Singlet Fermionic Dark Matter with Dark $Z$

We present a fermionic dark matter model mediated by the hidden gauge boson. We assume the QED-like hidden sector which consists of a Dirac fermion and U(1)$_X$ gauge symmetry, and introduce an additional scalar electroweak doublet field with the U(1)$_X$ charge as a mediator. The hidden U(1)$_X$ symmetry is spontaneously broken by the electroweak symmetry breaking and there exists an massive extra neutral gauge boson in this model which is the mediator between the hidden and visible sectors. Due to the U(1)$_X$ charge, the additional scalar doublet does not couple to the Standard Model fermions, which leads to the Higgs sector of type I two Higgs doublet model. The new gauge boson couples to the Standard Model fermions with couplings proportional to those of the ordinary $Z$ boson but very suppressed, thus we call it the dark $Z$ boson. We study the phenomenology of the dark $Z$ boson and the Higgs sector, and show the hidden fermion can be the dark matter candidate.


I. INTRODUCTION
The Standard Model (SM) provides a consistent description of known elementary particles and interactions. The CERN Large Hadron Collider (LHC) has discovered the Higgs boson to complete the SM field contents [1,2]. Still the majority of matter in our Universe is, however, the dark matter (DM) beyond the reach of our knowledge. Thus the existence of hidden sectors is an exciting possibility as an explanation of many problems beyond the SM including the DM.
If the DM is a fermion of the SM gauge singlet, a mediator field would connect the DM to the SM sector with renormalizable couplings. One of the minimal choice for the mediator field is a real singlet scalar which is coupled to the singlet fermionic dark matter (SFDM) with the Yukawa type interaction and to the SM through the quadratic term of the Higgs field, the only massive couplings in the SM lagrangian. Various aspects of such kind of minimal models, so called Higgs portal, has been studied in extensive literatures [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. If there is a gauge symmetry in the hidden sector, a vector field could be the mediator between the DM and the SM fields. When the hidden gauge symmetry is U(1), the corresponding gauge field can be coupled to the SM fields through the kinetic mixing with the field strength of the SM U(1) gauge interaction. Then the vector field has a vectorlike couplings to the SM sector and is usually called a dark photon. The hidden U(1) gauge symmetry is spontaneously broken in the hidden sector to yield the dark photon mass.
In this work, we consider an alternative way to connect the hidden sector including fermionic DM without the kinetic mixing to the SM. We introduce an additional scalar fielld which is the SM doublet and has the U(1) X charge to connect the hidden U(1) X gauge field to the SM fields. The new scalar doublet does not couple to the SM fermions due to the U(1) X charge, but couples to the SM Higgs doublet in the scalar potential as well as the SU(2) L gauge fields. Thus the Higgs sector is same as that of the two Higgs doublet model (2HDM) of type I. It is pointed out in Ref. [19] that an additional U(1) gauge symmetry can explain the type I 2HDM flavour structure instead of the discrete symmetry. The U(1) X gauge boson gets the mass via the electroweak symmetry breaking (EWSB) in this model and is mixed with the Z boson. Since the new gauge boson, Z ′ , is mixed with only the Z boson, its couplings to the SM fermions are same as the Z boson couplings except for involving the suppression factor. Thus we call it a dark Z boson. The dark Z boson mass should be of the EW scale or less, and actually expected to be much light. We anticipate that the couplings of the dark Z to the SM should be very small due to constraints from lots of low energy neutral current (NC) experiments. We consider the ρ-parameter, the atomic parity violation of Cs atom, and the rare decays of K and B mesons as experimental constraints in this work. Note that the new gauge coupling need not be extremely small to suppress the dark Z couplings to the SM sector, if the Higgs doublet mixing ∼ 1/ tan β could be small enough.
The SFDM carries the U(1) X charge and is connected to the SM through the dark Z after the EWSB. Since we have no restrictions on the U(1) X charge of the SFDM, the interaction strength of the SFDM ∼ g X X ψ is a new free parameter to fit the observed relic density and the DM-nucleon cross sections under the bounds from direct detection experiments of the DM. We show that our SFDM mediated by the dark Z can be a good DM candidate satisfying the stringent experimental constraints on the dark Z, DM and Higgs phenomenology. This paper is organized as follows. We describe the model in section 2. Presented are the experimental constraints on the dark Z boson from the ρ parameter, the atomic parity violation of C s atom, and decays of K and B mesons in section 3. The dark matter phenomenology is studied in section 4 and the Higgs sector phenomenology in section 5. We discuss the predictions for the future experiments and conclude in section 6.

