Two loop electroweak corrections to $\bar B\rightarrow X_sγ$ and $B_s^0\rightarrow μ^+μ^-$ in the B-LSSM
Jin-Lei Yang Tai-Fu Feng Hai-Bin Zhang Rong-Fei Zhu Shu-Min Zhao Xiu-Yi Yang









^aDepartment of Physics, Hebei University, Baoding, 071002, China

^bDepartment of Science, University of Science and Technology Liaoning, Anshan, 114051, China



The rare decays B̅→ X_sγ and B_s^0→μ^+μ^- are important to research new physics beyond standard model. In this work, we investigate two loop electroweak corrections to B̅→ X_sγ and B_s^0→μ^+μ^- in the minimal supersymmetric extension of the SM with local B-L gauge symmetry (B-LSSM), under a minimal flavor violating assumption for the soft breaking terms. In this framework, new particles and new definition of squarks can affect the theoretical predictions of these two processes, with respect to the MSSM. Considering the constraints from updated experimental data, the numerical results show that the B-LSSM can fit the experimental data for the branching ratios of B̅→ X_sγ and B_s^0→μ^+μ^-. The results of the rare decays also further constrain the parameter space of the B-LSSM.




12.60.Jv, 13.20.He

Two loop electroweak corrections to B̅→ X_sγ and B_s^0→μ^+μ^- in the B-LSSM
    Jin-Lei Yang^a[JLYangJL@163.com],
Tai-Fu Feng^a[fengtf@hbu.edu.cn],
Hai-Bin Zhang^a[hbzhang@hbu.edu.cn], Rong-Fei Zhu^a, Shu-Min Zhao^a, Xiu-Yi Yang^b
     
==========================================================================================================================================================





§ INTRODUCTION

The study on B physics is one of the most promising windows to detect the new physics beyond the standard model (SM), since the theoretical evaluations on the relevant physical quantities are not seriously affected by the uncertainties due to the QCD effects. Recently, the average experimental data on the branching ratios of B̅→ X_sγ and B_s^0→μ^+μ^- are shown as <cit.>

    Br(B̅→ X_s γ)=(3.49±0.19)× 10^-4,
          Br(B_s^0→μ^+μ^-)=(2.9_-0.6^+0.7)×10^-9.

The SM predicts the B̅→ X_sγ and B_s^0→μ^+μ^- branching ratios to be <cit.>

    Br(B̅→ X_s γ)=(3.36±0.23)× 10^-4,
          Br(B_s^0→μ^+μ^-)=(3.23±0.27)×10^-9,

which are in agreement with the experimental results very well. So, the precise measurements on the rare B-decay processes constrain the new physics beyond SM strictly.

In extensions of the SM, the supersymmetry is considered as one of the most plausible candidates. Actually, the analyses of constraints on parameters in the minimal supersymmetric extension of the SM (MSSM) are discussed in detail <cit.>. The authors of Refs. <cit.> present the calculation of the rate inclusive decay B→ X_sγ in the two-Higgs doublet model (THDM). The supersymmetric effect on B→ X_sγ is discussed in Refs. <cit.> and the next-to-leading order (NLO) QCD corrections are given in Ref. <cit.>. The branching ratio for B_s→ l^+l^- in THDM and supersymmetric extensions of the SM has been calculated in Refs. <cit.>. The hadronic B decays <cit.> and CP-violation in these processes <cit.> have also been discussed. The authors of Ref. <cit.> have discussed possibility of observing supersymmetric effects in rare decays B→ X_sγ and B→ X_s e^+ e^- at the B-factory. The supersymmetric effects on these processes are very interesting and studies on them may shed some light on the general characteristics of the supersymmetric model. A relevant review can be found in Refs. <cit.>.



The minimal supersymmetric extension of the SM with local B-L gauge symmetry (B-LSSM) <cit.> is based on the gauge symmetry group SU(3)_C⊗ SU(2)_L⊗ U(1)_Y⊗ U(1)_B-L, where B stands for the baryon number and L stands for the lepton number respectively. Besides accounting elegantly for the existence and smallness of the left-handed neutrino masses, the B-LSSM also alleviates the aforementioned little hierarchy problem of the MSSM <cit.>, because the exotic singlet Higgs and right-handed (s)neutrinos <cit.> release additional parameter space from the LEP, Tevatron and LHC constraints. The invariance under U(1)_B-L gauge group imposes the the R-parity conservation which is assumed in the MSSM to avoid proton decay. And R-parity conservation can be maintained if U(1)_B-L symmetry is broken spontaneously <cit.>. Furthermore, it could help to understand the origin of R-parity and its possible spontaneous violation in the supersymmetric models <cit.> as well as the mechanism of leptogenesis <cit.>. Moreover, the model can provide much more candidates for the Dark Matter comparing that with the MSSM <cit.>. In this work, we analyze two loop electroweak corrections to B̅→ X_sγ and B_s^0→μ^+μ^- in the B-LSSM. In this framework, new couplings and particles make new contributions to both of these processes with respect to the MSSM. The numerical results of the rare decays also further constrain the parameter space of the model.

Our presentation is organized as follows. In Sec. II, the main ingredients of B-LSSM are summarized briefly, including the superpotential, the general soft breaking terms, the Higgs sector and so on. Sec. III contains the effective Hamilton for B̅→ X_sγ and B_s^0→μ^+μ^-. The numerical analyses are given in Sec. IV, and Sec. V gives a summary. The tedious formulae are collected in Appendices.




§ THE B-LSSM

In the B-LSSM, one enlarges the local gauge group of the SM to SU(3)_C⊗ SU(2)_L⊗ U(1)_Y⊗ U(1)_B-L, where the U(1)_B-L can be spontaneously broken by the chiral singlet superfields η̂_1 and η̂_2. In literatures there are several popular versions of B-LSSM. Here we adopt the version described in Refs. <cit.> to proceed our analysis, because this version of B-LSSM is encoded in SARAH <cit.> which is used to create the mass matrices and interaction vertexes of the model. Besides the superfields of the MSSM, the exotic superfields of the B-LSSM are three generations right-handed neutrinos ν̂_i^c∼(1, 1, 0, 1) and two chiral singlet superfields η̂_1∼(1,1,0,-1), η̂_2∼(1,1,0,1). Meanwhile, quantum numbers of the matter chiral superfields for quarks and leptons are given by

    Q̂_i=([  Û_i; D̂_i ])∼(3, 2, 1/6, 1/6),    L̂_i=([ ν̂_i;  Ê_i ])∼(1, 2, -1/2, -1/2),
          Û^c_i∼(3, 1, -2/3, -1/6),   D̂^c_i∼(3, 1, 1/3, -1/6),    Ê^c_i∼(1, 1, 1, 1/2),

with i=1,2,3 denoting the index of generation. In addition, the quantum numbers of two Higgs doublets is assigned as

    Ĥ_̂1̂=([ H_1^1; H_1^2 ])∼ (1, 2, -1/2, 0),   Ĥ_̂2̂=([ H_2^1; H_2^2 ])∼(1, 2, 1/2, 0).


The corresponding superpotential of the B-LSSM is written as

    W=W_MSSM+W_(B-L).

Here, W_MSSM is the superpotential of the MSSM, and W_(B-L) is the sector involving exotic superfields,

    W_(B-L)=Y_ν, ijL̂_̂îĤ_̂2̂ν̂^c_j-μ' η̂_1 η̂_2
    +Y_x, ijν̂_i^c η̂_1 ν̂_j^c,

where i, j are generation indices. Correspondingly, the soft breaking terms of the B-LSSM are generally given as

    ℒ_soft=ℒ_MSSM+[-M_BB^'λ̃_B^'λ̃_B -
    1/2M_B^'λ̃_B^'λ̃_B^' -B_μ^'η̃_1 η̃_2 +T_ν^ij H_2 ν̃_i^c L̃_j+T_x^ijη̃_1 ν̃_i^c ν̃_j^c
    
          
    +h.c.]-m_η̃_1^2 |η̃_1|^2-m_η̃_2^2 |η̃_2|^2-m_ν̃,ij^2(ν̃_i^c)^* ν̃_j^c,

with λ_B, λ_B^' denoting the gaugino of U(1)_Y and U(1)_(B-L), respectively.
ℒ_MSSM is the soft breaking terms of the MSSM.

