The extended BLMSSM with a 125 GeV Higgs boson and dark matter

The European Physical Journal C, Apr 2018

To extend the BLMSSM, we not only add exotic Higgs superfields \((\Phi _{NL},\varphi _{NL})\) to make the exotic lepton heavy, but also introduce the superfields (Y,\(Y^\prime \)) having couplings with lepton and exotic lepton at tree level. The obtained model is called as EBLMSSM, which has difference from BLMSSM especially for the exotic slepton (lepton) and exotic sneutrino (neutrino). We deduce the mass matrices and the needed couplings in this model. To confine the parameter space, the Higgs boson mass \(m_{h^0}\) and the processes \(h^0\rightarrow \gamma \gamma \), \(h^0\rightarrow VV, V=(Z,W)\) are studied in the EBLMSSM. With the assumed parameter space, we obtain reasonable numerical results according to data on Higgs from ATLAS and CMS. As a cold dark mater candidate, the relic density for the lightest mass eigenstate of Y and \(Y'\) mixing is also studied.

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The extended BLMSSM with a 125 GeV Higgs boson and dark matter

Eur. Phys. J. C The extended BLMSSM with a 125 GeV Higgs boson and dark matter Shu-Min Zhao 1 Tai-Fu Feng 1 Guo-Zhu Ning 1 Jian-Bin Chen 0 Hai-Bin Zhang 1 Xing Xing Dong 1 0 College of Physics and Optoelectronic Engineering, Taiyuan University of Technology , Taiyuan 030024 , China 1 Department of Physics, Hebei University , Baoding 071002 , China To extend the BLMSSM, we not only add exotic Higgs superfields ( N L , ϕN L ) to make the exotic lepton heavy, but also introduce the superfields (Y ,Y ) having couplings with lepton and exotic lepton at tree level. The obtained model is called as EBLMSSM, which has difference from BLMSSM especially for the exotic slepton (lepton) and exotic sneutrino (neutrino). We deduce the mass matrices and the needed couplings in this model. To confine the parameter space, the Higgs boson mass mh0 and the processes h0 → γ γ , h0 → V V , V = (Z , W ) are studied in the EBLMSSM. With the assumed parameter space, we obtain reasonable numerical results according to data on Higgs from ATLAS and CMS. As a cold dark mater candidate, the relic density for the lightest mass eigenstate of Y and Y mixing is also studied. 1 Introduction The total lepton number (L) and baryon number (B) are good symmetries because neutrinoless double beta decay or proton decay has not been observed. In the standard model (SM), L and B are global symmetries [1,2]. However, the individual lepton numbers Li = Le, Lμ, Lτ are not exact symmetries at the electroweak scale because of the neutrino oscillation and the neutrinos with tiny masses [3,4]. In the Universe, there is matter-antimatter asymmetry, then the baryon number must be broken. With the detection of the light Higgs h0(m0h = 125.1 GeV) [5,6], the SM succeeds greatly and the Higgs mechanism is compellent. Beyond the SM, supersymmetry [7,8] provides a possibility to understand the light Higgs. The minimal supersymmetric extension of the SM (MSSM) [9] is one of the favorite models, where the light Higgs mass at tree level is mthree = m Z | cos 2β| [10–12]. The one loop corrections to Higgs mass mainly come from fermions and sfermions, that depend on the virtual particle masses and the couplings with the Higgs. There are many papers about the gauged B and L models, although most of them are non-supersymmetric [13,14]. Extending MSSM with the local gauged B and L, one obtains the so called BLMSSM, which was proposed by the authors in Refs. [10–12]. The proton remains stable, as B and L are broken at the TeV scale. Therefore, a large desert between the electroweak scale and grand unified scale is not necessary. In BLMSSM, the baryon number is changed by one unit, at the same time the lepton number is broken in an even number. R-parity in BLMSSM is not conserved, and it can explain the matter-antimatter asymmetry in the Universe. There are some works for Higgs and dark matters [15–17] in the BLMSSM [18,19]. In the framework of BLMSSM, the light Higgs mass and the decays h0 → γ γ and h0 → V V , V = (Z , W ) are studied in our previous work [19]. Some lepton flavor violating processes and CP-violating processes are researched with the new parameters in BLMSSM [20,21]. In BLMSSM, the exotic leptons are not heavy, because their masses just have relation with the parameters Ye4 υd , Ye5 υu . Here υu