Feynman Rules for Neutrinos and New Neutralinos in the BLMSSM*

Chinese Physics C Vol. 40, No. 9 (2016) 093103

Feynman rules for neutrinos and new neutralinos in the BLMSSM *

Xing-Xing Dong(Â33)1;1) Shu-Min Zhao(ëä¬)1;2) Hai-Bin Zhang(Ü°R)1 Fang Wang()3 Tai-Fu Feng(¾F)1,2

1Department of Physics and Technology, Hebei University, Baoding 071002, China 2State Key Laboratory of Theoretical Physics (KLTP), Institute of Theoretical Physics, Chinese Academy of Sciences, 100190 Beijing, China 3Department of Electronic and Information Engineering, Hebei University, Baoding 071002, China

Abstract: 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.

Keywords: supersymmetry, Feynman rules, mass matrices PACS: 13.15.+g, 12.60.-y DOI: 10.1088/1674-1137/40/9/093103

1 Introduction explain the discovery from neutrino oscillations. There- fore, theoretical physicists have extended the MSSM to In quantum ﬁeld theory, the Standard Model (SM) account for the light neutrino masses and mixings. is a theory concerning the electromagnetic weak and As an extension of the MSSM which considers the strong interactions. Though the lightest CP-even Higgs local gauged baryon (B) and lepton (L) symmetries, the

(mh0 126 GeV) was detected by the LHC, the SM is BLMSSM is spontaneously broken at the TeV scale [17– unable'to explain some phenomena and falls short of be- 20]. In the BLMSSM, the lepton number is broken in ing a complete theory of fundamental interactions. In an even number while baryon number can be changed the neutrino sector, the observations of solar and atmo- by baryon number violating operators through one unit. spheric neutrino oscillations [1–4] are not incorporated The BLMSSM can not only account for the asymme- in the SM, which provides clear evidence for physics be- try of matter-antimatter in the universe but also explain yond the SM. Furthermore, the authors think that a well- the data from neutrino oscillation experiments [21–23]. motivated dark matter candidate emerges from the neu- Compared with the MSSM, the BLMSSM includes many trino sector [5–7]. new fields such as new quarks, new leptons, new Higgs, Physics beyond the SM has drawn physicists’ atten- and the superfields Xˆ and Xˆ0 [24–26]. In this work, we tion for a long time. One of the most appealing theories mainly study the Feynman rules for the neutrino and to describe physics at the TeV scale is the minimal super- new neutralinos in the BLMSSM. symmetric extension of the Standard Model (MSSM) [8– In the BLMSSM, the light neutrinos get mass from 11]. The MSSM includes necessary additional new par- the seesaw mechanism, and proton decay is forbid- ticles that are superparters of those in the SM. The den [17–20]. Therefore, it is not necessary to build a large right-handed neutrino superfields can extend the next-to- desert between the electroweak scale and grand unified minimal supersymmetric standard model (NMSSM), and scale. This is the main motivation for the BLMSSM. these superfields only couple with the singlet Higgs [12– Many possible signals of the MSSM at the LHC have 15]. In R-parity [16] conserved MSSM, the left-handed been studied by the experiments. However, with the light neutrinos are still massless, leading to a failure to broken B and L symmetries, the predictions and bounds

Received 19 February 2016, Revised 16 May 2016 ∗ Supported by Major Project of NNSFC (11535002) and NNSFC (11275036), Natural Science Foundation of Hebei Province (A2016201010), and Foundation of Hebei Province (BR2-201), and the Natural Science Fund of Hebei University (2011JQ05, 2012-242), Hebei Key Lab of Optic-Electronic Information and Materials, Midwest Universities Comprehensive Strength Promotion Project 1) E-mail: [email protected] 2) E-mail: [email protected] Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

