Lepton specific two-Higgs-doublet model based on a $U(1)_X$ gauge symmetry with dark matter

We discuss a two Higgs doublet model with extra $U(1)_X$ gauge symmetry where lepton specific (type-X) structure for Yukawa interactions is realized by charge assignment of fields under the $U(1)_X$. Extra charged leptons are introduced to cancel gauge anomaly associated with extra gauge symmetry. In addition, we introduce scalar fields as dark matter candidates to which we assign $Z_2$ odd parity for guaranteeing stability of them. We then analyze phenomenology of the model such as scalar potential, muon anomalous magnetic dipole moment, collider physics associated with $Z'$ boson from $U(1)_X$, and dark matter physics. Carrying out numerical analysis we search for phenomenologically viable parameter region.


I. INTRODUCTIONS
The standard model (SM) of particle physics has been very successful to explain experimental results and its particle contents are confirmed completely by the discovery of the Higgs boson at the Large Hadron Collider (LHC). Although the SM is quite successful, there can be a new physics beyond the SM (BSM) accommodating with the experimental data and it is motivated by several issues such as existence of dark matter(DM) and non-zero mass of neutrinos which can not be explained within the SM. Furthermore the existence of new physics would induce interesting phenomenology such as flavor physics and new particle signatures at collider experiments.
One of the interesting extension of the SM is two Higgs doublet model (THDM) in which a second Higgs doublet is introduced. In general, THDM has flavor changing interactions through Yukawa interactions of both quarks and leptons, which are strongly constrained by various experiments searching for flavor violating processes. In many approaches softly broken Z 2 symmetry is introduced to restrict Yukawa interactions to avoid flavor changing neutral current(FCNC). One can also apply an extra U (1) gauge symmetry to control Yukawa interactions associated with two Higgs doublets. In such a scenario rich phenomenology would be induced from scalar bosons from Higgs sector as well as Z boson from extra U (1) symmetry. In fact many works have been carried out in a scheme of THDM with extra U (1) symmetry motivated by several issues such as absence of FCNC [1], neutrino mass [2][3][4][5][6][7], flavor physics [8][9][10][11][12], dark matter(DM) [13][14][15][16][17][18] and collider physics [19][20][21]. Also extra U (1) could be originated from string theory [22].
In this work, we construct a model based on an extra U (1) X gauge symmetry which can realize lepton specific (type-X) THDM. The type-X THDM is one of the interesting scenario in THDM in which one Higgs doublet only couples to quarks while the others only couples to leptons [23]. Interestingly one can obtain sizable contribution to muon anomalous magnetic moment (muon g − 2) from the structure of Yukawa coupling where the deviation from the SM prediction is [24][25][26][27]; ∆a µ = (26.1 ± 8) × 10 −10 , . ( It is the 3.3σ deviation with a positive value, and recent theoretical analysis further indicates 3.7σ deviation [28]. Moreover, several upcoming experiments such as Fermilab E989 [29] and J-PARC E34 [30] will provide the result with more precision in future. To explain the discrepancy a lot of studies have been carried out within type-X [31][32][33][34][35][36][37][38], muon specific [39] and general (type-III) THDM [41,42]. We then investigate muon g − 2 in our model taking into account constraints from the SM Higgs measurements. In addition, we introduce a scalar dark matter (DM) candidate in our model which is stabilized by discrete Z 2 symmetry and its interaction with muon can also contribute to muon g−2. The relic density of DM is estimated to search for parameters accommodating with the observed value imposing constraint from direct detection experiments. We also discuss possibility of indirect detection experiments.
This paper is organized as follows. In Sec. II, we introduce our model and formulate mass spectrum and interactions. In Sec. III, we discuss phenomenology of the model such as constraints from scalar potential, muon g − 2, Z boson production at the LHC, and dark matter physics. Finally we give summary and discussion.