and υd are the vacuum expectation values (VEVs) of two Higgs doublets Hu and Hd . In general, the Yukawa couplings Ye4 and Ye5 are not large parameters, so the exotic lepton masses are around 100 GeV. The light exotic leptons may lead to that the BLMSSM is excluded by high energy physics experiments in the future. To obtain heavy exotic leptons, we add two exotic Higgs superfields to the BLMSSM, and they are SU( 2 ) singlets N L and ϕN L , whose VEVs are υN L and υ¯ N L [22]. The exotic leptons and the superfields N L , ϕN L have Yukawa couplings, then υN L and υ¯ N L give contributions to the diagonal elements of the exotic lepton mass matrix. So the exotic leptons turn heavy and should be unstable. In the end, the super fields Y and Y are also introduced. At tree level, there are couplings for lepton-exotic lepton-Y (Y ). It is appealing that this extension of BLMSSM produces some new cold dark matter candidates, such as the lightest mass eigenstate of Y and Y mixing. The four-component spinor Y˜ is made up of the superpartners of Y and Y . In this extended BLMSSM (EBLMSSM), we study the lightest CP even Higgs mass with the one loop corrections. The Higgs decays h0 → γ γ and h0 → V V , V = (Z , W ) are also calculated here. Supposing the lightest mass eigenstate of Y and Y mixing as a cold dark matter candidate, we study the relic density. After this introduction, in Sect. 2, we introduce the EBLMSSM in detail, including the mass matrices and the couplings different from those in the BLMSSM. The mass of the lightest CP-even Higgs h0 is deduced in the Sect. 3. The Sect. 4 is used to give the formulation of the Higgs decays h0 → γ γ , h0 → V V , V = (Z , W ) and dark matter relic density. The corresponding numerical results are computed in Sect. 5. The last section is used for the discussion and conclusion. 2 Extend the BLMSSM The local gauge group of the BLMSSM [10–12] is SU (3)C ⊗ SU ( 2 )L ⊗ U ( 1 )Y ⊗ U ( 1 )B ⊗ U ( 1 )L . In the BLMSSM, the exotic lepton masses are obtained from the Yukawa couplings with the two Higgs doublets Hu and Hd . The VEVs of Hu and Hd are υu and υd with the relation υu2 + υd2 = υ ∼ 250 GeV. Therefore, the exotic lepton masses are not very heavy, though they can satisfy the experiment bounds at present. In the future, with the development of high energy experiments, the experiment bounds for the exotic lepton masses can improve in a great possibility. Therefore, we introduce the exotic Higgs superfields N L and ϕN L with nonzero VEVs to make the exotic lepton heavy. The heavy exotic leptons should be unstable, then the superfields Y, Y are introduced accordingly. These introduced superfields lead to tree level couplings for lepton-exotic lepton-Y (Y ). In EBLMSSM, we show the superfields in the Table 1. The superpotential of EBLMSSM is shown here WE B L M SSM = WM SSM + WB + WL + WX + WY , WL = λL Lˆ 4 Lˆ 5ϕˆN L + λE Eˆ4c Eˆ5 ˆ N L c + λN L Nˆ 4c Nˆ 5 ˆ N L + μN L ˆ N L ϕˆN L + Ye4 Lˆ 4 Hˆd Eˆ4 + Yν4 Lˆ 4 Hˆu Nˆ 4c c + Ye5 Lˆ c5 Hˆu Eˆ5 + Yν5 Lˆ c5 Hˆd Nˆ 5 + Yν Lˆ Hˆu Nˆ c + λN c Nˆ c Nˆ cϕˆL + μL ˆ L ϕˆL , WY = λ4 Lˆ Lˆ 5Yˆ + λ5 Nˆ c Nˆ 5Yˆ + λ6 Eˆ c Eˆ5Yˆ + μY Yˆ Yˆ . ( 1 ) c WM SSM is the superpotential of MSSM. WB and WX are same as the terms in BLMSSM [19]. WY includes the terms beyond BLMSSM, and they include the couplings of leptonexotic lepton-Y (l I − L − Y ). Therefore, the heavy exotic leptons can decay to leptons and mass eigenstates of Y and Y mixing whose lighter one can be a dark matter candidate. From WY , one can also obtain the coupling of lepton-exotic slepton-Y˜ (l I − L˜ − Y˜ ), where Y˜ is the four component spinor composed by the superpartners of Y and Y . The new couplings of l I − L −Y and l I − L˜ −Y˜ can give one loop corrections to lepton anormal magnetic dipole moment (MDM). They may compensate the deviation between the experiment value and SM prediction for muon MDM. The parameter μY can be complex number with non-zero imaginary part, which is a new source of CP-violating. Therefore, the both new couplings produce one loop diagrams contributing to the lepton electric dipole moment (EDM). Further more, if λ4 in λ4 Lˆ Lˆ c5Yˆ is a matrix and has non-zero elements relating with lepton flavor, this term can enhance the lepton flavor violating effects. In the whole, WY enriches the lepton physics to a certain degree, and these subjects will be researched in our latter