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= λ Qˆ Qˆc Φˆ +λ Uˆ cUˆ ϕˆ +λ Dˆ cDˆ ϕˆ for the collider experiments should be changed. From WB Q 4 5 B U 4 5 B D 4 5 B ˆ ˆ ˆ ˆ c ˆ ˆ ˆ c the decays of right handed neutrinos [19, 20, 27], we +µBΦBϕˆB +Yu4 Q4HuU4 +Yd4 Q4HdD4 can look for lepton number violation at the LHC. Sim- ˆc ˆ ˆ ˆc ˆ ˆ +Yu5 Q HdU5 +Yd5 Q HuD5, ilarly from the decays of squarks and gauginos, we can 5 5 also detect baryon number violation at the LHC. For ex- ample, the channels with multi-tops and multi-bottoms ˆ ˆ ˆc ˆ ˆ ˆ c ˆc ˆ ˆ L = Ye4 L4HdE4 +Yν4 L4HuN4 +Ye5 L5HuE5 may be caused by the baryon number violating decays W ˆc ˆ ˆ ˆ ˆ ˆ c ˆ c ˆ c of gluinos [19, 20]. +Yν5 L5HdN5 +Yν LHuN +λNc N N ϕˆL After this introduction, we brieﬂy summarize the +µLΦˆLϕˆL, main contents of the BLMSSM in Section 2. The mass ˆ ˆc ˆ ˆ c ˆ ˆ 0 matrices for the particles are collected in Section 3. Sec- X = λ1QQ5X +λ2U U5X W tions 4 and 5 are respectively devoted to the related cou- c 0 0 +λ3Dˆ Dˆ 5Xˆ +µX XˆXˆ . (2) plings of neutralinos and neutrinos beyond the MSSM. We give some discussion and conclusions in Section 6. The soft breaking terms soft of the BLMSSM are generally shown as [17, 18, 28L]

