Two loop electroweak corrections to $\bar B\rightarrow X_s\gamma$ and $B_s^0\rightarrow \mu^+\mu^-$ in the B-LSSM

The rare decays $\bar B\rightarrow X_s\gamma$ and $B_s^0\rightarrow \mu^+\mu^-$ are important to research new physics beyond standard model. In this work, we investigate two loop electroweak corrections to $\bar B\rightarrow X_s\gamma$ and $B_s^0\rightarrow \mu^+\mu^-$ 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 $\bar B\rightarrow X_s\gamma$ and $B_s^0\rightarrow \mu^+\mu^-$. The results of the rare decays also further constrain the parameter space of the B-LSSM.

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 [14][15][16][17][18][19][20][21]. The authors of Refs. [22][23][24] present the calculation of the rate inclusive decay B → X s γ in the two-Higgs doublet model (THDM). The supersymmetric effect on B → X s γ is discussed in Refs. [25][26][27][28][29][30][31] and the next-to-leading order (NLO) QCD corrections are given in Ref. [32]. The branching ratio for B s → l + l − in THDM and supersymmetric extensions of the SM has been calculated in Refs. [33][34][35][36][37][38]. The hadronic B decays [39] and CP-violation in these processes [40] have also been discussed. The authors of Ref. [41] have discussed possibility of observing supersymmetric effects in rare decays B → X s γ 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. [42,43].
The minimal supersymmetric extension of the SM with local B − L gauge symmetry (B-LSSM) [44,45] 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 [46], because the exotic singlet Higgs and right-handed (s)neutrinos [47][48][49][50][51][52][53][54][55] 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 [56]. Furthermore, it could help to understand the origin of R-parity and its possible spontaneous violation in the supersymmetric models [57][58][59] as well as the mechanism of leptogenesis [60,61]. Moreover, the model can provide much more candidates for the Dark Matter comparing that with the MSSM [62][63][64][65]. In this work, we analyze two loop electroweak corrections toB → X s γ and B 0 s → 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 forB → X s γ and B 0 s → µ + µ − . The numerical analyses are given in Sec. IV, and Sec. V gives a summary. The tedious formulae are collected in Appendices.

II. 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 superfieldsη 1 andη 2 . In literatures there are several popular versions of B-LSSM. Here we adopt the version described in Refs. [66][67][68][69] to proceed our analysis, because this version of B-LSSM is encoded in SARAH [70][71][72][73][74] 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 neutrinosν c i ∼(1, 1, 0, 1) and two chiral singlet superfieldsη 1 ∼ (1, 1, 0, −1),η 2 ∼ (1, 1, 0, 1). Meanwhile, quantum numbers of the matter chiral superfields for quarks and leptons are given bŷ with i = 1, 2, 3 denoting the index of generation. In addition, the quantum numbers of two Higgs doublets is assigned aŝ The corresponding superpotential of the B-LSSM is written as Here, W M SSM is the superpotential of the MSSM, and W (B−L) is the sector involving exotic superfields, where i, j are generation indices. Correspondingly, the soft breaking terms of the B-LSSM are generally given as with λ B , λ B ′ denoting the gaugino of U(1) Y and U(1) (B−L) , respectively. L M SSM is the soft breaking terms of the MSSM.  [75][76][77][78][79][80][81]. In practice, it turns out that it is easier to work with non-canonical covariant derivatives instead of offdiagonal field-strength tensors. However, both approaches are equivalent [82]. Hence in the following, we consider covariant derivatives of the form 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 where R is a 2 × 2 orthogonal matrix. Choosing R in a proper form, one can write the coupling matrix as where g 1 corresponds to the measured hypercharge coupling which is modified in B-LSSM as given along with g B and g Y B in Refs. [83]. In addition, we can redefine the U(1) gauge 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): For convenience, we define u 2 = u 2 1 + u 2 2 , v 2 = v 2 1 + v 2 2 and tan β ′ = u 2 u 1 in analogy to the ratio of the MSSM VEVs (tan β = v 2 v 1 ). Meanwhile, the charged and neutral gauge bosons acquire the nonzero masses as Compared the MSSM, this new gauge boson Z ′ makes new contribution to the process In addition, the charged Higgs mass can be written as In the basis (ReH 1 1 , ReH 2 2 , Reη 1 , Reη 2 ), the tree level mass squared matrix for Higgs bosons is given by 1+tan β 2 + n 2 tan β − 1 4 g 2 x 2 tan β 1+tan 2 β − n 2 where g 2 = g 2 1 + g 2 2 + g 2 Y B , T = 1 + tan 2 β 1 + tan 2 β ′ , n 2 = ReBµ u 2 , N 2 = ReBµ ′ u 2 , and x = v u , respectively. These new extra singletsη 1,2 and the corresponding pseudo-scalar Higgs boson make new contributions to the process B 0 s → 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 [84][85][86] where α 3 is the strong coupling constant, M S = √ mt 1 mt 2 with mt 1,2 denoting the stop masses,Ã t = A t − µ cot β 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 where m 0 h 1 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 where, It can be noted that tan β ′ and new gauge coupling constants g B , g Y B in the B-LSSM can affect the mass matrix of up type squarks. Since up type squarks appear in the loops of the processesB → X s γ and B 0 s → µ + µ − , new definition of them affects the predictions of Br(B → X s γ) and Br(B 0 s → µ + µ − ) by influencing their masses and the corresponding couplings.
