Neutrino masses in the minimal gauged $(B-L)$ supersymmetry

We present the radiative corrections to neutrino masses in a minimal supersymmetric extension of the standard model with local $U(1)_{B-L}$ symmetry. At tree level, three tiny active neutrinos and two nearly massless sterile neutrinos can be obtained through the seesaw mechanism. Considering the one-loop corrections to the neutrino masses, the numerical results indicate that two sterile neutrinos obtain ${\rm KeV}$ masses and the small active-sterile neutrino mixing angles. The lighter sterile neutrino is a very interesting dark matter candidate in cosmology. Meanwhile the active neutrinos mixing angles and mass squared differences agree with present experimental data.


I. INTRODUCTION
The discovery of Higgs boson on the Large Hadron Collider (LHC) [1,2] indicates that the Higgs mechanism to break electroweak symmetry has an experimental cornerstone now.
The experiments of atmospheric and solar neutrino oscillation give the neutrino data at least three types of neutrinos which have sub-eV masses, but the standard model (SM) of particle physics cannot account for the origin of these tiny masses naturally.
The supersymmetric extension of the SM is a rather popular choice. The discrete symmetry R-parity is defined through R = (−1) 3(B−L)+2S , where B, L, and S are baryon number, lepton number, and the spin of the particle, respectively [6]. In the minimal supersymmetry extension of SM (MSSM) with local U(1) B−L symmetry, the nonzero vacuum expectation values (VEVs) of the right-handed sneutrinos evoke the (B − L) symmetry and R-parity spontaneously broken simultaneously [7][8][9][10][11][12]. At tree level, the MSSM with local U(1) B−L symmetry can generate three active neutrinos to interpret the neutrino oscillation through the seesaw mechanism; meanwhile, the model predicts that there are two sterile neutrinos.
Nevertheless, two sterile neutrinos have far below eV masses at tree level [10][11][12][13]. Sterile neutrinos with KeV scale masses are a well-motivated dark matter candidate for two reasons.
First, fermionic dark matter cannot have an arbitrarily small mass, since in dense regions it cannot be packed within an infinitely small volume for the Pauli principle. Second, sterile neutrinos have a small mixing with the active neutrinos which would enable a dark matter particle to decay into an active neutrino and a photon [14]. The Tremaine-Gunn bound indicates that a sterile neutrino mass must be greater than about 0.4 KeV [14,15]. A strong bound on a sterile neutrino mass and mixing angle comes from the nondetection results of the monoenergetic X-rays by the decay of sterile neutrino [14]. Recently two groups reported evidence for a 3.55 KeV emission line which could be from the decay of a 7.1 KeV sterile neutrino with sin 2 (2θ) ∼ 10 −10 or 10 −11 [16,17], which is just below the previous X-ray bound. This observation is being fiercely discussed [18][19][20][21].
In this work, we investigate the origin of neutrino masses in the minimal gauged (B − L) supersymmetry. There are five light neutrinos (three light active and two almost massless sterile neutrinos) at tree level, which agrees with the results in Refs. [10][11][12][13]. The one-loop corrections to the light neutrinos are important to account for relevant experimental data [22,23]. In this article we present the one-loop radiative corrections to neutrino masses and relevant mixing matrix in the MSSM with local U(1) B−L symmetry. The numerical results indicate that there is parameter space to give two sterile neutrinos KeV masses and the small active-sterile neutrino mixing angles. The lighter sterile neutrino is a very interesting dark matter candidate in cosmology. Meanwhile, the mass squared differences and mixing angles of active neutrinos coincide with the experimental data from the solar and atmospheric oscillations [5].
Our presentation is organized as follows. In Sec. II, we briefly summarize the main ingredients of the MSSM with local U(1) B−L symmetry and then present the mass matrix for neutralinos and neutrinos at tree level in Sec. III. In Sec. IV, we analyze one-loop radiative corrections to the mass matrix. The numerical analysis for two possibilities on the neutrino mass spectrum (NO and IO) is given in Sec. V, and Sec. VI gives a summary.

