Fermionic and scalar dark matter with hidden $\mathrm{U}(1)$ gauge interaction and kinetic mixing

We explore the Dirac fermionic and complex scalar dark matter in the framework of a hidden $\mathrm{U}(1)_\mathrm{X}$ gauge theory with kinetic mixing between the $\mathrm{U}(1)_\mathrm{X}$ and $\mathrm{U}(1)_\mathrm{Y}$ gauge fields. The $\mathrm{U}(1)_\mathrm{X}$ gauge symmetry is spontaneously broken due to a hidden Higgs field. The kinetic mixing provides a portal between dark matter and standard model particles. Besides, an additional Higgs portal can be realized in the complex scalar case. Dark matter interactions with nucleons are typically isospin-violating, and direct detection constraints can be relieved. Although the kinetic mixing has been stringently constrained by electroweak oblique parameters, we find that there are several available parameter regions predicting an observed relic abundance through the thermal production mechanism. Moreover, these regions have not been totally explored in current direct and indirect detection experiments.


I. INTRODUCTION
The standard model (SM) with SU(3) C × SU(2) L × U(1) Y gauge interactions has achieved a dramatic success in explaining experimental data in particle physics. Nonetheless, the SM must be extended for taking into account dark matter (DM) in the Universe, whose existence is established by astrophysical and cosmological experiments [1][2][3][4]. The standard paradigm assumes dark matter is thermally produced in the early Universe, typically requiring some mediators to induce adequate DM interactions with SM particles.
Inspired by the gauge interactions in the SM, it is natural to imagine dark matter participating a new kind of gauge interaction. The simplest attempt is to introduce an additional U(1) X gauge symmetry with a corresponding gauge boson acting as a mediator [5]. In order to minimize the impact on the interactions of SM particles, one can assume that all SM fields do not carry U(1) X charges [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Thus, such a U(1) X gauge interaction belongs to a hidden sector, which also involves dark matter and probably an extra Higgs field generating mass to the U(1) X gauge boson via the Brout-Englert-Higgs mechanism [24][25][26] 1 . It is easy to make the theory free from gauge anomalies by assuming the DM particle is a Dirac fermion or a complex scalar boson. Gauge symmetries allow a renormalizable kinetic mixing term between the U(1) X and U(1) Y field strengths [29], which provides a portal connecting DM and SM particles.
In this paper, we focus on DM models with a hidden U(1) X gauge symmetry, which is spontaneously broken due to a hidden Higgs field. We assume that the DM particle is a SU(3) C × SU(2) L × U(1) Y gauge singlet, but carries a U(1) X charge. Because of the kinetic mixing term, the U(1) X and U(1) Y gauge fields mix with each other, modifying the electroweak oblique parameters S and T at tree level [30,31]. In the mass basis, electrically neutral gauge bosons include the photon, the Z boson, and a new Z boson. The Z and Z bosons couple to both the DM particle and SM fermions, based on the kinetic mixing portal. As a result, DM couplings to protons and neutron are typically different [9,10,12,13,17,18], leading to isospin-violating DM-nucleon scattering [32] in direct detection experiments.
In this framework, specifying different spins of the DM particle and various U(1) X charges in the hidden sector would lead to different DM models. The simplest case is to consider Dirac fermionic DM, whose phenomenology has been studied in Refs. [8,10,20,22]. Firstly, we revisit this case, investigating current constraints from electroweak oblique parameters, DM relic abundance, and direct and indirect detection experiments. Nonetheless, it is not easy to accommodate the constraints from relic abundance and direct detection, except for some specific parameter regions. The main reason is that DM annihilation in the early Universe due to the kinetic mixing portal alone is generally too weak, tending to overproduce dark matter.
Therefore, we go further to consider the case of complex scalar DM, which could have quartic couplings to both the SM and hidden Higgs fields. Consequently, the DM particle can also communicate with the SM fermions mediated by two Higgs bosons, which are mass eigenstates mixed with the SM and hidden Higgs bosons. Such an additional Higgs portal can help enhance DM annihilation. Moreover, it can also adjust the DM-nucleon couplings and weaken the direct detection constraint. Thus, it should be easier to find viable parameter regions in the complex scalar DM case.
