Feynman rules for neutrinos and new neutralinos in the BLMSSM

In a supersymmetric extension of the Standard Model where baryon and lepton numbers are local gauge symmetries (BLMSSM), we deduce the Feynman rules for neutrinos and new neutralinos. We briefly introduce the mass matrices for the particles and the related couplings in this work, which are very useful to research the neutrinos and new neutralinos.


Introduction
In quantum field theory, the Standard Model (SM) is a theory concerning the electromagnetic weak and strong interactions. Though the lightest CP-even Higgs (m h 0 126 GeV) was detected by the LHC, the SM is unable to explain some phenomena and falls short of being a complete theory of fundamental interactions. In the neutrino sector, the observations of solar and atmospheric neutrino oscillations [1][2][3][4] are not incorporated in the SM, which provides clear evidence for physics beyond the SM. Furthermore, the authors think that a wellmotivated dark matter candidate emerges from the neutrino sector [5][6][7].
Physics beyond the SM has drawn physicists' attention for a long time. One of the most appealing theories to describe physics at the TeV scale is the minimal supersymmetric extension of the Standard Model (MSSM) [8][9][10][11]. The MSSM includes necessary additional new particles that are superparters of those in the SM. The right-handed neutrino superfields can extend the next-tominimal supersymmetric standard model (NMSSM), and these superfields only couple with the singlet Higgs [12][13][14][15]. In R-parity [16] conserved MSSM, the left-handed light neutrinos are still massless, leading to a failure to explain the discovery from neutrino oscillations. Therefore, theoretical physicists have extended the MSSM to account for the light neutrino masses and mixings.
As an extension of the MSSM which considers the local gauged baryon (B) and lepton (L) symmetries, the BLMSSM is spontaneously broken at the TeV scale [17][18][19][20]. In the BLMSSM, the lepton number is broken in an even number while baryon number can be changed by baryon number violating operators through one unit. The BLMSSM can not only account for the asymmetry of matter-antimatter in the universe but also explain the data from neutrino oscillation experiments [21][22][23]. Compared with the MSSM, the BLMSSM includes many new fields such as new quarks, new leptons, new Higgs, and the superfieldsX andX [24][25][26]. In this work, we mainly study the Feynman rules for the neutrino and new neutralinos in the BLMSSM.
In the BLMSSM, the light neutrinos get mass from the seesaw mechanism, and proton decay is forbidden [17][18][19][20]. Therefore, it is not necessary to build a large desert between the electroweak scale and grand unified scale. This is the main motivation for the BLMSSM. Many possible signals of the MSSM at the LHC have been studied by the experiments. However, with the broken B and L symmetries, the predictions and bounds for the collider experiments should be changed. From the decays of right handed neutrinos [19,20,27], we can look for lepton number violation at the LHC. Similarly from the decays of squarks and gauginos, we can also detect baryon number violation at the LHC. For example, the channels with multi-tops and multi-bottoms may be caused by the baryon number violating decays of gluinos [19,20].
After this introduction, we briefly summarize the main contents of the BLMSSM in Section 2. The mass matrices for the particles are collected in Section 3. Sections 4 and 5 are respectively devoted to the related couplings of neutralinos and neutrinos beyond the MSSM. We give some discussion and conclusions in Section 6.
The BLMSSM superpotential is given by [28] where W MSSM represents the superpotential of the MSSM. The concrete forms of W B , W L and W X read as follows The soft breaking terms L soft of the BLMSSM are generally shown as [17,18,28] where L MSSM soft represent the soft breaking terms of MSSM, and λ B and λ L are the gauginos of Therefore, the local gauge symmetry SU In Ref. [28], the mass matrices of exotic Higgs, exotic quarks and exotic scalar quarks are obtained. In the BLMSSM, because of the introduced su-perfieldsN C , the tiny masses of the light neutrinos are produced. Another result is six scalar neutrinos in the BLMSSM.

