Dark matter in Inert Doublet Model with one scalar singlet and $U(1)_X$ gauge symmetry

The study of the abundance of dark matter is carried out in an extension of the standard model with an additional gauge symmetry $U(1)_X$. The considered extension includes two doublet and one complex singlet of scalar fields. The dark matter candidate arises from the second doublet scalar field meanwhile the singlet and first doublet scalar fields provide additional portals to the relic density. We also analyze in detail the stability of the dark matter candidate through the introduction of the discrete $Z_2$ symmetry or with the same gauge symmetry $U(1)_X$. We find constraints on the model parameter space which are in agreement with (i) the most up-to-date experimental results reported by CMS and ATLAS collaborations, namely, signal strengths $\mathcal{R}_{x\bar{x}}$; (ii) upper limit on WIMP-nucleon cross section imposed by XENON1T collaboration and (iii) upper limit on the production cross-section of a $Z^{\prime}$ gauge boson times the branching ratio of the $Z^{\prime}$ boson decaying into $\ell^-\ell^+$, with $\ell=e,\,\mu$. Subsequently these constraints are used to compute the relic density and we find that also satisfy it. Considering all constraints, we find regions that include light, intermediate and heavy dark matter candidate mass.

of two free parameters is considered. After considering the experimental restriction for the relic density and the Higgs boson mass of 125 GeV, that model allows a DM candidate with mass of the order of 500 GeV. Another more complete but equally simple model is the Inert Higgs Doublet Model (IDM) [29]. This model contains a neutral scalar particle with the conditions to play the role of WIMP [30]. The IDM shows a strong dependence on the mass splitting in order to find a viable mass region of the DM candidate. For small mass splitting the mass is of the order of 500 − 1000 GeV while other values of the mass splitting can allow masses of the order of 30 − 80 GeV. Supersymmetry provides a WIMP candidate through the lightest neutralino [31]. In universal extra dimension models, the lightest Kaluza-Klein partner is stable and a possible DM candidate [32].
In this work, we consider a simple but interesting model with an additional gauge symmetry U (1) X which includes two doublets and one complex singlet of scalar fields. One doublet is inert, of which we identify a degree of liberty as a DM candidate, meanwhile the other doublet is the usual SM doublet. The additional singlet scalar is used to break down the U (1) X symmetry and to add new portals [33] through a Z gauge boson and a scalar boson, both predicted by the model. Stability of the DM candidate is ensured by imposing a Z 2 symmetry or by the U (1) X symmetry. Models with extra U (1) X symmetries as extensions of the SM has many motivations. For example, grand unified and superstring theories contain additional U (1) X factors in the effective low energy limit. Supersymmetric and nonsupersymmetric extensions include theoretical and phenomenological aspects such as flavor physics, neutrino physics and DM [34][35][36][37]. Extended models with a U (1) X gauge symmetry also have a important phenomenology because predict a heavy vector boson Z derived from the spontaneous symmetry breaking (SSB) [38], [39]. In addition, the U (1) X symmetries can be incorporated in the extended models to suppress FCNCs while being free from triangle anomalies adding new fermions. On the other hand, a research, by one of us, with similar approach can be consulted in the ref. [40], in which the mass of DM is allowed for values of the order of 1.3 GeV to 70 GeV depending on the assignment of the free parameters associated with the U (1) symmetry. The experimental data from LEP and relic density observation are considered to find an allowed mass of DM candidate of the order of 70 GeV in a scenario that assigns the parameters of the model as Higgs-phobic type, in which the Z boson provides the channel of annihilation for DM suppressing the participation of the Higgs channel. The decay signal of Higgs diphoton also imposes strong restrictions through recent data from the CERN-LHC collider [41]. When it is combined with the observed value of DM relic density, an allowed mass region is obtained such that 5 GeV ≤ m χ ≤ 62 GeV for values of the order of 0.02 ∼ 0.08 of the quartic coupling between doublets and singlet scalar in a model with U (1) gauge symmetry [38]. In reference [42] we include extensive literature to be consulted.
The organization of our research is as follows. In Sec. II we give a general view of the model. Sec.III is focused to constraint on the free model parameters. Sec. IV is devoted to the analysis of the relic density, we present our results and an analysis of them. Finally, in the Sec. V conclusions are presented.