This paper is organized as follows. In Sec. II, we review the hidden U(1) X gauge theory with kinetic mixing, and study the constraint from electroweak oblique parameters. In Sections III and IV, we discuss a Dirac fermionic DM model and a complex scalar DM model, respectively, and investigate the constraints from the relic abundance observation, and direct and indirect detection experiments. Finally, we give the conclusions in Sec. V.

II. HIDDEN U(1) X GAUGE THEORY
In this section, we briefly review the hidden U(1) X gauge theory with the kinetic mixing between the U(1) X and U(1) Y gauge fields. Furthermore, we investigate the constraints from electroweak oblique parameters.
We assume that the U(1) X gauge symmetry is spontaneously broken by a hidden Higgs fieldŜ with U(1) X charge q S = 1. Now the Higgs sector involveŜ and the SM Higgs doublet H. The corresponding Lagrangian respecting the SU(2) L × U(1) Y × U(1) X gauge symmetry reads [20] The covariant derivatives are given by 3) denote the SU(2) L gauge fields and T a = σ a /2 are the SU(2) L generators.ĝ,ĝ , and g X are the corresponding gauge couplings.
BothĤ andŜ acquire non-zero VEVs, v and v S , driving spontaneously symmetry breaking. The Higgs fields in the unitary gauge can be expressed aŝ Vacuum stability requires the following conditions, The mass-squared matrix for (H, S), can be diagonalized by a rotation with an angle η. The transformation between the mass basis (h, s) and the gauge basis (H, S) is given by with the mixing angle η ∈ [−π/4, π/4]. The physical masses of scalar bosons h and s satisfy Note that h is the 125 GeV SM-like Higgs boson. If λ HS vanishes, h is identical to the SM Higgs boson.
The mass-squared matrix for the gauge fields (B µ , W 3 µ ,Ẑ µ ) generated by the Higgs VEVs reads Taking into account the kinetic mixing and the mass matrix diagonalization, the transformation between the mass basis (A µ , Z µ , Z µ ) and the gauge basis (B µ , W 3 µ ,Ẑ µ ) is given by [12,31]  with Here the weak mixing angleθ W satisfieŝ The rotation angle ξ is determined by Note that A µ and Z µ correspond to the photon and Z boson, and Z µ leads to a new massive vector boson Z . The photon remains massless, while the masses for the Z and Z bosons are given by [10] which will be useful in the following discussions.
The W mass is m W =ĝv/2, only contributed by the VEV ofĤ, as in the SM. Moreover, the charge current interactions of SM fermions at tree level are not affected by the kinetic mixing, remaining a form of where the charge current is On the other hand, the neutral current interactions become where the electromagnetic current is j µ EM = f Q f ef γ µ f , with e =ĝĝ / ĝ 2 +ĝ 2 and Q f denoting the electric charge of a SM fermion f . The neutral current coupled to Z is given by with T 3 f denoting the third component of the weak isospin of f and j µ DM ∝ g X represents the U(1) X current of dark matter, which will be discussed in the following sections. Such a current is coupled to Z due to the kinetic mixing. Furthermore, the neutral current coupled to Z can be expressed as Note that the photon couplings to SM fermions at tree level remain the same forms as in the SM. The electroweak gauge couplingsĝ andĝ are related to the electric charge unit e throughĝ = e/ŝ W andĝ = e/ĉ W , where e = √ 4πα can be determined by the MS fine structure constant α(m Z ) = 1/127.955 at the Z pole [33].
In the SM, the weak mixing angle satisfies at tree level. Based on this relation, one can define a "physical" weak mixing angle θ W via the best measured parameters α, G F , and m Z [31,34]. In the hidden U(1) X gauge theory, nonetheless, we have a similar relation Therefore, the hatted weak mixing angleθ W is related to Making use of Eq. (20), we arrive at [10] Hereafter we adopt a free parameter set From these free parameters, we can derive other parameters based on the above expressions. As a result, bothŝ W and t ξ become functions of s ε and m Z . The relations between the free and induced parameters are further described in Appendix A. Current Higgs signal strength measurements at the LHC have given a constraint on the scalar mixing angle η as |s η | 0.37 at 95% confidence level (C.L.) [35]. We will choose appropriate values for s η in the following numerical analyses.

