The one loop corrections to the neutrino masses in BLMSSM

The neutrino masses and mixings are studied in the model which is the supersymmetric extension of the standard model with local gauged baryon and lepton numbers(BLMSSM). At tree level the neutrinos can obtain tiny masses through the See-Saw mechanism in the BLMSSM. The one-loop corrections to the neutrino masses and mixings are important, and they are studied in this work with the mass insertion approximation. We study the numerical results and discuss the allowed parameter space of BLMSSM. It can contribute to study the neutrino masses and to explore the new physics beyond the standard model(SM).

For the mixing pattern, there are two large mixing angles and one small mixing angle. To explain the results in Eq.(1), a theory beyond the SM is necessary. Therefore, the neutrino sector is a natural testing ground for the new models beyond the SM.
For the new physics, the supersymmetric extension of the SM is a popular choice. The discrete symmetry known as R-parity is defined as R p = (−1) L+3B+2S with L(B), S denoting the lepton (baryon) number and the spin of the particle [4]. The minimal supersymmetric extension of the SM (MSSM) [5] has been studied for many years by theoretical physicists.
MSSM with R-parity conservation has some short comings, where neither µ problem nor the observed neutrino masses can be explained. R-parity violation can be obtained from L breaking, B breaking, both L and B breaking. In general, to explain the neutrino experiment data, the lepton number should be broken.
In R-parity violating supersymmetric models with generic soft supersymmetry breaking terms [6], neutrinos and neutralinos mix together at tree level. Therefore, one neutrino gets small mass through the see-saw mechanism [7]. The loop diagrams including lepton number violating effects provide masses to the other neutrinos. Taking into account the one loop effects, the authors research the neutrino masses and mixings. In µνSSM three right-handed neutrino superfields are introduced [8,9], and it can solve the µ problem. In this model [10], the neutrino masses and mixings are studied at one loop level, and significative results are obtained.
The BLMSSM is the minimal supersymmetric extension of the SM with local gauged B and L, which is spontaneously broken at TeV scale [11]. This model was first proposed by the authors in Ref. [12]. So BLMSSM is R-parity violating and can explain the asymmetry of matter-antimatter in the universe. In BLMSSM, the authors study the lightest CP-even Higgs mass and the decays h 0 → γγ, h 0 → ZZ(W W ) [13]. The one loop and two loop Barr-Zee type contributions to muon MDM and charged lepton flavor violating processes are also discussed [14]. Considering the CP-violation, we study neutron EDM, lepton EDM and B 0 −B 0 mixing in this model [15].
In the BLMSSM, because of the introduced right-handed neutrino fields, the three light neutrinos obtain tiny masses at tree level through the see-saw mechanism, which is shown in our previous work [16]. Here, using the mass insertion approximation we consider one loop corrections to the neutrino mass mixing matrix. The one loop corrections are important, especially for the light neutrinos.
After this introduction, in Section 2 we briefly introduce the BLMSSM. With mass insertion method, the one loop corrections to neutrino mass matrix are shown in Section 3.
Section 4 is devoted to the numerical analysis. The summary is given out in Section 5.

II. SOME CONTENT OF BLMSSM
The gauge symmetry of BLMSSM is SU ( pared with MSSM, BLMSSM includes many new fields [11]: 1. the exotic quarks (Q 4 ,Û c 4 ,D c 4 ,Q c 5 ,Û 5 ,D 5 ) used to cancel B anomaly, 2. the exotic leptons (L 4 ,Ê c 4 ,N c 4 ,L c 5 ,Ê 5 ,N 5 ) used to cancel L anomaly, 3. the exotic HiggsΦ L ,φ L introduced to break L spontaneously with nonzero vacuum expectation values (VEVs), 4. the exotic HiggsΦ B ,φ B introduced to break B spontaneously with nonzero VEVs, 5. the superfieldsX andX ′ used to make the exotic quarks unstable, 6. the right-handed neutrinos N c R introduced to provide tiny masses of neutrinos through see-saw mechanism.
The lightest mass eigenstate of the mixed mass matrix forX andX ′ could be a dark matter candidate. These new fields are shown particularly in the table1.