II. THE SUPERSYMMETRIC MODEL WITH LOCAL U (1) B−L SYMMETRY
When U(1) B−L is a local gauge symmetry, one can enlarge the local gauge group of the In the model proposed in Refs. [10][11][12], three exotic superfields for right-handed neutrinos areN c i ∼ (1, 1, 0, 1). Meanwhile, quantum numbers of the matter chiral superfields for quarks and leptons are given bŷ with I = 1, 2, 3 denoting the index of generation. In addition, the quantum numbers of two Higgs doublets are assigned aŝ The superpotential of the MSSM with local U(1) B−L symmetry is written as [13,[24][25][26] with W M SSM denoting the superpotential of the MSSM. Correspondingly, the soft breaking terms of the MSSM with local U(1) B−L symmetry are generally given as In this formula L M SSM sof t is the soft breaking terms of the MSSM, and λ BL denotes the gaugino of U(1) B−L . After the SU(2) L doublets H u , H d ,L I , and SU(2) L singlets,Ñ c I obtain the nonzero VEVs, the R-parity is broken spontaneously, and the local gauge symmetry SU(2) L ⊗ U(1) Y ⊗ U(1) B−L breaks down to the electromagnetic symmetry U(1) e . Then, the tree level masses of neutral and charged gauge bosons are, respectively, formulated as with abbreviations υ 2 Nα . In addition, g 2 , g 1 and g BL denote the gauge couplings of SU(2) L , U(1) Y , and U(1) B−L , respectively.
To satisfy present electroweak precision observations, we assume the mass of a neutral [12]. After the electroweak symmetry is broken spontaneously, the couplings between the left-handed neutrinos and neutralinos are (1/ Because of the TeV scale seesaw suppression, the Yukawa couplings (Y N ) IJ and nonzero VEVs v L I of left-handed sneutrino are sufficiently small for tiny active neutrino masses, (Y N ) IJ ≤ 10 −6 and v L I ≤ 10 −3 GeV [10,12]. Ignoring the terms which are negligible and assuming that the 3 × 3 matrices m 2 L , m 2Ñ c are real, we simplify the minimization conditions as [13] υ u {µ 2 + m 2 Hu + Note that the first two minimization conditions for H 0 u and H 0 d are not greatly altered from those in the MSSM, the third condition originates from the linear terms of υ L I , and the last equation implies that the vector (υ N 1 , υ N 2 , υ N 3 ) is an eigenvector of 3 × 3 mass squared matrix m 2Ñ c with eigenvalue −m 2 Z BL /2. Considering the last minimization condition in Eq. (8), we formulate the symmetric 3 × 3 matrix as with ξ 2 . This is the mixing matrix of the right-handed sneutrinos, the reasons for choosing it are shown in Appendix A.
Hu ), we can obtain the neutral fermion mass terms in the Lagrangian where the mass matrix for neutralinos and neutrinos M N is given by where M N denotes the 4 × 4 mass matrix for neutralinos in the MSSM. The concrete N , and A N are with the row indices of matrix I, J = 1, 2, 3 and the column indices of matrix I ′ , J ′ = 1, 2, 3, respectively. The eigenvalues of the 4 × 4 mass matrix M N are given as where ∆ BL = m 2 BL + g 2 BL υ 2 N , and then we can obtain m N 3,4 about TeV region for N has four eigenvalues which are zero order approximations of the U(1) B−L gaugino and three right-handed neutrinos masses.
However, there are only two nonzero eigenvalues, and the other two eigenvalues are zero.
Defining the 4 × 4 orthogonal matrix one obtains where η ± BL = 1 ± m BL ∆ BL , and U T N M (0) Using Eqs. (11) and (14), we formulate the submatrices in Eq. (15), respectively, as where the abbreviations are with the indices i = 1, 2, 3. The abbreviations are suppressed by the tiny (Y N ) ij and υ L i , and they are very small.
