Dark Mater Interactions From An Extra U(1) gauge symmetry with kinetic mixing and Higgs charge

We investigate fermionic dark matter interactions with standard model particles from an additional $\mathrm{U}(1)_\mathrm{X}$ gauge symmetry, assuming kinetic mixing between the $\mathrm{U}(1)_\mathrm{X}$ and $\mathrm{U}(1)_\mathrm{Y}$ gauge fields as well as a nonzero $\mathrm{U}(1)_\mathrm{X}$ charge of the Higgs doublet. For ensuring gauge-invariant Yukawa interactions and the cancellation of gauge anomalies, the standard model fermions are assigned $Y$-sequential $\mathrm{U}(1)_\mathrm{X}$ charges proportional to the Higgs charge. Although the Higgs charge should be small due to collider constraints, it is useful to decrease the effective cross section of dark matter scattering off nucleons by two orders of magnitude and easier evade from direct detection bounds. After some numerical scans performed in the parameter space, we find that the introduction of the Higgs charge can also enhance the dark matter relic density by at least two orders of magnitude. When the observed relic density and the direct detection constraints are tangled, at the case where the resonance effect is important for dark matter freeze-out, the Higgs charge can expand physical windows to some extent by relieving the tension between the relic density and the direct detection.

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 for DM production assumes that dark matter is thermally produced in the early Universe, typically requiring some mediators to induce adequate DM interactions with SM particles.
One of the simple attempts is to assume that the DM particle carries a U(1) X charge associated with an additional U(1) X gauge symmetry with the corresponding gauge boson acting as a mediator [5].In order to minimize the impact on the interactions of SM particles, one may assume that all SM fields do not carry U(1) X charges .Thus, the kinetic mixing between the U(1) X and U(1) Y gauge fields [30,31] induces DM interactions with SM particles.However, the DM interaction with such a kinetic mixing portal could be too weak to achieve the relic abundance via the freeze-out mechanism [32][33][34], or too strong to escape from DM direct detection, because a single mixing parameter may be insufficient to satisfy both requirements.Therefore, it could be useful to introduce more parameters.
In this paper, we assume the SM Higgs field also carries a U(1) X charge [35], which is very small for keeping the new Z ′ gauge boson weakly coupled to the SM sector.Because of the kinetic mixing term and the Higgs U(1) X charge, the U(1) X and U(1) Y gauge fields mix with each other, while one electrically neutral gauge boson remains massless, which is the photon.For ensuring the gauge invariance of the SM Yukawa couplings, the SM fermions should also be charged under U(1) X .In order to cancel chiral gauge anomalies, we assume that the fermions carry Y -sequential U(1) X charges [35][36][37][38][39], which are also very small because they should be proportional to the Higgs U(1) X charge.Such a case is different from those conventionally proposed [12,38,[40][41][42][43][44], since the latter usually have O(1) charges to lift some kind of physical processes.It is also notable that this case is similar to that for the U -boson [45,46], in the sense that the U(1) X gauge couplings to SM particles are much weaker than that to dark matter.Now there is one more free parameter, i.e., the Higgs U(1) X charge, to affect the Z ′ couplings to SM particles.It is necessary to investigate its compact on DM phenomenology.
In this context, we study a Dirac fermionic DM particle [8,10,20,22], and observe that the DM couplings to protons and neutrons are typically different [9,10,12,13,17,18], leading to isospin-violating DM-nucleon scattering [47] in direct detection experiments.It is not obvious whether the correct DM relic abundance can be achieved or not until we carry out some numerical scans.We find that the presence of the extra parameter can accommodate wider ranges of the U(1) X gauge coupling and the DM particle mass.
This paper is organized as follows.In Sec.II, we introduce the U(1) X gauge theory where the Higgs doublet carries a U(1) X charge, and discuss the induced interactions of SM fermions.In Sec.III, we study Dirac fermionic DM charged under U(1) X , and explore the effective DM-nucleon scattering cross-section for direct detection as well as the DM relic abundance via numerical scans.Finally, we gives the conclusions in Sec.IV.