B. Constraint from electroweak oblique parameters
Because of the kinetic mixing, the electroweak oblique parameters S and T [36,37] are modified at tree level. Therefore, electroweak precision measurements have put a significant constraint on the kinetic mixing parameter s ε .
In the effective Lagrangian formulation of the electroweak oblique parameters, the Zf f neutral current interactions can be expressed as [34] with Comparing to Eqs. (25), (26), and (30), we find that Utilizing these expressions, we obtain S and T as functions of s ε and m Z .
Assuming U = 0, a global fit of electroweak precision data from the Gfitter Group gives [38] with a correlation coefficient of 0.91. Using this result, we derive upper limits on s ε at 95% C.L., as shown in Fig. 1. For a light Z (r 1), s ε is bounded by s ε 0.0165. For m Z ∼ 1 TeV, the upper limit increases to s ε ∼ 0.42. For ε 1, S and T can be approximated as When r ∼ 1, the (1 − r) factors in the denominators greatly enlarge |S| and |T |. Therefore, the upper bound on s ε significantly decreases as m Z closes to m Z .

III. DIRAC FERMIONIC DARK MATTER
In this section, we discuss a model where the DM particle is a Dirac fermion χ with U(1) X charge q χ [8,10,20,22]. The Lagrangian for χ reads where D µ χ = (∂ µ − iq χ g XẐ µ )χ and m χ is the χ mass. In this case, the DM neutral current appearing in Eqs. (25) and (27) is Thus, DM can communicate with SM fermions through the mediation of Z and Z bosons, based on the kinetic mixing portal. Through the thermal production mechanism, the number densities of χ and its antiparticleχ should be equal, leading to a symmetric DM scenario. Both χ andχ particles constitute dark matter in the Universe. Below we study the phenomenology in DM direct detection, as well as relic abundance and indirect detection.

A. Direct detection
In such a Dirac fermionic DM model, DM-quark interactions mediated by Z and Z bosons could induce potential signals in direct detection experiments. As DM particles around the Earth have velocities ∼ 10 −3 , these experiments essentially operate at zero momentum transfers. In the zero momentum transfer limit, only the vector current interactions between χ and quarks contribute to DM scattering off nuclei in detectors. Such interactions can be described by an effective Lagrangian (see, e.g., Ref. [39]) with q = d, u, s, c, b, t, and From Eqs. (25) and (27), the vector current couplings of quarks to Z and Z bosons can be expressed as The DM-quark interactions give rise to the DM-nucleon interactions, which can be described by an effective Lagrangian where N represents nucleons. As the vector current counts the numbers of valence quarks in the nucleon, we have (42), (43), and (A1), we find that The second expression means that χn scattering vanishes in the zero momentum transfer limit.
A simple way to understand this is to realize that the kinetic mixing term −s εB µνẐ µν /2 contributes a s ε Q 2 factor to the scattering amplitude, where Q µ is the 4-momentum of the mediator, i.e., the momentum transfer. Note that theB µ field is related to the photon field A µ byB µ =ĉ W A µ −ŝ WẐ µ . Thus, χq scattering can be represented by two Feynman diagrams, as depicted in Fig. 2. In the zero momentum transfer limit, i.e., Q 2 → 0, the s ε Q 2 factor only picks up the 1/Q 2 pole of the photon propagator in the first diagram, while the second diagram vanishes becauseẐ µ is massive. Therefore, χq scattering is essentially induced by the photon-mediated electromagnetic current j µ EM . Since the neutron has no net electric charge, we arrive at G V χn = 0, resulting in vanishing χn scattering. As G V χn = 0 = G V χp , isospin is violated in DM scattering off nucleons. Thus, the convention way for interpreting data in direct detection experiments, which assumes isospin conservation, is no longer suitable for our model. Now we confront this issue following the strategy in Refs. [32,40].