II. INERT DOUBLET MODEL PLUS A COMPLEX SINGLET SCALAR (ISDM)
The proposed extension of the SM incorporates a local gauge symmetry U (1) X to the SM gauge symmetry G SM = SU (3) C ⊗SU (2) L ⊗U (1) Y . The gauge boson associated to U (1) X will provide an additional channel for the production, annihilation and scattering diagrams for DM. The singlet scalar is responsible for breaking of the U (1) X and will also provide another channels. The DM candidate arises from the second doublet scalar field. Two possible options to control the stability of the DM candidate are considered: (i) a discrete symmetry Z 2 or (ii) with the U (1) X symmetry. Then, the main objectives of U (1) X symmetry are reduce the interactions to achieve the stability of the DM candidate [43] and add portals in the production, annihilation and scattering diagrams of DM.

A. Scalar sector
The scalar fields and their assignments under the G SM ⊗ U (1) X group are given by: where two first entries denote the representation under SU (3) C and SU (2) L meanwhile the hypercharge and charge under U (1) X are written in the last two entries. The scalar fields are written as follows: The spontaneous symmetry breaking (SSB) is achieved as The vacuum expectation value (VEV) of the SM-like doublet is υ = 246 GeV and for the singlet is υ x which is a free parameter until now. Note that Φ 2 must have its VEV equal to zero to guarantee the stability of the DM candidate.

Scalar potential of the ISDM
The most general, renormalizable and gauge invariant potential is The M 13 and M 23 matrix elements must be removed to avoid mixing between DM candidate and neutral scalars. We consider two possibilities to leave stable the DM candidate, the first one is with a discrete symmetry Z 2 while the second one is through the gauge symmetry of the group U (1) X . The last one can be achieved under an appropriate selection of the value of the charges under U (1) X . The values of the U (1) X charges for the doublets are chosen as x 1 = x 2 when a discrete symmetry is included. On the other hand, if x 1 = x 2 then the U (1) X symmetry guarantees the stability for the DM candidate. No matter what the case the M 13 and M 23 matrix elements are eliminated.
The terms proportional to Φ † 1 Φ 2 in the potential are invariant under U (1) X , then it is necessary to introduce a discrete symmetry to eliminate them. The Z 2 symmetry is the simplest option according to the model with two doublets. The parameters µ 2 12 and λ 6,7,12x must be, in general, different from zero unless one doublet is even and the other is odd under the Z 2 symmetry. The proper assignment is Φ 1 → Φ 1 and Φ 2 → −Φ 2 , while the rest of the fields are even as Φ 1 . Under last assignment for the doublet, the M 13 = M 23 = 0 and the mass matrix for the neutral scalar, Eq. (4), can be diagonalized by and where tan (ii) when Im[λ 5 ] = 0, which there is CP conservation coming from this parameter, then r 2 = 0 and tan α 2 = 0. Therefore the masses for the scalars are by considering the previous approximation on r 1 and tan α 1 , we observe that while the charged scalar H ± , DM and pseudoscalar masses are given, respectively, by: The model allows the χ → H ± W ∓ decay, whose χH ± W ∓ coupling is shown in the table I. To avoid the instability of the DM candidate, besides invoking the symmetries Z 2 and U (1) X , we demand that the masses must satisfy To achieve this, from eq. (14) we get the following constraint −λ 4 < 2|λ 5 | (λ 4 < 2|λ 5 |). In the special case when λ 5 = 0, we have the degeneration case: m χ = m A .
The DM candidate can also be stable without an additional discrete symmetry. This can be achieved when x 1 = x 2 . In this case, not only the same parameters in the potential Eq. (3) , as previous case, must be eliminated to guarantee the invariance but also the λ 5 must be vanish. Note that the λ 5 is responsible for the splitting between the neutral scalars. From Eq.(14) there will be degeneration in the scalar masses m H3 = m H4 . There is not SSB in the direction of Φ 2 and the masses are proportional to µ 2 2 > 0. The mixing between scalars φ 1 and s x is also as Eq. (9); however, φ 2 and η 2 are not mixing since M 34 = 0 in Eq. (8). Therefore, this scalar sector can be obtain as limit case of the previos scalar sector for λ 5 = 0.