II. THE MODEL
We consider the QED-like hidden sector which consists of a SM gauge singlet Dirac fermion and the U(1) X gauge field. No fields in the SM lagrangian carry the U(1) X gauge charge and no kinetic mixing with the SM U(1) Y gauge field is assumed. We introduce an additional scalar field as a mediator between the hidden sector and the visible sector, which is the SM SU(2) doublet and carries the U(1) X charge. The field contents of two Higgs doublets H 1 and H 2 , and the hidden fermion ψ based on the gauge group SU(3) c × SU(2) L × U(1) Y × U(1) X are given by where the U(1) X charge of H 1 is fixed to be 1/2 for convenience and that of ψ is a free parameter.
Since the additional scalar doublet H 1 does not couple to the SM fermions due to the U(1) X charge, the visible sector lagrangian of our model looks like the 2HDM of type I except for the extra U(1) X gauge interaction for H 1 . We write the Higgs sector lagrangian as where V (H 1 , H 2 ) is the Higgs potential and L Y the Yukawa interactions of the SM fermions.
The covariant derivative is defined by where X is the hidden U(1) X charge operator and the A µ X corresponding gauge field. The Higgs potential is given by Note that the H † 1 H 2 quadratic term and the quartic term with λ 5 coupling are forbidden by the U(1) X gauge symmetry.
After the EWSB, the expectation values of two Higgs doublets arise, H i = (0, v i / √ 2) T with i = 1, 2, and the gauge bosons get masses as where Diagonalizing the mass matrix with the Weinberg angle θ W between W 3 and B, we get the massless mode, the photon, and diagonalization with the additional mixing angle θ X between A X and the ordinary Z mode follows to get the physical masses such as, where s W = sin θ W = g ′ / g 2 + g ′ 2 , s X = sin θ X and with tan β = v 2 /v 1 . We find the new mixing angle θ X is negative. Then the neutral gauge boson masses are Note that only two mixing angles are required to diagonalize the neutral gauge boson mass matrix in this model.
We write the NC interactions in terms of the physical states of the gauge bosons: where the electric charge is defined by Note that g L and g R are common with Z ′ and Z but the Z ′ couplings involve the suppression factor, − sin θ X . This is the reason why we call Z ′ the dark Z.
The structure of the Higgs sector is almost same as that of the type I 2HDM. The only difference is that the pseudoscalar Higgs boson does not exist in this model due to being the longitudinal mode of the dark Z. Thus there are only two additional Higgs bosons in this model, a neutral CP-even Higgs boson and a charged Higgs boson.
The physical CP-even neutral Higgs bosons h 1 , h 2 are defined by where ρ i are the neutral components of the doublets, 2) T , and the mixing angle α is defined by The masses are obtained by The heavier mode h 2 is the SM Higgs and h 1 is the extra neutral Higgs boson with relevant values of parameters as will be shown later.
The charged Higgs boson masses are diagonalized to get the physical mode H ± by, where the mixing angle is β in this case. One of the diagonalized masses is given by for H ± and the other is 0 for G ± . The massless mode G ± is the Goldstone mode eaten up to be the longitudinal mode of the W ± boson. We write the Yukawa interactions for the charged Higgs boson with the short-hand notation where V CKM are the corresponding quark mixings.

III. DARK Z PHENOMENOLOGY
The NC interactions with the dark Z boson are constrained by various experiments. Apart from the new Higgs masses and mixings, the independent model parameters are (g X , tan β) in our model lagrangian. Instead in this analysis, we present the results in terms of the observables (m Z ′ , −s X ).

A. The ρ parameter
We consider the precision test on the electroweak sector using the ρ parameter. The ρ parameter is defined by the ratio of W and Z boson masses, ρ ≡ m 2 W /m 2 Z c 2 W , and should be 1 at tree level in the SM. In this model, we have m W = gv/2 as in the SM at tree level.
But the Z boson mass is shifted such that and then the inverse of the ρ parameter is in the leading order of s 2 X . The deviation ∆ρ from the unity is defined by then the leading contribution to ∆ρ in this model is given by The correction ∆ρ is related to T parameter as [20] ∆ρ = α(m Z ) T of which values are and α (5) −1 (m Z ) = 127.955 ± 0.010 obtained in Ref. [25]. Then we have bounds for ∆ρ as −0.00039 < ∆ρ < 0.001485.
Applying this bound to ∆ρ X , we show the excluded regions in (m Z ′ , −s X ) plane in Fig. 1.
The green (grey) region denotes too small ∆ρ and the cyan (light grey) region too large ∆ρ.
Note that the region below the lower thin curve denotes the breakdown of the perturbativity,

