,4 are to satisfy the following conditions: in the Φ = 0 direction. They correspond to the ones in the usual 2HDMs without λ 5 . Following the conditions and Eq. (A.11) in Appendix A, the masses of scalars satisfy In the ordinary 2HDMs with softly broken Z 2 symmetry, sizable λ 5 is allowed and the conditions (3.1) and (3.2) should be modified by the replacements, m 2 In the Φ = 0 direction, the vacuum-stability conditions for λ Φ , λ 1 and λ 2 are where the directions of H 1 and H 2 fields in the last four conditions are the same as those of H 1 and H 2 fields in Eq. (3.1).
We also impose the perturbativity bounds λ i ≤ 4π on the quartic Higgs couplings and the tree-level unitarity conditions whose expressions are given in Ref. [37][38][39]. These will make theoretical constraints on the quartic couplings in the scalar potential (2).

Constraints from various experiments
The charged Higgs boson mass is constrained by the LEP experiments. It depends on the decay channel of the charged Higgs boson, and we take the model-independent bound m h + 80 GeV [40] in this work. We also impose a recent bound on the charged Higgs and tan β coming from the top quark decay from the LHC experiments [41][42][43]. We note that the flavor bound which mainly comes from the b → sγ experiments is tan β 1 in the type-I 2HDM [44].
Recently the BABAR Collaboration reported about 3.4σ deviation from the SM prediction in the B → D ( * ) τ ν decays [45]. This deviation cannot be accommodated with the ordinary 2HDM with MFV in the Yukawa sector. It turned out that 2HDMs which violate MFV might account for the discrepancy. The chiral U (1) ′ model with flavored Higgs doublets which slightly breaks the NFC criteria in the right-handed up-type quark sector [16] is one of such examples. Since the 2HDMs with U (1) H hold the MFV hypothesis, they cannot be accommodated with the deviation in B → D ( * ) τ ν. In this work, we do not consider these experiments seriously since the experimental results are not well settled down. In the future, if this deviation would be confirmed at Belle or Belle II, it might exclude our 2HDMs as well as the ordinary 2HDMs.
EWPOs in the LEP experiments which are usually parametrized by Peskin-Takeuchi parameters S, T , and U [46] provides strong bounds on the parameters in the Higgs potential. If new physics has no direct couplings to the SM fermions, their effects at the LEP energy scale would appear only through the self energies of SU (2) L gauge bosons. This is the case of the usual Type-I 2HDM. However, in our model there exists a new U (1) H gauge boson, which may couple to the SM fermions. In this case, one must consider all observables at the Z pole at the one-loop level instead of S, T , and U [47]. However if the new gauge boson is decoupled from the EW scale physics, S, T , and U will provide well-defined constraints on the 2HDMs with U (1) H . We will discuss this bound in a few next subsections.

Tree-level ρ parameter
If the Higgs doublets are charged under the extra gauge symmetry, the extra symmetry would also be broken along with EW symmetry breaking. Then there appears the mass mixing between the Z boson and the extra massive gauge boson. In the 2HDMs with U (1) H , the mixing between Z and Z H is generated as in Eq. (2.17). This mass mixing could allow the Z boson mass to deviate significantly from the SM prediction, and thus will strongly be constrained by the ρ parameter, which the SM predicts to be one at the tree level.
Assuming ξ ≪ 1, the tree-level ρ parameter is described as The mixing also changes the Z boson couplings with the SM fermions and the factor is estimated as 1 − ξ 2 /2. The bounds on the tree-level mixing have been discussed in Refs. [48][49][50]. As we will see in Fig. 1 (a), we can derive the bounds on g H , tan β, and M Z H in the case with (q H 1 , q H 2 ) = (1, 0), when we require that the tree-level contributions to the ρ parameter and the decay width of the Z boson, which are functions of the Z-boson couplings, are within the error of the SM predictions: ρ = 1.01051 ± 0.00011 and Γ Z = 2.4961 ± 0.0010 GeV [51]. The tree-level deviations may also affect the S, T , and U parameters, but they actually become negligible because of the requirement for the stringent bound from Z ′ search at the LHC, as we discuss in the next section.