The presence of two Abelian groups gives rise to a new effect absent in the MSSM or other SUSY models with just one Abelian gauge group: the gauge kinetic mixing. It results from the invariance principle which allows the Lagrangian to include a mixing term between the strength tensors of gauge fields associated with the U(1) gauge groups, -κ__Y,BLA__μ^' YA^'μ, BL. Here, A__μ^' Y, A^'μ, BL denote the gauge fields associated with the two U(1) gauge groups, κ__Y,BL is an antisymmetric tensor which includes the mixing of U(1)_Y and U(1)_B-L gauge fields. This mixing couples the B-L sector to the MSSM sector, and even if it is set to zero at M_GUT, it can be induced through renormalization group equations (RGEs) <cit.>.  In practice, it turns out that it is easier to work with non-canonical covariant derivatives instead of off-diagonal field-strength tensors. However, both approaches are equivalent <cit.>. Hence in the following, we consider covariant derivatives of the form

    D_μ=∂_μ-i([  Y, B-L ])
    ([    g__Y,  g__YB^'; g__BY^',   g__B-L ])
    ([  A__μ^' Y; A__μ^' BL ]) ,

where Y, B-L corresponding to the hypercharge and B-L charge respectively. As long as the two Abelian gauge groups are unbroken, we still have the freedom to perform a change of the basis

    D_μ=∂_μ-i([  Y, B-L ])
    ([    g__Y,  g__YB^'; g__BY^',   g__B-L ])R^TR
    ([  A__μ^' Y; A__μ^' BL ]) ,

where R is a 2×2 orthogonal matrix. Choosing R in a proper form, one can write the coupling matrix as

    ([    g__Y,  g__YB^'; g__BY^',   g__B-L ])
    R^T=([ g__1, g__YB;    0,  g__B ]) ,

where g__1 corresponds to the measured hypercharge coupling which is modified in
B-LSSM as given along with g__B and g__YB in Refs. <cit.>. In addition, we can redefine the U(1) gauge fields

    R([  A__μ^' Y; A__μ^' BL ])
    =([  A__μ^Y; A__μ^BL ]) .



The local gauge symmetry SU(2)_L⊗ U(1)_Y⊗ U(1)_B-L breaks down to the electromagnetic symmetry U(1)_em as the Higgs fields receive vacuum expectation values (VEVs):

    H_1^1=1/√(2)(v_1+ ReH_1^1+i  ImH_1^1),
          H_2^2=1/√(2)(v_2+ ReH_2^2+i  ImH_2^2),
          η̃_1=1/√(2)(u_1+ Reη̃_1+i  Imη̃_1),
            η̃_2=1/√(2)(u_2+ Reη̃_2+i  Imη̃_2) .

For convenience, we define u^2=u_1^2+u_2^2,  v^2=v_1^2+v_2^2 and tanβ^'=u_2/u_1 in analogy to the ratio of the MSSM VEVs (tanβ=v_2/v_1). Meanwhile, the charged and neutral gauge bosons acquire the nonzero masses as

    m_W^2=1/4g__2^2v^2,
                  m_Z,Z^'^2=1/8[(g__1^2+g_2^2+g__YB^2)v^2+4g__B^2u^2 
                         ∓√((g__1^2+g__2^2+g__YB^2)^2v^4
    +8(g__YB^2-g__1^2-g__2^2)g__B^2v^2u^2+16g__B^4u^4) ].

Compared the MSSM, this new gauge boson Z' makes new contribution to the process B_s^0→μ^+μ^-. In addition, the charged Higgs mass can be written as

    M_H^±^2=4B_μ(1+tanβ^2)+g_2^2 v_1^2tanβ(1+tanβ^2)/4tanβ.


In the basis (ReH_1^1, ReH_2^2, Reη̃_1, Reη̃_2),
the tree level mass squared matrix for Higgs bosons is given by

    M_h^2=u^2×
          ([        1/4g^2 x^2/1+tanβ^2+n^2tanβ        -1/4g^2x^2tanβ/1+tan^2β-n^2                    1/2g__Bg__YBx/T              -1/2g__Bg__YBxtanβ'/T; [6pt]
    -1/4g^2 x^2tanβ/1+tan^2β-n^2 1/4g^2tan^2β x^2/1+tanβ^2+n^2/tanβ                1/2g__Bg__YBxtanβ/T           1/2g__Bg__YBxtanβtanβ'/T;              [6pt]
    1/2g__Bg__YBx/T                1/2g__Bg__YBxtanβ/T          g__B^2/1+tan^2β'+tanβ'N^2         -g__B^2tanβ'/1+tan^2β'-N^2;        [6pt]
    -1/2g__Bg__YBxtanβ'/T           1/2g__Bg__YBxtanβtanβ'/T        -g__B^2tanβ'/1+tan^2β^'-N^2  g__B^2tan^2β'/1+tan^2β'+N^2/tanβ' ]),

where g^2=g__1^2+g__2^2+g__YB^2, T=√(1+tan^2β)√(1+tan^2β'),
n^2= ReBμ/u^2, N^2= ReBμ^'/u^2, and x=v/u, respectively. These new extra singlets η̃_1,2 and the corresponding pseudo-scalar Higgs boson make new contributions to the process B_s^0→μ^+μ^-, with respect to the MSSM.

Including the leading-log radiative corrections from stop and top particles, the mass of the SM-like Higgs boson can be written as <cit.>

    Δ m_h^2=3m_t^4/2π v^2[(t̃+1/2+X̃_t)+1/16π^2(3m_t^2/2v^2-32πα_3)(t̃^2
    +X̃_t t̃)],
          t̃=logM_S^2/m_t^2,     X̃_t=2Ã_t^2/M_S^2(1-Ã_t^2/12M_S^2),

where α_3 is the strong coupling constant, M_S=√(m_t̃_1m_t̃_2) with m_t̃_1,2 denoting the stop masses, Ã_t=A_t-μβ with A_t=T_u,33 being the trilinear Higgs stop coupling and μ denoting the Higgsino mass parameter. Then the SM-like Higgs mass can be written as

    m_h=√((m_h_1^0)^2+Δ m_h^2),

where m_h_1^0 denotes the lightest tree-level Higgs boson mass.


Meanwhile, due to the gauge kinetic mixing, additional D-terms contribute to the mass matrices of the squarks and sleptons, and up type squarks affect the subsequent analysis. On the basis (ũ_L, ũ_R), the mass matrix of up type squarks can be written as

    m_ũ^2=
    ([                        m_uL, 1/√(2)(v_2 T_u^†-v_1μ Y_u^†);  1/√(2)(v_2 T_u-v_1μ^* Y_u),                         m_uR ]),

where,

    m_uL=m_q̃^2+1/24[2g__B(g__B+g__YB)(u _2^2-u_1^2)+g__YB(g__B+g__YB)(v_2^2-v_1^2)
                  +(-3g__2^2+g__1^2)(v_2^2-v_1^2)]+v_2^2/2Y_u^† Y_u,
          m_uR=m_ũ^2+1/24[g__B(g__B+4g__YB)(u _1^2-u_2^2)+g__YB(g__B+4g__YB)(v_1^2-v_2^2)
                  +4g__1^2(v_1^2-v_2^2)]+v_1^2/2Y_u^† Y_u.

It can be noted that tanβ' and new gauge coupling constants g__B, g__YB in the B-LSSM can affect the mass matrix of up type squarks. Since up type squarks appear in the loops of the processes B̅→ X_sγ and B_s^0→μ^+μ^-, new definition of them affects the predictions of Br(B̅→ X_sγ) and Br(B_s^0→μ^+μ^-) by influencing their masses and the corresponding couplings.






§ THEORETICAL CALCULATION ON BR(B̅→ X_SΓ) AND BR(B_S^0→Μ^+Μ^-)

The effective Hamilton for the transition b→ s at hadronic scale can be written as

    H_eff=-4G_F/√(2)V_ts^∗ V_tb[C_1𝒪^c_1+C_2𝒪_2^c+∑_i=3^6𝒪_i+∑_i=7^10(C_i𝒪_i+C'_i𝒪'_i)
                  +∑_i=S,P(C_i𝒪_i+C'_i𝒪'_i)],

where 𝒪_i(i=1, 2,...,10, S, P) and 𝒪'_i(i=7, 8,...,10, S, P) are defined as <cit.>

    O__1^u=(s̅__Lγ_μ T^au__L)(u̅__Lγ^μ T^ab__L) ,   O__2^u=(s̅__Lγ_μ u__L)(u̅__Lγ^μ b__L) ,
    
           O__3=(s̅__Lγ_μ b__L)∑_q(q̅γ^μ q) ,   O__4=(s̅__Lγ_μ T^ab__L)∑_q(q̅γ^μ T^aq) ,
    
           O__5=(s̅__Lγ_μγ_νγ_ρ b__L)∑_q(q̅γ^μγ^νγ^ρ q) ,   O__6=(s̅__Lγ_μγ_νγ_ρ T^ab__L)∑_q(q̅γ^μγ^νγ^ρ T^aq) ,
    
           O__7=e 16π^2m__b(s̅__Lσ__μνb__R)F^μν ,   O__7'=e 16π^2m__b(s̅__Rσ__μνb__L)F^μν ,  
           O__8=g__s 16π^2m__b(s̅__Lσ__μνT^ab__R)G^a,μν ,   O__8'=g__s 16π^2m__b(s̅__Rσ__μνT^ab__L)G^a,μν ,  
           O__9=e^2 g__s^2(s̅__Lγ_μ b__L)l̅γ^μ l ,   O__9'=e^2 g__s^2(s̅__Rγ_μ b__R)l̅γ^μ l ,  
           O__10=e^2 g__s^2(s̅__Lγ_μ b__L)l̅γ^μγ_5 l ,   O__10'=e^2 g__s^2(s̅__Rγ_μ b__R)l̅γ^μγ_5 l ,  
           O__S=e^216π^2m__b(s̅__Lb__R)l̅l ,   O__S'=e^216π^2m__b(s̅__Rb__L)l̅l ,  
           O__P=e^216π^2m__b(s̅__Lb__R)l̅γ_5l ,   O__P'=e^216π^2m__b(s̅__Rb__L)l̅γ_5l ,

where g_s denotes the strong coupling, F^μν are the electromagnetic  field strength tensors, G^μν are the gluon field strength tensors, T^a (a=1,...,8) are SU(3) generators.