B. The atomic parity violation
The parity violation of the atomic spectra is observed due to the Z boson exchanges. The precise measurement of the atomic parity violation (APV) provides a strong constraint on the exotic NC interactions. We derive the effective lagrangian for the corresponding process at the quark level.
The APV is described by the weak charge of the nuclei defined by where Z (N) is the number of protons (neutrons) in the atom and the nucleon couplings are defined by g p AV ≡ 2g u AV + g d AV and g n AV ≡ g u AV + 2g d AV . In the SM, g p AV ≈ −1/2 + 2s 2 W and g n AV ≈ 1/2 lead to Q SM W ≈ −N + Z(1 − 4s 2 W ) at tree level, which is shifted by the dark Z contribution as in the leading order of s X . The SM prediction of the Cs atom is [21,22] Q SM W = −73.16 ± 0.05, (27) and the present experimental value is [23] Q exp W = −73.16 ± 0.35, which yields the bound at 90 % CL [24]. This constraint is shown as the upper thick line of the (m Z ′ , −s X ) plane in Fig. 1 The region above the line is excluded.

C. Rare meson decays
The flavour physics have been a good laboratory of the new physics. Davoudiasl et al. [24] suggest that the flavour-changing neutral current (FCNC) decays of K and B mesons provide strong constraints on the dark Z model. Here, we follow their analysis to constrain our model.
The FCNC interactions of the dark Z boson s → dZ ′ and b → sZ ′ derive K → πZ ′ and B → K(K * )Z ′ decays, and sequential decays of Z ′ into lepton pairs lead to rare decays K → πll and B → Kll.

The experimental measurements for K mesons
Br(K + → π + e + e − ) = (3.00 ± 0.09) × 10 −7 , and for B mesons Br(B → Kl + l − ) = (4.51 ± 0.23) × 10 −7 , are obtained [25]. Then the strongest constraints are derived [24] m Z m Z ′ s X ≤ 0.001 Although being not manifest in the analysis, the DM mass affects these constraints. If the DM mass is less than the half of Z ′ mass, then 100% of the Z ′ will decay into the DM and the Br(K + → π + l + l − ) and Br(B → Kl + l − ) constraints do not work.
The final result is depicted in Fig. 1, where ∆ρ, APV, and rare meson decays constraints are presented altogether. The violet (dark grey) region denotes the excluded points by the constraints given in Eq. (33). Finally we find that the rare meson decays provide the strongest constraints on m Z ′ and s X . We also see that the dark Z is rather light, m Z ′ ≤ 2 GeV, and the coupling is very small, | sin θ X | ≤ 5 × 10 −4 as expected.

IV. DARK MATTER PHENOMENOLOGY
Our hidden sector consists of a Dirac fermion with a U(1) X gauge symmetry. The hidden sector lagrangian is QED-like where and X is the U(1) X charge operator for ψ. We show that the singlet fermion ψ can be a DM candidate. Using Eq. (6), ψ has vectorial interactions with Z and Z ′ bosons, We have two additional parameters, m ψ and the U(1) X charge for the DM phenomenology.
The SFDM contribution to the relic abundance density Ω is obtained from global fits of various cosmological observations. We can read the present value of Ω of the cold nonbaryonic DM as from measurements of the anisotropy of the cosmic microwave background (CMB) and of the spatial distribution of galaxies [25]. Such precise value provides a stringent constraint on the model parameters. We calculate Ω and the DM-nucleon cross section using the micrOMEGAs [26] with the allowed values of parameters (m Z ′ , sin θ X ) given in the previous section. Figure [31,32]. We survey the DM mass, 2 < M ψ < 3 in GeV, and find that the cross section is generically small, σv < 10 −34 cm 3 s −1 due to the small mixing |s X |. Thus our model is safe for the present bounds from the indirect search of the DM.