II. U(1) X GAUGE THEORY
In this section, we introduce the U(1) X gauge theory with kinetic mixing between the U(1) X and U(1) Y gauge fields.We assign a small U(1) X charge to the SM Higgs doublet and the SM Yukawa interactions are gauge-invariant only if the SM fermions have appropriate U(1) X charges, which are chosen to be Y -sequential, i.e., obey the same relations as their U(1) Y charges, so that the theory remains free from chiral anomalies.
A. U(1) X gauge theory with U(1) X -charged SM Higgs doublet We denote the U(1) Y and U(1) X gauge fields as Bµ and Ẑ′ µ , respectively.Their gaugeinvariant kinetic terms in the Lagrangian read where the field strengths are Bµν The sin ϵ term is a kinetic mixing term, which makes the kinetic Lagrangian (1) in a noncanonical form.
We assume that the U(1) X gauge symmetry is spontaneously broken [48][49][50] by a Higgs field Ŝ with U(1) X charge x S = 11 .Now the Higgs sector involves Ŝ and the SM Higgs doublet Ĥ.The corresponding Lagrangian respecting the SU(2) L × U(1) Y × U(1) X gauge symmetry is [20] The covariant derivatives are given by where W a µ (a = 1, 2, 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 SU(2) L , U(1) Y , U(1) X gauge couplings, respectively.The hypercharge Y H = 1/2 for Ĥ is the same as in the SM.
The presence of ζ and Y S terms is notable here.They generally reflect a U(1) X charge of the SM Higgs doublet Ĥ and the U(1) Y charge of the exotic Higgs field Ŝ.Some studies thought this ζ charge can be absorbed into g X by a scaling, but in this work it is found to be an independent parameter.It will predict a different phenomenology as shown in the following.Before going to any detailed analysis, it is also necessary to point out that, in comparison to the Higgs charges introduced in Refs.[12,38,41,42] which were usually ∼ O(1), the magnitude of ζ in this work is expected to be very small, such that Ẑ′ would have a weak connection to SM particles.Nonetheless, compared to the size of the kinetic mixing parameter sin ϵ, the value of ζg X is not necessarily smaller.In fact, it is invited to balance the effect from the former.
Both Ĥ and Ŝ acquire nonzero vacuum expectation values (VEVs), v and v S , driving the spontaneously symmetry breaking of gauge symmetries.The Higgs fields in the unitary gauge can be expressed as Vacuum stability requires the following conditions: There is a transformation from the gauge basis (H, S) to the mass basis (h, s), with This can be regarded as a generalization of the simplest Higgs structure realized in Ref. [40].
Note that M 2 23 is present only for ζ ̸ = 0.The transformation from the gauge basis ( Bµ , W 3 µ , Ẑ′ µ ) to the mass basis (A µ , Z µ , Z ′ µ ) can be expressed as [12]    Bµ with to make the kinetic terms canonical and the mass-squared matrix diagonalized 2 .V (ϵ) is a 3-dimensional extension to a GL(2, R) transformation among ( Bµ , Ẑ′ µ ) [40], which makes the kinetic Lagrangian (1) canonical.The kinetic mixing parameter ϵ should satisfy ϵ ∈ (−1, 1) to ensure correct signs for the canonical kinetic terms.Note that the A µ and Z µ field correspond to the photon and the Z boson, and the Z ′ µ field leads to a new neutral massive vector boson Z ′ .After the mass diagonalization, the photon mass squared reads In order to keep the photon massless, it is natural for Y S and the weak mixing angle θW to satisfy Thus, the exotic Higgs field Ŝ cannot be charged under U(1) Y .Furthermore, the vanishing of the Z-Z ′ mass term M 2 ZZ ′ determine the rotation angle ξ to be 3 with Therein m Z ′ and m Z are the physical masses of the vector bosons Z ′ and Z, respectively, while C Z is a small correction originated from nonvanishing ϵ and ζ.The details are given 2 Through this text, s, c, and t denote the sine, cosine, and tangent functions, with the subscript denoting the argument.In particular, we define ŝW ≡ sin θW and ĉW ≡ cos θW .
3 It is easy to find this /cϵ is identical to Eq. ( 19) in Ref. [25] for ζ = 0.This equation has an uninterested solution of t 2ξ = 0 at ζ = g ′ s ϵ /(2g X ), which will be ignored hereafter since it will lead to the vanishing of the Z ′ couplings to SM fermions.

Fermions
in the appendix.It is notable that because of the existence of ζ, such a mixing represented by the angle ξ does not vanish in the limit ϵ → 0.

B. SM Fermions under U(1) X
Because the Higgs doublet Ĥ carries a U(1) X charge ζ, the SM fermions should also have appropriate U(1) X charges to keep the SM Yukawa couplings respecting the U(1) X gauge symmetry.Thus, the covariant derivatives of the SM quark fields in the gauge basis can be expressed as where i = 1, 2, 3 is the generation index.Therein x L q , x R u , and x R d are the U(1) X charges of the left-handed quark doublet, the right-handed up-type quark singlet, and the left-handed down-type quark singlet, respectively.Y q,u,d is the U(1) Y hypercharges as in the SM.