For a nucleus A constituted by Z protons and (A − Z) neutrons, the spin-independent (SI) χA scattering cross section assuming a point-like nucleus is where is the reduced mass of χ and A. Note that theχA scattering cross section σχ A is identical to σ χA . If isospin is conserved, i.e., is the χp scattering cross section with µ χp denoting the reduced mass of χ and p. Results in direct detection experiments are conventionally reported in terms of a normalized-to-nucleon cross section σ Z N for SI scattering, assuming isospin conservation for detector material with atomic number Z. Therefore, in the isospin conservation case, we have σ Z N = σ χp , and hence a relation σ Z N = σ χA µ 2 χp /(A 2 µ 2 χA ) [40]. Currently, the direct detection experiments utilizing two-phase xenon as detection material, including XENON1T [41], PandaX [42], and LUX [43], are the most sensitive in the 5 GeV m χ 10 TeV range for SI scattering. Among them, XENON1T gives the most stringent constraint. Here we would like to reinterpret its result for constraining our model. Since xenon (Z = 54) has several isotopes A i , the event rate per unit time can be expressed as [32] where η i is the fractional number abundance of A i in nature, and I A i is a factor depending on astrophysical, nuclear physics, and experimental inputs 2 . For xenon, we have A i = {128, 129, 130, 131, 132, 134, 136}, corresponding to η i = {1.9%, 26%, 4.1%, 21%, 27%, 10%, 8.9%}, respectively [32].
Experimentally, the normalized-to-nucleon cross section for SI scattering is determined in the isospin conservation case, where the relation σ Z N = σ χp holds. This leads to In the isospin violation case, however, σ Z N is not identical to σ χp , which is given by For a realistic situation, I A i just varies mildly for different A i , and thus we can approximately 2 The definition of I Ai can be found in Ref. [32].  [41]. The solid blue lines correspond to the mean value of the DM relic abundance, Ω DM h 2 = 0.120, measured by the Planck experiment [44], while the blue shaded areas indicate DM overproduction. The orange shaded areas are excluded at 95% C.L. by the Fermi-LAT observations of dwarf galaxies [45].
assume that all I A i are equal [32]. Therefore, the relation between σ Z N and σ χp becomes This is the expression we should use when comparing the model prediction with the experimental results in terms of the normalized-to-nucleon cross section.
In our model, G V χn = 0, and the above expression reduces to Therefore, σ Z N is smaller than σ χp , and experimental bounds are typically relaxed. In the following numerical calculations, we adopt q χ = 1 for simplicity. Thus, m χ is the only extra free parameter. We use the 90% C.L. upper bound on σ Z N from the XENON1T experiment [41] to obtain the exclusion region in the m χ -g X plane with fixed parameters m Z = 500 GeV, m s = 100 GeV, s ε = 0.01, and s η = 0.1, as shown in Fig. 3. The U(1) X gauge coupling is constrained as g X 0.2 − 0.55 in the mass range 100 GeV ≤ m χ ≤ 800 GeV.

B. Relic abundance and indirect detection
In the early Universe, χ andχ particles would be produced in equal numbers via the thermal mechanism. The total DM relic abundance is essentially determined by the total χχ annihilation cross section at the freeze-out epoch. The possible χχ annihilation channels include ff , W + W − , h i h j , Z i Z j , and h i Z j , with h i ∈ {h, s} and Z i ∈ {Z, Z }. All these channels are mediated via s-channel Z and Z bosons. In addition, the Z i Z j channels are also mediated via t-and u-channel χ propagators.
Some numerical tools are utilized to evaluation the prediction of the DM relic abundance in our model. Firstly we use a Mathematica package FeynRules [46] to generate model files, which encode the information of particles, Feynman rules, and parameter relations. Then we interface the model files to a Monte Carlo generator MadGraph5 aMC@NLO [47]. Finally we invoke a MadGraph plugin MadDM [48][49][50] to calculate the relic abundance. In the calculation, all possible annihilation channels are included, and the particle decay widths are automatically computed inside MadGraph.
From the measurement of cosmic microwave background anisotropies, the Planck experiment derives an observation value of the DM relic abundance, Ω DM h 2 = 0.120 ± 0.001 [44]. In Fig. 3, the solid blue lines are corresponding to the mean value of Ω DM h 2 predicted by the model. In the blue shaded areas, the model predicts overproduction of dark matter, contradicting the cosmological observation. On the other hand, a relic abundance lower than the observation value is not necessarily considered to be ruled out, as χ andχ particles could only constitute a fraction of dark matter, or there could be extra nonthermal production of χ andχ in the cosmological history.