Particle mass matrices
Lepton neutralinos are made up of λ L (the superpartner of the new lepton boson), ψ Φ L and ψ ϕ L (the superpartners of the SU (2) L singlets Φ L and ϕ L ). The mass mixing matrix of lepton neutralinos is shown in the basis (iλ L , ψ Φ L , ψ ϕ L ) [29,30]. Then 3 lepton neutralino masses are obtained from diagonalizing the mass mixing matrix χ 0 L i (i = 1, 2, 3) are the mass eigenstates of the lepton neutralinos.
λ B (the superpartner of the new baryon boson), ψ Φ B and ψ ϕ B (the superpartners of the SU (2) L singlets Φ B and ϕ B ) mix together producing 3 baryon neutralinos. Using Z N B one can diagonalize the mass mixing matrix M BN , and obtain 3 baryon neutralino masses, The mass eigenstates of the baryon neutralinos are represented by χ 0 B i (i = 1, 2, 3). In this work, because neutrinos are Majorana particles, we can use the following expression. In the base (ψ ν I L , ψ N cI R ), the formulas for neutrino mixing and mass matrix are shown as χ 0 Nα denotes the mass eigenstates of the neutrino fields mixed by the left-handed and right-handed neutrinos.
The introduced super-fieldsN c lead to six sneutrinos. From the superpotential and the soft breaking terms in Eqs. (2,3), we deduce the mass squared matrix of sneutrinos (Mñ) in the baseñ T = (ν,Ñ c ). To obtain the mass eigenstates of sneutrinos, Zν is used for the rotation.
The superfields Φ 0 B and ϕ 0 B mix together, and their mass squared matrix is with m Z B representing the mass of the neutral U (1) B gauge boson Z B . Z φ B is the rotation matrix to diagonalize the mass squared matrix M 2 EB and H 0 B i (i = 1, 2) denotes the mass eigenstates of the baryon Higgs.
In the same way, we obtain the mass squared matrix for (Φ 0 L , ϕ 0 L ) Here, m Z L is the mass of the neutral U (1) L gauge boson Z L . Z φ L is used to obtain mass eigenvalues for the matrix M 2 EL . H 0 L i (i = 1, 2) are the lepton Higgs mass eigenstates.

New MSSM neutralino couplings
From the superpotential W L in Eq. (2) and the interactions of gauge and matter multiplets ig , we deduce the couplings of MSSM neutralinoexotic lepton-exotic sleptons: The matrices U L and W L are used to diagonalize the exotic charged lepton mixing matrix [24][25][26], and L 4,5 are the mass eigenstates of the exotic charged leptons. The exotic slepton mass eigenstates are denoted byÑ 4,5 andẼ 4,5 with the rotation matrices Zν 4,5 and Zẽ 4,5 . In the MSSM, there are couplings for MSSM neutralinoneutrino-sneutrino which should be transformed into BLMSSM with the rotations of the neutrinos and sneutrinos in Eqs. (7,8).
In W L there is a new term Y νLĤuN c that can give corrections to the couplings of MSSM neutralinoneutrino-sneutrino. These new couplings are suppressed by the tiny neutrino Yukawa Y ν , In the same way, the couplings of MSSM neutralinoexotic quark-exotic squark are obtained: In the mass basis the exotic quarks are t and b , and their rotation matrices are W t , U t , W b and U b .Ũ and D are the exotic scalar quarks with their diagonalizing matrices U and D.

Lepton neutralino couplings
At tree level, lepton neutralinos not only have relations with leptons and sleptons, but also act with neutrinos and sneutrinos: The couplings for lepton neutralino-exotic lepton-exotic slepton and lepton neutralino-exotic neutrino-exotic sneutrino read as From the interactions of gauge and matter multiplets, we write down the couplings of lepton neutralino-lepton neutralino-lepton Higgs

Baryon neutralino couplings
Baryon neutralinos interact with quarks and squarks, and their couplings are in the following form: Similarly the couplings of baryon neutralino-exotic quark-exotic squark are deduced here: Besides the baryon neutralino-baryon neutralino-baryon Higgs couplings, there are also interactions among baryon neutralinos and X fields:

Couplings of neutrinos beyond the MSSM
Because of the non-zero masses and mixing angles of light neutrinos, physicists are interested in neutrino physics which implies lepton number violation in the Universe. In the MSSM, the neutrino couplings are obtained, so we deduce the neutrino couplings beyond the MSSM in this work. From the supperpotential W L and the interactions of gauge and matter multiplets, we obtain L 1 (ν) and L 2 (ν). L 1 (ν) includes the neutrino couplings with Higgs: 1. neutrino-neutrino-neutral CP -odd Higgs; 2. neutrino-neutrino-neutral CP -even Higgs; 3. neutrinolepton-charged Higgs; 4. neutrino-neutrino-lepton Higgs L 2 (ν) is composed of the couplings: 1. neutrinosneutrino-MSSM neutralino; 2.

Conclusion
In this work, we have briefly introduced the main content of the BLMSSM, which is an extension of the MSSM with local gauged B and L. In this model, there are new neutralinos and right handed neutrinos compared with the MSSM. We deduced the Feynman rules for neutrinos and neutralinos, and these can be used to further study neutrino masses and neutralinos in the BLMSSM. We also showed the mass matrices of particles such as lepton neutralinos and baryon neutralinos. Diagonalizing the corresponding mass mixing matrices one can obtain 3 lepton neutralino and 3 baryon neutralino masses.