V. HIGGS PHENOMENOLOGY
An additional Higgs doublet is introduced and extra scalar particles exist in our model.
Since the CP-odd scalar mode is eaten up by the dark Z boson, there exists no CP-odd scalar in this model. This is the noticeable difference from the ordinary 2HDM particle contents.
As a result the new particles are an neutral Higgs boson and a pair of charged Higgs bosons.
Most of the phenomenology of the Higgs sector is governed by quartic couplings of the Higgs potential. Hence we just discuss two issues on the Higgs phenomenology, the charged Higgs search and the Higgs invisible decays here.
To begin with we investigate the scalar masses. The masses are calculated with the perturbativity conditions on the quartic couplings |λ i | < 4π and the vacuum stability conditions [33], We find that h 1 is very light in this model since v 1 ≪ v 2 . Hence h 2 should be the SM Higgs boson. If we fix the mass of h 2 to be 125.18 ± 0.16 GeV, the h 1 mass is less than 1.2 GeV.
The charged Higgs boson mass is determined by λ 4 solely in this model and has the upper bound ∼ 616 GeV due to the perturbativity bound of λ 4 . We note that these features are very insensitive to the parameter set allowed in the previous analysis.
For the analysis of ∆ρ in the previous section, we consider only the dark Z contributions.
By the way, the additional Higgs bosons also contribute to the ρ parameter such as [34] ∆ρ where m 1 is the mass of h 1 and m ± the charged Higgs boson mass. Since h 1 is very light NS crucially depends only on the charged Higgs mass. If m ± ≥ 120 GeV, ∆ρ (1) NS exceeds 0.001485 of the experimental upper limit given in Eq. (23) and no parameter set can satisfy the ∆ρ. On the other hand, ∆ρ (1) NS is very sensitive to m ± and it does not play a role of constraints if m ± is just slightly smaller than 120 GeV, e.g. 119 GeV. (The cyan region of Fig. 1 is overlapped by other constraints.) Therefore we demand 80 GeV < m ± < 120 GeV in this model. The model-independent lower bound is given in [25].
The recent analysis of the CMS data for H ± → τ ± ν and H + → tb channels at √ s = 13 TeV with an integrated luminosity of 35.9 fb −1 shows the allowed values of tan β and the charged Higgs mass m ± in the type I 2HDM [35][36][37]. Since our charged Higgs is rather light, m ± < 120 GeV, it cannot decay into the top quark. For such a light charged Higgs, the CMS data provides the exclusion region tan β < 2.5 for all values of m ± by the b → sγ and H ± → τ ν decays. (See Fig. 5 in [37].) However the allowed parameter space given in Fig.   1 corresponds to very large tan β, numerically tan β > 500 and the present LHC bound for H ± is not relevant to our model.
Since the dark Z boson is light in this model, the Higgs boson decays into the dark Z pair are possible which contributes to the Higgs invisible decay modes. However, the h 2 Z ′ Z ′ coupling is suppressed by sin 2 θ X or g 2 X cos β sin α. Since sin θ X ∼ g X cos 2 β and sin α ∼ cos β, the decay rate Γ(h → Z ′ Z ′ ) is suppressed by the factor sin 2 θ X or less compared with Γ(h → ZZ). Thus the h → Z ′ Z ′ contribution to the Higgs invisible decay is much smaller than the current limit Br(h → invisible) < 0.22 by the CMS [38].

VI. CONCLUDING REMARKS
We have constructed the SFDM model mediated by the dark Z boson. The hidden U(1) gauge boson does not couple to the SM sector directly in this model, but interacts with the SM through the Higgs mixing with an additional Higgs doublet involving the hidden U(1) charge. The Higgs mixing induces the Z − Z ′ mixing, and the mixing angle depends upon the Higgs mixing angle, β and the hidden gauge coupling g X . Such a dark Z boson is severely constrained by various the electroweak data and thus the Z − Z ′ mixing angle θ X is very small. Then the allowed parameter space by the experiments favors the very large tan β region. The mass of the dark Z is approximately the VEV of the second Higgs doublet v 1 = v cos β and consequently it is rather light, < 2 GeV.
In this model, our DM is a SM singlet fermion and mediated by the dark Z boson. We find that it can satisfy the observed relic abundance from the CMB. Since the dominant channels of the DM annihilation in the early universe are s−channel at the dark Z resonance region, t−channel at the ψψ → Z ′ Z ′ opening region, and the Higgsstrahlung into Z ′ h 1 region, the DM mass is same order as that of the dark Z mass, ∼ GeV and less.
The coupling strength of the dark Z to the SM matter is very small, − sin θ X < 5 × 10 −4 .
If the DM mass is larger than the half of the dark Z mass, (Actually in that case, the DM mass is almost same as the dark Z mass to satisfy the relic density.) the dark Z boson might live long. Then the proposed intensity frontier experiments, e.g. SHiP [39], FASER [40], MATHUSLA [41] etc. will have the chance to probe the dark Z boson directly in the future.