Defining the 11 × 11 approximated orthogonal transformation matrix Z N [13] via the seesaw mechanism, we finally write the effective mass matrix for five light neutrinos (three active and two sterile) as In order to accommodate naturally the experimental data on neutrino oscillation and Z invisible decay width in this framework, one can find that only one possibility M LL is reasonable [10]. In fact, from Eq. (B2) this point implies To guarantee the decoupling of two tiny sterile neutrinos from the active neutrinos, we choose the Yukawa coupling for right-handed neutrinos as Only including the tree level contributions to the light neutrino mass matrix in Eq. (19), we diagonalize the effective neutrino mass matrix m ef f and then obtain three light lefthanded neutrinos and two nearly massless sterile neutrinos [10][11][12][13]. Recently, it has been shown in Refs. [14,16,17] that sterile neutrinos with KeV scale masses are interesting dark matter candidates in the Universe. The one-loop radiative corrections are important, especially for the light neutrinos [27,28]. We consider the one-loop radiative corrections to the mass matrix of the neutrinos in Eq. (15) and expect that two sterile neutrinos acquire their physical masses at KeV level in the following.

IV. THE RADIATIVE CORRECTIONS ON MASSES OF NEUTRINOS
A. The radiative corrections on masses of sterile neutrinos In this model, there is a large mixing between three right-handed neutrinos and a (B −L) gaugino. At leading order, this mixing induces two heavy Majorana fermions with masses about TeV scale and two light sterile neutrinos which acquire their tiny masses by a seesaw mechanism. Here, we consider one-loop radiative corrections to the masses of two light sterile neutrinos. From interactions of the gauge and matter multiplets ig , we can obtain the couplings involving sterile neutrinos. In the Majorana case, the most general form for N α → N β transition reads The invariance of CPT transformation requires The radiative corrections from real and image components of scalar right-handed neutri- where UÑ O , UÑ E can be found in Appendix B, and U N is given in Eq. (14). In a similar way, the radiative corrections from a (B − L) gauge boson are written as with ε = 2 − D 2 , where Λ denotes the energy scale of new physics, and Λ = 2 TeV in our numerical analysis.
The generic expression for the right-handed Majorana neutrinos and the (B − L) gaugino self-energy must be symmetric in its indices α, β, and the result of one-loop corrections to the mass matrix in the modified dimensional reduction (DR) scheme [22] is written as whereΣ denotes the renormalized self-energy in the DR scheme.
Using the concrete expression of U N in Eq. (14) where two-point scalar functions B 0 and B 1 are renormalized in the DR scheme, denoted byB 0 andB 1 , respectively. The corrections to the masses of sterile neutrino have nothing to do with small (Y N ) ij and υ L i . As α = 1, 2, β = 3, 4 and α, β = 3, 4, the results of one-loop corrections to the mass matrix can be found in Appendix C.
with α EW = e 2 /4π, α BL = g 2 BL /4π and δ(m 2 ν ) ij LL = ( Here, U χ 0 denotes the orthogonal matrix of a neutralino mass matrix, and U ± denotes the orthogonal matrix of a chargino mass matrix in the MSSM. We also adopt the abbreviations with x i = m 2 i /Λ 2 . Similarly, we derive the corrections from virtual sneutrino-neutralino to the mixing matrix N at a one-loop level as [29,31]

∆A
(2) with s 2 α = sin 2 α, c 2 α = cos 2 α, and (N (2) Additionally, the radiative corrections to the mixing between left-and right-handed neutrinos are proportional to Y N υ L i or A N υ L i and can be ignored safely.
Considering those one-loop corrections, the mass matrix in Eq. (15) is rewritten as Using the seesaw mechanism, the effective mass matrix for five light neutrinos (three active and two sterile) at the one-loop level is We can obtain five eigenvalues by diagonalizing the effective mass matrix m ′ ef f . The corrections to the sterile neutrinos are much larger than the corrections to the active neutrinos which are suppressed by the tiny parameters, so the three light eigenvalues are active neutrinos, and other two relatively heavy eigenvalues are sterile neutrinos. Under the assumption , the corrected effective mass matrix of three left-handed neutrinos is Using the "top-down" method [32,33] in the effective mass matrix m ′ ef f ν L , we diagonalize the Hermitian matrix The eigenvalues of the 3 × 3 effective mass squared matrix H are given as To formulate the expressions of a concise form, one can define the notations For the three active neutrino mixing, there are two possible solutions on the neutrino mass spectrum. The normal ordering (NO) spectrum is and the neutrino mass spectrum with the inverted ordering (IO) is From the mass squared matrix H and three eigenvalues one can get the orthogonal matrix U ν of H [32,33]. Correspondingly, the mixing angles among three active neutrinos are determined by It is important to calculate the active-sterile neutrino mixing angles which are strongly constrained by X-ray observations [14]. There are several mixing angles θ σI , where I is the sterile neutrino flavor and σ is the active neutrino flavor. We define θ 2 σI = (M ′ LR ν ) 2 σI /m 2 rI , where m 2 rI are the sterile neutrino masses. There are two sterile neutrinos; then, I = 1, 2. We define the active-sterile neutrino mixing angle as [34]

V. NUMERICAL RESULTS
The neutrino oscillation experimental data [5]  In this section, we analyze the numerical results for the mixing angles and mass squared differences of active neutrinos varying with υ N 2 , m BL , and tan β, assuming neutrino mass spectrum with normal ordering (NO) and inverted ordering (IO). Meanwhile, we discuss the numerical results of two sterile neutrino masses and the active-sterile neutrino mixing angles varying with these parameters.