MSSM 2 c∗ c 2 † 2 Main content of the BLMSSM = (mνc ) N˜ N˜ m Q˜ Q˜ Lsoft Lsoft − ˜ IJ I J − Q˜4 4 4 m2 U˜ c∗U˜ c m2 D˜ c∗D˜ c m2 Q˜c†Q˜c Extending the MSSM with local gauged baryon (B) − U˜4 4 4 − D˜ 4 4 4 − Q˜5 5 5 2 ˜ ∗ ˜ 2 ˜ ∗ ˜ 2 ˜† ˜ and lepton (L) numbers, one obtains the BLMSSM, m ˜ U U5 m ˜ D D5 m˜ L L4 − U5 5 − D5 5 − L4 4 and at the TeV scale the local gauge symmetries m2 N˜ c∗N˜ c m2 E˜c∗E˜c m2 L˜c†L˜c are spontaneously broken. In this section, we briefly − ν˜4 4 4 − e˜4 4 4 − L˜5 5 5 2 ˜ ∗ ˜ 2 ˜∗ ˜ 2 ∗ review some features of the BLMSSM. SU(3) mν˜5 N5 N5 me˜5 E5 E5 mΦB ΦBΦB C − − − SU(2) U(1) U(1) U(1) [19, 20] is the basic⊗ 2 ∗ 2 ∗ 2 ∗ L Y B L mϕB ϕBϕB mΦL ΦLΦL mϕL ϕLϕL gauge symmetry⊗ ⊗of the BLMSSM.⊗ The exotic leptons − − − c c mBλBλB +mLλLλL +h.c. L4 (1,2, 1/2,0,L4), E4 (1,1,1,0, L4), N4 − ∼ − c ∼ − ∼ (1,1,0,0, L4), L5 (1,2,1/2,0, (3 + L4)), E5 (1,1 ˜ ˜ c ˜ ˜ c − ∼ − ∼ − + Au4 Yu4 Q4HuU4 +Ad4 Yd4 Q4HdD4 1b,0,3 + L4), N5 (1,1,b0,0,3 + L4) and the exoticb ∼ c n ˜c ˜ ˜c ˜ quarks Q (3b,2,1/6,B ,0), U (3,1, 2/b3, B ,0), +Au5 Yu5 Q5HdU5 +Ad5 Yd5 Q5HuD5 4 ∼ 4 4 ∼ − − 4 Dc (3,1,1/3b, B ,0), Qc (3,2, 1/6, (1 + B ),0), +A λ Q˜ Q˜cΦ +A λ U˜ cU˜ ϕ 4 ∼ − 4 5 ∼ − − 4 BQ Q 4 5 B BU U 4 5 B U (3,b1,2/3,1 + B ,0), D b(3,1 1/3,1 + B ,0) are c 5 4 5 4 +A λ D˜ D˜ ϕ +B µ Φ ϕ +h.c. inbtro∼duced to cancel L andb ∼B anomalies− respectively. BD D 4 5 B B B B B o Theb exotic Higgs superfieldsb ΦL (1,1,0,0, 2), ϕL ˜ ˜c ˜ ˜ c ∼ − ∼ + Ae4 Ye4 L4HdE4 +Aν4 Yν4 L4HuN4 (1,1,0,0,2) and ΦB (1,1,0,1,0), ϕB (1,1,0, 1,0) ∼ ∼ − n ˜c ˜ ˜c ˜ are introduced respectively to bbreak lepton numberband +Ae5 Ye5 L5HuE5 +Aν5 Yν5 L5HdN5 ˜ ˜ c ˜ c ˜ c baryon number spbontaneously. The exoticb Higgs super- +AνYνLHuN +Aνc λνc N N ϕL fields Φ , ϕ and Φ , ϕ acquire nonzero vacuum expec- L L B B +B µ Φ ϕ +h.c. tation values (VEVs), then the exotic leptons and exotic L L L L quarksbobtain masses.b The model also includes the su- ˜ ˜c o ˜ c ˜ 0 b b 0 + A1λ1QQ5X +A2λ2U U5X perfields X (1,1,0,2/3+B4,0) and X (1,1,0, (2/3+ ∼ ∼ − n c 0 0 B4),0) to make heavy exotic quarks unstable. Further- +A3λ3D˜ D˜ 5X +BX µX XX +h.c. , (3) more, theblightest mass eigenstate canb be a dark matter 0 o candidate, while X and X mix together. Anomaly can- where MSSM represent the soft breaking terms of MSSM, Lsoft cellation requires the emergence of new families. How- and λB and λL are the gauginos of U(1)B and U(1)L, re- ever there is no flabvour violationb at tree level since they spectively. The SU(2)L doublets Hu, Hd and SU(2)L do not mix with the SM fermions and there are no Lan- singlets ΦB, ϕB, ΦL, ϕL acquire the nonzero VEVs υu, υd dau poles at the low scale due to the new families. and υB, υB, υL, υL respectively,

The BLMSSM superpotential is given by [28] + Hu Hu = 1 0 0 , = + + + , (1)  (υu +Hu +iPu )  WBLMSSM WMSSM WB WL WX √2   where MSSM represents the superpotential of the 1 0 0 W (υd +Hd +iPd ) MSSM. The concrete forms of B, L and X read Hd = √2 , W W W  −  as follows Hd   093103-2 Chinese Physics C Vol. 40, No. 9 (2016) 093103

