Scalar neutrino dark matter in $U(1)_X$SSM

$U(1)_X$SSM is the extension of the minimal supersymmetric standard model(MSSM) and its local gauge group is $SU(3)_C\times SU(2)_L \times U(1)_Y \times U(1)_X$. To obtain this model, three singlet new Higgs superfields and right-handed neutrinos are added to MSSM. In the framework of $U(1)_X$SSM, we study the Higgs mass and take the lightest CP-even sneutrino as a cold dark matter candidate. For the lightest CP-even sneutrino, the relic density and the cross section for dark matter scattering off nucleon are both researched. In suitable parameter space of the model, the numerical results satisfy the constraints of the relic density and the cross section with the nucleon.


I. INTRODUCTION
From the cosmological observations, astronomers are sure about the existence of dark matter in the universe, whose contribution is about five times that of visible matter [1,2].
Various luminous objects (stars, gas clouds globular clusters, or entire galaxies), moving faster than expectations [3,4], are the earliest and the most compelling evidences for dark matter [5][6][7][8]. Dark matter must be electrically and color neutral, and can only take part in weak interactions. Dark matter is stable and has a long life-time [9,10]. At present, the mass and interaction properties of the dark matter are unknown. Though the standard model(SM) successfully predicts the detection of the CP-even Higgs(125.1GeV) [11,12], it can not explain the relic density of dark matter in the universe.
As a result, there must exist new physics beyond the SM. There are several dark matter candidates: axions, sterile neutrinos, primordial black holes and weakly interacting massive particles (WIMPs) [9,14,15]. WIMP, in particular, ranks among the most popular candidates for dark matter, whose detection is crucial for both distinguishing new physics models and understanding the nature of dark matter. The direct detection for dark matter is studying the recoil energy of nuclei caused by the elastic scattering of a WIMP off a nucleon.
The neutralino in the minimal supersymmetric standard model (MSSM) has been extensively studied [16] as one of the favorite dark matter candidates. However, the left-handed sneutrino meets severe troubles because the cross section for elastic scattering off nuclei exceeds the experimental limit by several orders with the exchange of vector boson Z [17].
In this work, we extend the MSSM to the U(1) X SSM, whose local gauge group is [40][41][42]. In comparison with the MSSM, our model U(1) X SSM has more superfields: U(1) X gauge field, righ-handed neutrinos, three SU(2) L singlet Higgs superfieldsη,η,Ŝ and their superpartners. The vacuum expectation value(VEV) ofη produces masses of the right-handed neutrinos. The righ-handed neutrinos and left-handed neutrinos mix together through Y ννlĤu . Therefore, light neutrinos obtain tiny masses through the seesaw mechanism. The lightest sneutrino can be a new dark matter candidate different from the case in MSSM. Moreover, PAMELA [43] claims an excess in the electron/positron flux and no excess in the proton/antiproton flux [44]. Thus, the idea that dark matter carries lepton number is intriguing. The little hierarchy problem in MSSM is relieved in U(1) X SSM by the right-handed neutrinos, sneutrinos and additional We introduce the U(1) X SSM in detail in section II. Supposing the lightest CP-even sneutrino as a dark matter candidate, we study its relic density in section III. Section IV is devoted to research the direct detection for sneutrino elastic scattering off the nuclei. The numerical results for Higgs masses, relic density for dark matter and its direct detection are all presented in section V. Sec. VI is devoted to the discussions and conclusions.
important and they are taken into account to get 125 GeV Higgs mass [45,46].
The superpotential for this model reads: There are two Higgs doublets and three Higgs singlets, whose explicit forms are shown in the follow, v u , v d , v η , vη and v S are the corresponding VEVs of the Higgs superfields H u , H d , η,η and S. Here, we define tan β = v u /v d and tan β η = vη/v η . The definition ofν L andν R is The soft SUSY breaking terms are The particle content and charge assignments for U(1) X SSM are shown in the Table 1.
We use Y Y for representing the U(1) Y charge and Y X for representing the U(1) X charge.
According to the textbook [47], the SM is anomaly free. The details regarding the absence of anomaly within the U(1) X SSM model can be summarized as follows: 1. The anomaly of three SU(2) L gauge bosons vanishes as in the SM and the condition of three SU(3) C gauge bosons is similar.
2. The anomalies containing one SU(3) C boson or one SU(2) L boson are proportional to The anomalies of three U(1) gauge bosons are divided into four types 6. The gravitational anomaly with one U(1) gauge boson is The anomalies that do not relate to U(1) X are very similar as the SM condition and can be proved free easily. The anomalies including U(1) X are also proved free, which are more complicated than those of SM. In the end, this model is anomaly free.
The presence of two Abelian groups U(1) Y and U(1) X in U(1) X SSM has a new effect absent in the MSSM with just one Abelian gauge group U(1) Y : the gauge kinetic mixing.
This effect can also be induced through RGEs, even if it is set to zero at M GU T .
The covariant derivatives of this model have the general form [39,[48][49][50] Here, A ′Y µ and A ′X µ denote the gauge fields of U(1) Y and U(1) X , while Y and X represent the hypercharge and X charge respectively. We can perform a basis transformation, because the two Abelian gauge groups are unbroken. The following formula can be obtained with a correct matrix R [39,49,50] So the U(1) gauge fields are redefined as The interesting thing is that the gauge bosons A X µ , A Y µ and V 3 µ mix together at the tree level, and the mass matrix is shown in the basis ( with v 2 = v 2 u + v 2 d and ξ 2 = v 2 η + v 2 η . To diagonalize the mass matrix in Eq. (9), an unitary matrix including two mixing angles θ W and θ ′ W is used here We deduce sin 2 θ ′ W as The new mixing angle θ ′ W appears in the couplings involving Z and Z ′ . The exact eigenvalues of Eq. (9) are calculated [39,49,50] The Higgs potential is deduced here To simplify the following discussion, we suppose that the parameters ters. The VEVs of the Higgs satisfy the following equations The mass squared matrix for CP-odd Higgs in the basis (σ d , σ u , σ η , ση, σ s ) is diagonalized by Z A . The neutral CP-even Higgs φ d , φ u , φ η , φη and φ S mix together at the tree level and they form 5 × 5 mass squared matrix which is diagonalized by Z H . Their concrete forms are collected in the Appendix. As discussed in the MSSM, the loop corrections to the lightest CP-even Higgs mass are known to be large. Therefore, we include the leading-log radiative corrections from stop and top particles [45,46]. The mass of the lightest Higgs boson can be written as with m 0 h 1 representing the lightest tree-level Higgs boson mass. The concrete form of ∆m 2 α 3 is the strong coupling constant. MT = √ mt 1 mt 2 and mt 1,2 are the stop masses.Ã t = A t − µ cot β and A t is the trilinear Higgs stop coupling.
The neutrino mass matrix is deduced in the base (ν L ,ν R ) and it is diagonalized by the matrix Z ν through the formula The mass matrix for CP-even sneutrino (φ l , φ r ) reads To obtain the masses of sneutrinos, we use Z R to diagonalize M 2 ν R . The mass matrix for CP-odd sneutrino (σ l , σ r ) is also deduced here Using the matrix Z I , we can diagonalize the mass squared matrix of the sneutrino M 2 ν I . In the same way, we deduce the mass matrixes for slepton and neutralino, and show them in the Appendix.
Here, we show some needed couplings in this model. The CP-odd Higgs bosons interact withν I andν R , whose concrete form is We also deduce the vertexes ofν To save space in the text, the remaining vertexes are placed in Appendix.