works. Because of the introduction of the superfields N L , ϕN L , Y and Y , the soft breaking terms are written as E B L M SSM Lsof t = LsBoLf Mt SSM − m2 N L ∗N L N L −m2ϕN L ϕ∗N L ϕN L + AL L λL L˜ 4 L˜ c5ϕN L + AL E λE e˜4e˜5 N L + AL N λN L ν˜4cν˜5 N L c +BN L μN L N L ϕN L + h.c. + A4λ4 L˜ L˜ c5Y + A5λ5 N˜ cν˜5Y + A6λ6e˜ce˜5Y + BY μY Y Y + h.c. . ( 2 ) B L M SSM is the soft breaking terms of BLMSSM, Here Lsof t whose concrete form is in our previous work [19]. The SU (2)L doublets Hu , Hd acquire the nonzero VEVs υu , υd . The SU (2)L singlets B , ϕB , L , ϕL , N L , ϕN L obtain the nonzero VEVs υB , υ B , υL , υ L , υN L , υ N L respectively. Hu = Hd = 1 B = √2 1 ϕB = √2 Hu+ √12 υu + Hu0 + i Pu0 √12 υd + Hd0 + i Pd0 Hd− υB + 0 B + i PB0 , 0 0 υ B + ϕB + i P B , U ( 1 )B 1/3 −1/3 −1/3 0 0 0 B4 0 L + i PL0 , ϕN L = √12 υ N L + ϕ0N L + i P0N L . 0 N L + i PN0 L , Here, we define tan β = υu /υd , tan βB = υ¯ B /υB , tan βL = υ¯ L /υL and tan βN L = υ¯ N L /υN L . The VEVs of the Higgs satisfy the following equations |μ|2 − g12 +8 g22 (υu2 − υd2) + m2Hd + Re[Bμ] tan β = 0, ( 3 ) ( 4 ) |μ|2 + g12 +8 g22 (υu2 − υd2) + m2Hu + Re[Bμ] cot β = 0, |μB |2 + g22B (υB2 − υ¯ B2 ) + m2 B − Re[BB μB ] tan βB = 0, ( 6 ) |μB |2 − g22B (υB2 − υ¯ B2 ) + m2ϕB − Re[BB μB ] cot βB = 0, ( 7 ) ( 8 ) ( 9 ) |μL |2 − 2g2L VL2 + m2 L − Re[BL μL ] tan βL = 0, |μL |2 + 2g2L VL2 + m2ϕL − Re[BL μL ] cot βL = 0, |μN L |2 − 3g2L VL2 + m2 N L − Re[BN L μN L ] tan βN L = 0, ( 10 ) ( 11 ) |μN L |2 + 3g2L VL2 + m2ϕN L − Re[BN L μN L ] cot βN L = 0, with VL2 = υ2L − υL2 + 23 (υ2N L − υN2 L ). Here, the Eqs. ( 8 ) and ( 9 ) are similar as the corresponding equations in BLMSSM, but Eqs. ( 8 ) and ( 9 ) have relation with the new parameters υN L and υ¯ N L . We obtain the new Eqs. ( 10 ) and ( 11 ) through ∂ ∂VN L and ∂∂ϕVN L , with V denoting the Higgs scalar potential. Here we deduce the mass matrices in the EBLMSSM. Compared with BLMSSM, the superfields N L and ϕN L are introduced and they give corrections to the mass matrices of the slepton, sneutrino, exotic lepton, exotic neutrino, exotic slepton and exotic sneutrino. That is to say, in EBLMSSM, the mass matrices of squark, exotic quark, exotic squark, baryon neutralino, MSSM neutralino, X and X˜ are same as those in the BLMSSM, and their concrete forms can be found in our previous works [23–25]. Though the mass squared matrices of slepton and sneutrino in EBLMSSM are different from those in BLMSSM, we can obtain the slepton and sneutrino mass squared matrices in EBLMSSM easily just 2 2 using the replacement υ L − υL → VL2 for the BLMSSM results. In the BLMSSM, the issue of Landau pole has been discussed in detail by the authors of Refs. [10–12]. Their conclusion is that there are no Landau poles at the low scale due to the new families. In EBLMSSM, the parts of quark (squark), exotic quark (exotic squark) are same as those in BLMSSM. Therefore, the Landau pole conditions for the Yukawa couplings of quark (squark), exotic quark (exotic squark) have same behaviors of BLMSSM. The added superfields ( N L , ϕN L , Y, Y ) do not have couplings with the gauge fields of SU ( 3 )C , SU ( 2 )L , U ( 1 )Y and U ( 1 )B . So the characters of gauge couplings g1, g2, g3 and gB in BLMSSM and EBLMSSM are same. The different parts between BLMSSM and EBLMSSM are the terms including N L , ϕN L , Y and Y . The new terms in the superpotential WL are λL Lˆ 4 Lˆ 5ϕˆN L + λE Eˆ4c Eˆ5 ˆ N L + c λN L Nˆ 4c Nˆ 5 ˆ N L + μN L ˆ N L ϕˆN L and they have corresponding relations with λQ Qˆ 4 Qˆ c5 ˆ B +λU Uˆ4cUˆ5ϕˆ B +λD Dˆ 4c Dˆ 5ϕˆ B + μB ˆ B ϕˆ B in WB by the replacements Lˆ 4 ↔ Qˆ 4, Lˆ c5 ↔ Qˆ c5, Eˆ c 4 4 ↔ Dˆ 4c, Nˆ 5 ↔ Dˆ 5, ˆ N L ↔ 4 ↔ Uˆ c, Eˆ5 ↔ Uˆ5, Nˆ c ϕˆ B , ϕˆN L ↔ ˆ B . The corresponding relations for WY = λ4 Lˆ Lˆ c5Yˆ + λ5 Nˆ c Nˆ 5Yˆ + λ6 Eˆ c Eˆ5Yˆ + μY Yˆ Yˆ and WX = λ1 Qˆ Qˆ c5 Xˆ + λ2Uˆ cUˆ5 Xˆ + λ3 Dˆ c Dˆ 5 Xˆ + μX Xˆ Xˆ are obvious with Lˆ ↔ Qˆ , Lˆ c5 ↔ Qˆ c5, Eˆ c ↔ Uˆ c, Eˆ5 ↔ Uˆ5, Nˆ c ↔ Dˆ c, Nˆ 5 ↔ Dˆ 5, Xˆ ↔ Yˆ , Xˆ ↔ Yˆ . From this analysis, the Landau pole conditions of gauge coupling gL and Yukawa couplings of exotic leptons should possess similar peculiarities of gauge coupling gB and Yukawa couplings of exotic quarks. In conclusion, similar as BLMSSM, there are no Landau poles in EBLMSSM at the low scale because of the new families. The concrete study of Landau poles for the couplings should use renormalization