Bound from Z ′ search in the collider experiments
Extra neutral gauge bosons are strongly constrained by Z ′ searches at high energy colliders. In our models, Z H can couple with the SM fermions through the Z-Z H mixing, even if we choose the charged assignment that the SM fermions are not charged under U (1) H .
If Z H couples with leptons, especially electron and muon, Z H would be produced easily at LEP and the coupling and mass of Z H are strongly constrained by the experimental results, which are consistent with the SM prediction with very high accuracy. If Z H is heavier than the center-of-mass energy of LEP (209 GeV), we could derive the bound on the effective coupling of Z H [52][53][54]. The lower bound on M Z H /g H would be O(10) TeV [53,54]. If Z H is lighter than 209 GeV, the upper bound of Z H coupling would be O(10 −2 ) to avoid conflicts with the data of e + e − → f − f + (f = e, µ) [51,53,54].
Furthermore, there will be strong bounds from hadron colliders, if quarks are charged under U (1) H . The upper bounds on the Z H production at the Tevatron and LHC are investigated in the processes, pp(p) → Z H X → f fX [51,52,55,56], and the stringent bound requires O(10 −3 ) times smaller couplings than the Z-boson couplings for We could avoid these strong constraints, in the case that all particles except for one Higgs doublet are not charged under U (1) H . Actually the model in the first row of Table 1 is this case. Z H couples with the SM fermions only through the Z-Z H mixing, so that the mixing should be sufficiently small. In the following sections, we focus on the fermiophobic U (1) H charge assignment and require the (conservative) bound sin ξ 10 −3 , according to Ref. [56]. The small mixing especially contributes to the T parameter as αT ∼ ρ − 1, but it will not affect our results.
In the 2HDM with U (1) H , Z H can decay to Z and scalars, so that the strong bound, sin ξ 10 −3 , will be relaxed if the branching ratio of the Z H decay into Z and scalars is almost one. In the following sections, we study the region with M Z H ≤ 1TeV , and the additional branching ratio is at most 0.1 in that region. If we assume that there are extra particles charged under U (1) H and Z H mainly decays to the extra particles, the larger value for sin ξ could be allowed. We note that the constraint from the Z ′ search in the dijet production at the LHC can easily be avoided by the bound on the mixing angle ξ.
In the region of M Z H > 1 TeV, the constraints from the Z ′ search are relaxed and the constraint on g H cos β from the ρ parameter and Γ Z becomes stronger as we will see in Fig. 1 (a).

S, T , and U parameters at the one-loop level
Here, we introduce S, T , and U parameters in the 2HDMs with the U (1) H gauge boson and Φ at the one-loop level. They involve only gauge interactions of scalars, so that the results could be applied to other types of 2HDMs [35]. The EWPOs in 2HDMs with extra scalars have been calculated in Refs. [57,58].
In order to calculate the S, T , and U parameters, we define mass eigenstates {H + l }, {H l }, and {A l } of Higgs bosons in terms of mixing angles β, α, and α 1,2 , Each mixing angle is given in Eqs. (2.7), (2.9), and (2.11). Let us discuss the constraints on the loop corrections to the EWPOs in terms of the S, T , and U parameters defined as [51] where α(M 2 Z ) is the fine-structure constant at the scale, M Z , and (s W , c W ) = (sin θ W , cos θ W ) are defined by the Weinberg angle, θ W . S 2HDM , T 2HDM and U 2HDM are the parameters in the ordinary 2HDMs, which could be found in Refs. [59,60]. The new gauge boson Z H and the extra scalar bosonh in our model make new one-loop contributions to the vacuum polarizations of gauge fields, denoted by (∆Π W W,ZZ ). Their explicit expressions up to the O(ξ) corrections are given by which are used for phenomenological analyses of the EWPOs. We have defined a new parameter γ for convenience:  Fig. 1 (a), the gray region satisfies the collider bound, sin ξ ≤ 10 −3 , mainly from the Drell-Yan process at the LHC and the dashed line corresponds to the upper limit on the constrains coming from the ρ parameter and Γ Z . In the region M Z H 1 TeV, the collider bound is stronger than the bound from the ρ parameter and Γ Z . We note that we include the one-loop corrections involving Z H to S, T , and U , where 126 GeV ≤ M Z H ≤ 1000 GeV and 0 ≤ |g H | ≤ 4π. The tree-level contribution to the T parameter is also considered but it just yields the deviation, |∆T | 0.01.