In Fig. 3, the kinetic mixing parameter we adopt, s ε = 0.01, is rather small. Thus, DM annihilation for m χ 230 GeV is commonly suppressed, leading to DM overproduction. Nonetheless, the Z -pole resonance effect at m χ ∼ m Z /2 = 250 GeV significantly enhances the annihilation cross section, giving rise to a narrow available region. Moreover, the sZ and Z Z annihilation channels opening for m χ (m s + m Z )/2 and m χ m Z also greatly enhance the total annihilation cross section, because they are basically dark sector processes that are not suppressed by s ε . As a result, the solid blue curve with m χ 280 GeV can give a correct relic abundance.
In addition, DM annihilation at present day could give rise to high energy γ rays from the radiations and decays of the annihilation products. Nonetheless, the Fermi-LAT experiment has reported no such signals in the continuous-spectrum observations of fifteen DM-dominated dwarf galaxies around the Milky Way with six-year data, leading to stringent bounds on the DM annihilation cross section [45].
We further utilize MadDM to calculate the total velocity-averaged DM annihilation cross section σ ann v at a typical average velocity in dwarf galaxies, 2×10 −5 . Then the Fermi-LAT 95% C.L. upper limits on the annihilation cross section in the bb channel [45] are adopted to constrain σ ann v . This should be a good approximation, because the γ-ray spectra yielded from the dominant annihilation channels in our model would be analogue to that from the bb channel [51]. The orange shaded areas in Fig. 3 are excluded by the Fermi-LAT data.
In Fig. 3, we can see that the relic abundance observation tends to disfavor small g X , while the direct and indirect detection experiments tend to disfavor large g X . This leaves only two surviving regions. One is a narrow strip around m χ ∼ m Z /2 due to the Z -pole resonance annihilation, while the other region lies in 300 GeV m χ 450 GeV, where the sZ annihilation channel opens. Now we explore more deeply into the parameter space. Inspired by the above observation, we investigate the Z resonance region with a fixed relation m Z = 2.05m χ , and demonstrate the result in Fig. 4(a). Other parameters are chosen to be m s = 100 GeV, s ε = 0.01, and s η = 0.1. We find that the correct relic abundance corresponds to two curves, one around m χ ∼ 10 − 30 GeV and one around m χ ∼ 1 TeV. A large area between the two curves predicts a lower relic abundance. Nonetheless, the direct and indirect detection experiments have excluded a region with m χ 160−400 GeV, which involves the first curve. The second curve is totally allowed.
Furthermore, we change the fixed relation to be m Z = 0.9(2m χ − m s ), with which the sZ annihilation channel always opens, and present the result in Fig. 4(b). The correct relic abundance is corresponding to a curve with g X ∼ 0.23 − 0.41 in the 100 GeV ≤ m χ ≤ 1 TeV range, which is not excluded by the Fermi-LAT data. Nonetheless, the XENON1T experiment has excluded a region with m χ 270 − 400 GeV.

IV. COMPLEX SCALAR DARK MATTER
For the Dirac fermionic DM model in the previous section, DM interactions with SM particles are only induced by the kinetic mixing portal. Thus, the interaction strengths and types are limited. As a result, it is not easy to simultaneously satisfy the direct detection and relic abundance constraints, except for some particular regions. This motivates us to study complex scalar DM with an additional Higgs portal in this section.