II. MODEL SETUP
In this section, we introduce our model and formulate mass spectrum and interactions.
This model has extra U (1) X gauge symmetry and exotic charged leptons E R(L) with U (1) X charge −1(0) are introduced to cancel gauge anomalies. In scalar sector, we introduce two Higgs doublets H 1 and H 2 whose U (1) X charges are 0 and 1 respectively, and complex SM singlet scalars φ, χ and χ with U (1) X charge −1, 1 and 0. We also impose Z 2 parity where E L(R) , χ and χ are odd and the other fields are even, and neutral scalar χ and χ can be our DM candidate [40]. Here we consider two DM candidates χ and χ where the former has gauge interaction associated with U (1) X and the other is gauge singlet. The full charge assignment of fields are summarized in Table I. Scalar fields in our model are written as Also to obtain vanishing VEV of χ(χ ), we require M 2 χ(χ ) and couplings associated with χ(χ ) to be positive.
After spontaneous symmetry breaking, we obtain mass matrix for charged scalar such The mass matrix can be diagonalized as in the THDM and mass eigenstates are  where tan β = v 2 /v 1 , G ± is Nambu-Goldstone(NG) boson absorbed by W ± and H ± is physical charged Higgs boson. The mass of charged Higgs boson is given by where v = v 2 1 + v 2 2 . The mass matrix for Z 2 even and CP odd scalar bosons is obtained as We can diagonalize the mass matrix by rotating the basis as follows: where G 0 1 and G 0 2 are massless NG bosons and these degrees of freedom are absorbed by Z and Z bosons. The physical CP-odd scalar boson A 0 has non-zero mass of We thus find that A 0 becomes massless in the limit of µ → 0.
The Z 2 even and CP-even scalar sector has three physical degrees of freedom {h 1 , h 2 , φ R } and the mass matrix is given by This mass matrix can be diagonalized by an orthogonal matrix R with three Euler parameters {α 1 , α 2 , α 3 } which is written as and mass eigenstates are obtained such that We write parameters in scalar potential {m 1 , m 2 , µ, λ 1 , λ 2 , λ 3 , λ 4 , λ φ , λ φH 1 , λ φH2 } by phys-ical masses and VEVs such that Here we formulate masses of Z 2 odd scalar fields . For simplicity we assume µ χ η M 2 χ and ignore χ-χ mixing. Then masse eigenvalues of them are given by where the real and imaginary part of χ(χ ) have the same mass, and we write them as m χ and m χ . Here the mass degeneracy of real and imaginary part is due to the requirement of invariance under phase transformation and smallness of µ χ parameter in the scalar potential as we assumed above. Thus our DM is identified as complex scalar bosons.

B. Yukawa interactions
The Yukawa interactions in our model are controlled by U (1) X gauge symmetry, and one obtains lepton specific (type-X) structure for two Higgs doublet scalars and terms associated with exotic charged leptons: where we omit flavor indices. We can derive the SM fermion masses the same as the THDM.
In addition the masses of exotic leptons E is given by Then rewriting scalar fields by mass eigenstates the Yukawa interactions become where V ud indicates an element of CKM matrix. The coefficients associated with neutral scalar bosons, y f Φ , are summarized in Table. II while interactions associated with charged Higgs are the same as the type-X THDM. In our model neutrino mass is generated as Dirac type and mass matrix is simply given by m ν = y ν v 2 / √ 2 from Yukawa interaction Eq. (25).
Note that neutrino ν L in Eq. (27) corresponds to flavor eigenstate.

C. Gauge sector
Here we formulate mass eigenvalues and corresponding eigenstates in our gauge sector 1 .
After symmetry breaking gauge bosons obtain masses from kinetic term of scalar fields where g, g and g are gauge couplings associated with SU (2) L , U (1) Y and U (1) X . The mass of W boson is given by m W = gv/2 with mass eigenstate W ± µ = (W 1 µ ∓ iW 2 µ ) as in the SM. On the other hand mass matrix for neutral gauge bosons becomes 3 × 3 such that Rotating (W 3 µ , B µ ) T by Weinberg angle θ W , we identify massless photon field A µ as where c W (s W ) = cos θ W (sin θ W ) whose definition is the same as in the SM. Then we obtain 2 × 2 mass matrix in the basis of (Z µ , B µ ) such that where the elements are given by The mass eigenvalues are and the mass eigenstates are obtained such that The mixing between Z and Z is sufficiently small in our parameter region of interest and we ignore the effect of the mixing in the following analysis.
The gauge interactions among Z and fermions are given by We also obtain Z -scalar-scalar gauge interactions such that where c β (s β ) = cos β(sin β). In addition the h 0 V V and H 0 V V interactions are given by Note that we reproduce THDM interaction in the limit of α 1 → α, α 2 → 0 and α 3 → 0 as

III. CONSTRAINTS AND PHENOMENOLOGY
In this section, we discuss experimental constraints and phenomenologies in the model.
We first investigate constraints from Higgs sector such as stability and perturbativity bound in the potential, in order to search for allowed parameter region. Then muon anomalous magnetic moment is estimated applying the allowed parameter sets. We also explore collider phenomenology and dark matter physics.