1 0 0 k0 ΦB = (υB +ΦB +iPB), 3i 0 0 Bi ψϕ = Z k , χ = . (6) √2 B NB Bi Bi ¯0 kBi ! 1 0 0 ϕB = (υB +ϕB +iP B), √2 The mass eigenstates of the baryon neutralinos are rep- 0 1 0 0 resented by χBi (i = 1,2,3). ΦL = (υL +Φ +iP ), √2 L L In this work, because neutrinos are Majorana particles, we can use the following expression. In the base 1 0 0 (ψνI ,ψNcI ), the formulas for neutrino mixing and mass ϕL = (υL +ϕL +iP L). (4) L R √2 matrix are shown as Therefore, the local gauge symmetry SU(2) U(1) L ⊗ Y ⊗ vu IJ U(1)B U(1)L is broken down to the electromagnetic sym- 0 (Yν) ⊗ √ metry U(1)e. In Ref. [28], the mass matrices of exotic T 2 ZNν  ZNν Higgs, exotic quarks and exotic scalar quarks are ob- vu T IJ v¯L IJ (Yν ) (λ c ) tained. In the BLMSSM, because of the introduced su-  √ √ N   2 2  perfields Nˆ C , the tiny masses of the light neutrinos are   produced. Another result is six scalar neutrinos in the = diag(mνα ), α = 1,...,6. Iα 0 (I+3)α 0 ψ I = Z k , ψ cI = Z k , BLMSSM. νL Nν Nα NR Nν Nα k0 3 Particle mass matrices χ0 = Nα . (7) Nα ¯0 kNα ! Lepton neutralinos are made up of λL (the superpartner of the new lepton boson), ψ and ψ (the super- ΦL ϕL χ0 denotes the mass eigenstates of the neutrino partners of the SU(2) singlets Φ and ϕ ). The mass Nα L L L fields mixed by the left-handed and right-handed neu- mixing matrix of lepton neutralinos is shown in the basis trinos. (iλ ,ψ ,ψ ) [29, 30]. Then 3 lepton neutralino masses L ΦL ϕL The introduced super-fields Nˆ c lead to six sneutrinos. are obtained from diagonalizing the mass mixing matrix From the superpotential and the soft breaking terms in M by Z , LN NL Eqs. (2,3), we deduce the mass squared matrix of sneu- T c trinos ( n) in the base n˜ = (ν˜,N˜ ). To obtain the mass 2ML 2vLgL 2v¯LgL M˜ − eigenstates of sneutrinos, Zν˜ is used for the rotation. MLN = 2v g 0 µ ,  L L − L  2v¯LgL µL 0  − −  g2 +g2   2 ∗ 1 2 2 2 2 2 1i 0 2i 0 n˜ (ν˜I ν˜J ) = (vd vu)δIJ +gL(vL iλL = ZNL kLi , ψΦL = ZNL kLi , M 8 − 2 2 vu † 2 0 ν k vL)δIJ + (Yν Y )IJ +(mL˜ )IJ , 3i 0 0 Li − 2 ψϕL = ZNL kL , χL = . (5) i i k¯0 2 Li ! 2 ˜ c∗ ˜ c 2 2 2 vu † n˜ (NI NJ ) = gL(vL vL)δIJ + (Yν Yν)IJ χ0 (i = 1,2,3) are the mass eigenstates of the lepton neu- M − − 2 Li 2 † 2 c tralinos. +2vL(λNc λN )IJ +(mN˜ c )IJ v v λB (the superpartner of the new baryon boson), ψΦB L L +µL (λNc )IJ (ANc )IJ (λNc )IJ , and ψϕB (the superpartners of the SU(2)L singlets ΦB √2 − √2 v and ϕB) mix together producing 3 baryon neutralinos. 2 ˜ c ∗ d † n˜ (ν˜I NJ ) = µ (Yν)IJ vuvL(Yν λNc )IJ Using ZNB one can diagonalize the mass mixing matrix M √2 − MBN , and obtain 3 baryon neutralino masses, vu + (AN )IJ (Yν)IJ . √2 2MB vBgB v¯BgB † 2 2 2 2 2 2 − Zν˜ n˜ Zν˜ = (mN˜ 1 ,mN˜ 2 ,mN˜ 3 ,mN˜ 4 ,mN˜ 5 ,mN˜ 6 ). (8) MBN = v g 0 µ . M  − B B − B  v¯BgB µB 0 0 0  −  The superfields ΦB and ϕB mix together, and their 1i 0 2i 0  iλB = ZNB kBi , ψΦB = ZNB kBi , mass squared matrix is

2 2 2 2 2 2 m cos βB +m 0 sin βB, (m +m 0 )cosβB sinβB 2 ZB AB ZB AB EB = 2 2 2 2 2 2 , M (m +m 0 )cosβB sinβB, m sin βB +m 0 cos βB ZB AB ZB AB !