where we choose the hadron scale µ b = 2.5 GeV and use the SM contribution at NNLO level where where C (1,..,4) are Wilson coefficients of the process b → sγ corresponding to Fig. 1, Fig. 2, and the concrete expressions are collected in Appendix A.
Eq.(A3) indicates that the corrections to the Wilson coefficients from Fig. 2 are Wilson coefficients of the process b → sg in the B-LSSM. Compared with the SM, the corresponding Feynman diagrams are shown in Fig. 1(2, 3), Fig. 2(1, 2, 3), then C 8g,N P (µ EW ) and where Q u = 2/3. And the concrete expressions of Here, the superscripts (1, ..., 11, W W ) corresponding to the new physics corrections in Fig. 3 and Fig. 2(1), and the concrete expressions of these Wilson coefficients can be found in Appendix B. The Wilson coefficients at hadronic energy scale from the SM to next-to-nextto-logarithmic (NNLL) accuracy are shown in Table I. In addition, The Wilson coefficients in Eq. (27) are calculated at the matching scale µ EW , then evolved down to hadronic scale µ ∼ m b by the renormalization group equations: Correspondingly, the evolving matrices are approached as where the anomalous dimension matrices can be read from Ref. [99] as Then, the squared amplitude can be written as and where f B 0 s = (227 ± 8)MeV denote the decay constants, M B 0 s = 5.367GeV denote the masses of neutral meson B 0 s . The branching ratio of B 0 s → µ + µ − can be written as Meanwhile the CKM matrix is The updated experimental data [100] on searching Z ′ indicates M Z ′ ≥ 4.05TeV at 95% Confidence Level (CL). Due to the contributions of heavy Z ′ boson are highly suppressed, we choose M Z ′ = 4.2TeV in our following numerical analysis. And Refs. [101,102] give us an upper bound on the ratio between the Z ′ mass and its gauge coupling at 99% CL as Then the scope of g B is limited to 0 < g B ≤ 0.7. Additionally, the LHC experimental data also constrain tan β ′ < 1.5 [67]. Considering the constraints from the experiments [1], for those parameters in Higgsino and gaugino sectors, we appropriately fix M 1 = 500GeV, M 2 = 600GeV, µ = 700GeV, µ ′ = 800GeV, M BB ′ = 500GeV, M BL = 600GeV, for simplify.