 §.§ Rare decay B̅→ X_sγ




Compared with the SM, the main one loop Feynman diagrams contributing to the process B̅→ X_sγ in the B-LSSM are shown in Fig. <ref>. The picture shows that, new definition of the up type squarks affects the prediction of the process B̅→ X_sγ, with respect to the MSSM. In addition, since the two loop electroweak corrections from closed squark loop are highly suppressed by heavy squark masses, we consider the corrections from closed fermion loop, and the corresponding Feynman diagrams are shown in Fig. <ref>. From the picture, we can see that, compared with the MSSM, new neutralinos in the B-LSSM make contributions to the process B̅→ X_sγ.




Then the branching ratio of B̅→ X_sγ in the B-LSSM can be written as

    Br(B̅→ X_sγ)=R(|C_7γ(μ_b)|^2
    +N(E_γ)) ,

where the overall factor R=2.47×10^-3, and the nonperturbative contribution
N(E_γ)=(3.6±0.6)×10^-3<cit.>. C_7γ(μ_b) is defined by

    C_7γ(μ_b)=C_7γ,SM(μ_b)+C_7,NP(μ_b),

where we choose the hadron scale μ_b=2.5 GeV and use the SM contribution at NNLO level C_7γ,SM(μ_b) = -0.3689 <cit.>. The Wilson coefficients for new physics at the bottom quark scale can be written as <cit.>

    C_7,NP(μ_b)≈0.5696
    C_7,NP(μ_EW)+0.1107 C_8,NP(μ_EW),

where

    C_7,NP^NP(μ_EW)=C_7,NP^(1)(μ_EW)+C_7,NP^(2)(μ_EW)+
    C_7,NP^(3)(μ_EW)+C_7,NP^(4)(μ_EW)+
                         C_7,NP^'(1)(μ_EW)+C_7,NP^'(2)(μ_EW)+C_7,NP^'
    (3)(μ_EW)+C_7,NP^'(4)(μ_EW)+
                         C_7,NP^WW(μ_EW)+C_7,NP^WH(μ_EW),
          C_8,NP(μ_EW)=C_8g,NP(μ_EW)+C_8g,NP^'(μ_EW)+C_8,NP^WW(μ_EW)+C_8,NP^WH(μ_EW),

 where C_7,NP^(1,..,4)(μ_EW), C_7,NP^WW(μ_EW), C_7,NP^WH(μ_EW) are Wilson coefficients of the process b→ sγ corresponding to Fig. <ref>, Fig. <ref>, and the concrete expressions are collected in Appendix <ref>. Eq.(<ref>) indicates that the corrections to the Wilson coefficients from Fig. <ref>(2, 3) are suppressed. In addition, C_8g,NP(μ_EW), C_8g,NP^'(μ_EW), C_8,NP^WW(μ_EW), C_8,NP^WH(μ_EW) are Wilson coefficients of the process b→ sg in the B-LSSM. Compared with the SM, the corresponding Feynman diagrams are shown in Fig. <ref>(2, 3), Fig. <ref>(1, 2, 3), then C_8g,NP(μ_EW) and C_8g,NP^'(μ_EW) at electroweak scale are
 
    C_8g,NP(μ_EW)=[C_7,NP^(2)(μ_EW)+C_7,NP^(3)(μ_EW)]/Q_u+C_8,NP^WW(μ_EW)+
    C_8,NP^WH(μ_EW), 
          C_8g,NP^'(μ_EW)=C_8g,NP(μ_EW)(L↔ R),

where Q_u=2/3. And the concrete expressions of C_8,NP^WW(μ_EW), C_8,NP^WH(μ_EW) are collected in Appendix <ref>. Compared with the MSSM, new neutralinos make contributions to the process B̅→ X_sγ. In addition, tanβ', g__B and gauge kinetic mixing g__YB in the B-LSSM might influence the theoretical prediction on Br(B̅→ X_sγ) through affecting the up type squarks masses and the corresponding couplings.




 §.§ Rare decay B_s^0→μ^+μ^-


The main Feynman diagrams contributing to the process B_s^0→μ^+μ^- in the B-LSSM are shown in Fig. <ref>. Compared with the MSSM, Fig. <ref> shows that new definition of up type squarks, new gauge boson Z', new scalar and pseudo-scalar Higgs bosons make new contributions to the processes B_s^0→μ^+μ^- in the B-LSSM.

In addition, since the contributions from Fig. <ref>(2, 3) are suppressed, we also consider the two loop electroweak corrections from Fig. <ref>(1), a virtual photon, Z boson or Z' boson is attached in all possible ways. At the electroweak energy scale μ_EW, the corresponding Wilson coefficients can be written as

    C__S,NP(μ__ EW)=√(2)s__Wc__W/4m_be^3V_ts^*V_tb[C__S,NP^(1)(μ__ EW)+C__S,NP^(2)(μ__ EW)+C__S,NP^(3)(μ__ EW)+C__S,NP^(4)(μ__ EW)
                         +C__S,NP^(6)(μ__ EW)+C__S,NP^(9)(μ__ EW)+C__S,NP^(11)(μ__ EW)],
          C__S,NP^'(μ__ EW)=C__S,NP(μ__ EW)(L↔ R),
    
          C__P,NP(μ__ EW)=√(2)s__Wc__W/4m_be^3V_ts^*V_tb[C__P,NP^(1)(μ__ EW)+C__P,NP^(2)(μ__ EW)+C__P,NP^(3)(μ__ EW)+C__P,NP^(4)(μ__ EW)
                         +C__P,NP^(6)(μ__ EW)+C__P,NP^(9)(μ__ EW)+C__P,NP^(11)(μ__ EW)],
          C__P,NP^'(μ__ EW)=-C__P,NP(μ__ EW)(L↔ R),
    
          C__9,NP(μ__ EW)=√(2)s__Wc__Wg__s^2/64π^2e^3V_ts^*V_tb[C__9,NP^(5)(μ__ EW)+C__9,NP^(6)(μ__ EW)+C__9,NP^(7)(μ__ EW)+C__9,NP^(8)(μ__ EW)
                         +C__9,NP^(9)(μ__ EW)+C__9,NP^(10)(μ__ EW)+C__9,NP^WW(μ__ EW)] ,
          C__9,NP^'(μ__ EW)=C__9,NP(μ__ EW)(L↔ R),
    
          C__10,NP(μ__ EW)=√(2)s__Wc__Wg__s^2/64π^2e^3V_ts^*V_tb[C__10,NP^(5)(μ__ EW)+C__10,NP^(6)(μ__ EW)+C__10,NP^(7)(μ__ EW)+C__10,NP^(8)(μ__ EW)
                         +C__10,NP^(9)(μ__ EW)+C__10,NP^(10)(μ__ EW)+C__10,NP^WW(μ__ EW)] ,
          C__10,NP^'(μ__ EW)=-C__10,NP(μ__ EW)(L↔ R).

Here, the superscripts (1,...,11,WW) corresponding to the new physics corrections in Fig. <ref> and Fig. <ref>(1), and the concrete expressions of these Wilson coefficients can be found in Appendix <ref>. The Wilson coefficients at hadronic energy scale from the SM to next-to-next-to-logarithmic (NNLL) accuracy are shown in Table I.

In addition, The Wilson coefficients in Eq.(<ref>) are calculated at the matching scale μ_EW, then evolved down to hadronic scale μ∼ m_b by the renormalization group equations:

    C__NP(μ)=U(μ,μ_0)C__NP(μ_0)
     ,
    
          C^'__NP(μ)=U^'(μ,μ_0)
    C^'__NP(μ_0)

with

    C__NP^T=(C__1,NP, ⋯, C__6,NP,
    C__7,NP^eff, C__8,NP^eff, C__9,NP^eff-Y(q^2), 
    C__10,NP^eff)
     ,
    
          C__NP^', T=(C__7,NP^', eff, 
    C__8,NP^', eff, C__9,NP^', eff, 
    C__10,NP^', eff) .