III. THE COUPLING
Because in the BLMSSM neutrinos are Majorana particles, we can use the following expressions for the neutrinos. In the base (ψ ν I L , ψ N cI R ), the formulae for mass mixing matrix and mass eigenstates are shown here [15].
However, the six new neutralinos do not mix with the four MSSM neutralinos. λ L (the superpartners of the new lepton boson) and ψ Φ L , ψ ϕ L (the superpartners of the SU(2) L singlets Φ L , ϕ L ) mix and they produce three lepton neutralinos. In the basis (iλ L , ψ Φ L , ψ ϕ L ), the mass mixing matrix of lepton neutralinos is [15] To get the mass eigenstates for lepton neutralinos, we use the rotation matrix Z N L to diagonalize the mass mixing matrix in Eq. (8).
Similarly, three baryon neutralinos are produced from λ B (the superpartners of the new baryon boson) and ψ Φ B , ψ ϕ B (the superpartners of the SU(2) L singlets Φ B , ϕ B ). We show the mass mixing matrix of baryon neutralinos here in the basis To obtain three baryon neutrino masses, we use the rotation matrix Z N B to diagonalize the mass mixing matrix in Eq. (9).
From the supperpotential W L and the interactions of gauge and matter multiplets , we obtain the couplings with light neutrinos at tree level.
In the same way, the couplings related to heavy neutrinos are also obtained

IV. THE ONE LOOP CORRECTIONS TO NEUTRINO MASS MATRIX
The neutrino Yukawa couplings (Y ν ) IJ , (I, J = 1, 2, 3) are much smaller than the other couplings. For Eqs. (10) can be neglected safely, because they are suppressed by Y ν compared with the other terms. The Y ν in the neutrino mass mixing matrix at tree level is not neglected. One can find that Z Nν is the function of Y ν from Eq. (7). That is to say, Z Nν is relevant to the chiral symmetry breaking terms.
Using the mass insertion approximation [17], we deduce the neutrino mass corrections from the virtual slepton-chargino at one loop level The one loop function I 0 with Λ representing the energy scale of the new physics and x i = m 2 i Λ 2 for i = 1, 2, 3. In Eq. (12), it seems that the results have nothing to do with the chiral symmetry breaking terms. In fact, Eq. (12) includes Z Nν which is the function of nonzero Y ν . Therefore, Eq. (12) includes the chiral symmetry breaking terms and gives corrections to the neutrino mass mixing matrix.
In the same way, the neutrino mass corrections from the virtual sneutrino-lepton neutralino and sneutrino-neutralino are obtained The virtual Higgs-charged lepton and exotic Higgs-neutrino can also give the contributions The definition of I 0 In the flavor basis at tree level the neutrino mass mixing matrix is With the rotation matrix Z Nν , the masses of neutrinos are gotten by the formula We use the matrix Z T Nν in the leading order of ς, which is defined as . All the elements in ς are very small (ς IJ ≪ 1), because they are suppressed by the tiny neutrino Yukawa Y ν . It is a good approximation to adopt Z T Nν in the following form [10] Z We use the matrices S and R defined in Eq. (18) In this condition, M seesaw ν is expressed as The one loop corrections are calculated in the mass basis at tree level, which is Here, we obtain the sum of one loop corrections from Eqs. (12,14,15) For neutrino mass mixing matrix, to get the sum of tree level results and one loop level corrections, we rotate ∆M N into the flavor basis (ψ ν I L , ψ N cI R ). Therefore the sum can be expressed as Obviously, the matrix M sum N in Eq.(22) including the one loop corrections also possesses a seesaw structure. Similar as Eq. (20), at one loop level we obtain the corrected effective light neutrino mass matrix in the following form [10] M ef f Using the " top-down " method [18], from the one loop corrected effective light neutrino mass matrix M ef f ν we get the Hermitian matrix One can diagonalize the 3 × 3 matrix H to gain three eigenvalues The concrete forms of the parameters in Eq. (25) For the neutrino mass spectrum, there are two possibilities in the 3-neutrino mixing case.
The neutrino mass spectrum with normal ordering (NO) is We also write down the neutrino mass spectrum with inverted ordering (IO) From the mass squared matrix H, one gets the normalized eigenvectors The concrete forms of X I , Y I , Z I for I = 1, 2, 3 are shown here The mixing angles among three tiny neutrinos can be defined as follows