D. Gauge bosons interactions
The kinetic terms for the gauge symmetries U (1) Y and U (1) X are where,B µν andẐ 0 µν are the field strength tensors defined byF µν = ∂ µFν − ∂ νFµ forF ν =B ν ,Ẑ 0ν [53,54]. The mixing term betweenB µν andẐ 0µν is allowed by the gauge invariance. However, this mixing term can be eliminated by the field redefinition where the fields with hat notation contain the kinetic mixing term and ε must be small to be in agreement with the experiment. After SSB, the gauge bosons in the mass states are The ε is assumed to be ε << cos θ W in order to ignore terms higher or equal to O(ε 2 ). In fact, the ε is constrained experimentally with values smaller than 10 −3 [55]. The interaction between gauge and scalar fields is where the covariant derivative D µ for neutral gauge bosons is defined as where g x and Q i are the coupling constant and the charge for U (1) X , respectively. When the SSB is achieved not only the mass terms are generated but also mixing terms are obtained, where and meanwhile the Z mass retains the same value set by the SM, In order to cancel the mixing term the following rotation is required where the mixing angle ξ satisfy the expression tan 2ξ = , and has been constrained to |ξ| < 10 −3 [56]. I: ISDM couplings involved in the calculations of this work. We define λ345 = λ3 + λ4 + 2λ5. For Z fifi coupling we consider the limit when the kinetic mixing term ε → 0.
Coupling Expression

E. Fermion interactions
The most general structure for the Yukawa couplings among fermions and scalar is where Y 0f a are the 3 × 3 Yukawa matrices, for f = u, d, l. q L and l L denote the left handed fermion doublets under SU (2) L , while u R , d R , l R correspond to the right handed singlets. The zero superscript in fermion fields stands for the interaction basis. The DM stability is lost if the couplings Y 0f 2ij appear in the Eq. (27). These Yukawa couplings can be eliminated by the correct assignment of values for charges under the Z 2 and U (1) X symmetries, as previously done.
In the case of discrete Z 2 symmetry with x 1 = x 2 , the couplings Y 0f 2ij must be equal to zero in order to respect the Z 2 symmetry. The couplings Y 0f 1ij are allowed if the assignment of the U (1) X charges for the fermions satisfy x 1 − x L + x R = 0, where x L and x R are the U (1) X charges of left-handed and right-handed fermions, respectively.
In the case of x 1 = x 2 we set the U (1) X charges such that x 2 − x L + x R = 0 in order to eliminated the couplings Y 0f 2ij in Eq. (27). Obviously, the Φ 1 also satisfaces with x 1 − x L + x R = 0 to provide the masses of the fermions as in SM. Feynman rules of ISDM are shown in the table I.

III. CONSTRAINT ON FREE MODEL PARAMETERS
In this section we obtain the experimentally allowed regions for the free model parameters involved in the relic density by considering the most up-to-date experimental collider results reported by CMS [44] ans ATLAS [45] collaborations, namely, signal strengths, denoted by R xx . In this work we consider the production of H i via gluon fusion and we use the narrow width approximation, then, R xx can be written as follows: where Γ(H i → gg) is the decay width of H i into gluon pair, with H i = h ISDM and h SM , here h ISDM is the SM-like Higgs boson coming from ISDM and h SM is the SM Higgs boson; B(H i → xx) is the branching ratio of H i decaying into a xx, where xx = bb, τ − τ + , µ − µ + , W W * , ZZ * , γγ. In addition to measurements of colliders, we use the most-up-date upper limit on WIMP-nucleon cross section, for the spin independent case, reported by XENON1T collaboration [46] and whose value for a DM candidate mass of 30 GeV is given by: Finally, in order to constrain the Z gauge boson mass, m Z , the upper limit on the production cross-section of a Z gauge boson times the branching ratio of the Z decaying into − + [47], with = e, µ, was considered. The upper limit as a function of representative values of the Z gauge boson mass is shown in the table II.On the other hand, the free parameters of the ISDM involved in our analysis are the following: • Mixing angle α 1 .
• Vacuum expectation value of the complex scalar singlet, v x .
• Z gauge boson mass, m Z .