In Fig. 1 (b), the gray region is allowed for g hV V and m A in the ordinary type-I 2HDM, where α 1 = α 2 = 0 and Z H and Φ are decoupled. If the pseudoscalar mass is heavy, g hV V should be close to one so that the Higgs signal around 126 GeV should be SM-like. The red points are allowed in the 2HDM with U (1) H with sin ξ ≤ 10 −3 . We note that the small g hV V region is also allowed due to an extra factor cos α 1 in g hV V . The small g hV V would reduce the production rate of the SM-like Higgs boson and the partial decay width of h to the EW gauge bosons.
In Fig. 2, we show the bounds on the mass differences among m A , m H and m H + . In Fig. 2 (a), m A is less than 700 GeV, and the (dark) gray region satisfies 126 GeV ≤ m A < 300 GeV (300 GeV ≤ m A < 700 GeV). In Fig. 2 (b), m A is within 700 GeV ≤ m A ≤ 1000 GeV. The gray region is allowed for all the constraints in the ordinary type-I 2HDM with α 1 = α 2 = 0 and λ 5 = 0. As we see in Appendix A, we can realize such small mixings assuming very small λ 1 , λ 2 and µ or very large v Φ .
The light blue region corresponds to the ordinary 2HDM with non-zero λ 5 (|λ 5 | ≤ 1). In the case of the 2HDMs with λ 5 = 0, the vacuum stability requires the relation (3.2). On the other hand, non-zero λ 5 modifies the relation and, especially, negative λ 5 pushes the lower bound on m H down, so that the wider region is allowed in Figs. 1 and 2.
As we see in Fig. 2, each scalar mass could become different. However, it seems that at least two of them should be close to each other in the typical 2HDM with small λ 5 . The heavier pseudoscalar mass requires the smaller mass difference.
In our 2HDM with h and Z H , the strict bounds could be evaded because of the contributions of the extra particles. The red and blue points are allowed in the type-I 2HDM with U (1) H and the additional constraints, g hV V ≥ 0.9 and g htt ≥ 0.9, are imposed on the blue points. Here g htt is the h-t-t coupling normalized to the SM coupling and it is given by g htt = cos α 1 cos α/ sin β in the type-I 2HDM with U (1) H . Once Φ is added and h Φ mixes with h 1 and h 2 , the relation (3.2) is discarded, so that the red (blue) points exist outside of the gray region, when h Φ and Z H reside in the O(100) GeV scale. In particular, the predictions of the masses of the CP-even scalars are modified, so that m H − m A would have larger allowed region, compared with m H + − m A . Even if the SM Higgs search limits the normalized h-V -V and h-t-t couplings, the mass difference could not be constrained strongly as shown in the region of the blue points.