A. NO spectrum
In order to fit the experimental data on active neutrino mass squared differences and mixing angles in this scenario, we choose the VEVs of left-handed sneutrinos and the Yukawa couplings of right-handed neutrinos, respectively, as Correspondingly the theoretical predictions on active neutrino mixing angles, mass squared differences, the sum of the active neutrino masses, two sterile neutrino masses m r1 , m r2 , and the active-sterile neutrino mixing angles θ 2 1 , θ 2 2 are derived as when υ N 1 = 3 GeV, υ N 2 = 7.7 GeV, m BL = 1.08 TeV, and tan β = 20.
Assuming neutrino mass spectrum with NO and taking υ N 1 = 3 GeV, m BL = 1.08 TeV, and tan β = 20, we depict the active neutrino mixing angles varying with the VEV υ N 2 of right-handed sneutrinos in Fig. 1(a), where the solid line denotes sin 2 θ 23 versus υ N 2 , the dashed line denotes sin 2 θ 12 versus υ N 2 , and the dotted line denotes sin 2 θ 13 versus υ N 2 .
With the increasing of υ N 2 , theoretical predictions of these mixing angles vary gently. In this region of υ N 2 , the three mixing angles satisfy the experiment bounds simultaneously [5]. Using the same choice on parameter space, we draw the mass squared differences of active neutrinos varying with υ N 2 in Fig. 1(b), where the solid line denotes ∆m 2 A versus υ N 2 , and the dashed line denotes ∆m 2 ⊙ versus υ N 2 . With the increasing of υ N 2 , ∆m 2 A and ∆m 2 ⊙ decreases slowly. The effective mass matrix for active neutrinos depends on υ N 2 through the term ζ i ζ j /Λ ζ ≃ (2μ 4 υ 2 u m BL )/(Λm4υ 2 N ), and υ N 2 has relatively small influence on υ N (υ N 2 ≪ υ N ). Additionally, we study the two sterile neutrino masses m r1 , m r2 varying with υ N 2 in Fig.   1(c), where the solid line denotes m r1 versus υ N 2 , and the dashed line denotes m r2 versus υ N 2 . It shows that two sterile neutrinos obtain KeV scale masses. When υ N 2 ≤ 10 GeV, m r1 increases steeply with the increasing of υ N 2 , and m r2 changes mildly with υ N 2 . However, when υ N 2 ≥ 10 GeV, the dependence of m r1 on υ N 2 is not obvious, and m r2 increases quickly with the increasing of υ N 2 . This is because the two sterile neutrinos obtain masses from the one-loop corrections. The corrections depend on U N , m Nα , and m 2 Z BL , which are all related to υ N 2 . Under the same choice on parameter space, the numerical result of the active-heavier sterile neutrino mixing angle sin 2 θ 2 changes gently about 10 −10 or 10 −11 . We only study the active-lighter sterile neutrino mixing angle sin 2 θ 1 varying with υ N 2 in Fig. 1(d). Considering the restrictions of X-ray line searches on the mixing angle, the applicable range of υ N 2 is about from 5 to 12 GeV [14]. When υ N 2 = 7.7 GeV, the sterile neutrino mass m r1 is about 7.13 KeV with the mixing angle sin 2 θ 1 ∼ 10 −11 which can explain the observed X-ray line at 3.5 KeV [16,17]. So, the lighter sterile neutrino can be a dark matter candidate. In this model, the U(1) B−L gaugino mass m BL also affects the final numerical results of the neutrino sector. Taking tan β = 20 and υ N 1 = 3 GeV, υ N 2 = 7.7 GeV; we plot the active neutrino mixing angles varying with m BL in Fig. 2(a), where the solid line