A. Constraints from Higgs sector
Here we discuss constraints on our parameters such as neutral scalar mixing {α 1 , α 2 , α 3 }, scalar boson masses and tan β taking into account unitarity, stability and perturbativity bounds for the Higgs sector as well as the experimental measurements of SM Higgs coupling strength. The constraints from unitary and perturbativity are given by [45] where a 1,2,3 are the solution of the following equation We also obtain constraints from stability condition for scalar potential such that [46][47][48] λ 1,2,φ > 0, 2 λ 1 λ 2 + λ 3 + λ 4 > 0,  where we have applied 2σ region of observed values in refs. [49,50].
Here we scan out parameters to search for allowed parameter region, such that where we can take range of mixing angle in [π/2, π/2] without loss of generality. The allowed parameter regions are shown in Fig. 1 where we take m A 0 as a scanning parameter in left side plots and m A 0 = 40 GeV is chosen in right side plots. We find that relations among mixing angle α 1 < 0 and α 2 = −α 3 are required to satisfy the constraints. Furthermore correlations of parameters |α 2 | ∼ π/2 and m A 0 ∼ m ξ 0 are preferred to obtain large tan β.
On the other hand value of η is not strongly constrained and does not correlate with the other parameters. We can thus take η as almost free parameter.

B. Muon g − 2
Here we estimate muon g − 2 in our model. Firstly we have contributions from loop diagrams with Z 2 even scalar bosons at one-and two-loop level. The two-loop Barr-Zee type diagrams can provide sizable contributions to muon g − 2 and the formula is given in refs. [41,51]. We find that sum of contributions to muon g − 2 from loop diagrams associated with φ = {h 0 , H 0 , ξ 0 , A 0 } is at most O(10 −10 ) when we apply allowed parameter region satisfying constraints discussed in previous subsection. This behavior is due to the negative contribution from two loop diagram associated with ξ 0 . We thus need the other contribution to explain muon g − 2 in the model.
In fact, we have a contribution to muon g − 2 from one loop diagram in which χ and E propagate inside loop [52,53]. This contribution is estimated as where r µ E i = m 2 µ /M 2 E i and r χ E i = m 2 χ /M 2 E i . In Fig. 2, we show ∆a µ from χ -E loop contribution as a function of Yukawa coupling Y χ where we assumed three generations of E have the same mass and all Y χ i2 has the same value. We also assume Y χ i1 ( constraints from lepton flavor violation processes. Thus ∆a µ 10 −9 can be realized with sizable Yukawa coupling Y χ when the masses of χ and E are around electroweak scale.

C. Collider physics
In this subsection, we discuss collider physics mainly focusing of Z boson production at the LHC. Our Z boson can be produced byqq → Z process since right-handed quarks   [57] to search for allowed parameter region. We then obtain allowed parameter space on (M Z , η) plane where we also scanned tan β whose values are indicated by color gradient. It is found that large η region is allowed since U (1) X gauge coupling is small due to the relation g M Z /η. Furthermore tan β dependence is small since Z-Z mixing is always very small in the parameter region.
In addition we estimate forward backward asymmetry (AFB) for tt final state from Z decay which is defined by where N (∆|y| > (<)0) indicates number of events with corresponding sign of ∆|y| = |y t |−|yt| for rapidities of top and anti-top quarks y t and yt. We find that ∆A