In the complex scalar DM model, we introduce a complex scalar field φ with U(1) X charge q φ = 1/4. The Lagrangian related to φ reads where D µ φ = (∂ µ − iq φ g XẐ µ )φ. We assume that the φ field does not develop a VEV, and thus the scalar boson φ and its antiparticleφ are stable, serving as DM particles. AfterĤ andŜ gain their VEVs, the mass squared of φ is given by The DM neutral current in Eqs. (25) and (27) is leading to φ couplings to the Z and Z bosons. Besides, φ also couples to the scalar bosons h and s, described by the Lagrangian Note that the choice of q φ = 1/4 forbids some interaction terms that could lead to φ decays or mass splitting between the real and imaginary components of φ, such asŜ †Ŝ †Ŝ † φ, S †Ŝ † φ,Ŝ †Ŝ † φφ,Ŝ † φφ, andŜ † φφφ terms. In other words, a global U(1) symmetry φ → e iθ φ is accidentally preserved in the Lagrangian (54), and φ remains a complex scalar boson after the spontaneously symmetry breaking of U(1) Y × U(1) X . Now DM interactions with SM fermions are not only mediated by the Z and Z bosons from the kinetic mixing portal, but also mediated by the h and s bosons as a Higgs portal. Assuming φ andφ particles are thermally produced in the early Universe, we arrive at a symmetric DM scenario, i.e., the present number densities of φ andφ are equal. However, as we will see soon, the φA andφA scattering cross sections are not identical in general. A. Direct detection φq andφq scatterings, which are relevant to direct detection, are mediated by the Z and Z vector bosons (kinetic mixing portal) as well as by the h and s scalar bosons (Higgs portal). The corresponding Feynman diagrams are depicted in Fig. 5. In the zero momentum transfer limit, DM-quark interactions can be described by an effective Lagrangian (see, e.g., Ref. [52]) Similar to Eq. (41), the vector current effective coupling due to the kinetic mixing portal is with g q Z and g q Z defined in Eqs. (42) and (43). The scalar-type effective coupling induced by the Higgs portal is At the nucleon level, the effective Lagrangian reads Analogous to the Dirac fermionic DM case, the vector current effective couplings for the proton and neutron are G V φp = 2G V φu + G V φd and G V φn = G V φu + 2G V φd . Similar to Eqs. (45), we have Once again, G V φn vanishes because the neutron does not carry electric charge. On the other hand, the scalar-type effective couplings for nucleons are given by [1] The form factors f N q in the first term are related to light quark contributions to the nucleon mass, defined by m N f N q = N |m qq q|N . Their values are f p u = 0.020 ± 0.004, f p d = 0.026 ± 0.005, f n u = 0.014±0.003, f n d = 0.036±0.008, f p s = f n s = 0.118±0.062 [53]. The second term with the form factor f N is contributed by the heavy quarks at loop level. An approximate relation G S φp G S φn numerically holds [17]. This means that the scalar-type interactions are roughly isospin-conserving.
The φN andφN scattering cross sections due to the Lagrangian (61) are obtained as with The only difference between the Feynman diagrams for the φq andφq scatterings in Fig. 5 is the arrow direction of the φ line, which affects the relative signs between the contributions from the vector current and scalar-type interactions. This explains the different signs in the above f φN and fφ N expressions [17,54]. Since G V φn = 0, we have f φn = fφ n = G S φn /(2m φ ).
In Fig. 6, we demonstrate f φp , fφ p , and f φn as functions of g X for the fixed parameters m φ = 500 GeV, m Z = 1000 GeV, m s = 250 GeV, s ε = 0.1, s η = 0.01, λ Hφ = 0.1, and λ Sφ = −0.1. For g X 0.03, f φp , fφ p , and f φn are rather close to each other. The reason is that the relation G S φp G S φn holds and the contributions from G V φp is negligible for small g X . From Eq. (A3), we know that v S ∝ 1/g X . Consequently, as g X increases, G S φp and G S φn decrease, and hence f φp , fφ p , and f φn decrease till g X ∼ 0.03, where they close to zero. At g X ∼ 0.03, the contributions from the h and s mediators roughly cancel each other out, and thus G S φp and G S φn basically vanish. After this point, the contributions from G V φp become more and more important, pushing f φp up but lowering fφ p down.
Note that f φn = fφ n leads to σ φn = σφ n . Nonetheless, σ φp and σφ p are not identical in general. Consequently, the φA andφA scattering cross sections are different. In the symmetric DM scenario, the average point-like SI cross section of φ andφ particles scattering off nuclei with mass number A is given by Since has no cross terms of the form G V φN G S φN , the interference between the vector current and scalar-type interactions actually cancels out for symmetric DM [17]. For several isotopes A i with the same atomic number Z, the event rate per unit time in a direct detection experiment becomes The experimental reports in terms of the normalized-to-nucleon cross section σ Z N actually correspond to the assumption f φp = fφ p = f φn = fφ n , where the relation σ Z N = σ φp holds. This leads to an expression similar to Eq. (50), In the realistic situation for our model, the above assumption does not satisfied, and the  relation between σ Z N and σ φp becomes Here we have assumed that all I A i are equal.