denotes sin 2 θ 23 versus υ N 2 , the dashed line denotes sin 2 θ 12 versus υ N 2 , and the dotted line denotes sin 2 θ 13 versus υ N 2 . With the increasing of m BL , the theoretical prediction on the mixing angle sin 2 θ 12 depends on m BL mildly, and the mixing angles sin 2 θ 23 , sin 2 θ 13 increase steeply. Using the same choice on parameter space, we plot the mass squared differences of active neutrinos varying with m BL in Fig. 2(b), where the solid line denotes ∆m 2 A versus υ N 2 , and the dashed line denotes ∆m 2 ⊙ versus υ N 2 . It shows that ∆m 2 A raises steeply with the increasing of m BL , and ∆m 2 ⊙ diminishes quickly with the increasing of m BL . The effective mass matrix for active neutrinos depends on m BL through the term N ); therefore, the numerical evaluations on sin 2 θ 23 , sin 2 θ 13 , ∆m 2 A and ∆m 2 ⊙ depend on m BL strongly. From those numerical results on these parameter spaces, we find that the updated experiment data require m BL ∼ 1.08 TeV. Additionally we study the masses of two sterile neutrinos varying with m BL in Fig. 2(c). The numerical results indicate that both m r1 and m r2 depend on m BL mildly. The active-sterile neutrino mixing angle sin 2 θ 2 changes gently about 10 −10 with the increasing of m BL . We study the active-sterile neutrino mixing angle sin 2 θ 1 varying with υ N 2 in Fig. 2(d). With increasing of m BL , the mixing angle sin 2 θ 1 increases quickly.
Taking m BL = 1.08 TeV, υ N 1 = 3 GeV, and υ N 2 = 7.7 GeV, we draw the active neutrino mixing angles varying with tan β in Fig. 3(a), where the solid line denotes ∆m 2 A versus υ N 2 , and the dashed line denotes ∆m 2 ⊙ versus υ N 2 . With the increasing of tan β, theoretical predictions on the mixing angle sin 2 θ 12 varies gently, and the mixing angles sin 2 θ 23 , and sin 2 θ 13 decrease slowly. Using the same choice on parameter space, we draw the mass squared differences of active neutrinos varying with tan β in Fig. 3(b), where the solid line denotes ∆m 2 A versus υ N 2 , and the dashed line denotes ∆m 2 ⊙ versus υ N 2 . It shows that ∆m 2 ⊙ varies mildly and ∆m 2 A decreases steeply with the increasing of tan β. Additionally, we study the masses of two sterile neutrinos versus tan β in Fig. 3(c); the numerical result implies that two sterile neutrino masses depend on tan β gently. This is because two sterile neutrinos obtain masses mainly from the radiative corrections, and from Eq. (29) the corrections on masses of two sterile neutrinos are almost not dependent on tan β. For the same reason, the active-sterile neutrino mixing angle has barely changed with tan β. The numerical result of sin 2 θ 2 is about 10 −10 . The numerical result of sin 2 θ 1 is about 10 −11 in Fig. 3(d).
Assuming neutrino mass spectrum with normal ordering, two sterile neutrinos obtain KeV scale masses. Both KeV sterile neutrinos were produced in the early Universe via oscillations. The lighter sterile neutrino forms dark matter, but the oscillation mechanism cannot produce enough of these neutrinos to act as all dark matter for given m r1 and sin 2 θ 1 [37,38]. The heavier sterile neutrino can decay into the lighter one to enrich the production of the sterile neutrino DM.