D. Dark matter physics
Here we analyze DM physics such as relic density and constraint from direct/indirect detection experiments. In our model, DM candidates are Z 2 odd scalar bosons χ(χ ) and its interactions relevant to annihilation processes are given by where we ignored Z-Z mixing effect since it is negligibly small, and mass eigenstates for scalar fields are obtained applying Eqs. (7), (10) and (14). The scalar bosons and Z decay into SM particles via interactions given in Secs. II A and II C. Note that χ(χ ) decays into R,I ) state for m χ(χ ) > m χ (χ) via the interaction with coupling µ χ so that only the lighter state among χ and χ is the DM. Then we estimate relic density of our DM for each scenario given below applying micrOMEGAs 4.3.5 [60] by implementing relevant interactions.
Note that in scenario (2) we will get the same behavior if we exchange role of χ and χ so that we only consider the case in which χ is DM. Under these scenarios, we estimate the relic density of DM.
In addition to the relic density, we need to take into account constraints from DM direct detection experiments. In our model DM can interact with nucleon through scalar and Z exchange when DM is χ. Then we can estimate the DM-nucleon scattering cross section, in non-relativistic limit, such that where m N is nucleon mass, µ N χ = m N m χ /(m N + m χ ) and f N is effective Nucleon-Higgs coupling [61,62].
In our numerical analysis below, we adopt micrOMEGAs 4.3.5 in estimating σ N −χ and the experimental constraints are imposed [63]. When DM is χ only scalar mediating interaction contribute to DM-nucleon scattering where we can obtain the contribution by exchanging χ to χ for couplings in Eq. (54).
We perform parameter scan for each scenarios to search for parameter region realizing observed relic density of DM. Firstly we set following parameter ranges for all scenarios: where the range of α i is chosen as indicated by the constraints from scalar sector discussed above. The other parameters are set for scenario (1) In addition, we impose LHC constraint on {m Z , g } parameter space discussed in the previous section and we scan these values within allowed region. For scenario (2), we chose Note that we assume χ and χ masses are not degenerated, and co-annihilation processes are not taken into account in relic density calculation.
In Fig. 5, we show allowed parameter region giving relic density, 0.11 < Ωh 2 < 0.13, in scenario (1) where horizontal(vertical) axis corresponds to M χ (M Z ) and color gradient indicate the value of g . It is found that the observed relic density can be obtained around M Z ∼ 2M χ since the annihilation cross section is enhanced by resonant effect [64][65][66][67][68].
We also show allowed parameter region for scenario (2) in Fig. 6 where horizontal(vertical) axis indicates M χ (M H 0 ) and color gradient shows M ξ 0 . In this case we obtain allowed region for M χ < M H 0 (or M ξ 0 ) since the relic density is explained by the process χχ → H 0 H 0 (ξ 0 ξ 0 ). In addition, the allowed parameter region for scenario (3) is shown in Fig. 7 where horizontal(vertical) axis indicates Y χ 12 (Y χ 22 ) and color gradient shows Y χ 32 . We find that required values of Yukawa couplings Y χ i2 are O(1) scale which is also required to obtain sizable ∆a µ . annihilation cross section by Fermi-LAT observation [69,70]. For scenario (3), current DM annihilation is small since the cross section is suppressed by DM velocity since it is P-wave dominant process. We thus estimate DM annihilation cross section in current universe for scenario (1) and (2) using micrOMEGAs 4.3.5. In Fig. 8 and 9, we respectively show the DM annihilation cross section in the current universe for scenario (1) and (2). We find that the cross section in scenario (1) is smaller than that in scenario (2) since DM-DM-Z coupling include derivative and the cross section is suppressed by momentum factor. Therefore the scenario (2) is the most sensitive case for indirect detection where the shown parameter region is still allowed by the current measurements [69,70], and it can be tested by in future data. For illustration, we also estimate spectrum of γ-ray from DM annihilation in scenario (2) where we adopt two benchmark points(BPs) given in Table III and use micrOMEGAs.
The spectrum for BP1 and BP2 are shown in left and right plot of Fig. 10 where we applied three angular regions characterized by galactic latitude b and longitude l. We find that dΦ γ /dE has broad range and its value is larger for smaller energy region since γ-ray comes from radiation from charged particle in final states in DM annihilation.

IV. SUMMARY AND CONCLUSIONS
We have constructed a two Higgs doublet model with extra U (1) X gauge symmetry in which lepton specific (type-X) structure is realized by charge assignment of Higgs doublets, quarks and leptons. In addition exotic charged leptons E are also introduced to cancel gauge anomalies. We have also introduced discrete Z 2 symmetry under which exotic charged leptons have odd parity, in order to restrict exotic charged lepton interactions. Furthermore the SM singlet scalars χ and χ with Z 2 odd parity are added as our dark matter candidate where χ is charged under U (1) X while χ is not charged.
We analyzed scalar sector formulating mass eigenstates and relation among parameters in the scalar potential. Then allowed parameter regain is explored by investigating constraints from scalar sector such as stability and peturbativity bound in the potential. We have also  (2) where we applied three angular regions.
estimated muon g − 2 applying the allowed parameter sets. It has been found that the contributions from loop diagrams with Z 2 even scalar bosons can not be sizable to explain muon g − 2 discrepancy. To explain muon g − 2 we should rely on contribution from loop diagrams with Z 2 odd particle and it can give sufficiently large muon g − 2 with sizable Yukawa coupling associated with exotic leptons and dark matter.
The collider physics has been also discussed focusing on Z boson production at the LHC. Our Z boson has leptophobic interactions and pp → Z → tt process provides the strongest constraint if Z mass is heavier than 2m t . We have estimated the Z production cross section and discussed its constraints. In addition we have discussed tt asymmetry for the pp → Z → tt process.
Finally we have analyze dark matter physics such as relic density and constraint from direct/indirect detection experiments. In our analysis, we have considered several scenarios: (1) DM is χ and Z interaction is dominant, (2) DM is χ (or χ ) and scalar portal interaction is dominant, (3) DM is χ and Yukawa interaction with exotic leptons is dominant. Then allowed parameter region for each case have been searched for taking into account observed relic density and direct detection constraints. We then find all the cases can realize the observed relic density by choosing parameters relevantly. In addition we have discussed possibility of indirect detection estimating DM annihilation cross section at the current