A. Constraint on mixing angle α1
Because the coupling g ISDM hP P = cos α 1 · g SM hP P , with P = f i , W , we can to extract experimentally allowed regions for cos α 1 = c α1 from R xx ; however the x = b, τ, µ, channels give very weak bounds on c α1 . In addition, because g ISDM hP P , with P = Z, γ, is a function of additional model parameters, we find that R W W * is a stringent way for limiting c α1 . In the fig. 1 we show the c α1 − R W W * plane, where the dark area is the allowed region, to 2σ, by R W W * . We note that the allowed interval for c α1 is between ∼ 0.99-1. This is to be expected since c α1 must be closed to the unit in order to have small deviations of the SM couplings. In particular, when c α1 = 1 the SM is recovered. The graph were generated via SpaceMath [49]. A dominant portal arise through χχ → h → bb process, which is highly sensitive to c α1 because the hχχ and hbb couplings are proportionals to c α , see table I. From now on we will consider c α1 = 0.99.

B. Constraint on the Z gauge boson mass m Z
In order to constrain the Z gauge boson mass, we now turn to analyze the Z production cross-section times the branching ratio of Z decaying into − + (σ Z B Z ), with = e, µ. The ATLAS and CMS collaborations [47], [48] searched for a new resonant and non-resonant high-mass phenomena in dilepton final states at √ s = 13 TeV with an integrated luminosity of 36.1 fb −1 and 36 fb −1 , respectively, nevertheless no significant deviation from the SM prediction was observed. Lower limits on the resonant mass was reported depending on specific models. In the table II we show some relevant values of the σ Z B Z as a function of m Z . Figure 2 shows σ Z B Z as a function of the Z gauge boson mass for g x = 0.4, 0.5 and 2m Z /v. The last value is related to the coupling of the SM Z gauge boson to fermions.
We observe that σ Z B Z above ∼ 10 −4 pb excludes m Z < 3 TeV for g x = 0.4, while m Z < 3.4 TeV for g x = 0.5 are excluded. Finally, we explored the case in which g x = g Z = 2m Z /v and observe a behavior similar to reported in the ref. [47].

C. Constraint on vx, gx
The ρ parameter allows a large value for v x due to that the hypercharge and isospin of the complex singlet vanish. This can be dangerous since several physical quantities coming from ISDM depend on v x , but at the same time this must be in accordance with the constrains imposed on the Z gauge boson mass. In the fig. 3 we show the g x − v x plane, in which allowed regions by R ZZ * and the upper limit on WIMP-nucleon cross section, σ SI (χN → χN ), are displayed. We generate the Feynman rules of the ISDM via LanHEP [50] and later we evaluate σ SI (χN → χN ) through CalcHep [51]. We observe that the consistent zone with both R ZZ * and σ SI (χN → χN ) allow values for v x in the interval from ∼ 11 to ∼ 16 TeV for g x = 0.4, while v x 33 TeV are exclude.

D. Constraint on the charged scalar boson mass m H ±
As far as the charged scalar boson mass is concerned we use R γγ in order to constrain it. In addition to the SM contributions, the h → γγ decay receives contributions at one-loop level of charged scalar bosons predicted by the ISDM. The b → sγ process imposes strong restrictions on the charged scalar boson, however this particle arise from the inert doublet and therefore does not interact directly with fermions, so it is not a way to restrict the charged scalar boson mass. Figure 4 shows the m H ± − R γγ plane. We can to observe that R γγ imposes a lower bound on the charged scalar boson mass of, at 1σ (2σ), 330 GeV m H ± (130 GeV m H ± ) by considering λ 2x = 0. On the other hand, for λ 2x = 0.005, bounds less restrictives than previous case are found. At 1σ (2σ) 170 GeV m H ± (75 GeV m H ± ). Finally, the total decay width of the Higgs boson [52] exclude 60 GeV m H ± . The choice of values for λ 2x are motivated because they favor the understanding of relic density within the framework of the ISDM. We generated random values for λ 3 between 0.01 − 0.0105 (0.0297 − 0.03) for λ 2x = 0.005 (λ 2x = 0) for the same reason.
In the table III we present a summary of the values for the model parameters used in our calculations.