For those parameters in the soft breaking terms, we set B ′ µ = 5 × 10 5 GeV 2 , ml = mẽ = 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 [103,104] and the third generation squark masses are not constrained by the LHC as strong as the first two generations, and affect the SMlike Higgs mass. Therefore we take mq = mũ = diag(2, 2, mt)TeV, and the discission about the observed Higgs signal in Ref. [105] limits mt > ∼ 1.5TeV. For simplify, we also choose T u 1,2 = 1TeV. 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 B → X s γ and B 0 s → µ + µ − depend essentially on A t , tan β and charged Higgs mass M H ± . In order to see how A t , tan β, M H ± affect the theoretical evaluations of Br(B → X s γ) and Br(B 0 s → µ + µ − ) in the B-LSSM, we assume that g B = 0.4, g Y B = −0.5, tan β ′ = 1.1 in the following analysis. Then taking mt = 1.6TeV, M H ± = 1.5TeV, we plot Br(B → X s γ) and Fig. 4, for tan β = 5(solid line), tan β = 20(dashed line) and tan β = 35(dotted line), respectively. The gray area denotes the experimental 1σ bounds in Eq.(1). In Fig. 4(a), we can see that Br(B → X s γ) decreases with the increasing of A t , and Br(B → X s γ) will be easily coincides with experimental data within one standard deviation when A t is positive. Meanwhile, Fig. 4(b) shows that Br(B 0 s → µ + µ − ) favor A t in the ranges −3.6TeV < ∼ A t < ∼ 1.4TeV as tan β = 5, −1.2TeV < ∼ A t < ∼ 0.2TeV as tan β = 20 and −0.8TeV < ∼ A t < ∼ 0TeV as tan β = 35, which also coincide with the experimental data on Br(B → X s γ). It can be noted that when tan β is large, the range of A t is limited strongly by the experimental data on Br(B 0 s → µ + µ − ).  of Br(B → X s γ) can reach around 3%, which produces a more precise prediction on the processB → X s γ, and we cannot neglect the corrections with this magnitude. In Fig. 5(b), these two lines coincide with each other, which indicates the two loop corrections to Br(B 0 s → 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 tan β = 20, we plot the SM-like Higgs mass m h versus A t in Fig. 6 for mt = 1.5TeV(solid line), mt = 2.5TeV(dashed line) and mt = 3.5TeV(dotted line). The gray area denotes the experimental 3σ interval. To keep the SM-like Higgs mass around 125GeV, we need A t ≈ ±1.8TeV as mt = 1.5TeV. When mt = 2.5TeV, we require that A t in the range 5TeV. And the allowed range of A t is −2.5TeV < ∼ A t < ∼ 2.5TeV when mt = 3.5TeV.
Since the large charged Higgs mass does not affect the SM-like Higgs mass signally, we can choose A t = −2.5TeV, −0.5TeV and 1.5TeV for mt = 3.5TeV, to keep the SM- g Y B acutely when tan β ′ is large.
Then we take tan β ′ = 1.2, and plot Br(B → X s γ) and Br(B 0 s → µ + µ − ) varying with g B in Fig. 9, for g 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 Y B do not affect Br(B → X s γ) obviously. And Br(B 0 s → µ + µ − ) can exceed the experimental 1σ upper bound easily when g B is small and |g Y B | is large. In addition, with the decreasing of |g Y B |, Br(B 0 s → 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 Y B affect Br(B 0 s → µ + µ − ) mainly in two ways. Firstly, g B and g Y B 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 0 s → µ + µ − by influencing the new contributions from Z ′ gauge boson, new scalar and pseudoscalar Higgs bosons in Fig. 3.

V. 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(B → X s γ) and Br(B 0 s → 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 processB → X s γ can reach around 3%, which produces a more precise prediction on the processB → X s γ. Nevertheless, the corrections from two loop diagrams to the process and Br(B 0 s → µ + µ − ) obviously. And when tan β is large, A t is limited strongly by the experimental data for Br(B 0 s → µ + µ − ). In addition, tan β ′ , g Y B and g B can also affect theoretical predictions on Br(B 0 s → µ + µ − ) obviously.
Appendix A: The Wilson coefficients of the processB → X s γ.
The one loop Wilson coefficients corresponding to b → sγ can be written as where S dentes CP-even and CP-odd Higgs, C L,R abc 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 , the concrete expressions for I k (k = 1, ..., 4) can be given as: Assuming m χ ± i , m χ 0 j ≫ m W , then the two loop Wilson coefficients corresponding to b → sγ can be simplified as , 1, where m F runs all m χ ± i , m χ 0 j , and Appendix B: The Wilson coefficients of the process B 0 s → µ + µ − .
The Wilson coefficients corresponding to b → sµ + µ − can be written as , where V denotes photon γ, Z boson, Z ′ boson. And Ct V t , Cχ0