Correspondingly, the evolving matrices are approached as

    U(μ,μ_0)≃1-[12β_0lnα__s(μ)α__s(μ_0)]γ^(0)T ,
    
          U^'(μ,μ_0)≃1-[12β_0lnα__s(μ)α__s(μ_0)]γ^'^(0)T ,

where the anomalous dimension matrices can be read from Ref. <cit.> as

    γ^(0)=([        -4        83         0       -29         0         0   -208243    173162  -2272729         0;        12         0         0        43         0         0     41681      7027   1952243         0;         0         0         0      -523         0         2    -17681      1427  -6752243         0;         0         0      -409     -1009        49        56   -152243   -587162  -2192729         0;         0         0         0     -2563         0        20   -627281    659627 -84032243         0;         0         0     -2569       569       409       -23   4624243    477281 -37856729         0;         0         0         0         0         0         0       323         0         0         0;         0         0         0         0         0         0      -329       283         0         0;         0         0         0         0         0         0         0         0         0         0;         0         0         0         0         0         0         0         0         0         0;           ])
     ,
    
          γ^'^(0)=([  323    0    0    0; -329  283    0    0;    0    0    0    0;    0    0    0    0;      ]) .


Then, the squared amplitude can be written as

    |ℳ_s|^2=16G_F^2|V_tbV_ts^*|^2M_B_s^0^2[|F_S^s|^2+|F_P^s+2m_μF_A^s|^2],

and

    F_S^s=α_EW(μ_b)/8πm_b M_B_s^0^2/m_b+m_sf_B_s^0(C_S-C_S'),
          F_P^s=α_EW(μ_b)/8πm_b M_B_s^0^2/m_b+m_sf_B_s^0(C_P-C_P'),
          F_A^s=α_EW(μ_b)/8πf_B_s^0[C_10^eff(μ_b)-C_10^' eff(μ_b)],

where f_B_s^0=(227±8) MeV denote the decay constants, M_B_s^0=5.367  GeV denote the masses of neutral meson B_s^0.


The branching ratio of B_s^0→μ^+μ^- can be written as

    Br(B_s^0→μ^+μ^-)=τ_B_s^0/16π|ℳ_s|^2/M_B_s^0√(1-4m_μ^2/M_B_s^0^2),

with τ_B_s^0=1.466(31) ps denoting the life time of meson.







§ NUMERICAL ANALYSES

In this section, we present the numerical results of the branching ratios of
rare B-decays B̅→ X_sγ and B_s^0→μ^+μ^-. The relevant SM input parameters are
chosen as m_W=80.385 GeV, m_Z=90.19 GeV, α_em(m_Z)=1/128.9, α_s(m_Z)=0.118,
 m_b=4.65 GeV, m_s=0.095 GeV. Since the tiny neutrino masses basically do not affect Br(B̅→ X_sγ) and Br(B_s^0→μ^+μ^-), we take Y_ν=Y_x=0 approximately. The SM-like Higgs mass is <cit.>

    m_h=125.09±0.24 GeV.

Meanwhile the CKM matrix is

    ([         0.97417          0.2248      4.09×10^-3;     [6pt]
    -0.22           0.995      4.05×10^-2; [6pt]
    8.2×10^-3        -4×10^-2           1.009 ]).


The updated experimental data <cit.> on searching Z^' indicates M_Z^'≥4.05 TeV at 95% Confidence Level (CL). Due to the contributions of heavy Z' boson are highly suppressed, we choose M_Z'=4.2 TeV in our following numerical analysis. And Refs. <cit.> give us an upper bound on the ratio between the Z^' mass and its gauge coupling at 99% CL as

    M_Z^'/g__B≥6 TeV .

Then the scope of g__B is limited to 0<g__B≤0.7. Additionally, the LHC experimental data also constrain tanβ^'<1.5 <cit.>. Considering the constraints from the experiments <cit.>, for those parameters in Higgsino and gaugino sectors, we appropriately fix M_1=500 GeV, M_2=600 GeV, μ=700 GeV, μ^'=800 GeV, M_BB^'=500 GeV, M_BL=600 GeV, for simplify. For those parameters in the soft breaking terms, we set  B_μ'=5×10^5  GeV^2, m_l̃=m_ẽ=T_ν=T_x=diag(1, 1, 1)  TeV. In addition, the first two generations of squarks are strongly constrained by direct searches at the LHC <cit.> and the third generation squark masses are not constrained by the LHC as strong as the first two generations, and affect the SM-like Higgs mass. Therefore we take m_q̃=m_ũ=diag(2, 2, m_t̃) TeV, and the discission about the observed Higgs signal in Ref. <cit.> limits m_t̃≳1.5 TeV. For simplify, we also choose T_u_1,2=1 TeV. As a key parameter, T_u_3=A_t affects SM-like Higgs mass and the following numerical calculation.




In the scenarios of the MSSM, the new physics contributions to the branching ratios of B̅→ X_sγ and B_s^0→μ^+μ^- depend essentially on A_t, tanβ and charged Higgs mass M_H^±. In order to see how A_t, tanβ, M_H^± affect the theoretical evaluations of Br(B̅→ X_sγ) and Br(B_s^0→μ^+μ^-) in the B-LSSM, we assume that g__B=0.4, g__YB=-0.5, tanβ'=1.1 in the following analysis. Then taking m_t̃=1.6 TeV, M_H^±=1.5 TeV, we plot Br(B̅→ X_sγ) and Br(B_s^0→μ^+μ^-) versus A_t in Fig. <ref>, for tanβ=5(solid line), tanβ=20(dashed line) and tanβ=35(dotted line), respectively. The gray area denotes the experimental 1σ bounds in Eq.(<ref>). In Fig. <ref>(a), we can see that Br(B̅→ X_sγ) decreases with the increasing of A_t, and Br(B̅→ X_sγ) will be easily coincides with experimental data within one standard deviation when A_t is positive. Meanwhile, Fig. <ref>(b) shows that Br(B_s^0→μ^+μ^-) favor A_t in the ranges -3.6 TeV≲ A_t≲1.4 TeV as tanβ=5, -1.2 TeV≲ A_t≲0.2 TeV as tanβ=20 and -0.8 TeV≲ A_t≲0 TeV as tanβ=35, which also coincide with the experimental data on Br(B̅→ X_sγ). It can be noted that when tanβ is large, the range of A_t is limited strongly by the experimental data on Br(B_s^0→μ^+μ^-).

Then, in order to see the difference between two loop and one loop corrections to the processes, we take tanβ=20, and plot Br(B̅→ X_sγ), Br(B_s^0→μ^+μ^-) versus A_t in Fig. <ref>. The solid and dashed line denote two loop and one loop predictions respectively.

Fig. <ref>(a) shows that, the relative corrections from two loop diagrams to one loop corrections of Br(B̅→ X_sγ) can reach around 3%, which produces a more precise prediction on the process B̅→ X_sγ, and we cannot neglect the corrections with this magnitude. In Fig. <ref>(b), these two lines coincide with each other, which indicates the two loop corrections to Br(B_s^0→μ^+μ^-) are negligible compared with the one loop corrections. In the analysis of the numerical results, we use the more precise two loop predictions.





In addition, we also need to consider the constraint of the SM-like Higgs mass. Taking tanβ=20, we plot the SM-like Higgs mass m_h versus A_t in Fig. <ref> for m_t̃=1.5 TeV(solid line), m_t̃=2.5 TeV(dashed line) and m_t̃=3.5 TeV(dotted line). The gray area denotes the experimental 3σ interval. To keep the SM-like Higgs mass around 125 GeV, we need A_t≈±1.8 TeV as m_t̃=1.5 TeV. When m_t̃=2.5 TeV, we require that A_t in the range -2.5 TeV≲ A_t≲1.6 TeV or 1.6 TeV≲ A_t≲2.5 TeV. And the allowed range of A_t is -2.5 TeV≲ A_t≲2.5 TeV when m_t̃=3.5 TeV.



Since the large charged Higgs mass does not affect the SM-like Higgs mass signally, we can choose A_t=-2.5 TeV, -0.5 TeV and 1.5 TeV for m_t̃=3.5 TeV, to keep the SM-like Higgs mass around 125 GeV. Then we plot Br(B̅→ X_sγ) and Br(B_s^0→μ^+μ^-) versus M_H^± in Fig. <ref>, where the solid line, dashed line and dotted line denote A_t=-2.5 TeV, -0.5 TeV, 1.5 TeV, respectively. The gray area denotes the experimental 1σ bounds.

It is obvious that A_t affect the numerical results negligibly compared with M_H^±, because large m_t̃ suppresses the effects of A_t. In addition, Fig. <ref>(a) shows that Br(B̅→ X_sγ) decreases along with the increasing of M_H^±, because the contributions from charged Higgs diagrams decay like 1/M_H^±^4<cit.>. And the experimental data on Br(B̅→ X_sγ) favor M_H^± lying in the ranges M_H^±≳0.7 TeV. Meanwhile, in Fig. <ref>(b), we can see that for small M_H^±, the new physics could contribute with large corrections to the branching ratio of B_s^0→μ^+μ^-, and the experimental data on Br(B_s^0→μ^+μ^-) limits that M_H^±≳0.8 TeV.