In Fig. 7(a), we display the DM-nucleon scattering cross section σ Z N as a function of g X for the same fixed parameters adopted in Fig. 6. For g X 0.015 and g X 0.22, σ Z N exceed the upper bound at m φ = 500 GeV from the XENON1T experiment [41]. Nonetheless, there is a dip at g X ∼ 0.03, evading the experimental constraint. We can understand this result through the following analysis.
The behavior of σ Z N is essentially controlled by the two terms inside the curly bracket of the first line in Eq. (68). They can be approximately estimated by the following quantities, whereĀ = 131.293 is the atomic weight for xenon. Note that F 1 and F 2 are the contributions from the φ andφ particles, respectively. In Fig. 7(b), we show F 1 , F 2 , and their sum as functions of g X . We find that both the F 1 and F 2 curves have dips around g X ∼ 0.03, because f φp , f φn , and fφ p are all close to zero around g X ∼ 0.03, as shown in Fig. 6. The two dips lead to a dip at g X ∼ 0.03 in the F 1 + F 2 curve, explaining the dip in Fig. 7(a).
Additionally, the F 2 curve has a second dip at g X ∼ 0.7. The reason is that the ratio f φn /f φp closes to −Z/(Ā − Z) −0.7 [32] at g X ∼ 0.7, and the two terms inside the square bracket of the F 2 expression cancel each other out. Nonetheless, this dip has no manifest effect in F 1 + F 2 , since F 2 is much larger than F 1 at g X ∼ 0.7. The F 1 + F 2 curve basically catches the behavior of σ Z N in Fig. 7(a). We utilize Eq. (70) to derive the direct detection constraint. In Fig. 8, the red shaded areas are excluded at 90% C.L. by the XENON1T experiment [41] in the m φ -g X plane with fixed parameters m Z = 1000 GeV, m s = 250 GeV, s ε = 0.1, s η = 0.01, λ Hφ = 0.01, and λ Sφ = −0.01. As discussed for Figs. 6 and 7, a region around g X ∼ 0.03 corresponds to a rather small σ Z N and evades the XENON1T constraint. Moreover, for m φ 20 GeV the constraint becomes weaker and weaker as m φ increases. This is mainly because the G S φN /(2m φ ) terms in f φN and fφ N are suppressed by m φ .

B. Relic abundance and indirect detection
Now we discuss the constraints from relic abundance observation and indirect detection. Analogous to Dirac fermionic DM, the possible φφ annihilation channels include ff , W + W − , h i h j , Z i Z j , and h i Z j , with h i ∈ {h, s} and Z i ∈ {Z, Z }. Nonetheless, these annihilation processes are not only induced by the kinetic mixing portal, but also by the Higgs portal. In Fig. 8, the solid blue lines correspond to the correct relic abundance, while the blue shaded areas predict DM overproduction. The orange shaded areas are excluded at 95% C.L. by the Fermi-LAT experiment [45].
There are several available regions for the relic abundance observation. Firstly, two available strips around m φ ∼ m h /2 = 62.5 GeV are related to resonant annihilation at the h pole. These strips cannot meet each other because the hφφ coupling (λ Sφ s η v S + λ Hφ c η v) approaches zero at g X ∼ 0.04. Nonetheless, the upper strip is excluded by XENON1T, while a section of the lower strip is free from experimental constraints.
In addition, both the ZZ annihilation channel opening for m χ m Z and the resonance of the s boson at m φ ∼ m s /2 = 125 GeV contribute to a narrow available region with 90 GeV m φ 150 GeV. Only a small fraction of this region evades the constraints from XENON1T and Fermi-LAT. Moreover, a broad available region with 170 GeV m φ 1 TeV is contributed by the sZ and ss annihilation channels opening for m φ 170 GeV and m φ 250 GeV, respectively. This region circumvents the XENON1T constraint but faces the Fermi-LAT constraint.
The annihilation processes contributing to the above available regions are primarily induced by the Higgs portal. Nonetheless, there is another available strip with g X 0.4 at m φ ∼ m Z /2 = 500 GeV corresponding to the resonant annihilation at the Z pole, which is induced by the U(1) X gauge interaction and the kinetic mixing portal. For g X < 0.6, this strip is free from the direct and indirect detection constraints.