group equation which is tedious, and we shall research this issue in our future work. 2.1 The mass matrices of exotic lepton (slepton) and exotic neutrino (sneutrino) in EBLMSSM In BLMSSM, the exotic lepton masses are not heavy, because they obtain masses only from Hu and Hd . The VEVs of N L and ϕN L are υN L and υ¯ N L , that can be large parameters. So, the EBLMSSM exotic leptons are heavier than those in BLMSSM. The mass matrix for the exotic leptons reads as mass − Le = e¯4R , e¯5R − √12 λL υ N L , √12 Ye5 υu − √12 Ye4 υd , √12 λE υN L × e4L e5L + h.c. × ν4L ν5L + h.c. ( 12 ) ( 13 ) Obviously, υ N L and υN L are the diagonal elements of the mass matrix in the Eq. ( 12 ). It is easy to obtain heavy exotic lepton masses with large υ N L and υN L . If we take υ N L and υN L as zero, the mass matrix is same as that in BLMSSM. In fact, our used values of υ N L and υN L are at TeV order, which produce TeV scale exotic leptons. Heavy exotic leptons have strong adaptive capacity to the experiment bounds. The exotic neutrinos are four-component spinors, whose mass matrix is mass − Lν = ν¯4R , ν¯5R √12 λL υ N L , − √12 Yν5 υd √12 Yν4 υu , √12 λN L υN L Similar as the exotic lepton condition, heavy exotic neutrinos are also gotten. In BLMSSM, the exotic sleptons of 4 generation and 5 generation do not mix, and their mass matrices are both 2×2. In EBLMSSM, the exotic sleptons of 4 generation and 5 generation mix together, and their mass matrix is 4 × 4. With the base (e˜4, e˜4c∗, e˜5, e˜5c∗), we show the elements of exotic 2 in the following form. slepton mass matrix ME˜ M2E˜ (e˜5c∗e˜5c) = λ2L υ¯ N22 L + υ2u2 |Ye5 |2 + ML2˜ 5 2 2 − g1 −8 g2 (υd2 − υu ) − g2L (3 + L4)VL2, 2 M2E˜ (e˜5∗e˜5) = λ2E υN22 L + υ2u2 |Ye5 |2 + Me˜25 g2 2 + 41 (υd2 − υu ) + g2L (3 + L4)VL2, 2 2 υ¯ N L M2E˜ (e˜4∗e˜4) = λL 2 2 2 + g1 −8 g2 (υd2 − υu ) 2 υ2 + 2d |Ye4 |2 + M 2 L˜ 4 + g2L L4VL2, 2 2 υN L M2E˜ (e˜4c∗e˜4c) = λE 2 g2 2 − 41 (υd2 − υu ) υ2 + 2d |Ye4 |2 + M 2 e˜4 − g2L L4VL2, υN L M2E˜ (e˜4∗e˜5) = υd Ye∗4 λE 2 M2E˜ (e˜5e˜5) = μ∗ √υd Ye5 + Ae5 Ye5 √ , c υu 2 2 + λL Ye5 υ¯ N L vu , 2 c υ¯ N L M2E˜ (e˜4e˜5) = μ∗N L λE √2 c υN L M2E˜ (e˜4e˜5) = −μ∗N L √2 − AL E λE υ√N2L , λL + AL L λL υ¯√N L , 2 M2E˜ (e˜4e˜4) = μ∗ √υu Ye4 + Ae4 Ye4 √ , c υd 2 2 M2E˜ (e˜5ce˜4c∗) = −Ye5 λE υu υN L − λL Ye∗4 υ¯ N L vd . 2 2 ( 14 ) In Eq. ( 14 ), the non-zero terms M2E˜ (e˜4e˜5c), ME˜ 2 (e˜4∗e˜5), tMon2E˜m(e˜i5xcei˜n4c∗g)oafndgeMneEr˜ations 4 and 5. These mixing terms all c 2 (e˜4e˜5) are the reason for the exotic slepinclude the parameters υN L and υ¯ N L . It shows that this mixing is caused basically by the added Higgs superfields N L and ϕN L . Using the matrix Z E˜ , we obtain mass eigenstates witIhntthheefsoarmmeulwa aZy†E,˜tMhe2Ee˜xZoE˜tic=sndeiuatgri(nmo2Em˜1 ,asms2Es˜ q2,umar2eE˜d3 ,mma2Et˜r4i)x. is also obtained 2 2 υ¯ N L M2N˜ (ν˜5c∗ν˜5c) = λL 2 2 2 − g1 +8 g2 (υd2 − υu ) 2 υ2 + 2d |Yν5 |2 + M 2 L˜ 5 − g2L (3 + L4)VL2, 2 2 υ¯ N L M2N˜ (ν˜4∗ν˜4) = λL 2 2 2 + g1 +8 g2 (υd2 − υu ) 2 υ2 + 2u |Yν4 |2 + M 2 L˜ 4 + g2L L4VL2, 2 M2N˜ (ν˜5∗ν˜5) = λ2N L υN2 L + g2L (3 + L4)VL2 2 M2N˜ (ν˜4c∗ν˜4c) = λ2N L υN2 L − g2L L4VL2 υ2 + 2d |Yν5 |2 + Mν˜25 , υ2 + 2u |Yν4 |2 + Mν˜24 , M2N˜ (ν˜5ν˜4c∗) = λN L Yν5 c υN L υd − λL Yν∗4 υ¯ N L υu , 2 2 M2N˜ (ν˜5ν˜5) = μ∗ √υu Yν5 + Aν5 Yν5 √ , c υd 2 2 υ¯ N L M2N˜ (ν˜4cν˜5) = μ∗N L λN L √ 2 υN L − AL N λN √2 , c υN L M2N˜ (ν˜4ν˜5) = μ∗N L √2 υ¯ N L υd M2N˜ (ν˜4∗ν˜5) = λL Yν5 2 λL − AL L λL υ¯√N L , 2 − υu υN L 2 λN L Yν∗4 , M2N˜ (ν˜4ν˜4c) = μ∗ √υd Yν4 + Aν4 Yν4 √υu . ( 15 ) 2 2 For the exotic sneutrino, the mixing of generations 4 and 5 is similar as that of exotic slepton. In the base (ν˜4, ν˜4c∗, ν˜5, ν˜5c∗), we get the mass squared matrix of the exotic sneutrino, and foobrtmaiunlathZe†N˜mMas2sN˜ ZeiNg˜ e=nstdaiteasg(bmy2N˜th1,emm2N˜a2t,rimx 2N˜Z3N,˜ mth2N˜r4o)u.gh the 2.2 The lepton neutralino mass matrix in EBLMSSM In EBLMSSM, the superfields ( L , ϕL , N L , ϕN L ) have their SUSY superpartners (ψ L , ψϕL , ψ N L , ψϕN L ). They mix with λL , which is the superpartner of the new lepton type gauge boson Z μL. Therefore, we deduce their mass matrix in the base (i λL , ψ L , ψϕL , ψ N L , ψϕN L ) ML = ⎜⎜⎜ ⎜ ⎝ ⎛ 2ML 2υL gL −2υ¯ L gL 3υN L gL −3υ¯ N L gL 2υL gL −2υ¯ L gL 3υN L gL −3υ¯ N L gL ⎞ 0 −μL 0 0 ⎟ −μL 0 0 0 ⎟⎟ . 