IV. RELIC DENSITY
Once the free model parameters were bounded by several observables, we now turn to analyze if the model can help us to understand the relic density, which is a measure of the present quantity of DM particles that remains after of frezee-out process. The PLANCK collaboration [6] reported the observed value for non-baryonic matter, where h is the Hubble constant in units of 100 km s.M pc . This can be attributed to DM candidate of the WIMP type, as in this work. As we said above, the ISDM has the scalar fields Φ 1 and Φ X with VEV's different to zero, which means that their physical states of the neutral scalars can not be DM candidates. However, the Φ 2 has VEV equal to zero and the Yukawa interactions are absent, as was discuss in sec. II. Therefore, the physical states, the neutral scalar, χ, and the pseudoscalar boson, A, could be proposed as DM candidates. Our analysis is based on selecting χ as a DM candidate, two different candidates are not considered simultaneously; although the model allows it. A difference between χ and A are the CP states and their masses when λ 5 = 0. The scalar χ is CP-even, meanwhile the pseudoscalar A is CP-odd. The relic density can be obtained by solving the Boltzmann equation for the number density rate which is given by where n is the DM number density and a is a scale factor. All information about the model is contained in the thermally averaged cross section, defined as here |M | 2 is the moduli square average amplitude of the dark matter annihilation process, χχ → SM SM. We obtain the Ωh 2 using the package micrOMEGAs [57]. The model information required by micrOMEGAs was generated with the package LanHep [50]. In the fig. 5 we present a scattering plot of the relic density as a function of the DM candidate mass, m χ . We show three escenarios to note the sensitivity of the relic density on λ 2x . These scenarios are classified by their random value intervals: In all cases we use the random values for λ 345 ∼ O(10 −2 − 10 −1 ) and for m χ from 1 to 3000 GeV. The R 3 scenario is the most favored and contain the especial case when λ 2x = 0; for this value (hχχ) ISDM ∼ (hχχ) IDM coupling. Nevertheless, in the ISDM two new portals constribute to relic density, namely, Z gauge boson and the neutral scalar S; both arise from the complex singlet, S. We observe that, depending on λ 345 and λ 2x , masses from a few GeV to ∼ 2 TeV are in agree with the results of the PLANCK collaboration [6] for the relic density. It is worth mentioning that we include pair of photons in the final state in the process of annihilation of the dark particles, i.e., χχ → h → γγ. Figure 6 shows the same as in 5 but for representatives values of λ 345 in the interval 0.0297 − 0.03 (0.01 − 0.0105) with λ 2x = 0 (λ 2x = 0.005). The scattering process σ SI (χN → χN ) excludes an important region of allowed values of g x and v x for λ 345 > 0.0105 (λ 345 > 0.03) by assuming λ 2x = 0 (λ 2x = 0.005). Under these considerations we find intervals for the DM candidate masses:

V. CONCLUSIONS
In this work we have studied an extension of the SM that includes two doublets and one complex singlet. The DM candidate proposed arises of the inert Higgs doublet. To ensure the stability of the DM candidate we consider two scenarios: a Z 2 discrete symmetry and the U (1) X symmetry. We explored the allowed regions for free model parameters of the ISDM taking into account the most up-to-date experimental collider and astrophysical results. We found that the signal strength R W W * is very stringent, allowing an interval for the mixing angle 0.99 cos α 1 1. From the analysis of the σ(pp → Z ) production cross-section times B(Z → − + ), with = e, µ, m Z 3 TeV for g x = 0.4 are excluded. We found that light O(10) GeV, intermediate O(100) GeV, and heavy O(2) TeV DM candidate masses, depending mainly on λ 345 and λ 2x , are in agreement with upper limit on σ SI (χN → χN ) and relic density reported by XENON1T and PLANCK collaborations, respectively. This model presents an improvement in the mass region due to the two portals associated with the Z gauge boson, and a scalar boson S, both predicted by the ISDM, which are absent in models as IDM. We find that the results for the case Z 2 and U (1) X show a similar behavior because the charges x, x 1 and x 2 did not play a relevant role in the processes analyzed in this research. Moreover, an important difference between the two scenarios is the presence of λ 5 , which was taken as λ 5 1. We find that the allowed interval for the DM candidate mass is highly sensitive to λ 345 and λ 2x . For this all, the ISDM is a viable model for the study of dark matter. In addition, the ISDM has a rich phenomenology through processes involving to Z , S and H ± bosons that could be tested at hadron colliders. In addition, besides impose the Z 2 and U (1) X , we find restrictions for some parameters of the model that prohibit the decay χ → W ± H ∓ , which naturally predicts the ISDM.