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2 2 0 1i 0 0 2i 0 vBt = υB +υB, mZB = gBvBt , ΦB = ZφB HBi , ϕB = ZφB HBi , (9) q

with mZB representing the mass of the neutral U(1)B denotes the mass eigenstates of the baryon Higgs. gauge boson ZB. ZφB is the rotation matrix to diago- In the same way, we obtain the mass squared matrix nalize the mass squared matrix 2 and H 0 (i = 1,2) for (Φ0 , ϕ0 ) MEB Bi L L

2 2 2 2 2 2 m cos βL +m 0 sin βL, (m +m 0 )cosβL sinβL 2 ZL AL ZL AL EL = 2 2 2 2 2 2 , M (m +m 0 )cosβL sinβL, m sin βL +m 0 cos βL ZL AL ZL AL !

2 2 0 1i 0 0 2i 0 vLt = υL +υL, mZL = 2gLvLt , ΦL = ZφL HLi , ϕL = ZφL HLi . (10) q

Here, mZL is the mass of the neutral U(1)L gauge bo- ¯0 1i∗ 3j 1k +N i+3 Yν5 WN ZN Zν˜5 son ZL. ZφL is used to obtain mass eigenvalues for the 2 0 matrix EL . HLi (i = 1,2) are the lepton Higgs mass M 1 e 2j e 1j 1i∗ 2k eigenstates. + Z Z W Zν PL √2 s N − c N N ˜5 4 Couplings of neutralinos beyond the ∗ 2k 3j∗ 2i∗ 0 ˜ k +Yν5 Zν˜5 ZN UN PR χj N5 +h.c. (11) MSSM

The matrices UL and WL are used to diagonalize the 4.1 New MSSM neutralino couplings 0 exotic charged lepton mixing matrix [24–26], and L4,5 are the mass eigenstates of the exotic charged leptons. From the superpotential L in Eq. (2) and the inter- The exotic slepton mass eigenstates are denoted by N˜ 0 W √ a a ∗ 4,5 actions of gauge and matter multiplets ig 2Tij (λ ψj Ai ˜0 − and E with the rotation matrices Zν˜4 5 and Ze˜4 5 . In λ¯aψ¯ A ), we deduce the couplings of MSSM neutralino- 4,5 , , i j the MSSM, there are couplings for MSSM neutralino- exotic lepton-exotic sleptons: neutrino-sneutrino which should be transformed into BLMSSM with the rotations of the neutrinos and sneu- 2 4 trinos in Eqs. (7, 8). 0 0˜0 0 1 e 2j (χ l l ) = χ¯j ZN ˆ ˆ ˆ c L √2 s In L there is a new term YνLHuN that can i,k=1 j=1 W X X give corrections to the couplings of MSSM neutralino- e + Z1j U 1iZ1k∗ +Y U 1iZ3j Z2k∗ P neutrino-sneutrino. These new couplings are suppressed c N L e˜4 e4 L N e˜4 L by the tiny neutrino Yukawa Yν, ∗ 1k∗ 3j∗ 2i n 0 ∗ + Y Z ZN W (χ χ N˜ ) e4 e˜4 L L N 3 4 6 IJ Iα 4i (J+3)j∗ e 1j∗ 2i 2k∗ 0 k∗ √2 Z W Z P L E˜ = χ¯Nα Yν ZNν ZN Zν˜ PL N L e˜4 R i+3 4 − − c I,J=1 i=1 j=1 X X X 0 1i 4j 2k∗ IJ∗ Ij∗ 4i∗ (J+3)α∗ 0 j∗ χ¯ Yν4 U ZN Zν +Yν Zν Z Z P χ N˜ +h.c. (12) − j N ˜4 ˜ N Nν R i 1 e 2j e 1j 1i 1k∗ + ( Z Z )U Zν PL In the same way, the couplings of MSSM neutralino- √2 s N − c N N ˜4 exotic quark-exotic squark are obtained: ∗ 1k∗ 4j∗ 2i 0 k∗ ˜ 2 4 +Yν4 Zν˜4 ZN WN PR Ni+3N4 0 0 0 0 0 ∗ 3i t (χ q q˜ ) = χ¯i YU5 U3kZN U2j L ¯0 4j 1i∗ 1k j=1 i,k=1 L i+3 Ye5 Z W Z X X − N L e˜5 0 e 2√2 1i t ∗ 1 e 2i 1 e 2j e 1j 1i∗ 2k ZN U2j U4k ZN Z + Z W Z PL − c 3 − √2 s −√2 s N c N L e˜5 0 0 1 e 1i t ∗ t 4i ∗ ∗ k 4j∗ i∗ 1 2 + ZN U1j U1k YU4 U1j ZN U2k PL + Ye5 Ze˜5 ZN UL 3 c − 0 e 1j∗ 2i∗ 1k 0 k ∗ t 3i∗ ∗ 1 e 2i∗ +√2 Z U Z PR χ E˜ + Y W Z U + Z c N L e˜5 j 5 U5 1j N 4k √2 s N