On a necessary condition
the SM Yukawa interactions of quarks and the Higgs doublet respect the U(1) X gauge symmetry.For SM leptons, a similar argument leads to ζ = x L l − x R l .On the other side, in order to cancel the chiral anomalies, all these U(1) X charges are further bounded.In this work, we make a simple choice to assume the U(1) X charges of SM fermions proportional to their U(1) Y charges.This is the so-called Y -sequential charges [37], as listed in Tab.I.
The charge current interactions of SM fermions at tree level are not affected by the kinetic or mass mixing, keeping the SM form of where the charge current is The neutral current interactions are given by Here j µ EM = f Q f e f γ µ f the electromagnetic current with e ≡ ĝĝ ′ / ĝ2 + ĝ′ 2 .Q f is the electric charge of a fermion f in the mass basis.The Z neutral current is with T 3 f corresponding to the third component of the weak isospin of f and The Z ′ neutral current is with It is remarkable that, at the limit ϵ → 0, the corrections to the interactions between the SM fermions and the Z boson will be proportional to ζ, just like their couplings to Z ′ .Recall that t ξ (and hence s± ξ and c± ξ ) implicitly depends on ζ, so Eqs.( 23) and ( 25) explicitly demonstrate that ζ cannot be absorbed into a redefinition of g X .

C. Parameterization and constraints
The discussions in the above subsections indicate that not all the presented parameters are independent.It is necessary to define a convenient scheme for later calculation.Firstly, the photon couplings to SM fermions remain the same forms as in the SM at tree level, with the electric charge unit e = √ 4πα can be determined by the MS fine-structure constant α(m Z ) = 1/127.955at the Z pole [53].The mass of the W boson receive contribution only from the Higgs doublet VEV v in the form m W = ĝv/2, leading to an expression of v from the Fermi constant 2 ) −1 .The electroweak gauge couplings ĝ and ĝ′ are related to e through ĝ = e/ŝ W and ĝ′ = e/ĉ W , but the Weinberg angle θW is corrected by new physics.In the U(1) X gauge theory, it is straightforward to get a relation at tree level, ŝ2 Comparing to its SM counterpart ) and utilizing Eq. (A1) in the appendix, we have where C Z is defined in Eq. (A2).Therefore, the hatted weak mixing angle θW can be expressed as a correction added to its SM counterpart, while the latter are determined by the best-measured parameters α, G F , and m Z [53,54].
The rotation angle ξ can be represented as a function of fundamental parameters like g X , m Z ′ , ϵ, and ζ.With the procedure described in the appendix, one can find an approximate solution as From this equation, one can inversely solve ζ as a function of t ξ .Thus, t ξ can be regarded as a free parameter, while ζ becomes a induced parameter.Fortunately, the procedure can be traded in an exact way as detailed in the appendix.It is obvious that t ξ is of more convenience as a free parameter for phenomenological discussions.Hereafter, we adopt a free parameter set as {g X , m Z ′ , t ϵ , t ξ , m s , s η }.
From these free parameters, we can derive all other parameters based on the above expressions 4 .These free parameters are constrained by the measurements of the Z f f vector and axialvector couplings, where the LEP-II precise measurements is most important.The quantities like Γ Z , A 5 will be recalculated in our model and confirmed within the experimental limits from Tab. 10.5 in Ref. [53].The measurements at the Z pole further require that the correction to the Weinberg angle s 2 W need to be sufficiently small, rendering the couplings of gauge bosons close to their SM values.Moreover, the searches for the Z ′ boson at the LHC [55,56] have put constraints on the Z ′ f f couplings.The mixing angle η between the two Higgs bosons will be set sufficiently small (≤ 0.1), so no deviation is expected in the Higgs phenomena.

III. DIRAC FERMIONIC DARK MATTER
We are interested in the connection between the Z ′ boson and dark matter phenomenology.In this section, we discuss the case that the DM particle is a Dirac fermion χ with a U(1) X charge q χ [8,10,20,22].The Lagrangian for χ reads where )χ and m χ is the χ mass.Thus, the DM neutral current appearing in Eqs. ( 23) and ( 25) is Thus, the Z and Z ′ bosons mediate the interaction between DM and SM fermions.The number densities of χ and its antiparticle χ yielded by the freeze-out mechanism should be equal.Both χ and χ fermions make up dark matter in the Universe.Below, we study the phenomenology of DM direct detection, as well as relic abundance and indirect detection.q χ = 1 will be adopted in the following calculation.