In order to see the effects of g__B, g__YB and tanβ' which are new parameters in the B-LSSM compared with MSSM, we take tanβ=20, m_t̃=1.6 TeV, A_t=-0.5 TeV, M_H^±=1.5 TeV and g__B=0.2. Then we plot Br(B̅→ X_sγ) and Br(B_s^0→μ^+μ^-) varying with tanβ' in Fig. <ref>, for g__YB=-0.6( solid  line), g__YB=-0.5( dashed  line), g__YB=-0.4( dotted  line), respectively. Considering the constraint from concrete Higgs boson mass, the allowed range of tanβ' is 1.12<tanβ'<1.5. The gray area denotes the experimental 1σ bounds.

The picture shows that the numerical results of Br(B̅→ X_sγ) depend on g__YB or tanβ' weakly in our chosen parameter space. Meanwhile, Br(B_s^0→μ^+μ^-) depends on tanβ' more acutely when |g__YB| is lager, and can exceed the experimental 1σ upper bound easily with the increasing of tanβ' when g__YB=-0.6, which indicates that the effect of tanβ' to the process is influenced by the strength of gauge kinetic mixing strongly. tanβ' affects Br(B_s^0→μ^+μ^-) mainly through the new definition of up type squarks, new scalar and pseudoscalar Higgs bosons. In addition, Br(B_s^0→μ^+μ^-) increases with the increasing of |g_YB| and depend on g_YB acutely when tanβ' is large.


Then we take tanβ'=1.2, and plot Br(B̅→ X_sγ) and Br(B_s^0→μ^+μ^-) varying with g__B in Fig. <ref>, for g__YB=-0.6( solid  line), g__YB=-0.5( dashed  line), g__YB=-0.4( dotted  line), respectively. Considering the constraints from the concrete Higgs boson mass, the allowed range of g__B is 0.13<g__B<0.7. The gray area denotes the experimental 1σ bounds.

It can be noted that g__B and g__YB do not affect Br(B̅→ X_sγ) obviously. And Br(B_s^0→μ^+μ^-) can exceed the experimental 1σ upper bound easily when g__B is small and |g__YB| is large. In addition, with the decreasing of |g__YB|, Br(B_s^0→μ^+μ^-) depends on g__B negligibly, which indicates that the effect of g__B to the process is influenced by the strength of gauge kinetic mixing strongly. g__B and g__YB affect Br(B_s^0→μ^+μ^-) mainly in two ways. Firstly,
g__B and g__YB affect the up type squark masses and the corresponding rotation matrix, which appears in the couplings involve the up type squarks. Secondly, they affect the process B_s^0→μ^+μ^- by influencing the new contributions from Z' gauge boson, new scalar and pseudoscalar Higgs bosons in Fig. <ref>.








§ SUMMARY

Rare B-meson decays offer high sensitivity to new physics beyond SM. In this work, we study the two loop electroweak corrections to the branching ratios Br(B̅→ X_sγ) and Br(B_s^0→μ^+μ^-) in the framework of the B-LSSM under a minimal flavor violating assumption. Considering the constraint from the observed Higgs signal and updated experimental data, the numerical analyses indicate that the corrections from two loop diagrams to the process B̅→ X_sγ can reach around 3%, which produces a more precise prediction on the process B̅→ X_sγ. Nevertheless, the corrections from two loop diagrams to the process B_s^0→μ^+μ^- are negligible, compared with the one loop corrections. Meanwhile, the new physics can fit the experimental data for the rare decay B̅→ X_sγ and B_s^0→μ^+μ^-, and also further constrain the parameter space of the model. Under our assumption on parameters of the considered model, A_t, tanβ, M_H^± affect the theoretical predictions on Br(B̅→ X_sγ) and Br(B_s^0→μ^+μ^-) obviously. And when tanβ is large, A_t is limited strongly by the experimental data for Br(B_s^0→μ^+μ^-). In addition, tanβ', g__YB and g__B can also affect theoretical predictions on Br(B_s^0→μ^+μ^-) obviously.




The work has been supported by the National Natural Science Foundation of China (NNSFC) with Grants No. 11535002,  No. 11705045, and No. 11647120, Natural Science Foundation of Hebei province with Grants No. A2016201010 and No. A2016201069, Foundation of Department of Education of Liaoning
province with Grant No. 2016TSPY10, Youth
Foundation of the University of Science and Technology
Liaoning with Grant No. 2016QN11, Hebei Key Lab of Optic-Eletronic Information and Materials, and the Midwest Universities Comprehensive Strength Promotion
project.







§ THE WILSON COEFFICIENTS OF THE PROCESS B̅→ X_SΓ. 

The one loop Wilson coefficients corresponding to b→ sγ can be written as

    C_7,NP^(1)(μ_EW)=∑_H^-_i,u_js_W^2/2e^2V^*_tsV_tb{1/2C_H^-_is̅ u_j^R C_H^-_i b u̅_j^L[-I_3(x_u_j,x_H^-_i)+I_4(x_u_j,x_H^-_i)]+
                        m_u_j/m_bC_H^-_is̅ u_j^L C_H^-_i b u̅_j^L[-I_1(x_u_j,x_H^-_i)+I_3(x_u_j,x_H^-_i)]},
          C_7,NP^(2)(μ_EW)=∑_H^-_j,u_is_W^2/3e^2V^*_tsV_tb{1/2C_H^-_js̅ u_i^R C_H^-_j b u̅_i^L[-I_1(x_u_i,x_H^-_j)+2I_3(x_u_i,x_H^-_j)
                        -I_4(x_u_i,x_H^-_j)]+m_u_i/m_bC_H^-_js̅ u_i^L C_H^-_j b u̅_i^L[I_1(x_u_i,x_H^-_j)-I_2(x_u_i,x_H^-_j)
                        -I_3(x_u_i,x_H^-_j)]},
          C_7,NP^(3)(μ_EW)=∑_U^+_i,χ^-_js_W^2/3e^2V^*_tsV_tb{1/2C_U^+_is̅χ^-_j^R C_U^+_i b χ̅^-_j^L[-I_3(x_χ^-_j,x_U^+_i)+I_4(x_χ^-_j,x_U^+_i)]+
                        m_χ^-_j/m_bC_U^+_is̅χ^-_j^L C_U^+_i b χ̅^-_j^L[-I_1(x_χ^-_j,x_U^+_i)+I_3(x_χ^-_j,x_U^+_i)]},
          C_7,NP^(4)(μ_EW)=∑_U^+_j,χ^-_is_W^2/2e^2V^*_tsV_tb{1/2C_U^+_is̅χ^-_i^R C_U^+_j b χ̅^-_i^L[-I_1(x_χ^-_i,x_U^+_j)+2I_2(x_χ^-_i,x_U^+_j)
                        -I_4(x_χ^-_i,x_U^+_j)]+m_χ^-_i/m_bC_U^+_js̅χ^-_i^L C_U^+_j b χ̅^-_i^L[I_1(x_χ^-_i,x_U^+_j)-I_2(x_χ^-_i,x_U^+_j)
                        -I_3(x_χ^-_i,x_U^+_j)]},
          C_7^' NP(a)(μ_EW)=C_7^' NP(a)(μ_EW)(L↔ R), (a=1,2,3,4),

where S dentes CP-even and CP-odd Higgs, C_abc^L,R denotes the constant parts of the interaction vertex about abc, which can be got through SARAH, and a, b, c denote the interactional particles. L and R in superscript denote the left-hand part and right-hand part. Denoting x_i=m_i^2/m_W^2, the concrete expressions for I_k(k=1,...,4) can be given as:

    I_1(x_1,x_2)=1+ ln x_2/(x_2-x_1)+x_1  ln x_1-x_2  ln x_2/(x_2-x_1)^2,
          I_2(x_1,x_2)=-1+ ln x_1/(x_2-x_1)-x_1  ln x_1-x_2  ln x_2/(x_2-x_1)^2,
          I_3(x_1,x_2)=1/2[3+2 ln x_2/(x_2-x_1)-2x_2+4x_2 ln x_2/(x_2-x_1)^2-2x_1^2 ln x_1/(x_2-x_1)^3+2x_2^2 ln x_2/(x_2-x_1)^3],
          I_4(x_1,x_2)=1/4[11+6 ln x_2/(x_2-x_1)-15+18x_2 ln x_2/(x_2-x_1)^2+6x_2^2+18x_2^2 ln x_2/(x_2-x_1)^3+
                    6x_1^3 ln x_1-6x_2^3 ln x_2/(x_2-x_1)^4].