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1 e 0 0 1i∗ 2k∗ 2j 0 ˜ k∗ 1i∗ t ∗ ∗ ∗ 4i∗ t ZNL Zν˜4 WN PR)Nj+3N4 + ZN W1j U3k YU4 U1kZN W2j − 3 c − √ 0 1i 1j∗ 2k + 2(3+L4)gLχ¯Li (ZNL Ze˜5 UL PL 0 e 2√2 1i∗ t ∗ 0 ∗ 1i∗ 1k 2j∗ 0 j∗ + Z W U P t ˜ Z W Z PR)L E˜ c 3 N 2j 2k R j+3Uk − NL L e˜5 k+3 5 0 1i 1j∗ 2k +√2(3+L )g χ¯ (Z Zν U P 0 4 L Li NL ˜5 N L 0 b 3i ∗ +χ¯i Yd4 U1j ZN D2k 1i∗ 1k 2j∗ 0 j∗ Z W Zν P )N N˜ +h.c. (15) − NL N ˜5 R k+3 5 1 e 1 e 0 + Z2i Z1i U b D∗ √2 s N − 3 c N 1j 1k From the interactions of gauge and matter multiplets, 0 e √2 0 Y D∗ Z4iU b + Z1iU b D∗ P we write down the couplings of lepton neutralino-lepton − d5 3j N 2k c 3 N 2j 4k L neutralino-lepton Higgs 0 1 e + Y ∗ D∗ Z3i∗W b Z2i∗ d4 1k N 2j − √2 s N 3 2 0 0 0∗ √ 1i 3j 2k∗ 0 0 (χLχLHL ) = 2 2gL ZNL ZNL ZφL 1 e 1i∗ b ∗ ∗ b 4i∗ ∗ L ZN W1j D3k Yd5 W1j ZN D4k i,j=1 k=1 −3 c − X X 2j 1k∗ 0 0 0∗ Z Z χ¯ PLχ H +h.c. (16) e √2 0 NL φL Li Lj Lk Z1i∗W b D∗ P b0 ˜∗ +h.c. (13) − − c 3 N 2j 2k R j+3Dk In the mass basis the exotic quarks are t0 and b0, and 4.3 Baryon neutralino couplings 0 0 0 0 their rotation matrices are W t ,U t ,W b and U b . ˜ and ˜ are the exotic scalar quarks with their diagonalizingU Baryon neutralinos interact with quarks and squarks, Dmatrices U and D. and their couplings are in the following form: 4.2 Lepton neutralino couplings 3 6 0 √2 1i Ij∗ At tree level, lepton neutralinos not only have rela- (χ qq˜) = gBχ¯ 0 (Z Z PL L B 3 Bi NB U˜ tions with leptons and sleptons, but also act with neu- I,i=1 j=1 trinos and sneutrinos: X X 1i∗ (I+3)j∗ I ∗ √2 1i Ij 6 3 ˜ 0 ZNB ZU˜ PR)u Uj + gBχ¯B (ZNB ZD˜ PL (χ0 l˜l) = χ¯ λIJ − 3 i L L Nα − Nc α,j=1 I,J,j=1 X X 1i∗ (I+3)j∗ I ˜ ∗ ZNB ZD˜ PR)d Dj +h.c. (17) JI (I+3)α 3i (J+3)j∗ − +λNc ZNν ZNL Zν˜ √ 1i Iα Jj∗ Similarly the couplings of baryon neutralino-exotic + 2gLZNL ZNν Zν˜ δIJ PL quark-exotic squark are deduced here: 1i∗ (I+3)α∗ (J+3)j∗ j∗ √2gLZ Z Zν δIJ PR χ 0 N˜ − NL Nν ˜ Li 3 6 2 3 6 0 0 0 √ ¯0 1i Ij (χBq q˜ ) = 2gBt k+3 (1 + √2gLχ¯ 0 (Z Z PL L − Li NL L˜ i=1 j=1 k=1 i,I=1 j=1 XXX X X 0 0 +B )W t ∗U Z1i +λ U W t ∗Z2i Z1i∗Z(I+3)j∗P )eI L˜−∗ +h.c. (14) 4 1k 3j NB Q 1j 1k NB − NL L˜ R j 0 0 t ∗ 3i t ∗ 1i The couplings for lepton neutralino-exotic lepton-exotic +λU W2k ZNB U4j +B4U2j W2k ZNB PL slepton and lepton neutralino-exotic neutrino-exotic 0 0 sneutrino read as t ∗ 1i∗ t ∗ 1i∗ + B U U j Z +(1+B )U j U Z 3 2 4 1k 1 NB 4 4 2k NB 0 0 0 1i 1j 1k∗ (χ l ˜l ) = L √2g χ¯ 0 (Z U Z P L 4 L Li NL L e˜4 L L 0 0 i=1 j,k=1 ∗ t ∗ 2i∗ ∗ t ∗ 3i∗ 0 λ U U j Z λ U j U Z PR χ ˜j X X − Q 1k 3 NB − U 2 2k NB Bi U 1i∗ 2k∗ 2j 0 ˜k∗ ZNL Ze˜4 WL PR)Lj+3E4 − 0 0 b ∗ 2i 1i 1j 1k∗ √ ¯ √ 0 + 2gBb k+3 λQD1j W1k ZNB +L4 2gLχ¯L (ZN UN Zν˜4 PL i L