A. Direct detection
Only the vector current interactions between χ and quarks contribute to DM scattering off nuclei in the zero momentum transfer limit, at which the DM direct detection experiments essentially operate.In the context of effective field theory [57], the interactions between the DM fermion χ and SM quarks q can be described by with From Eqs. ( 23) and ( 25), the vector current couplings of quarks to the Z and Z ′ bosons are given by The effective Lagrangian for DM-nucleon interactions induced by the DM-quark interactions is where χd counts the contributions of valence quarks to the vector current interactions of nucleons.Following the strategy in Refs.[25,47,58], the effective spin-independent (SI) DM-nucleon cross section for isotope nuclei with atomic number Z can be recast as where σ χp is the DM-proton scattering cross section.
is the reduced mass of χ and a isotope nucleus with mass number A i and fractional number abundance η i .We use this expression to compare the model prediction to the experimental results expressed by the normalized-to-nucleon cross section.Such a setup typically leads to isospin violation in DM-nucleon scatterings.The case of ζ = 0 gives G V χn = 0 ̸ = G V χp [25].In the case of a nonzero ζ, however, we find that G V χn = 0 is no longer held.What is more interesting is that the presence of ζ is able to bring us a relative minus sign between the neutron coupling G V χn and the proton coupling G V χp .Eventually, a nonzero ζ may lead to destructive interference in the total cross section, which may help the model survive from the stringent direct detection constraints.
In Fig. 1, we show the σ SI χN dependence on sin ϵ for g X = 0.01, 0.1, 1 assuming liquid xenon as detection material with m χ = 120 GeV and m Z ′ = 500 GeV fixed.The black points correspond to ζ = 0, while the blue points are given by adjusting ζ for each sin ϵ to achieve a cancellation in σ SI χN .The calculation is double-checked by both the formula and the MadDM code [59,60].It is easily observed that σ SI χN can be decreased by two orders of magnitude for appropriate ζ.Thus, this model could easily survive in the recent direct detection experiments [61][62][63].

B. Relic abundance and numerical scan
The relic abundance of χ and χ particles are basically determined by their annihilation cross section at the freeze-out epoch.To investigate the effect of nonzero ζ in comparison to the case only with kinetic mixing, we compute the total χ χ annihilation cross section.The possible 2-body annihilation channels involve f f , W + W − , hh, ss, hs, Z (′) Z (′) , hZ (′) , and sZ (′) .All these channels are mediated via s-channel Z and Z ′ bosons.In the case of ζ = 0, all of these annihilation processes are controlled by a single parameter t ε , such that they are typically suppressed by the observation that σ SI χ N is very small.Here, we list two interaction vertices with larger contributions to the annihilation, where with g 0 = e 2 v/(2ŝ 2 W ĉ2 W ). One could notice the presence of the extra parameter ζ, which would mitigate the tension between direct detection and relic abundance.This can be confirmed by the relic density plotted with adjusted ζ in Fig. 2.
The calculation of the DM relic abundance in our model resorts to numerical procedures, where micrOmegas [64,65] is invoked, and Eq. ( 38) is coded into this framework after some double-checks.The attempts to globally explore all the allowed parameter regions is still restrained, for the numerical scans exhaust too much, especially when there are too many free parameters.In order to highlight the effect of nonzero ζ, the results in Ref. [25] for ζ = 0 can be taken as a typical reference.To this end, we prepare a scan over the model parameters, each round of the scan starts from a sampling of parameters {g X , s ϵ , s ξ } running from small to large6 .We just fix M Z ′ = 500 GeV, and χχ annihilation would meet the Z ′ resonance for m χ ∼ m Z ′ /2.
Given a point in this 3D space, the LEP and LHC constraints mentioned above are calculated at first (a failing parameter will be rejected hereafter), and then the effective DM- nucleon cross section is calculated and required to satisfy the LZ constraint [62].Finally, a survival point will be fed to the estimation of the relic abundance Ωh 2 comparing with the observed value Ω 0 h 2 = 0.1200 ± 0.0012 [66].
The scan started from m χ = (m h + m Z )/2 ≃ 115 GeV but the relic abundance is not satisfied for such low m χ .Up to m χ = 210 GeV, as shown in Fig. 2, the relic abundance Ωh 2 has almost (but not yet, log(Ω/Ω 0 ) ∼ 0.4) reached 0.12.Nonetheless, such a figure demonstrates that, for g X = 0.04, 0.08, 0.126, and 0.4, the obtained relic density for nonzero ζ (blue points) can be decreased by at least two orders of magnitude, comparing to the black points for ζ = 0.The first physical solution, which passes all the constraints mentioned above and satisfies |Ωh 2 − 0.12| ≤ 0.012, is found until m χ = 215 GeV, as shown in Tab.II, When the DM candidate become heavier than 260 GeV, which is the last row in this table, physical solutions will disappear again.In between, e.g., m χ ∼ 235 GeV, there are too many solutions to be recorded with g X running from 10 −1 to 10 −3 .We think it just reflects the the Z ′ resonance effect ( i.e. 2m c hi ≃ m Z ′ ) [67,68] for freeze-out DM.In the case of ζ = 0, the resonance region is around m χ ∼ 230-250 GeV, as shown in Fig. 4 of Ref. [25].The consequence of nonzero ζ is to extend the window to 215-260 GeV.
We also rescan around m χ ≃ 115 GeV but with m Z ′ = 2.05m χ and just show a few from many solutions in Tab.III.The relation m Z ′ = 2.05m χ makes sure that the Z ′ resonance effect is always important at the freeze-out epoch.In the ζ = 0 case, this solution is totally    rejected by the direct detection constraints, as shown in Fig. 5(a) of Ref. [25].But in this work, it is recovered with a tuning of ζ.
With micrOmegas, the γ-ray spectrum for DM indirect detection is also investigated for comparing with the upper bounds from Fermi-LAT γ-ray observations [69].At least upon the parameters passing the above procedures in Tabs.II and III, there is no excess observed.