V. NUMERICAL RESULTS
In this section, we discuss the numerical results for the neutrinos including three mixing angles and two mass squared differences. Using BLMSSM, we have studied several processes in our precious works, such as the lightest neutral CP-even Higgs mass and the charged lepton flavor violating processes. Because the masses of light neutrinos are very tiny at 10 −1 TeV order, the used parameters should be precise enough. In this work, the tiny neutrino Yukawa Y ν can give contributions to light neutrino masses at tree level through the see-saw mechanism. Therefore, Y ν are important parameters and should be considered earnestly.
We show the used parameters A. NO spectrum At first we study the NO spectrum with the supposition for neutrino Yukawa couplings and issue the numerical results for light neutrino masses and mixing angles |∆m 2 A | = 2.4707 × 10 −3 eV 2 , ∆m 2 ⊙ = 7.5344 × 10 −5 eV 2 , sin 2 θ 12 = 0.3221, sin 2 θ 13 = 0.0247, sin 2 θ 23 = 0.5240, The diagrams are plotted near this point that satisfies the experiment constraints for light neutrinos. are very small. In the right diagram of Fig.3, the dotted line(∆m 2 ⊙ ) is small in the range 6.4×10 −8 < Y nu13 < 6.9×10 −8 . The values of ∆m 2 A ×10 3 /eV 2 represented by the solid line vary from 2.5 to 8.0. Taking into account the neutrino experiment bounds, the appropriate Y nu13 value is around 6.4 × 10 −8 .

B. IO spectrum
With the neutrino mass spectrum being IO, the neutrino mass squared differences and mixing angles are also studied numerically here. Using the following parameters Here, we discuss the effects from the non-diagonal elements of (m 2Ñ ), and suppose  Fig.4. ∆m 2 ⊙ increases a little faster than −∆m 2 A with the increasing Nuf . In this parameter space, Nuf should be no more than 200GeV as shown in the Fig.4.
Here, we also discuss how the diagonal element (Y ν ) 33 = Y nu33 influences the theoretical predictions on the neutrino mixing angles and mass squared differences in the Fig.5. From the solid line(sin 2 θ 12 ), dotted line(10 × sin 2 θ 13 ) and dashed line(sin 2 θ 23 ) in the left diagram, we should take Y nu33 around 1.695×10 −6 . In the right diagram, both the solid line(−∆m 2 A ) and the dotted line(∆m 2 ⊙ ) reach small values near the point Y nu33 = 1.695 × 10 −6 . The non-diagonal element (Y ν ) 12 = Y nu12 can obviously influence the numerical results for the neutrinos. The mixing angles(sin 2 θ 12 , 10 × sin 2 θ 13 , sin 2 θ 23 ) versus Y nu12 are plotted by the solid, dotted and dashed line in the left diagram of Fig.6. We show the neutrino mass squared differences −∆m 2 A (the solid line) and ∆m 2 ⊙ (the dashed line) in the right diagram of Fig.6. From the both diagrams and neutrino experiment bounds, the appropriate value of Y nu12 should be around 9.6 × 10 −8 . As it is well known that the light neutrino masses are very tiny and there are five experiment constraints(three mixing angles and two mass squared differences), the obtained suitable parameter space is narrow.

VI. SUMMARY
The neutrino experiment data from both solar and atmospheric neutrino experiments show that neutrinos have tiny masses and three mixing angles including two large mixing angles and one small mixing angle. The SM can not solve the neutrino experiment data, and physicists consider SM should be the low energy effective theory of a large model. So, the SM should be extended. One of the supersymmetric extensions of the SM is BLMSSM which has local gauged B and L symmetries. In this model, we have studied some processes in our previous works [13][14][15]19]. In this work, with the mass insertion approximation the one loop corrections to the neutrino mixing matrix are researched.
In the BLMSSM, the tree level neutrino mass mixing matrix is obtained in our previous work [16]. Using the top-down method, we give the formulae for the neutrino masses and mixing angles.
For neutrino mass spectrum, both NO and IO conditions are discussed numerically. In our used parameter space, the obtained numerical results for the neutrino three mixing angles and two mass squared differences can account for the corresponding experiment data. Our results imply that the light neutrino masses are at the order of 10 −1 eV.
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