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0 0 B D W b ∗Z1i (1+B )W b ∗D Z1i lepton neutralino; 3. neutrino-slepton-chargino: − 4 2j 2k NB − 4 1k 3j NB 0 0 b ∗ 3i ∗ b ∗ 2i∗ 3 6 4 λDW2k ZNB D4j PL + λQU1k D3j ZNB 2 ν 0 IJ 4i Iα (J+3)j∗ − ( ) = χ¯i Yν ZN ZNν Zν˜ PL L − 0 0 I,J=1 α,j=1 i=1 ∗ b ∗ 3i∗ b ∗ 1i∗ X X X λDU2k D2j ZNB +(1+B4)D4j U2k ZNB − IJ∗ 4i∗ (J+3)α∗ Ij∗ 0 j∗ +Yν Z Z Zν P χ N˜ 0 N Nν ˜ R Nα b ∗ 1i∗ 0 ˜ +U D1j Z PR χ j +h.c. (18) 1k NB Bi D 3 1i Iα Jj∗ IJ JI 3i + √2g Z Z Zν δ (λ c +λ c )Z Besides the baryon neutralino-baryon neutralino-baryon L NL Nν ˜ IJ − N N NL i=1 Higgs couplings, there are also interactions among X baryon neutralinos and X ﬁelds: √ 1i IJ (I+3)α (J+3)j∗ 0 0 ˜ j∗ + 2gLZNL δ ZNν Zν˜ χ¯LiPLχNα N 3 2 2j 0 0 √ 1k∗ 2 (χBχBHB) = 2gB ZNB ZφB L IJ 2i (J+3)α Ij∗ 0 + ˜ i,j=1 k=1 Yν Z ZNν Z˜ χ¯ PLχ Lj +h.c. (21) X X − + L Nα i i=1 Z3j Z2k∗ Z1i χ¯0 P χ0 H 0∗ , X − NB φB NB Bj L Bi Bk These neutrino couplings are favourable for the study of 3 2 0 2 1i 0 1j∗ neutrinos in the BLMSSM. (χ ψX X˜) = +B √2g Z χ¯ [Z P L B 3 4 B NB Bi X L i=1 j=1 X X 6 Conclusion Z2j∗P ]ψ X˜ ∗ +h.c. (19) − X R X j In this work, we have brieﬂy introduced the main con- 5 Couplings of neutrinos beyond the tent of the BLMSSM, which is an extension of the MSSM MSSM with local gauged B and L. In this model, there are new neutralinos and right handed neutrinos compared with Because of the non-zero masses and mixing angles the MSSM. We deduced the Feynman rules for neutrinos of light neutrinos, physicists are interested in neutrino and neutralinos, and these can be used to further study physics which implies lepton number violation in the Uni- neutrino masses and neutralinos in the BLMSSM. We verse. In the MSSM, the neutrino couplings are obtained, also showed the mass matrices of particles such as lep- so we deduce the neutrino couplings beyond the MSSM ton neutralinos and baryon neutralinos. Diagonalizing in this work. From the supperpotential and the inter- the corresponding mass mixing matrices one can obtain WL actions of gauge and matter multiplets, we obtain 1(ν) 3 lepton neutralino and 3 baryon neutralino masses. L and 2(ν). 1(ν) includes the neutrino couplings with We deduced the new couplings of the MSSM L L Higgs: 1. neutrino-neutrino-neutral CP-odd Higgs; 2. neutralinos, including MSSM neutralino-exotic lepton- neutrino-neutrino-neutral CP-even Higgs; 3. neutrino- exotic slepton, MSSM neutralino-neutrino-sneutrino and lepton-charged Higgs; 4. neutrino-neutrino-lepton Higgs MSSM neutralino-exotic quark-exotic squark. The cou-