0 0 0 −μN L ⎠⎟ 0 0 0 −μN L The lepton neutralino mass eigenstates are four-component spinors X 0Li = (K L0i , K¯ L0i )T , and their mass matrix is diagonalized by the rotation matrix Z N L . The relations for the components are i λL = Z 1NiL K L0i , ψ L = Z 2NiL K L0i , ψϕL = Z 3NiL K L0i , ψ N L = Z 4NiL K L0i , ψϕN L = Z 5NiL K L0i . In BLMSSM, there are no ψ N L , ψϕN L , and the base of lepton neutralino is (i λL , ψ L , ψϕL ), whose mass matrix is 3 × 3. EBLMSSM extends this matrix to 5 × 5 including the BLMSSM results. 2.3 The Higgs superfields and Y in EBLMSSM The superfields L , ϕL , N L , ϕN L mix together and form 4 × 4 mass squared matrix, which is larger than the corre sponding 2 × 2 mass matrix in the BLMSSM. Diagonalizing the mass squared matrix, four CP even exotic Higgs are obtained. M2φ ( 0L 0L ) = 21 g2L 6υL2 − 2υ¯ L2 + 3(υN2 L − υ¯ N2 L ) 1 2 1 m2 , + 2 μL + 2 L M2φ (ϕL0 ϕL0 ) = 21 g2L 6υ¯ L2 − 2υL2 + 3(υ¯ N2 L − υN2 L ) ( 16 ) ( 17 ) + 2 μL + 21 m2ϕL , 1 2 , We use Zφ˜L to diagonalize the mass squared matrix in Eq. ( 18 ), and the relation between mass eigenstates and the comments are 0L = Zφ1˜iL HL0i , ϕL0 = Zφ2˜iL HL0i , 0 0 N L = Z 3i HL0i , ϕN L = Z 4i HL0i . φ˜L φ˜L In EBLMSSM, the conditions for the exotic CP odd Higgs P0, P¯L0 are same as those in BLMSSM, and they do not mix L with the added exotic CP odd Higgs PN0 L , P¯N0 L . Here, we show the mass squared matrix for the added exotic CP odd Higgs PN0 L , P¯N0 L . M2p( PN0 L PN0 L ) = 21 g2L 29 υN2 L − 29 υ¯ N2 L + 3(υL2 − υ¯ L2 ) 1 2 1 m2 + 2 μN L + 2 N L , M2p( P¯N0 L P¯N0 L ) = 21 g2L 29 υ¯ N2 L − 29 υN2 L + 3(υ¯ L2 − υL2 ) + 2 μN L + 21 m2ϕN L , 1 2 M2p( PN0 L P¯N0 L ) = BN L2μN L . ( 20 ) The scalar superfields Y and Y mix, and their mass squared matrix is deduced here. This condition is similar as that of X and X , then the lightest mass eigenstate of Y and Y can be a candidate of the dark matter. With SY = g2L (2 + L4)VL2, the concrete form for the mass squared matrix is shown here. To obtain mass eigenstates, the matrix ZY is used through the following formula, with the supposi tion m2Y1 < m2Y2 . ZY† |μY |2 + SY −μY BY −μ∗Y BY∗ |μY |2 − SY ZY = m2Y1 0 0 m2Y2 Y1 Y2 = ZY† Y Y ∗ ( 23 ) The superpartners of Y and Y form four-component Dirac spinors, and the mass term for superfields Y˜ is shown as − LmY˜ ass = μY Y˜¯ Y˜ , Y˜ = ψY ψ¯ Y The spinor Y˜ and the mixing of superfields Y, Y are all new terms beyond BLMSSM, that add abundant contents to lepton physics and dark matter physics. 2.4 Some couplings with h0 in EBLMSSM In EBLMSSM, the exotic slepton(sneutrino) of generations 4 and 5 mix. So the couplings with exotic slepton(sneutrino) are different from the corresponding results in BLMSSM. We deduce the couplings of h0 and exotic sleptons 4 − e2υ cos β 14−sW24csW2W2 (Z 1E˜i∗ Z 1E˜j − Z 4E˜i∗ Z 4E˜j ) AE4 Z 2i∗ Z 1 j μ∗ −υ cos β|Ye4 |2δi j − √2 E˜ E˜ − √2 Ye5 Z 4E˜i∗ Z 3E˜j 1 1 Z 1i∗Ye∗4 λE Z 3Ej υN L sin α . − 2 Ye∗4 Z 2E˜j λL Z 4E˜i∗υ¯ N L + 2 E˜ ˜ In Eq. ( 23 ), different from BLMSSM, there are new terms ( 21 λL Ye5 Z 3E˜j Z 3E˜i∗υ¯ N L − 21 Ye∗5 Z 4E˜j λE Z 2E˜i∗υN L ) cos α−( 21 Z 1E˜i∗ iYne∗g4 λoEfZg3Ee˜jnυeNraLti−on21sYe4∗4 Zan2E˜djλ5L Zsl4E˜eip∗υt¯oNnL. )Osbinvαiobueslsyid,ethsethsee mneixwterms include υN L and υ¯ N L , which are the VEVs of added Higgs superfields N L and ϕN L . In the same way, the cou plings of h0 and exotic sneutrinos are also calculated 4 ( 29 ) 3 The mass of h0 3 × vL Zφ1˜kL − v¯L Zφ2˜kL + 2 vN L Zφ3˜kN L HL0k Yi∗Y j . ing lepton Higgs HL0. Then the couplings of HL0Y Y ∗ and χ¯ N0 χ N0 HL0 are needed ( 24 ) ( 25 ) ( 26 ) ( 27 ) Similar as BLMSSM, in EBLMSSM the mass squared matrix for the neutral CP even Higgs are studied, and in the basis (Hd0, Hu0) it is written as 2 Meven = M121 + M122 + 11 M122 + 12 M222 + , where M121, M122, M222 are the tree level results, whose concrete forms can be found in Ref. [19] 1L1, B 11 + B 12 + B 22 + M SSM 11 M SSM 12 M SSM 22 11 = + 12 = + 1L2, 22 = + 2L2. ( 30 ) The MSSM contributions are represented by 1M1SSM , 1M2SSM and 2M2SSM . The exotic quark (squark) contributions denoted by 1B1, 1B2 and 2B2 are the same as those in BLMSSM [19]. However, the corrections 1L1, 1L2 and 2L2 from exotic lepton (slepton) are different from those in BLMSSM, because the mass squared matrices of exotic slepton and exotic sneutrino are both 4 × 4 and they relate with υN L and υ¯ N L . Furthermore, the exotic leptons and exotic neutrinos are heavier than those in BLMSSM, due to the introduction of N L and ϕN L . L G F Yν44 υ4 11 = 4√2π 2 sin2 β · 2.5 The couplings with Y For the dark matter candidate Y1, the necessary tree level couplings are deduced in EBLMSSM. We show the couplings (lepton-exotic lepton-Y ) and (neutrino-exotic neutrino-Y ) L = e¯I λ4WL1i ZY1 j∗ PR − λ6UL2i ZY2 j∗ PL Li+3Y j∗ 2 i, j=1 6 − α=1 i, j=1 3 I =1 3 2 6 X¯ Nα λ4 Z NIαν∗WN1i ZY1 j∗ PR 0 + λ5 Z (NIν+3)αU N2i ZY2 j∗ PL Ni+3Y j∗ + h.c. The new gauge boson Z L couples with leptons, neutrinos and Y , whose concrete forms are L = − gL Z μLe¯I γμeI − gL (2 + L4)Z μLYi∗i ∂μY j 2 ϕL gives masses to the light neutrinos trough the see-saw mechanism and L , ϕL , N L , ϕN L mix together produc G F Ye45 υ4 + 4√2π 2 sin2 β · At the LHC, h0 is produced chiefly from the gluon fusion (gg → h0). The one loop diagrams are the leading order (LO) contributions. The virtual t quark loop is the dominate contribution because of the large Yukawa coupling. Therefore, when the couplings of new particles and Higgs are large, they can influence the results obviously. For h0 → gg, the EBLMSSM results are same as those in BLMSSM, and are shown as [26–28] N P (h0 → gg) = with xa = m2h0 /(4ma2). Here, q and q are quark and exotic quark. While, q˜ and q˜ denote squark and exotic squark. The concrete expressions for gh0qq , gh0q q , gh0q˜q˜ , gh0q˜ q˜ (i = 1, 2) are in literature [19]. The functions A1/2(x ) and A0(x ) are[28] The decay h0 → γ γ obtains contributions from loop diagrams, and the leading order contributions are from the one loop diagrams. In the EBLMSSM, the exotic quark (squark) and exotic lepton (slepton) give new corrections to the decay width of h0 → γ γ . Different from BLMSSM, the exotic leptons in EBLMSSM are more heavy and the exotic sleptons of the 4 and 5 generations mix together. These parts should influence the numerical results of the EBLMSSM theoretical prediction to the process h0 → γ γ to some extent. The decay width of h0 → γ γ can be expressed as [29] ( 33 ) (34) where gh0W W = sin(β − α) and A1(x ) = − 2x 2 + 3x + 3(2x − 1)g(x ) /x 2. The formulae for h0 → Z Z , W W are , , (35) with gh0 Z Z = gh0W W and F (x ) is given out in Refs. [30– 32]. The observed signals for the diphoton and Z Z , W W channels are quantified by the ratios Rγ γ and RV V , V = (Z , W ), whose current values are Rγ γ = 1.16 ± 0.18 and RV V = 1.19+−00..2220 [33]. 4.2 Dark matter Y In BLMSSM, there are some dark matter candidates such as: the lightest mass eigenstate of X X mixing, X˜ the fourcomponent spinor composed by the super partners of X and X . They are studied in Ref. [18]. In EBLMSSM, the dark matter candidates are more than those in BLMSSM, because the lightest mass eigenstate of Y Y mixing and Y˜ are dark matter candidates. After U ( 1 )L is broken by L and N L , Z2 symmetry is left, which guarantees the stability of the dark matters. There are only two elements ( 1, −1 ) in Z2 group. This symmetry eliminates the coupling for the mass eigenstates of Y Y mixing with two SM particles. The condition for X is similar as that of Y , and it is also guaranteed by the Z2 symmetry. In this subsection, we suppose the lightest mass eigenstate of Y Y mixing in Eq. ( 21 ) as a dark matter candidate, and calculate the relic density. So we summarize the relic density constraints that any WIMP candidate has to satisfy. The interactions of the WIMP with SM particles are deduced from the EBLMSSM, then we study its annihilation rate and its relic density D by the thermal dynamics of the Universe. The annihilation cross section σ (Y1Y1∗ → anyt hi ng) should be calculated and can be written as σ vrel = a + bvr2el in the Y1Y1∗ center of mass frame. vrel is the twice velocity of Y1 in the Y1Y1∗ c.m. system frame. To a good approximation, the freeze-out temperature (TF ) can be iteratively computed from[15–17] m D xF = TF ln 0.038MPl m D(a + 6b/xF ) √g∗xF , with xF ≡ m D/ TF and m D = mY1 representing the WIMP mass. MPl = 1.22 × 1019 GeV is the Planck mass and g∗ is the number of the relativistic degrees of freedom with mass less than TF . The density of cold non-baryonic matter is D h2 = 0.1186 ± 0.0020 [33], whose formula is simplified as (36) D h2 1.07 × 109xF √g∗ MP L (a + 3b/xF )GeV . To obtain a and b in the σ vrel , we study the Y1Y1∗ dominate decay channels whose final states are leptons and light neutrinos: ( 1 ) Y1Y1∗ → Z L → l¯I l I ; ( 2 ) Y1Y1∗ → Z L → ν¯ I ν I ; ( 3 ) Y1Y1∗ → ϕL → ν¯ I ν I ; ( 4 ) Y1Y1∗ → L → l¯I l I ; ( 5 ) Y1Y1∗ → N → ν¯ I ν I . Using the couplings in Eqs. ( 26 ), ( 27 ), ( 28 ), we deduce the results of a and b (37) a = l=e,μ,τ 1 π | 2 i=1 0 χNα=νe,νμ,ντ 3 4 I =1 i=1 + × × 1 + π m Li (m2D + m2 ) Li g4L (2 + L4)2 8π 1 (4m2D − m2 i ) × λ4WL1i ZY11∗λ6UL2i Z 21∗ 2 Y | (ZY11∗ ZY11 − Z 21∗ ZY21) Y λN c Z (NIν+3)α Z (NIν+3)α Zφ2iL 3 3 vL Zφ1˜iL − v¯L Zφ2˜iL + 2 vN L Zφ3˜iL − 2 v¯N L Zφ4˜iL 2 2 3 m Ni i=1 I =1 (m2D + m2Ni ) ×λ4 Z NIαν∗WN1i ZY11∗λ5 Z (NIν+3)αU N2i ZY21∗ 2 5.1 h0 decays and m A0 , m H0 In this section, we research the numerical results. For the parameter space, the most strict constraint is that the mass of the lightest eigenvector for the mass squared matrix in Eq. ( 29 ) is around 125.1 GeV. To satisfy this constraint, we use mh0 = 125.1 GeV as an input parameter. Therefore, the CP odd Higgs mass should meet the following relation. m2A0 = m2h0 (m2Z − m2h0 + 11 + 22) − m2Z A + 122 − 11 22 −m2h0 + m2Z cos2 2β + B 1000 Au5 GeV 1000 0 2000 3000 1000 0 2000 3000 where A = sin2 β 11 + cos2 β 22 + sin 2β 12, B = cos2 β 11 + sin2 β 22 + sin 2β 12 . (40) To obtain the numerical results, we adopt the following parameters as Yu4 = 1.2Yt , Yu5 = 0.6Yt , Yd4 = Yd5 = 2Yb, gB = 1/3, λu = λd = 0.5, Au4 = Ad4 = Ad5 = Ae4 = Ae5 = Aν4 = Aν5 = 1 TeV, λQ = 0.4, gL = 1/6, m Q˜ 4 = m Q˜ 5 = mU˜4 = mU˜5 = m D˜ 4 = m D˜ 5 = mν˜4 = mν˜5 = 1 TeV, Ye5 = 0.6, υN L = υL = Ab = 3 TeV, tan βN L = tan βL = 2, λL = λN L = λE = 1, m L˜ = me˜ = 1.4δi j TeV, AL˜ = AL˜ = 0.5δi j TeV (i, j = 1, 2, 3), μB = 0.5 TeV, AB Q = ABU = AB D = μN L = AL L = AL E = AL N = 1 TeV, Yν4 = Yν5 = 0.1, m L˜ 4 = m L˜ 5 = m E˜4 = m E˜5 = m2 = 1.5 TeV, m D˜ 3 = 1.2 TeV, B4 = L4 = 1.5. (41) Here Yt and Yb are the Yukawa coupling constants of top quark and bottom quark, whose concrete forms are Yt = √2mt /(υ sin β) and Yb = √2mb/(υ cos β) respectively. To embody the exotic squark corrections, we calculate the results versus Au5 which has relation with the mass squared matrix of exotic squark. In the left diagram of Fig. 1, Rγ γ and RV V versus Au5 are plotted by the solid line and dashed line respectively with m Q˜ 3 = mU˜3 = 1.2 TeV, tan β = 1.4, At = 1.7 TeV, υB = 3.6 TeV, μ = −2.4 TeV, tan βB = 1.5 and Ye4 = 0.5. In the left diagram of Fig. 1, the solid line (Rγ γ ) and dashed line (RV V ) change weakly with the Au5 . When Au5 enlarges, Rγ γ is the increasing function and RV V is the decreasing function. During the Au5 region (−1700 to 1000) GeV, both Rγ γ and RV V satisfy the experiment limits. The dot-dashed line(dotted line) in the right diagram denotes the Higgs mass m0A(m0H ) varying with Au5 . The dot-dashed line and dotted line increase mildly with Au5 . The value of m0A is a little bigger than 500 GeV, while the value of m0H is very near 500 GeV. For the squark, we assume the first and second generations are heavy, so they are neglected. The scalar top quarks are not heavy, and their contributions are considerable. At is in the mass squared matrix of scalar top quark influencing the mass and mixing. The effects from At to the ratios Rγ γ , RV V , Higgs masses m A0 and m H0 are of interest. As m Q˜ 3 = 2.4 TeV, mU˜3 = 1.2 TeV, tan β = tan βB = 2.15, υB = 4.1 TeV, μ = −2.05 TeV, Ye4 = 0.5 and Au5 = 1 TeV. Rγ γ (solid line) and RV V (dashed line) versus At are shown in the left diagram of Fig. 2. While the right diagram of Fig. 2 gives out the Higgs masses m A0 (dot-dashed line) and m H0 (dotted line). In the At region (2–4.8) TeV, the Rγ γ varies from 1.25 to 1.34. At the same time, the RV V is in the range (1.2–1.38). The dot-dashed line and dotted line are very near. In the At region (3000–4000) GeV, the masses of Higgs A0 and H 0 are around 1000 GeV. In this parameter space, the allowed biggest values of A0 and H 0 masses can almost reach 1350 GeV. Ye4 is the Yukawa coupling constant that can influence the mass matrix of exotic lepton and exotic slepton. We use m Q˜ 3 = mU˜3 = 1.2 TeV, tan β = 2.3, tan βB = 1.77, At = 1.7 TeV, υB = 5.43 TeV, μ = −2.64 TeV, Au5 = 1 TeV and obtain the results versus Ye4 in the Fig. 3. In the left diagram, the Rγ γ (solid line) and RV V (dashed line) are around 1.3 and their changes are small during the Ye4 range (0.05–1). One can see that in the right diagram m A0 (dot-dashed line) and m H0 (dotted line) possess same behavior versus Ye4 . They are both decreasing functions of Ye4 and vary from 1500 to 500 GeV. In general, Ye4 effect to the Higgs masses m A0 and m H0 is obvious. 