The constraints from EWPOs could easily be applied to the other type 2HDMs by changing the experimental constraints on the charged Higgs mass. For example, b → sγ gives the lower bound on m H + 360 GeV in the type-II 2HDMs [44]. 4 Collider phenomenology of the Higgs bosons

Analysis strategies
In this section, we consider collider phenomenology of the Higgs bosons, in particular, focusing on the SM-like Higgs boson. For the calculation of the decay rates of the neutral Higgs bosons, we use the HDECAY [64] with corrections to Higgs couplings to the SM fermions and gauge bosons and with inclusion of the charged Higgs contribution to the h → γγ and h → Zγ decays. There are 10 parameters in the potential neglecting the Z H boson effects at the EW scale, and one of them is fixed by the SM-like Higgs boson mass m h ∼ 126 GeV. We choose the other 9 parameters as tan β, m A , dm H + , dm H , mh, α, α 1 , α 2 , and v φ , where dm H + (dm H ) = m H + (m H ) − m A is the mass difference between the charged Higgs (heavy Higgs) and pseudoscalar Higgs boson. In this analysis, we choose each parameter region as follows: In order to compare our models with the Higgs data at the LHC, we consider the signal strength µ for each decay mode i of the SM-like Higgs boson with the production tag j, which is defined by where σ(pp → h) j means the production cross section for the SM-like Higgs boson with the production tag j and Br(h → i) is the branching ratio of the SM-like Higgs boson decay into the i state. Here j = gg, V h, or V V h, which correspond to the gg fusion production, vector boson associated production, and vector boson fusion production tag, respectively. Finally i = γγ, W W , ZZ, or τ τ , depending on the decay channels. The search for the SM Higgs boson also constrains the mass and couplings of the heavy Higgs boson. In high mass region greater than 200 GeV, the main search mode is h → ZZ → 4l [65]. For the SM-like Higgs boson, the lower limit for the Higgs boson mass is about 650 GeV and 300 GeV for the gg fusion production and V V h + V h production, respectively. More detailed analysis is given in Ref. The larger mass-scale region could be considered, but they relate to the SM-Higgs signals indirectly through the bounds from the EWPOs and theoretical constraints, as we discuss in Sec. 3. Hence, they would not change our results in this section. in the gg fusion production, which varies according to m H . From the SM Higgs search for m H ≤ 200 GeV, we get the constraint on the signal strength µ ZZ gg < 0.1 ∼ 0.5 whose bound depends on m H . We impose these bounds on the heavy Higgs boson H.
In this work, we consider two distinct cases in our Type-I 2HDM with U (1) H gauge symmetry: • First, we consider the Type-I 2HDM with U (1) H , assuming that Z H is decoupled from the low energy Higgs physics. Then, the extra contribution is from only the extra Higgs scalar, and the effect is parametrized by m h and α 1,2 .
• Secondly, we consider the Type-I 2HDM with U (1) H , including Z H contribution. The charge assignment is fermiophobic by setting u = d = 0. In this case the Z H boson couples with the SM fermions only through the Z-Z H mixing, and it contributes to the EWPOs.
We compare each case with the ordinary type-I 2HDM by setting α 1,2 = 0 and omitting the singlet scalar Φ and Z H . We note there is no λ 5 term in the Higgs potential in this case, as we mentioned in the previous section.

2HDM with the extra singlet scalar
In this section, we consider the type-I 2HDM with the extra singlet scalar field, h Φ , where we assume that the imaginary part of Φ is eaten by Z H and the effects of the U (1) H gauge boson are small enough to be ignored. This could easily be achieved with an assumption of the heavy Z H mass and small g H , namely in the limit of large v Φ . We show the scattered plots for µ γγ gg and µ ZZ gg in Fig. 3(a), and for µ τ τ gg and µ W W gg in Fig. 3(b), respectively. The red points are allowed in the ordinary type-I 2HDM, whereas the blue points are consistent with the type-I 2HDM with h Φ , respectively. The skyblue and green regions are consistent with the Higgs signal strengths reported by CMS and ATLAS Collaborations within the 1σ range, respectively, where µ γγ gg,CMS = 0.70 +0.33 −0.29 , µ γγ gg,ATLAS = 1.6 ± 0.4, µ ZZ gg,CMS = 0.86 +0.32 −0.26 , and µ ZZ gg,ATLAS = 1.8 +0.8 −0.5 . Each signal strength at CMS is consistent with that at ATLAS within the 2σ's.
in the ordinary type-I 2HDM (red) and type-I 2HDM with h Φ (blue). The effect of Z H boson is assumed to be small enough to be ignored. The skyblue and green regions are the allowed ones at CMS and ATLAS in the 1σ level.
