Light neutralino dark matter in $U(1)_X$SSM

The $U(1)_X$ extension of the minimal supersymmetric standard model(MSSM) is called as $U(1)_X$SSM with the local gauge group $SU(3)_C\times SU(2)_L \times U(1)_Y \times U(1)_X$. $U(1)_X$SSM has three singlet Higgs superfields beyond MSSM. In $U(1)_X$SSM, the mass matrix of neutralino is $8\times8$, whose lightest mass eigenstate possesses cold dark matter characteristic. Supposing the lightest neutralino as dark matter candidate, we study the relic density. For dark matter scattering off nucleus, the cross sections including spin-independent and spin-dependent are both researched. In our numerical results, some parameter space can satisfy the constraints from the relic density and the experiments of dark matter direct detection.


I. INTRODUCTION
There are several existences of dark matter in the universe, and dark matter contribution is more important than the visible matter. The earliest and the most compelling evidences for dark matter are the luminous objects that move faster than one expects [1]. The other evidences for the dark matter can be found in Refs. [2,3]. Besides the gravitational interaction, dark matter can take part in weak interaction [4]. To keep the relic density of dark matter, dark matter should be stable and live long. People have paid much attention to dark matter for many years, but they have not known its mass and interaction property.
The non-baryonic matter density is Ωh 2 = 0.1186 ± 0.0020 [5], and the standard model (SM) can not explain this problem. It implies that there must be new physics beyond the SM.
Considering the shortcoming of SM, physicists extend it and obtain a lot of extended models. In these new models, the minimal supersymmetric standard model (MSSM) [7] is the favorite one, where the lightest neutralino can be dark matter candidate [8]. Furthermore, MSSM is also extended by people, and its U(1) extensions are interesting [9,10]. In this work, we extend the MSSM with the U(1) X gauge group [11]. On the base of MSSM, we add three right-handed neutrinos and three singlet Higgs superfieldsη,η,Ŝ. The righthanded neutrinos can not only give tiny masses to light neutrinos but also produce the lightest scalar neutrino possessing dark matter character. Our U(1) X extension of MSSM is called as U(1) X SSM [12,13], which relieves the so called little hierarchy problem that is in the MSSM. The baryon number violating operators are avoided because the U(1) X gauge symmetry breaks spontaneously. So, the proton is stable.
In U(1) X SSM, there are the terms µĤ uĤd and λ HŜĤuĤd .Ŝ is the singlet Higgs superfield and possesses a non-zero VEV (v S / √ 2). Therefore, U(1) X SSM has an effective µ ef f = µ + λ H v S / √ 2, which will probably relieve the µ problem. As discussed in Ref. [14], one can particularly put the µ term to zero by a redefinition v S √ 2 → v S √ 2 − µ λ H . The singlet S can improve the lightest CP-even Higgs mass at tree level. Then large loop corrections to the 125 GeV Higgs mass are not necessary. S can also make the second light neutral CP-even Higgs heavy at TeV order. This can easily satisfy the constraints for heavy Higgs from experiments(such as LHC). If we delete S, the second light neutral CP-even Higgs is light, whose mass varies from 150 GeV to 400 GeV. Considering the limit for the heavy neutral CP-even Higgs, we should make the second light neutral CP-even Higgs heavier. The singlet S can produce this effect.
In our previous work [13], the lightest CP-even scalar neutrino is supposed as dark matter candidate in the framework of U(1) X SSM. Its relic density and the cross section scattering from nucleus have been researched in detail. Some works of scalar neutrino dark matter can be found in Refs. [15][16][17][18]. Here, we study the lightest neutralino as dark matter candidate [19].
In the base of the neutral bino (B), wino (W 0 ) and higgsinos (H 0 d ,H 0 u ), the neutralino mass eigenstates in the MSSM has the parameters tan β, M 1 , M 2 and µ. The lower limit on the lightest neutralino χ 0 1 mass is about 46 GeV, which can be derived from the Large Electron Positron (LEP) chargino mass limit [20]. While, this limit increases to well above 100 GeV, in the constrained MSSM (cMSSM) [21]. In pMSSM, the authors research the lightest neutralino below 50 GeV satisfying the constraints from LHC and XENON100 [22].
Many people have studied the phenomenology of lightest neutralino in MSSM [8] and there are a lot of works of neutralino dark matter in several models. They enrich the dark matter research and give light to the direct research of dark matter.
After this introduction, some content of U(1) X SSM is introduced in section II. In section III, we suppose the lightest neutralino as a dark matter candidate and we study its relic density. The direct detection of the lightest neutralino scattering off nuclei is reseached in section IV, which includes both the spin-independent cross section and spin-dependent cross section. The numerical results of the relic density and cross sections of dark matter scattering are all calculated in section V. We give our discussion and conclusion in section VI. Some formulae are collected in the appendix.