1.6 3000 3500 At GeV 01500 2000 2500 3000 3500 At GeV 4000 4500 5000 m Q˜ 3 and mU˜3 are the diagonal elements of the squark mass squared matrix, and they should affect the results. Supposing m Q˜ 3 = mU˜3 = MQ , tan β = 2.1, tan βB = 2.24, At = 1.7 TeV, υB = 3.95 TeV, μ = −1.9 TeV, Ye4 = 0.6, Au5 = 1 TeV, we calculate the results versus MQ and plot the diagrams in the Fig. 4. It shows that in this figure the solid line, dashed line, dotted line and dot-dashed line are all stable. Rγ γ and RV V are around 1.2. At the same time m A0 and m H 0 are about 1 TeV. 5.2 Scalar dark matter Y1 Here, we suppose Y1 as a scalar dark matter candidate. In Ref. [33] the density of cold non-baryonic matter is D h2 = 0.1186 ± 0.0020. To obtain the numerical results of dark matter relic density, for consistency the used parameters in this subsection are of the same values as in Eq. (41) if they are supposed. Therefore, we just show the values of the parameters beyond Eq. (41). These parameters are taken as μY = 1500 GeV, λ5 = 1, μL = BL = BN L = 1 TeV, tan β = 1.4, BY = 940 GeV, m2 L = m2ϕL = m2 N L = m2ϕN L = 3 TeV2, Ye4 = 0.5. With the relation λ4 = λ6 = L m, we study relic density D and x F versus L m in the Fig. 5. In the right diagram of Fig. 5, the grey area is the experimental results in 3 σ and the solid line representing D h2 turns small with the increasing L m. During the L m region (0.7–1.4), D h2 satisfies the experi ment bounds of dark matter relic density. x F is stable and in the region (23.5–24). Taking Ye4 = 1.3, λ4 = λ6 = 1 and the other parameters being same as Eq. (42) condition, we plot the relic density(x F ) versus Ye5 in the left (right) diagram of the Fig. 6. In this parameter space, during Ye5 region (0.1–2.5), our theoretical results satisfy the relic density bounds of dark matter, and x F is very near 23.55. Generally speaking, both the solid line and dashed line are very stable. (42) 6 Discussion and conclusion Considering the light exotic lepton in BLMSSM, we add exotic Higgs superfields N L and ϕN L to BLMSSM in order 1.5 2hD0.10 F x F x to make the exotic leptons heavy. Light exotic leptons may be excluded by the experiment in the future. On the other hand, heavy exotic leptons should not be stable. So we also introduce the superfields Y and Y to make exotic leptons decay quickly. The lightest mass eigenstate of Y and Y mixing mass matrix can be a dark matter candidate. Therefore, the exotic leptons are heavy enough to decay to SM leptons and Y at tree level. We call this extended BLMSSM as EBLMSSM, where the mass matrices for the particles are deduced and compared with those in BLMSSM. Different from BLMSSM, the exotic sleptons of 4 and 5 generations mix together forming 4 × 4 mass squared matrix. EBLMSSM has more abundant content than BLMSSM for the lepton physics. To confine the parameter space of EBLMSSM, we study the decays h0 → γ γ and h0 → V V , V = (Z , W ). The CP even Higgs masses mh0 , m H0 and CP odd Higgs mass m0A are researched. In the numerical calculation, to keep mh0 = 125.1 GeV, we use it as an input parameter. In our used parameter space, the values of Rγ γ and RV V both meet the experiment limits. The CP odd Higgs mass m A0 is a little heavier than the CP even Higgs mass m H 0 . Generally speaking, both m A0 and m H 0 are in the region (500– 1500) GeV. Based on the supposition that the lightest mass eigenstate Y1 of Y and Y mixing possesses the character of cold dark matter, we research the relic density of Y1. In our used parameter space, D h2 of Y1 can match the experiment bounds. EBLMSSM has a bit more particles and parameters than those in BLMSSM. Therefore, EBLMSSM possesses stronger adaptive capacity to explain the experiment results and some problems in the theory. In our later work, we shall study the EBLMSSM and confine its parameter space to move forward a single step. Acknowledgements Supported by the Major Project of NNSFC (no. 11535002, no. 11605037, no. 11705045), the Natural Science Foundation of Hebei province with Grant no. A2016201010 and no. 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Shu-Min Zhao, Tai-Fu Feng, Guo-Zhu Ning, Jian-Bin Chen, Hai-Bin Zhang, Xing Xing Dong. The extended BLMSSM with a 125 GeV Higgs boson and dark matter, The European Physical Journal C, 2018, 324, DOI: 10.1140/epjc/s10052-018-5744-x