Assuming m_χ_i^±, m_χ_j^0≫ m__W, then the two loop Wilson coefficients corresponding to b→ sγ can be simplified as

    C_7,NP^WW(μ_EW)=∑_χ^±_i,χ^0_j-1/8π^2{ (|C_W^-χ̅^0_j χ^+_i^L|^2 +|C_W^-χ̅^0_j χ^+_i^R|^2)[ P(1/8,1/4,-1/48,-1/144,1,x_t)+
                        2/3P(-11/12,-29/72,-1/12,1/144,1,x_t)]+(C_W^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L*+C_W^-χ̅^0_j χ^+_i^L
                         C_W^-χ̅^0_j χ^+_i^R*)[ P(1/16,1/4,-1/16,1/144,1,x_t)+2/3P(-11/12,-29/72,
    -1/12,1/144,1,
                         x_t)]+1/8(C_W^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^R*-C_W^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L*)∂^2 ρ_2,1/∂ x_1^2(x_W,x_t)},
          C_8,NP^WW(μ_EW)=∑_χ^±_i,χ^0_j-3/8π^2{ (|C_W^-χ̅^0_j χ^+_i^L|^2 +|C_W^-χ̅^0_j χ^+_i^R|^2)P(-11/12,-29/72,-1/12,1/144,1,x_t)+
                        (C_W^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L*+C_W^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^R*)P(-1/12,-5/24,1/12,-1/144,1,x_t)+
                        (C_W^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L*-C_W^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^R*)P(1/16,7/24,0,0,1,x_t)},
          C_7,NP^WH(μ_EW)=∑_χ^±_i,χ^0_jC_H^-d̅ u^Lm_W^2/4√(2)π^2m_dm_fV_ud^*{[(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^L+C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^R)-
                         i(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^L+C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^R)](21/64-5/288+5/24J(m_W^2,
                         M_H^±^2,m_t^2))+[(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^R+C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L)-
                         i(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^R+C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L)](-1/144-1/24J(m_W^2,
                         M_H^±^2,m_t^2))-[(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^L-C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^R)-
                         i(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^L-C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^R)](16/144+1/6J(m_W^2,
                         M_H^±^2,m_t^2))-[(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^R-C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L)-
                         i(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^R-C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L)](1/72+1/12J(m_W^2,
                         M_H^±^2,m_t^2))},
          C_8,NP^WH(μ_EW)=∑_χ^±_i,χ^0_jC_H^-d̅ u^Lm_W^2/4√(2)π^2m_dm_fV_ud^*{[(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^L+C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^R)-
                         i(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^L+C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^R)](-1/8√(2)J(m_W^2,M_H^±^2,m_t^2))+
                        [(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^R+C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L)+
                         i(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^R+C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L)](-1/8√(2)J(m_W^2,M_H^±^2,m_t^2))+
                        [(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^L-C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^R)-
                         i(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^L-C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^R)](-1/4√(2)J(m_W^2,M_H^±^2,m_t^2))+
                        [(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^R+C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L)+
                         i(C_H^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^R+C_H^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L)](1/4√(2)J(m_W^2,M_H^±^2,m_t^2))},

where m_F runs all m_χ_i^±, m_χ_j^0, and

    ρ_i,j(x_1,x_2)=x_1^i ln^j x_1-x_2^i ln^j x_2/x_1-x_2,
          P(y_1,y_2,y_3,y_4,x_1,x_2)=y_1∂ρ_1,1(x_1,x_2)/∂ x_1+y_2∂^2 ρ_2,1(x_1,x_2)/∂ x_1^2+
                                 y_3∂^3 ρ_3,1(x_1,x_2)/∂ x_1^3+y_4∂^4 ρ_4,1(x_1,x_2)/∂ x_1^4,
          J(x_1,x_2,x_3)=ln m_F^2-ρ_2,1(x_1,x_3)-ρ_2,2(x_2,x_3)/x_1^2-x_2^2






§ THE WILSON COEFFICIENTS OF THE PROCESS B_S^0→Μ^+Μ^-. 

The Wilson coefficients corresponding to b→ sμ^+μ^- can be written as

    C__S,NP^(1)(μ__ EW)=∑__Ũ_i,χ^-_j,χ^-_k,S=h_l,A_l
    C_μ^-Sμ^+^L+C_μ^-Sμ^+^R/2(m_b^2-m_S^2)[C_Ũ_is̅χ^-_j^R C_χ̅^-_j S χ^-_k^L C_χ̅^-_k s Ũ_i^R G_2(x_Ũ_i,x_χ^±_j,x_χ^±_k)
                      +M_χ^±_jM_χ^±_k C_Ũ_is̅χ^-_j^R C_χ̅^-_j S χ^-_k^R C_χ̅^-_k s Ũ_i^R G_1(x_Ũ_i,x_χ^±_j,x_χ^±_k)]
                      +∑__H^-_i,u_j,u_k,S=h_l,A_lC_μ^-Sμ^+^L+C_μ^-Sμ^+^R/2(m_b^2-m_S^2)[C_H^-_is̅ u_j^R C_u̅_j S u_k^L C_u̅_k b H^-_i^R G_2(x_H̃^±_i,x_u_j,x_u_k)
                      +m_u_jm_u_kC_H^-_is̅ u_j^R C_u̅_j S u_k^R C_u̅_k b H^-_i^R G_1(x_H̃^±_i,x_u_j,x_u_k)],
          C__P,NP^(1)(μ__ EW)=∑__Ũ_i,χ^-_j,χ^-_k,S=h_l,A_l
    -C_μ^-Sμ^+^L+C_μ^-Sμ^+^R/2(m_b^2-m_S^2)[C_Ũ_is̅χ^-_j^R C_χ̅^-_j S χ^-_k^L C_χ̅^-_k s Ũ_i^R G_2(x_Ũ_i,x_χ^±_j,x_χ^±_k)
                      +M_χ^±_jM_χ^±_k C_Ũ_is̅χ^-_j^R C_χ̅^-_j S χ^-_k^R C_χ̅^-_k s Ũ_i^R G_1(x_Ũ_i,x_χ^±_j,x_χ^±_k)]
                      +∑__H^-_i,u_j,u_k,S=h_l,A_l-C_μ^-Sμ^+^L+C_μ^-Sμ^+^R/2(m_b^2-m_S^2)[C_H^-_is̅ u_j^R C_u̅_j S u_k^L C_u̅_k b H^-_i^R G_2(x_H̃^±_i,x_u_j,x_u_k)
                      +m_u_jm_u_kC_H^-_is̅ u_j^R C_u̅_j S u_k^R C_u̅_k b H^-_i^R G_1(x_H̃^±_i,x_u_j,x_u_k)],


    C__S,NP^(2)(μ__ EW)=∑_u_i,H^±_j,H^±_k,S=h_l,A_l1/2(m_b^2-m_S^2)m_u_iC_s̅ u_i H^±_j^RC_u̅_i b H^±_k^RC_S H^±_j H^±_kG_1(x_u_i,x_H^±_j,x_H^±_k)
                       (C_μ^-Sμ^+^L+C_μ^-Sμ^+^R)
                      +∑_χ^±_i,Ũ_j,Ũ_k,S=h_l,A_l1/2(m_b^2-m_S^2)m_χ^±_iC_s̅χ^±_i Ũ_j^RC_χ̅^±_i b Ũ_k^RC_S Ũ_j Ũ_kG_1(x_χ^±_i,x_Ũ_j,x_Ũ_k)
                       (C_μ^-Sμ^+^L+C_μ^-Sμ^+^R),
          C__p,NP^(2)(μ__ EW)=∑_u_i,H^±_j,H^±_k,S=h_l,A_l1/2(m_b^2-m_S^2)m_u_iC_s̅ u_i H^±_j^RC_u̅_i b H^±_k^RC_S H^±_j H^±_kG_1(x_u_i,x_H^±_j,x_H^±_k)
                       (-C_μ^-Sμ^+^L+C_μ^-Sμ^+^R)
                      +∑_χ^±_i,Ũ_j,Ũ_k,S=h_l,A_l1/2(m_b^2-m_S^2)m_χ^±_iC_s̅χ^±_i Ũ_j^RC_χ̅^±_i b Ũ_k^RC_S Ũ_j Ũ_kG_1(x_χ^±_i,x_Ũ_j,x_Ũ_k)
                       (-C_μ^-Sμ^+^L+C_μ^-Sμ^+^R),


    C__S,NP^(3)(μ__ EW)=∑_u_i,H^±_k,S=h_l,A_l-C_W^± S H^±_k/2(m_b^2-m_S^2)[C_s̅ W^± u_i^LC_u̅_i H^±_k b^RG_2(x_u_i,1,x_H^±_k)-2m_bm_u_iC_s̅ W^± u_i^L
                       C_u̅_i H^±_k b^LG_1(x_u_i,1,x_H^±_k)](C_μ^-Sμ^+^L+C_μ^-Sμ^+^R),
          C__P,NP^(3)(μ__ EW)=∑_u_i,H^±_k,S=h_l,A_l-C_W^± S H^±_k/2(m_b^2-m_S^2)[C_s̅ W^± u_i^LC_u̅_i H^±_k b^RG_2(x_u_i,1,x_H^±_k)-2m_bm_u_iC_s̅ W^± u_i^L
                       C_u̅_i H^±_k b^LG_1(x_u_i,1,x_H^±_k)](-C_μ^-Sμ^+^L+C_μ^-Sμ^+^R),