1) X and adding three Higgs singletsη,η,Ŝ, right-handed neutrinosν i to MSSM, one can obtain U(1) X SSM [13]. The introduction of right-handed neutrinos can explain the neutrino experiments. The mass squared matrix of CP-even Higgs is 5 × 5, because the CPeven parts of η,η, S mix with the neutral CP-even parts of H u , H d . We take into account the one loop corrections for the lightest CP-even Higgs with 125 GeV. The condition is similar for the CP-odd Higgs, whose mass squared matrix is also 5 × 5. The sneutrinos are departed into CP-even sneutrinos and CP-odd sneutrinos, whose mass squared matrixes are both 6 × 6. Here, we show the U(1) X charges of the MSSM superfields: v u and v d are the VEVs of the Higgs doublets H u and H d . While, v η , vη and v S are the VEVs of the Higgs singlets η,η and S. The angles β and β η are defined as tan β = v u /v d and tan β η = vη/v η .
The sneutrino fieldsν L andν R read as We show the superpotential and the soft breaking terms in U(1) X SSM Here, L M SSM sof t represents the soft breaking terms of MSSM. Obviously, the U(1) X SSM is more complicated than the MSSM. In our previous work, Y Y represents the U(1) Y charge and Y X denotes the U(1) X charge. We have proven that U(1) X SSM is anomaly free, and the details can be found in Ref. [13]. In U(1) X SSM, there are two Abelian groups U(1) Y and U(1) X , which cause the gauge kinetic mixing. This effect is the characteristic beyond MSSM and it can also be induced through RGEs.
We write the covariant derivatives of U(1) X SSM in the general form . Considering the fact that the two Abelian gauge groups are unbroken, we change the basis through a correct matrix R and redefine the U(1) gauge fields Different from MSSM, the U(1) X SSM gauge bosons A X µ , A Y µ and V 3 µ mix together at the tree level. In the basis (A Y µ , V 3 µ , A X µ ), the corresponding mass matrix reads as with v 2 = v 2 u + v 2 d and ξ 2 = v 2 η + v 2 η . One can diagonalize the above mass matrix by an unitary matrix including two mixing angles θ W and θ ′ W . θ W is Weinberg angle and θ ′ W is defined as The lightest neutralino is supposed as dark matter candidate, and we obtain the mass matrix of neutralino in the basis (λB,W 0 ,H 0 d ,H 0 u , λX,η,η,s). This is caused by the super partners of the added three Higgs singlets and new gauge boson, which mix with the MSSM neutralino superfields.
The concrete forms of A, B and C are This matrix is diagonalized by N It is too difficult to obtain exactly the analytic forms of the eigenvalues, eigenvectors and N for Mχ0. With some supposition, we can deduce the lightest neutralino mass and eigenvector approximately. Comparing with A and C, the matrix B is very small. In this condition, we can use Z T N to simplify Mχ0 with matrix ζ, whose elements are all small parameters of the order B/A.
ζ T can be calculated from the equation Aζ T + B − ζ T C = 0. If we take the simplest approximation, it is In this work, we suppose the lightest neutralino m χ 0 1 is different from the MSSM condition, that is to say m χ 0 1 dominantly comes from the matrix C and MSSM neutralinos in the matrix A are heavy. Therefore, we calculate the mass eigenstates of C, which is tedious to solve the common quartic equation with one unknown quantity. Considering the constraint that The eigenvalues of C are deduced to the leading order according to the small parameter We take M BL and mss are both positive parameters. To satisfy the constraint from m Z ′ , v η is large and bigger than v S . So it is easy to see that m χ 0 a and m χ 0 c are large values. Using mss ≫ M BL and µ η , m χ 0 d is smaller than m χ 0 b , so m χ 0 d is the lightest neutralino mass m χ 0 1 . For the lightest neutralino mass, we consider the correction at the order s m .
At the leading order, the eigenvector of m χ 0 d is Here, a 0 is a small parameter. The s m correction to eigenvector V In the whole, the eigenvector of the lightest neutralino is dominatly composed by the linear combination ofη andη.
In the MSSM, the lightest CP-even Higgs mass at tree level is no more than 90 GeV, and the loop corrections to the lightest CP-even Higgs mass can be large. Including the leading-log radiative corrections from stop and top particles [23], we write the mass of the lightest CP-even Higgs boson in the following form Here, m 0 h 1 represents the lightest Higgs boson mass at tree level and ∆m 2 h is shown analytically with α 3 denoting the strong coupling constant. The parameterÃ t isÃ t = A t − µ cot β with A t representing the trilinear Higgs stop coupling. MT = √ mt 1 mt 2 and mt 1,2 are the stop masses. To save space in the text, other used couplings are collected in the appendix.