3 6 2 plings of lepton neutralinos were obtained from the su- 1 IJ (J+3)α 2j 0 I + perpotential and the interactions of gauge and mat- (ν) = Yν Z Z χ¯ PLe H L Nν H Nα j ter multiplets: lepton neutralino-lepton-slepton, lep- I,J=1 α=1 j=1 X XX ton neutralino-exotic lepton-exotic slepton and lepton 6 2 i IJ Iα (3+J)β 2j 0 0 0 neutralino-lepton neutralino-lepton Higgs. The baryon Yν Z Z Z χ¯ P χ A − 2 Nν Nν H Nβ L Nα j neutralino couplings were deduced in the same way: α,β=1 j=1 X X baryon neutralino-quark-squark, baryon neutralino- 6 2 1 exotic quark-exotic squark, baryon neutralino-baryon IJ Iα (3+J)β 2j 0 0 0 ˜ Yν ZNν ZNν ZR χ¯N PLχNα Hj neutralino-baryon Higgs, and baryon neutralinos-X-ψ . − 2 β X α,β=1 j=1 X X Finally, we obtained the couplings of neutrinos beyond the MSSM, which can be divided asµneutrino-neutrino- 6 3 neutral CP-odd Higgs; neutrino-neutrino-neutral CP- IJ (I+3)α (J+3)β 2i 0 0 0 λ c ZNν ZNν Z χ¯ PLχ H even Higgs; neutrino-lepton-charged Higgs; neutrino- − N φL Nα Nβ Li α,β=1 i=1 X X neutrino-lepton Higgs; neutrino-sneutrino-MSSM neu- +h.c. (20) tralino; neutrino-sneutrino-lepton neutralino; and neutrino-slepton-chargino. The obtained couplings are 2(ν) is composed of the couplings: 1. neutrino- favourable for the study of neutrinos and processes re- sneutrino-MSSML neutralino; 2. neutrino-sneutrino- lating to neutralinos in the BLMSSM.

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