B. IO spectrum
With the active neutrino mass spectrum being IO spectrum, we choose the VEVs of left-handed sneutrinos and the Yukawa couplings of sterile neutrinos, respectively, as Correspondingly the theoretical predictions on active neutrino mixing angles, mass squared differences, the sum of the active neutrino masses, two sterile neutrino masses m r1 , m r2 , and the active-sterile neutrino mixing angles θ 2 1 , θ 2 2 are derived as when υ N 1 = 3 GeV, υ N 2 = 7.7 GeV, m BL = 1.08 TeV, and tan β = 20.
When the neutrino mass spectrum is IO, the manners of parameters υ N 2 , m BL , and tan β affecting the numerical results on the neutrino sector may differ from that of the neutrino mass spectrum with NO. Assuming neutrino mass spectrum with IO and taking υ N 1 = 3 GeV, m BL = 1.08 TeV, and tan β = 20, we depict the active neutrino mixing angles varying with υ N 2 in Fig. 4(a), where the solid line denotes sin 2 θ 23 versus υ N 2 , the dashed line denotes sin 2 θ 12 versus υ N 2 , and the dotted line denotes sin 2 θ 13 versus υ N 2 .
Obviously, theoretical predictions on those mixing angles vary slowly with the increasing of υ N 2 . Adopting the same choice on parameter space, we draw the mass squared differences of active neutrinos varying with υ N 2 in Fig. 4(b), where the solid line denotes ∆m 2 A versus υ N 2 , and the dashed line denotes ∆m 2 ⊙ versus υ N 2 . It shows that ∆m 2 A decreases gently with the increasing of υ N 2 , but ∆m 2 ⊙ decreases steeply. Taking into account the neutrino experiment bounds, the appropriate region of υ N 2 is υ N 2 ≤ 60 GeV. In addition, we study the masses of two sterile neutrinos varying with υ N 2 in Fig. 4(c), where the solid line denotes m r1 versus only study the active-lighter sterile neutrino mixing angle sin 2 θ 1 versus υ N 2 in Fig. 4(d). It shows that the mixing angle sin 2 θ 1 depends on υ N 2 strongly. Considering the restrictions of X-ray line searches on the mixing angle, the applicable range of υ N 2 is about from 5 to 10 GeV [14]. When υ N 2 = 7.7 GeV, the sterile neutrino mass m r1 is about 7.13 KeV with the mixing angle sin 2 θ 1 ∼ 10 −11 which can explain the observed X-ray line at 3.5 KeV [16,17]. Therefore, the lighter sterile neutrino can be a dark matter candidate. Taking tan β = 20, υ N 1 = 3 GeV, and υ N 2 = 7.7 GeV, we plot the active neutrino mixing angles varying with m BL in Fig. 5(a), where the solid line denotes sin 2 θ 23 versus υ N 2 , the dashed line denotes sin 2 θ 12 versus υ N 2 , and the dotted line denotes sin 2 θ 13 versus υ N 2 . With the increasing of m BL , the theoretical predictions on the mixing angles of active neutrinos sin 2 θ 23 and sin 2 θ 13 depend on m BL mildly, and the mixing angle sin 2 θ 12 decreases steeply.
Adopting the same choice on parameter space, we plot the mass squared differences of active neutrinos varying with m BL in Fig. 5(b), where the solid line denotes ∆m 2 A versus m BL and the dashed line denotes ∆m 2 ⊙ versus m BL . It shows that the mass squared differences of active neutrinos ∆m 2 A and ∆m 2 ⊙ increase steeply with the increasing of m BL . From those numerical results, we find that the updated experimental data require m BL ∼ 1.08 TeV.
In addition, we study the masses of two sterile neutrinos varying with υ N 2 in Fig. 5(c  Taking m BL = 1.08 TeV, υ N 1 = 3 GeV, and υ N 2 = 7.7 GeV, we draw the neutrino mixing angles varying with tan β in Fig. 6(a). With the increasing of tan β, theoretical predictions on those mixing angles vary gently. Adopting the same choice on parameter space, we draw the mass squared differences of active neutrinos varying with tan β in Fig. 6(b). It shows that ∆m 2 A (solid line) changes gently with the increasing of tan β; however, ∆m 2 ⊙ (dashed line) decreases rapidly with the increasing of tan β. In addition, we study the masses of two sterile neutrinos versus tan β in Fig. 6(c). It shows that the masses of two sterile neutrinos depend on tan β gently. The active-sterile neutrino mixing angles have barely changed with tan β. The numerical result of sin 2 θ 2 is about 10 −10 . The numerical result of sin 2 θ 1 is about 10 −11 in Fig. 6(d).