III. RELIC DENSITY
Supposing the lightest neutralino(χ 0 1 ) as dark matter candidate, we calculate the relic density. The constraint of dark matter relic density is severe, and the concrete value is Ω D h 2 = 0.1186 ± 0.0020 [5]. The χ 0 1 number density n χ 0 1 should satisfy the Boltzmann equation [24,25] dn χ 0 For χ 0 1 , we take into account self-annihilation and co-annihilation with another particle φ. At the temperature T F , the annihilation rate of χ 0 1 is approximately equal to the Hubble expansion rate, and the lightest neutralino freezes out. We suppose χ 0 1 is the lightest SUSY particle and m φ is larger than m χ 0 1 . The relevant formulae are [26] σv SA n χ 0 After we study the self-annihilation cross section σ(χ 0 1 χ 0 1 → anything) and co-annihilation cross section σ(χ 0 1 φ → anything), σv SA and σv CA are gotten. The annihilation results can be written as σv rel = a + bv 2 rel in the mass center frame. Here, v rel is the relative velocity of the two particles in the initial states. Using the following formula, we can approximately calculate the freeze-out temperature (T F ) [8,25,27] with M P l denoting the Planck mass.
The relativistic degrees of freedom with mass less than T F is represented by g * . The cold non-baryonic dark matter density is simplified in the following form [1,25,28].
It is well known that, the self-annihilation processes are dominant in general condition.
We show the researched concrete self-annihilation processes: Here i = 1, 2, 3 and h represents the lightest CP-even Higgs. The neutrinos in final state are just three light neutrinos not including heavy neutrinos.
For co-annihilation processes, if the mass of another particle is almost equal to the mass of χ 0 1 , they give considerable contributions to the annihilation cross section. a. The lightest neutralino χ 0 1 annihilates with heavier neutralinos χ 0 k (k = 2 . . . 8), whose final states are same as those produced by self-annihilation processes.
The quark level operators should be converted to the effective nucleus operators. To convert the operatorχ 0 1 χ 0 1q q, we use the following formulae [29] a q m qq q → f N m NN N, Integrating out heavy quark loops, the coupling to gluons is induced, which is included in f N . We show the concrete values of the parameters f (N ) T q [30], T s = 0.0447.
For the spin-independent operatorχ 0 1 χ 0 1q q, the scattering cross section reads as [29] σ = 1 with Z p denoting the number of proton, and A representing the number of atom.
The scattering cross section for the spin-dependent operatorχ 0 1 γ µ γ 5 χ 0 1q γ µ γ 5 q is shown as with J N is the number of angular momentum for the nucleus. The corresponding formula for one nucleon is

V. NUMERICAL RESULTS
To study the numerical results, we should take into account the experimental constraints.
One strict constraint from experiment is the mass(125 GeV) [31] of the lightest CP-even Higgs. Z ′ boson mass constraint is also important. The mass bounds for M Z ′ from LHC are more severe than the limits from the low energy data. In the Sequential Standard Model, the lower mass limit of Z ′ SSM is 4.5 TeV at 95% confidence level(CL). The Lower mass limits of the Z ′ boson in the left-right symmetric model and the (B-L) model [32] are respectively 4.1 TeV and 4.2 TeV. The upper bound on the ratio between M Z ′ and its gauge coupling is M Z ′ /g X ≥ 6 TeV at 99% CL [33,34]. Considering the LHC experimental data, tan β η should be smaller than 1.5 [35]. We take into account the above constraints and choose the parameters to satisfy the relation M Z ′ > 4.5 TeV [13].
Here, we also add other experiment limits. The considered mass limits for the particles beyond SM are [5]: 1 the mass limits for heavy neutral Higgs (H 0 , A 0 ) and charged Higgs (H ± ); 2 the mass limits for neutralino, chargino, sneutrino, scalar charged lepton, squark.
The decays of the lightest CP even Higgs (m h 0 = 125 GeV) such as h 0 → γ + γ, h 0 → Z + Z and h 0 → W + W are considered. The constraint from B → X s + γ is also taken into account. With new experiment data of muon g-2 from the Fermion National Accelerator Laboratory(FNAL) [36], the deviation between experiment and SM prediction is ∆a µ = a exp µ − a SM µ = 251(59) × 10 −11 and increases to 4.2σ. We study muon g-2 in U(1) X SSM in the previous work [37], and consider this limit here.