Below we study the phenomenology in the planes of other parameter pairs. The experimental constraints in the m φ -m Z plane are demonstrated in the two panels of Fig. 9 for g X = 0.01, s η = 0.01, and λ Hφ = λ Sφ = 0.1. In Fig. 9(a) with m s = 100 GeV and s ε = 0.01, Z is light (30 GeV ≤ m Z ≤ 60 GeV) and the vector current interactions are dominant in DM-nucleus scattering. Therefore, the XENON1T bound is more stringent for lighter Z , excluding up to m φ ∼ 1.35 TeV at m Z = 30 GeV. The correct relic abundance is corresponding to a curve with m φ ∼ 1 − 1.3 TeV, while the Fermi-LAT experiment excludes a region with m φ 800 GeV.
On the other hand, Z is heavy (400 GeV ≤ m Z ≤ 1.5 TeV) in Fig. 9(b) with m s = 250 GeV and s ε = 0.1, and thus the scalar-type interactions are important in direct detection.
Because g X is fixed, v S increases with m Z following Eq. (A3). As a result, the XENON1T constraint is stricter for heavier Z , excluding up to m φ ∼ 1.65 TeV at m Z = 1.5 TeV.
In this case, the Fermi-LAT constraint is even more stringent, ruling out a region with m φ 3.45 TeV. The observed relic abundance corresponds to a curve with m φ 2.5 TeV, which is not excluded by the detection experiments.
The experimental constraints are also displayed in m φ -λ Sφ plane with g X = 0.01 in Fig. 10(a), as well as in the g X -λ Sφ plane with m φ = 4 TeV in Fig. 10(b). The other parameters in both plots are fixed as m Z = 1 TeV, m s = 250 GeV, s ε = 0.1, s η = 0.01, and λ Hφ = 0.1. In Fig. 10(a), the relic abundance observation is corresponding to a curve with 0.0032 λ Sφ 0.0067 in the range of 500 GeV ≤ m φ ≤ 800 GeV. This curve totally evades the Fermi-LAT constraint, but is excluded for m φ 570 GeV by XENON1T. In Fig. 10(b), the correct relic abundance corresponds to two curves with 0.05 λ Sφ 0.35 and −0.4 λ Sφ −0.05 in the range of 0.004 ≤ g X ≤ 0.05. Both the direct and indirect detection experiments cannot exclude these two curves.

V. CONCLUSIONS
In this work, we have explored the phenomenology of Dirac fermionic and complex scalar DM with hidden U(1) X gauge interaction and kinetic mixing between the U(1) X and U(1) Y gauge fields. Besides the DM particle, the extra particles beyond the SM involve a massive neutral vector boson Z and a Higgs boson s originated from the Brout-Englert-Higgs mechanism that gives mass to the U(1) X gauge field. The measurement of the electroweak oblique parameters S and T puts a stringent constraint on the kinetic mixing parameter s ε if Z is not too heavy.
For the Dirac fermionic DM particle χ, the kinetic mixing term provides a portal for interactions with SM fermions, inducing potential signals in DM direct and indirect detection experiments. In such a case, the DM-nucleon interactions are isospin-violating. More specifically, χ scatters off protons, but not off neutrons at the zero momentum transfer limit. This leads to weaker direct detection constraints than those under the conventional assumption of isospin conservation.
Assuming DM is thermal produced in the early Universe, we have investigated the parameter regions that are consistent with the relic abundance observation. As the kinetic mixing parameter s ε has been bounded to be small, the available regions arise from the resonant annihilation at the Z pole or the sZ annihilation channel with dark sector interactions. These regions have not been totally explored in the XENON1T direct detection and Fermi-LAT indirect detection experiments.
For the complex scalar DM particle φ, the communications with SM particles are not only through the kinetic mixing portal, but also through the Higgs portal arising from the scalar couplings. The DM-nucleon scattering is still isospin-violating. Moreover, theφp scattering cross section is typically different from the φp scattering cross section. After a dedicated analysis, we have found that the XENON1T constraint can be significantly relaxed for particular parameters that leads to a cancellation effect between the h and s propagators.
For the relic abundance observation, our calculation has shown that there are several available regions, corresponding to the resonant annihilation at the h, s and Z poles, as well as the ZZ, sZ, and ss annihilation channels. Additionally, we have carried out further investigations in the parameter space. We have found that there are still a lot of parameter regions that predict an observed relic abundance but have not been excluded by the direct and indirect detection experiments.