III. RELIC DENSITY
In this section, we suppose the lightest mass eigenstate (ν R 1 ) of CP-even sneutrino mass squared matrix in Eq. (23) as a dark matter candidate and calculate the relic density. Any WIMP candidate has to satisfy the relic density constraints. Theν R 1 number density nνR 1 is governed by the Boltzmann equation [3,[51][52][53] ν R 1 can both self-annihilate and co-annihilate with another specy φ. When the annihilation rate ofν R 1 becomes roughly equal to the Hubble expansion rate, the species freeze out at the temperature T F , With the supposition M φ > MνR 1 [54] Then it becomes We study its annihilation rate σv SA ( σv CA ) and its relic density Ω D in the thermal history of the universe. To this end, the self-annihilation cross section σ(ν R 1ν R * 1 → anything) and co-annihilation cross section σ(ν R 1 φ → anything) should be calculated. In the center of mass frame, their results can be written as σv rel = a + bv 2 rel , with v rel denoting the relative velocity of the two particles in the initial states. It is a good approximation to calculate the freeze-out temperature (T F ) from the following formula [3,53,55,56] x M P l is the Planck mass 1.22 × 10 19 GeV. m D = mνR 1 denoting the WIMP mass and x F ≡ m D /T F . g * is the number of the relativistic degrees of freedom with mass less than T F . The formula for the density of cold non-baryonic matter can be simplified in the following form [3,9,53,57] and its value should be Ω D h 2 = 0.1186 ± 0.0020 [13].
The dominant processes for the self-annihilation are: 2, 3, h representing the lightest CP-even Higgs. ν i denote three light neutrinos. The studied co-annihilation processes read as:

IV. DIRECT DETECTION
The main scattering processes of CP-even sneutrinos off nucleons areν R + q →ν R + q andν R + q →ν I + q. For the first type processν R + q →ν R + q, the exchanged particles are CP-even Higgs. While, for the second type processν R + q →ν I + q, the exchanged particles are vector bosons Z and Z ′ . The CP-odd Higgs boson contributions are much smaller than the contributions from CP-even Higgs boson and can be neglected safely [58]. After some calculation, we obtain the operatorsν R * νRq q andν R * ∂ µν Rq γ µ q at the quark level.
To get the final results, we should convert the quark level coupling to the effective nucleon coupling. For the operatorν R * νRq q, the useful expressions are shown below [58] a q m qq q → f N m NN N, f N includes the coupling to gluons induced by integrating out heavy quark loops. The T q are collected here [59][60][61], It is easy to convert the operator b qν R * ∂ µν Rq γ µ q to b Nν R * ∂ µν RN γ µ N through the following formulas With the obtained f N , one gets the scattering cross section Here Z p is the number of proton, and A represents the number of atom.

V. NUMERICAL RESULTS
In this section, we study the numerical results. Z ′ boson properties are constrained by manifold low energy experiments [62,63]. The lower limits on the mass of Z ′ set by low energy data are about 1 TeV in some models.  [66]. The authors [67,68] give the upper bound(M Z ′ /g X ≥ 6 TeV) on the ratio between M Z ′ and its gauge coupling at 99% CL. tan β η is also constrained by the LHC experimental data and should be smaller than 1.5 [69]. In order to satisfy the constraints from LHC, we choose the parameters to make M Z ′ > 4.5 TeV, because the quoted number are valid in other models and do not apply directly. The constraints for supersymmetric particles, shown in Ref. [13], are also taken into account.
Considering the above constraints, we use the following parameters Here, we take T ν , T X and M ν as diagonal matrices, for example We list the remaining parameters which will vary in the following numerical analysis: Firstly, we research the lightest CP-even Higgs mass including the loop corrections and discuss the other CP-even Higgs masses. Secondly, the relic density of the lightest CPeven sneutrino is calculated numerically. At last, we study the cross section for the lightest sneutrino scattering off nucleon. and slepton. So, M 2 L33 influences the both type scalars. The allowed results are plotted by the dots in Fig. 3, where they are almost symmetric with respect to T ν33 = 0.
In the plan of M 2 E33 and M 2 ν33 , the allowed results in ±3σ sensitivity of Ω D h 2 are also researched by taking µ = 500 GeV, T ν33 = 1.6 TeV and M 2 L33 = 3 TeV 2 . We show these results by the dots in Fig. 4. The effect of M 2 E33 is small, because it influences the numerical results only by affecting the slepton mixing and masses. M 2 ν33 appears in the mass matrix of sneutrino, which can affect the lightest sneutrino mass and the mixing of sneutrino.
Therefore, M 2 ν33 is a sensitive parameter, and has obvious influence on mνR 1 and Ω D h 2 . The favorite region of M 2 ν33 is from 60000 to 68000 GeV 2 . This region of M 2 ν33 can also keep the lightest CP-even sneutrinoν R 1 as LSP. According to the parameter space under consideration, the lightest CP-even sneutrino mass is about 320 GeV. The other CP-even sneutrinos (ν R 2 . . .ν R 6 ) are all heavier than 1900 GeV. The masses of all CP-odd sneutrinos (ν I 1 . . .ν I 6 ) are larger than 1900 GeV. For the relic density in ±1σ sensitivity, the lightest neutralino is around 850 GeV. That is to sayν R 1 is the LSP, and can be the dark matter candidate.
If the mass of the virtual particle in s-channel is around 2M D , the resonance annihilation will occur. The resonance annihilation strongly affects the annihilation cross-section hence the relic density. In these numerical results, the mass of dark matter is M D ∼ 320 GeV.
The four virtual CP-even Higgs bosons in s-channel are all heavier than 2.5 TeV, and the lightest CP-even Higgs boson is about 125 GeV. It is obvious that 2M D is far from all the CP-even Higgs masses. So the resonance annihilation can not take place. Taking into account the constraint from the relic density, we calculate numerically the cross section of the sneutrino scattering off nucleon in this subsection. Within the considered parameter space, the lightest CP-even sneutrino is around 320 GeV. The experimental limit on direct detection for a dark matter of 320 GeV is about 2.5 × 10 −46 cm 2 for Xenon and about twice as large for PandaX [70,71]. Using the parameters v S = 3 TeV, m 2 ν33 = 250 2 GeV 2 , M 2 L33 = M 2 E33 = 3 TeV 2 that can satisfy the relic density constraint, we research the cross section of the sneutrino scattering off nucleon.