The SM point is µ γγ,ZZ,W W,τ τ gg = 1, which is in agreement with the CMS data, but the ATLAS data are consistent only at the 2σ level. In the ordinary 2HDM, the allowed points are in the regions of µ γγ gg 1.4 and 0.4 µ ZZ gg 1.1. In the 2HDM with h Φ the allowed region is wider in the gg → h → ZZ process: 0 µ ZZ gg 1.1. Both 2HDMs contain the SM point µ = 1, and the CMS data for µ γγ gg and µ ZZ gg , but only the edge of the allowed region is barely consistent with the ATLAS data in the 2σ level.
For µ τ τ gg both models predict a large allowed region from 0 (0.4) to 1.5 or larger so that it is difficult to constrain the parameters in the 2HDMs using only µ τ τ gg . In the ordinary 2HDM 0.4 µ W W gg 1 is allowed, whereas much wider region 0 µ W W gg 1 is allowed in the 2HDM with h Φ . The allowed region in the 2HDM with h Φ is much broader than that in the ordinary 2HDM.
As shown in Fig 3, the region of µ ZZ gg 0.4 and µ W W gg 0.4 is not allowed in the ordinary 2HDM. Hence, if it turns out that the two signal strengths were less than 0.4, one might be able to conclude that the 2HDM with h Φ is more favored than the ordinary 2HDM. However, if it turns out that each signal strength is close to the SM point, the 2HDM with h Φ cannot be distinguished from the ordinary 2HDM as well as the SM. The mixing with the extra CP-even singlet scalar decreases the two signal strengths, so that we could conclude that their upper bounds are µ γγ  Type-I 2HDM sin(β-α)=1 sin(β+α)=1 with h Φ ordinary Figure 5. sin α vs. tan β in the type-I ordinary 2HDM (red) and in the type-I 2HDM with h Φ (blue). The points are consistent with the CMS data for µ γγ gg and µ ZZ gg in the 1σ level. The black and green lines correspond to the cases sin(β − α) = 1 (SM limit) and sin(β + α) = 1, respectively. We note that the decay widths of the Higgs boson h into ZZ or W W are rescaled by g hV V = cos α 1 sin(β − α), while those into a fermion pair are by g hf f = cos α 1 cos α/ sin β. In the limit of small cos α or large sin β, the branching ratio of the h decay into a bb pair could get much smaller than the branching ratio in the SM and as a result, the branching ratios of the h decay into ZZ or W W could be much enhanced.
As shown in Fig 4, the region of µ τ τ V V h,V h 0.4 is not allowed in the ordinary 2HDM. Hence, if it turns out that the signal strengths are less than 0.4, one might conclude that the 2HDM with h Φ is more favored than the ordinary 2HDM. In the region of µ V V h,V h > 0.4 we cannot distinguish the 2HDM with h Φ from the ordinary 2HDM. If it turns out that each signal strength is close to the SM point, the 2HDM with h Φ cannot be distinguished from the ordinary 2HDM as well as the SM.
In Fig. 5, we depict the scattered plot for sin α and tan β. The red and blue points are consistent with the CMS data for µ γγ gg and µ ZZ gg at the 1σ level in the type-I ordinary 2HDM and in the type-I 2HDM with h Φ , respectively. The black line corresponds to the SM limit sin(β − α) = 1 while the green line to sin(β + α) = 1. In the ordinary 2HDM and the 2HDM with h Φ , the allowed points are scattered over the region | sin α| 0.8. The region | sin α| 0.8 is forbidden, since the coupling g hf f ∼ cos α/ sin β to the fermions becomes small for tan β > 1. In both models, the allowed regions contain the SM limit sin(β − α) = 1 and there is no distinction between the two models. There is no region which agrees with the ATLAS data for µ γγ gg and µ ZZ gg at the 1σ level, but one can obtain a in the ordinary type-I 2HDM (red) and type-I 2HDM with a Z H (blue). The skyblue and green regions are the allowed ones at CMS and ATLAS in the 1σ level. similar figure for the ATLAS data at the 2σ level.