IV. CONCLUSIONS
In this work, we introduce an extra U(1) X gauge symmetry, which is responsible for the interactions of Dirac fermionic DM with a U(1) X charge.In particular, we assume the SM Higgs doublet also carry a U(1) X charge ζ.In order to make the SM Yukawa interaction terms gauge-invariant and make the theory free from gauge anomalies, the SM fermions are assigned Y -sequential U(1) X charges proportional to ζ.The mixing between the U(1) X and U(1) Y gauge fields are induced by both kinetic mixing and the U(1) X charge of the SM Higgs doublet.Thus, the DM interactions with SM particles mediated by the Z boson and the new Z ′ boson are essentially controlled by the kinetic mixing parameter ϵ and the Higgs charge ζ.
After the analytical calculation, we have performed numerical scans with fixed m Z ′ over the parameter space of g X , s ϵ , and ζ.The new parameter ζ is found to invite destructive interference in the effective DM-nucleon cross section, and it can affect the relic density by about two orders of magnitude.Although the magnitude of ζ is quite small due to the constraints from LEP and LHC experimental data, the cancellation between the interactions originated from the kinetic mixing and the Higgs U(1) X charge indeed takes place, Therefore, it can definitely extend the physical windows where the resonance effect for relic density are important.Nonetheless, the introduction of ζ itself is not enough to make generic parameter regions work.close: The leading iteration simply starts from This is also the leading expression in many previous studies.Together with ŝ(1) W = s W , it can be utilized to obtain When Eqs. (A7) and (A6) are inserted back into Eq.(A3), one can get t 2ξ in Eq. ( 30).This expression can be expanded longer and longer when the round of iterations is further extended.These formulas are helpful to demonstrate the effects of a nonzero ζ.
Alternatively, since the rotation angle ξ takes more places in the Lagrangian, for example in the interactions among SM particles, especially in the fermion sector, while ζ only explicitly shows up in the Z ′ interactions in the Higgs sector, it is convenient to choose ξ as a free parameter and regard ζ as a derived parameter.Along this line, Eqs. ( 15) and (A2) are helpful in eliminating ζ (and even ŝW ): In such a choice one needs no more than the renormalization of the Weinberg angle θW via Eq.( 29) with a small correction from C Z : In the case where ζ is explicitly involved, it can be recalculated from Therein, ŝW (ĉ W ) should be replaced with the above formulation.It is possible that ζ could be not small when r becomes large.
It is also straightforward to iterate C Z ′ via Eq.(A5), solve and propagate the basic parameters and corrections to the Higgs sector via Utilizing the above relations, all the parameters in the model are calculable on the base of g X , m Z ′ , s ϵ , ζ, m s , and s η , together with the well-measured parameters G F , m Z , and α.

1 )
X gauge theory A. U(1) X gauge theory with U(1) X -charged SM Higgs doublet B. SM Fermions under U(1) X C. Parameterization and constraints III.Dirac fermionic dark matter A. Direct detection B. Relic abundance and numerical scan IV. Conclusions I. INTRODUCTION

TABLE I .
Y -sequential U(1) X charges for SM fermions in the gauge basis.

TABLE II .
[25]parameters corresponding to the correct relic density and consistent with the constraints.The unshown parameters take values from Ref.[25].

TABLE III .
The parameters corresponding to the correct relic density and consistent with the constraints for m χ = (m h + m Z )/2 ≃ 115 GeV and m Z ′ = 2.05m χ .