    C__S,NP^(4)(μ__ EW)=∑_u_i,H^±_j,S=h_l,A_l-C_W^± S H^±_j/2(m_b^2-m_S^2)C_s̅ H^±_j u_i^RC_u̅_i W^± b^RG_2(x_u_i,x_H^±_j,1)(C_μ^-Sμ^+^L+C_μ^-Sμ^+^R),
          C__S,NP^(4)(μ__ EW)=∑_u_i,H^±_j,S=h_l,A_l-C_W^± S H^±_j/2(m_b^2-m_S^2)C_s̅ H^±_j u_i^RC_u̅_i W^± b^RG_2(x_u_i,x_H^±_j,1)(-C_μ^-Sμ^+^L+C_μ^-Sμ^+^R),


    C__9,NP^(5)(μ__ EW)=∑__Ũ_i,χ^-_j,χ^-_k,V
    C_μ^-Vμ^+^L+C_μ^-Vμ^+^R/-2(m_b^2-m_V^2)[-1/2C_Ũ_is̅χ^-_j^R C_χ̅^-_j V χ^-_k^R C_χ̅^-_k s Ũ_i^L G_2(x_Ũ_i,x_χ^±_j,x_χ^±_k)
                      +M_χ^±_jM_χ^±_k C_Ũ_is̅χ^-_j^R C_χ̅^-_j V χ^-_k^L C_χ̅^-_k s Ũ_i^L G_1(x_Ũ_i,x_χ^±_j,x_χ^±_k)]
                      +∑__H̃^±_i,u_j,u_k,V
    C_μ^-Vμ^+^L+C_μ^-Vμ^+^R/-2(m_b^2-m_V^2)[-1/2C_H^±_is̅ u_j^R C_u̅_j V u_k^R C_u_k s H^±_i^L G_2(x_H^±_i,x_u_j,x_u_k)
                      +m_u_jm_u_k C_H^±_is̅ u_j^R C_u̅_j V u_k^L C_u̅_k s H^±_i^L G_1(x_H^±_i,x_u_j,x_u_k)],
          C__10,NP^(5)(μ__ EW)=∑__Ũ_i,χ^-_j,χ^-_k,V
    -C_μ^-Vμ^+^L+C_μ^-Vμ^+^R/-2(m_b^2-m_V^2)[-1/2C_Ũ_is̅χ^-_j^R C_χ̅^-_j V χ^-_k^R C_χ̅^-_k s Ũ_i^L G_2(x_Ũ_i,x_χ^±_j,x_χ^±_k)
                      +M_χ^±_jM_χ^±_k C_Ũ_is̅χ^-_j^R C_χ̅^-_j V χ^-_k^L C_χ̅^-_k s Ũ_i^L G_1(x_Ũ_i,x_χ^±_j,x_χ^±_k)]
                      +∑__H̃^±_i,u_j,u_k,V
    -C_μ^-Vμ^+^L+C_μ^-Vμ^+^R/-2(m_b^2-m_V^2)[-1/2C_H^±_is̅ u_j^R C_u̅_j V u_k^R C_u_k s H^±_i^L G_2(x_H^±_i,x_u_j,x_u_k)
                      +m_u_jm_u_k C_H^±_is̅ u_j^R C_u̅_j V u_k^L C_u̅_k s H^±_i^L G_1(x_H^±_i,x_u_j,x_u_k)],


    C__9,NP^(6)(μ__ EW)=∑__u_i,H^±_j,H^±_k,V
    C_μ^-Vμ^+^L+C_μ^-Vμ^+^R/4(m_b^2-m_V^2)C_s̅ u_i H^±_j^RC_u̅_i b H^±_k^LC_V H^±_j H^±_kG_2(x_u_i,x_H^±_j,x_H^±_k)
                      +∑__χ^±_i,Ũ_j,Ũ_k,V
    C_μ^-Vμ^+^L+C_μ^-Vμ^+^R/4(m_b^2-m_V^2)C_s̅χ^±_i Ũ_j^RC_χ̅^±_i b Ũ_k^LC_V Ũ_j Ũ_kG_2(x_χ^±_i,x_Ũ_j,x_Ũ_k),
          C__10,NP^(6)(μ__ EW)=∑__u_i,H^±_j,H^±_k,V
    -C_μ^-Vμ^+^L+C_μ^-Vμ^+^R/4(m_b^2-m_V^2)C_s̅ u_i H^±_j^RC_u̅_i b H^±_k^LC_V H^±_j H^±_kG_2(x_u_i,x_H^±_j,x_H^±_k)
                      +∑__χ^±_i,Ũ_j,Ũ_k,V
    -C_μ^-Vμ^+^L+C_μ^-Vμ^+^R/4(m_b^2-m_V^2)C_s̅χ^±_i Ũ_j^RC_χ̅^±_i b Ũ_k^LC_V Ũ_j Ũ_kG_2(x_χ^±_i,x_Ũ_j,x_Ũ_k),
          C__S,NP^(6)(μ__ EW)=∑__u_i,H^±_j,H^±_k,V
    C_μ^-Vμ^+^L+C_μ^-Vμ^+^R/-2(m_b^2-m_V^2)m_bm_u_iC_s̅ u_i H^±_j^RC_u̅_i b H^±_k^RC_V H^±_j H^±_kG_1(x_u_i,x_H^±_j,x_H^±_k)
                      +∑__χ^±_i,Ũ_j,Ũ_k,V
    C_μ^-Vμ^+^L+C_μ^-Vμ^+^R/-2(m_b^2-m_V^2)m_bm_χ^±_iC_s̅χ^±_i Ũ_j^RC_χ̅^±_i b Ũ_k^RC_V Ũ_j Ũ_kG_1(x_χ^±_i,x_Ũ_j,x_Ũ_k),
          C__P,NP^(6)(μ__ EW)=∑__u_i,H^±_j,H^±_k,V
    C_μ^-Vμ^+^L-C_μ^-Vμ^+^R/-2(m_b^2-m_V^2)m_bm_u_iC_s̅ u_i H^±_j^RC_u̅_i b H^±_k^RC_V H^±_j H^±_kG_1(x_u_i,x_H^±_j,x_H^±_k)
                      +∑__χ^±_i,Ũ_j,Ũ_k,V
    C_μ^-Vμ^+^L-C_μ^-Vμ^+^R/-2(m_b^2-m_V^2)m_bm_χ^±_iC_s̅χ^±_i Ũ_j^RC_χ̅^±_i b Ũ_k^RC_V Ũ_j Ũ_kG_1(x_χ^±_i,x_Ũ_j,x_Ũ_k),


    C__9,NP^(7)(μ__ EW)=∑__u_i,H^±_k,V
    C_μ^-Vμ^+^L+C_μ^-Vμ^+^R/2(m_b^2-m_V^2)m_u_iC_s̅ u_i W^±^LC_u̅_i b H^±_k^LC_V W^± H^±_kG_1(x_u_i,x_W,x_H^±_k),
          C__10,NP^(7)(μ__ EW)=∑__u_i,H^±_k,V
    -C_μ^-Vμ^+^L+C_μ^-Vμ^+^R/2(m_b^2-m_V^2)m_u_iC_s̅ u_i W^±^LC_u̅_i b H^±_k^LC_V W^± H^±_kG_1(x_u_i,x_W,x_H^±_k),


    C__9,NP^(8)(μ__ EW)=∑__u_i,H^±_j,V
    C_μ^-Vμ^+^L+C_μ^-Vμ^+^R/2(m_b^2-m_V^2)m_u_iC_s̅ u_i H^±_j^RC_u̅_i b W^±^LC_V W^± H^±_kG_1(x_u_i,x_H^±_j,x_W),
          C__10,NP^(8)(μ__ EW)=∑__u_i,H^±_j,V
    -C_μ^-Vμ^+^L+C_μ^-Vμ^+^R/2(m_b^2-m_V^2)m_u_iC_s̅ u_i H^±_j^RC_u̅_i b W^±^LC_V W^± H^±_kG_1(x_u_i,x_H^±_j,x_W),


    C__9,NP^(9)(μ__ EW)=∑__Ũ_i,χ^±_j,χ^±_k,ν̃_l
    -1/8C_s̅Ũ_i χ^±_j^RC_χ̅^±_j μ^+ ν̃_l^L(C_μ̅^-ν̃_lχ^±_k^LC_χ̅^±_k Ũ_i b^R+C_μ̅^-ν̃_lχ^±_k^RC_χ̅^±_k Ũ_i b^L)
                       G_4(x_Ũ_i,x_χ^±_j,x_χ^±_k,x_ν̃_l),
          C__10,NP^(9)(μ__ EW)=∑__Ũ_i,χ^±_j,χ^±_k,ν̃_l
    -1/8C_s̅Ũ_i χ^±_j^RC_χ̅^±_j μ^+ ν̃_l^L(C_μ̅^-ν̃_lχ^±_k^LC_χ̅^±_k Ũ_i b^R-C_μ̅^-ν̃_lχ^±_k^RC_χ̅^±_k Ũ_i b^L)
                       G_4(x_Ũ_i,x_χ^±_j,x_χ^±_k,x_ν̃_l),
          C__S,NP^(9)(μ__ EW)=∑__Ũ_i,χ^±_j,χ^±_k,ν̃_l
    -1/2M_χ^±_jM_χ^±_kC_s̅Ũ_i χ^±_j^RC_χ̅^±_j μ^+ ν̃_l^R(C_μ̅^-ν̃_lχ^±_k^LC_χ̅^±_k Ũ_i b^L+C_μ̅^-ν̃_lχ^±_k^RC_χ̅^±_k Ũ_i b^R)
                       G_3(x_Ũ_i,x_χ^±_j,x_χ^±_k,x_ν̃_l),
          C__P,NP^(9)(μ__ EW)=∑__Ũ_i,χ^±_j,χ^±_k,ν̃_l
    -1/2M_χ^±_jM_χ^±_kC_s̅Ũ_i χ^±_j^RC_χ̅^±_j μ^+ ν̃_l^R(-C_μ̅^-ν̃_lχ^±_k^LC_χ̅^±_k Ũ_i b^L+C_μ̅^-ν̃_lχ^±_k^RC_χ̅^±_k Ũ_i b^R)
                       G_3(x_Ũ_i,x_χ^±_j,x_χ^±_k,x_ν̃_l),