2HDM with the Z H boson: fermiophobic case
In this section, we discuss the 2HDM with U (1) H where the U (1) H gauge bosonẐ H is fermiophobic, assuming u = d = 0 as shown in Table 1. Then theẐ H boson does not couple with the SM fermions, but in the mass eigenstate the Z H boson, which is a mixture ofẐ andẐ H , can couple with the SM fermions, and the couplings of the Z boson is modified with Z H ordinary Figure 8. sin α vs. tan β in the type-I ordinary 2HDM (red) and in the type-I 2HDM with a Z H boson (blue). The points are consistent with the CMS data for µ γγ gg and µ ZZ gg in the 1σ level. The black and green lines correspond to the cases sin(β − α) = 1 (SM limit) and sin(β + α) = 1, respectively. by the mixing angle betweenẐ andẐ H .
In this model, we have 10 parameters except m h fixed to 126 GeV. The general model allows the mixing of h Φ , h 1 and h 2 as shown in Eq. (2.7). However, the analysis of the model is time-consuming and the general feature of mixing between two Higgs doublets and singlet fields would reduce signal strengths as in the previous section. Therefore, we consider no mixing case by setting α 1 = α 2 = 0 and compare our results with the typical 2HDM.
We choose the parameter regions as follows: We depict the scattered plots for µ γγ gg and µ ZZ gg in Fig. 6(a), and for µ τ τ gg and µ W W gg in Fig. 6(b), respectively. The red and blue points correspond to the ordinary Type-I 2HDM and Type-I 2HDM with the Z H boson, respectively. The skyblue and green regions are CMS and ATLAS bounds at the 1σ level. As shown in Fig. 6, the 2HDM with the Z H boson seems to have broader regions of the Higgs signal strengths than those in the ordinary 2HDM, but there is no essential difference. In case of the general mixing between the neutral Higgs bosons, we might be able to distinguish the 2HDM with the Z H boson from the ordinary 2HDM in some parameter spaces, especially in the region µ ZZ,W W gg 0.4. However, this region is inconsistent with the current measurements. Both 2HDMs are consistent with the CMS data at the 1σ level. However, it is difficult to increase µ γγ gg and µ ZZ gg to the ATLAS data in the present models. Therefore the 2HDM with the Z H boson are not in agreement with the ATLAS data at the 1σ level. Fig. 7 shows the scattered plots (a) for µ γγ V V h and µ ZZ V V h , (b) for µ τ τ V V h and µ W W V V h (c) for µ γγ V h and µ ZZ V h , and (d) for µ τ τ V h and µ W W V h , respectively. The red points are allowed in the ordinary type-I 2HDM while the blue ones are in the 2HDM with the Z H boson. In the 2HDM with the Z H boson, µ ZZ,W W V V h,V h could get much larger than the SM prediction as shown in the figures. If the mixing between the two Higgs doublet and singlet fields are allowed, broader region with smaller signal strengths would be allowed as in the 2HDM with h Φ discussed in the previous subsection. The SM points µ ZZ,W W V V h,V h = 1 are consistent with the (ordinary) 2HDMs at the 1σ level except for µ W W V V h . However the deviation in µ W W V V h is not statistically significant yet because of large experimental errors. In Fig. 8, we depict the scattered plot for sin α and tan β, where the red and blue points are consistent with the CMS data for µ γγ gg and µ ZZ gg in the 1σ level in the type-I ordinary 2HDM and in the type-I 2HDM with the Z H boson, respectively. The black line corresponds to the SM limit sin(β − α) = 1 while the green line to sin(β + α) = 1. As in the 2HDM with h Φ , the region | sin α| 0.8 is not allowed and there is no difference between the ordinary 2HDM and the 2HDM with U (1) H Higgs gauge symmetry in the type-I case even though the extra Z H boson contribution is taken into account. However, in the type-II 2HDMs, one could find apparent distinction between the 2HDMs without U (1) H Higgs gauge symmetry and with the gauge symmetry [35].

Conclusion
Discovery of a SM-like Higgs boson at the LHC has opened a new era in particle physics. It is imperative to answer the question if this new boson is the SM Higgs boson or one of Higgs bosons in an extended model with multi-Higgs fields. The 2HDM is one of the simplest models which extend the SM Higgs sector and is well motivated by MSSM, GUT, etc. In Ref. [7], it was suggested to replace the Z 2 symmetry in the ordinary 2HDM with U (1) H gauge symmetry, which can easily realize the NFC criterion with proper U (1) H charge assignments to the two Higgs doublets and the SM chiral fermions. The local U (1) H symmetry may be the origin of softly broken Z 2 symmetry which has been widely discussed so far.