VI. DISCUSSION AND CONCLUSION
The U(1) X SSM is the extension of MSSM, whose local gauge group is SU (  Taking into account the loop corrections, we study the lightest CP-even Higgs mass (SM-like) in the U(1) X SSM. Comparing with the MSSM, there are three additional Higgs superfields(η,η,Ŝ) in the U(1) X SSM, which is also discussed. With the assumption that the lightest CP-even sneutrino can be a cold dark matter candidate, the relic density of dark matter and the cross section of dark matter scattering off nucleon are both studied.
The virtual Higgs contributions to both the relic density and the scattering cross section are dominant. The numerical results imply that the parameters M 2 ν33 , M 2 L33 , T ν33 and µ are all important. The used parameter space is reasonable and satisfy the dark matter constraints from both the relic density and the scattering off nucleon. This work gives constraints to the parameter space of the U(1) X SSM and may be benefit for the future direct detection.

Acknowledgments
We are very grateful to Wei Chao the professor of Beijing Normal University for giving us some useful discussions and Tiago Adorno the professor of Hebei University for English rewriting. This work is supported by National Natural Science Foundation of China In the basis (φ d , φ u , φ η , φη, φ s ), the mass squared matrix of CP-even Higgs reads The explicit forms of the elements m φ d φ d etc in this mass matrix are shown Eq.(A3) is the CP-odd Higgs mass squared matrix, whose elements are The mass matrix for slepton with the basis (ẽ L ,ẽ R ) is diagonalized by Z E through the formula Z E m 2 e Z E, † = m diag 2,ẽ , The mass matrix for neutralino in the basis (λB,W 0 ,H 0 d ,H 0 u , λX,η,η,s) is, Here, we show the needed couplings in this model. The CP-even Higgs couple with CP-even sneutrinos The coupling of two CP-even Higgs and two CP-even sneutrinos reads as The other used vertexes including the couplings of: ×  Some other used couplings are shown as Z R * ia Z R * ja 1 2 g 2 2 (cos θ W cos θ ′ W ) 2 + 1 2 g 2 1 (sin θ W cos θ ′ W ) 2 +g 1 g 2 cos θ W sin θ W (cos θ ′ W ) 2 − g Y X sin θ ′ W cos θ ′ W (g 2 cos θ W + g 1 sin θ W ) (2g 1 sin θ W sin θ ′ W + (2g Y X + g X ) cos θ ′ W )γ µ P R lZ ′µ , L Zuu =ū − i 6 (3g 2 cos θ W cos θ ′ W − g 1 sin θ W cos θ ′ W + g Y X sin θ ′ W )γ µ P L + i 6 [−(4g Y X + 3g X ) sin θ ′ W + 4g 1 sin θ W cos θ ′ W ]γ µ P R uZ µ , L Z ′ uu =ū − i 6 (−3g 2 cos θ W sin θ ′ W + g 1 sin θ W sin θ ′ W + g Y X cos θ ′ W )γ µ P L − i 6 [(4g Y X + 3g X ) cos θ ′ W + 4g 1 sin θ W sin θ ′ W ]γ µ P R uZ ′µ ,