    C__9,NP^(10)(μ__ EW)=∑__u_i,H^±_j,H^±_k,ν̃_l1/8C_s̅ u_i H^±_j^RC_u̅_i b H^±_k^L(C_μ̅^-H^±_kν_l^LC_ν̅_l H^±_j μ^+^R+C_μ̅^-H^±_kν_l^RC_ν̅_l H^±_j μ^+^L)
                       G_4(x_u_i,x_H^±_j,x_H^±_k,x_ν_l),
          C__10,NP^(10)(μ__ EW)=∑__u_i,H^±_j,H^±_k,ν̃_l1/8C_s̅ u_i H^±_j^RC_u̅_i b H^±_k^L(C_μ̅^-H^±_kν_l^LC_ν̅_l H^±_j μ^+^R-C_μ̅^-H^±_kν_l^RC_ν̅_l H^±_j μ^+^L)
                       G_4(x_u_i,x_H^±_j,x_H^±_k,x_ν_l),


    C__S,NP^(11)(μ__ EW)=-∑__u_i,H^±_j,ν_l1/2C_s̅ u_i H^±_j^RC_u̅_i b W^±^RC_μ̅^-W^±ν_l^RC_ν̅_l H^±_j μ^+^L G_4(x_u_i,x_H^±_j,x_W,x_ν_l),
          C__P,NP^(11)(μ__ EW)=-∑__u_i,H^±_j,ν_l1/2C_s̅ u_i H^±_j^RC_u̅_i b W^±^RC_μ̅^-W^±ν_l^RC_ν̅_l H^±_j μ^+^L G_4(x_u_i,x_H^±_j,x_W,x_ν_l).


    C__9,NP^WW(μ__ EW)=∑__χ^+_i,χ^0_j,V(C_μ^-Vμ^+^R+C_μ^-Vμ^+^L)g_s^2/128π^4 s_W^2m_b^2/m_b^2-m_V^2{[(|C_W^-χ̅^0_j χ^+_i^L|^2 +|C_W^-χ̅^0_j χ^+_i^R|^2) 
                         P(1/8,1/4,-1/48,-1/144,1,x_t)+(|C_W^-χ̅^0_j χ^+_i^L|^2 -|C_W^-χ̅^0_j χ^+_i^R|^2)∂^1 ρ_1,1/∂ x_1(x_W,x_t)+
                        (C_W^-χ̅^0_j χ^+_i^RC_W^-χ̅^0_j χ^+_i^L*+C_W^-χ̅^0_j χ^+_i^LC_W^-χ̅^0_j χ^+_i^R*) P(1/16,1/4,-1/16,1/144,1,x_t)+
                        1/8(C_W^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^R*-C_W^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L*)∂^2 ρ_2,1/∂ x_1^2(x_W,x_t)]C_VW^-W^-+
                        [(|C_W^-χ̅^0_j χ^+_i^L|^2 +|C_W^-χ̅^0_j χ^+_i^R|^2)P(1/144,-1/12,-29/72,-11/12,1,x_t)+(C_W^-χ̅^0_j χ^+_i^R
                         C_W^-χ̅^0_j χ^+_i^L*+C_W^-χ̅^0_j χ^+_i^LC_W^-χ̅^0_j χ^+_i^R*) P(-1/144,1/12,-5/24,-1/12,1,x_t)]C_t̅Vt+
                        [(|C_W^-χ̅^0_j χ^+_i^L|^2 +|C_W^-χ̅^0_j χ^+_i^R|^2)P(3/16,3/16,0,0,1,x_t)+1/16(C_W^-χ̅^0_j χ^+_i^L
                         C_W^-χ̅^0_j χ^+_i^R*-C_W^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L*)∂^2 ρ_2,1/∂ x_1^2(x_W,x_t)](C_χ̅_j^0Vχ_j^0+C_χ̅_i^+Vχ_i^+)},


    C__10,NP^WW(μ__ EW)=∑__χ^+_i,χ^0_j,V(C_μ^-Vμ^+^R-C_μ^-Vμ^+^L)g_s^2/128π^4 s_W^2m_b^2/m_b^2-m_V^2{[(|C_W^-χ̅^0_j χ^+_i^L|^2 +|C_W^-χ̅^0_j χ^+_i^R|^2) 
                         P(1/8,1/4,-1/48,-1/144,1,x_t)+(|C_W^-χ̅^0_j χ^+_i^L|^2 -|C_W^-χ̅^0_j χ^+_i^R|^2)∂^1 ρ_1,1/∂ x_1(x_W,x_t)+
                        (C_W^-χ̅^0_j χ^+_i^RC_W^-χ̅^0_j χ^+_i^L*+C_W^-χ̅^0_j χ^+_i^LC_W^-χ̅^0_j χ^+_i^R*) P(1/16,1/4,-1/16,1/144,1,x_t)+
                        1/8(C_W^-χ̅^0_j χ^+_i^L C_W^-χ̅^0_j χ^+_i^R*-C_W^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L*)∂^2 ρ_2,1/∂ x_1^2(x_W,x_t)]C_VW^-W^-+
                        [(|C_W^-χ̅^0_j χ^+_i^L|^2 +|C_W^-χ̅^0_j χ^+_i^R|^2)P(1/144,-1/12,-29/72,-11/12,1,x_t)+(C_W^-χ̅^0_j χ^+_i^R
                         C_W^-χ̅^0_j χ^+_i^L*+C_W^-χ̅^0_j χ^+_i^LC_W^-χ̅^0_j χ^+_i^R*) P(-1/144,1/12,-5/24,-1/12,1,x_t)]C_t̅Vt+
                        [(|C_W^-χ̅^0_j χ^+_i^L|^2 +|C_W^-χ̅^0_j χ^+_i^R|^2)P(3/16,3/16,0,0,1,x_t)+1/16(C_W^-χ̅^0_j χ^+_i^L
                         C_W^-χ̅^0_j χ^+_i^R*-C_W^-χ̅^0_j χ^+_i^R C_W^-χ̅^0_j χ^+_i^L*)∂^2 ρ_2,1/∂ x_1^2(x_W,x_t)](C_χ̅_j^0Vχ_j^0+C_χ̅_i^+Vχ_i^+)},

where V denotes photon γ, Z boson, Z' boson. And C_t̅Vt, C_χ̅_j^0Vχ_j^0, C_χ̅_i^+Vχ_i^+ denote the vector parts of the corresponding interaction vertex. The concrete expressions for G_k(k=1,...,4) can be given as:

    G_1(x_1,x_2,x_3)=-1/m_W^2[x_1 ln x_1/(x_2-x_1)(x_3-x_1)+x_2 ln x_2/(x_1-x_2)(x_3-x_2)+x_3 ln x_3/(x_1-x_3)(x_2-x_3)],
          G_2(x_1,x_2,x_3)=-x_1^2 ln x_1/(x_2-x_1)(x_3-x_1)-x_2^2 ln x_2/(x_1-x_2)(x_3-x_2)-x_3^2 ln x_3/(x_1-x_3)(x_2-x_3),
          G_3(x_1,x_2,x_3,x_4)=1/m_W^4[x_1 ln x_1/(x_2-x_1)(x_3-x_1)(x_4-x_1)+x_2 ln x_2/(x_1-x_2)(x_3-x_2)(x_4-x_2)+
                          x_3 ln x_3/(x_1-x_3)(x_2-x_3)(x_4-x_3)x_4 ln x_4/(x_1-x_4)(x_2-x_4)(x_3-x_4)],
          G_4(x_1,x_2,x_3,x_4)=1/m_W^2[x_1^2 ln x_1/(x_2-x_1)(x_3-x_1)(x_4-x_1)+x_2^2 ln x_2/(x_1-x_2)(x_3-x_2)(x_4-x_2)+
                          x_3^2 ln x_3/(x_1-x_3)(x_2-x_3)(x_4-x_3)x_4^2 ln x_4/(x_1-x_4)(x_2-x_4)(x_3-x_4)].





99
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