In this paper, we performed detailed phenomenological analysis of the observed 126 GeV Higgs boson within the Type-I 2HDM with the U (1) H symmetry proposed in Ref. [7]. We added an extra complex scalar that breaks U (1) H spontaneously, in order to avoid the strong constraint on the mixing between the Z boson and the extra Z H boson from EWPOs. Our extension of 2HDMs predicts one extra gauge boson and one extra neutral scalar compared with the 2HDMs with Z 2 symmetry, and allows a large pseudoscalar mass according to the spontaneous U (1) H symmetry breaking.
Taking into account experimental constraints from the SM-Higgs search, EWPO etc., and theoretical constraints from perturbativity, unitarity, and vacuum stability, we studied the signal strengths in two different cases: • Case I: Type-I 2HDM with the extra scalar h Φ , assuming the U (1) H gauge boson is heavy enough to be decoupled at the EW scale. In this case, the Higgs sector includes an extra scalar which is a remnant from spontaneous U (1) H symmetry breaking, and the EWPOs will be affected. We found that the signal strengths in the 2HDMs with h Φ could be much smaller than those in the 2HDM with Z 2 symmetry in some channels. However, if the signal strengths are close to the SM prediction, it would be nontrivial to distinguish the 2HDM with h Φ from the 2HDM with Z 2 symmetry with Higgs signal strengths alone, especially when all the signal strengths are observed close to the SM values. In case the signal strengths are bigger than the SM prediction, the extra mixing of CP-even scalars does not help to save type-I 2HDM especially in h → V V .
• Case II: Type-I 2HDM with the Z H boson where the U (1) H boson is fermiophobic. This is the simplest solution to the U (1) H assignments to the SM chiral fermions listed in Table I. Then, Z H boson can couple with the SM fermions only through the mixing between theẐ andẐ H bosons. In general, the 2HDM with the Z H boson allows wider region compared with the 2HDM with Z 2 symmetry, but if the mixing between two Higgs doublets and singlet fields are ignored, there is no essential distinction in the allowed regions from the 2HDM with Z 2 symmetry. In particular, if the signal strengths turn out to be close to the SM prediction, the distinction would be nontrivial from the Higgs search alone. Direct search for extra U (1) H gauge boson and/or extra neutral scalar would be important in such a case.
• In either case, for a given µ γγ , the allowed regions for µ W W and µ ZZ are broader than the ordinary 2HDMs. And µ τ τ in Case I could be smaller than those predicted in the ordinary 2HDMs, but is similar in Case II. On the other hand, it would be difficult to distinguish the ordinary Type-I 2HDM from the model with local U (1) H gauge symmetry based on the observed 126 GeV Higgs signal strengths alone, if the data are close to the SM predictions. It would be essential to discover the extra scalar bosons and the new gauge boson Z H in order to tell one from the other.
In this work, we considered only the type-I 2HDMs with U (1) H gauge symmetry, which are the simplest since they are anomaly-free without any extra fermions as long as we choose suitable U (1) H charges for the SM chiral fermions as in Table I. In this anomaly-free case without extra fermions, it is difficult to enhance the signal strengths µ γγ gg for example. On the other hand, more general 2HDMs with U (1) H gauge symmetry would generically have gauge-anomaly, like in U (1) B or U (1) L models. This gauge anomaly can be cured by adding extra chiral fermions and/or vector-like fermions, which would contribute to the production and the decay of Higgs boson via extra colored and/or electrically charged new particles in the loop and thus could enhance µ γγ gg . It is straightforward to extend the present analysis to other type of 2HDMs with U (1) H gauge symmetry discussed in Ref. [7], in particular, Type-II 2HDM. These models would have richer structures and be more interesting in theoretical and phenomenological aspects, and we plan to report the phenomenological analysis on such models in future publications [35].