Assuming the neutrino mass spectrum with inverted ordering, two sterile neutrinos obtain KeV scale masses. The lighter sterile neutrino forms dark matter, and the heavier sterile neutrino can decay into the lighter one to enrich the production of the sterile neutrino DM.

VI. SUMMARY
We investigate the origin of neutrino masses in the MSSM with local U(1) B−L symmetry.
In this model sneutrinos all obtain nonzero VEVs. We constrain the relevant parameter space by the neutrino oscillation experimental data and the mass of the lightest CP-even Higgs.
At tree level, three left-handed neutrinos and two sterile neutrinos obtain masses through the seesaw mechanism, but the masses of two sterile neutrinos are very tiny. The oneloop radiation corrections to the mass matrix of neutralino-neutrino are also studied. Both NO spectrum and IO spectrum are studied. The active neutrino mass squared differences and mixing angles can account for the experimental data on neutrino oscillations. Because the one-loop radiative corrections to the left-handed neutrinos are suppressed by the tiny Yukawa couplings and the small nonzero VEVs of left-handed sneutrinos, the corrections to the active neutrinos are very small. The active neutrinos obtain mass mainly from tree level. When one-loop corrections are included, the numerical results show that there is parameter space to give two sterile neutrinos with KeV masses and the small active-sterile neutrino mixing angle. When υ N 2 = 7.7 GeV, m BL = 1.08 TeV, and tan β = 20, the mass of the heavier sterile neutrino m r2 is about 12.88 KeV, the mixing angle sin 2 θ 2 is about 10 −10 , the mass of lighter sterile neutrino m r1 is about 7.1 KeV, and the mixing angle sin 2 θ 1 is about 10 −11 . The lighter sterile neutrino can account for the observed X-ray line at 3.5 KeV [16,17]. Therefore, the lighter sterile neutrino can be a dark matter candidate.
However the oscillation mechanism does not produce enough of these neutrinos to act as all dark matter [36,37]. Nonresonant production contributes to the dark matter abundance Ω s h 2 ≈ 0.3( sin 2 2θ 10 −10 )( mr 100kev ) 2 [37]. Only 1% of dark matter is produced for the lightest sterile neutrino. In this model, the heavier sterile neutrino can decay into the lighter one to enrich the production of the sterile neutrino DM. However, this still does not produce enough dark matter and it may require other mechanisms (Shi-Fuller mechanism [38]) in the early Universe [14].

Acknowledgments
The work has been supported by the National Natural Science Foundation of There are three parameters in matrix m 2Ñ c . We make an approximation of this matrix to reduce the number of free parameters. A possible solution is that υ N 1 and υ N 2 are small and υ N 3 is large for large υ N (υ 2 N = 3 α=1 υ 2 Nα ). In this case, we can obtain (m 2Ñ c ) 33 ≃ −m 2 Z BL /2 from the last minimization condition in Eq. (8). Compared to (m 2Ñ c ) 33  should be small considering small υ N 1,2 and large υ N 3 . The number of matrix parameters is reduced to two. Then, we can have Eq.(9).
Using the minimization conditions, we derive the 3 × 3 mass squared matrix for neutral CP-odd scalars P 0 N I at tree level with I, J = 1, 2, 3 denoting the index of generation. Correspondingly the orthogonal 3 × 3 matrix from interaction eigenstates to mass eigenstates is written as UÑ O , and three masses of the CP-odd scalars are and the concrete expressions for ω A,B are Additionally the 3 × 3 mass squared matrix for neutral CP-even scalarsν R I is (UÑ E ) ia (UÑ E ) jaB0 (m 2 Nα , m 2 N δ , m 2 H Na ) As α, β = 3, 4, the result of one-loop corrections to the mass matrix is The corrections from real and image components of right-handed sneutrinos only appear as β = 3, 4, so their contributions to the sterile neutrino masses are relatively small.