Therefore, we use the following parameters M S = 2.7 TeV, T κ = 1.6 TeV, g Y X = 0.2, g X = 0.3, λ C = −0.08, λ H = 0.1, To simplify the numerical discussion, most of the parameters T ν , T X , T u etc. are supposed as diagonal matrices and we use the supposition A. The relic density of neutralino dark matter With the supposition that the lightest neutralino χ 0 1 is the lightest SUSY particle(LSP), we research the relic density of χ 0 1 . In this subsection, we adopt the parameters as M 2 Q11 = M 2 Q22 = M 2 D = 10TeV 2 and T d = 1TeV. M 1 is the mass of U(1) Y gaugino and appears in the neutralino mass matrix. Therefore, M 1 can affect neutralino masses and mixing to some extent. In the Fig.1, we plot the relic density in the banded gray area with ±3σ sensitivity. The relic density versus 1 is around 302 GeV, and the other SUSY particles are all much heavier than χ 0 1 . So, the self-annihilation processes are dominant. That is to say, the contributions from the co-annihilation processes are tiny. Because the masses of exchanged virtual particles are not near 2 * m χ 0 1 , the resonance annihilation affecting the relic density strongly can not take place.
In this parameter space, the masses of some SUSY particles that can co-annihilate with the lightest neutralino are collected here: the second light neutralino mass m χ 0 2 ∼ 800GeV, the lightest scalar neutrino mass (CP-even and CP-odd) mν ∼ 1600GeV, the lightest scalar lepton mass mL 1 ∼ 880GeV, the lightest chargino mass m χ ± 1 ∼ 780GeV, the lightest scalar quark mass mq 1 ∼ 1800GeV.  In the plane of M BL and M 1 , the numerical results of the relic density are researched as tan β = 9, µ = 0.5TeV, M 2 = 1TeV, M BB ′ = 0.4TeV. It is obvious that the allowed region in the Fig.6 is smaller than that in the Fig.4 and Fig.5. The points gather around the narrow band near M BL = 1000 GeV.
To find large parameter space satisfying the relic density, we plot the relic density versus the lightest neutralino mass M χ 0 1 with the parameters: 2 ≤ tan β ≤ 50, and −2TeV ≤ The results are shown in the Fig.7, where the gray band represents the relic density in three σ sensitivity. One can easily see that large reasonable parameter space in near M χ 0 1 ∼ 300GeV. In the M χ 0 1 region (120 GeV to 280 GeV), there are also reasonable parameter space for Ω D h 2 , but these parameter space are much smaller than the reasonable parameter space for M χ 0 1 ∼ 300GeV.

VI. DISCUSSION AND CONCLUSION
We extend MSSM with the U(1) X local gauge group and obtain the so called U(1) X SSM.
In the U(1) X SSM, there are several superfields beyond MSSM, such as right-handed neutrinos, three singlet Higgs superfieldsη,η,Ŝ. As discussed in MSSM, the lightest neutralino is studied in detail as dark matter candidate. While, both the lightest sneutrino and the lightest neutralino can be dark matter candidates in U(1) X SSM. Supposing the lightest CP-even sneutrino as LSP and dark matter candidate, we research its relic density and the scattering cross section off nucleus in our previous work [13]. U(1) X SSM has richer phenomenology than MSSM. To compare the scalar neutrino condition, we research the lightest neutralino as dark matter candidate in this work.
To calculate the relic density of χ 0 1 , we consider the self-annihilation and co-annihilation processes. In our used parameter space, the masses of the SUSY particles except χ 0 1 are all heavier enough than the mass of χ 0 1 . Therefore, self-annihilation processes are dominant and co-annihilation processes are suppressed by the exponential function. In the whole, this is the general condition. The resonance annihilation does not take place, because the masses of the exchanged virtual particles are not near 2 * m χ 0 1 . From our numerical results, we find that M BB ′ and M BL in the neutralino mass matrix are sensitive parameters for the relic density.
The reason is that both M BB ′ and M BL affect neutralino mixing. Large reasonable parameter space supports m χ 0 1 ∼ 300GeV, though m χ 0 1 can be smaller with reasonable parameter space. The obtained numerical results can well satisfy the experimental constraints from the relic density of dark matter. The cross section of χ 0 1 scattering off nucleus are also calculated in this work. The spin-independent and spin-dependent cross sections are at least one order smaller than their experimental constraints. This work makes up for the dark matter research [13], where just the lightest CP-even scalar neutrino is supposed as dark matter. The couplings of neutralino and gauge bosons are

Acknowledgments
Their concrete forms are shown as U and V are the rotation matrixes to diagonalize chargino mass matrix. The couplings χ 0 − χ 0 − Z and χ 0 − χ 0 − Z ′ contribute to the self-annihilation, and the coupling χ 0 − χ ± − W ± gives correction to co-annihilation. We deduce the coupling of neutralino-lepton- Z E is used to diagonalize the mass squared matrix of slepton.