$ { \bar{ B}\to{ {X_s \gamma }}}$ in BLMSSM

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Jian-Bin Chen, Meng Zhang, Li-Li Xing, Tai-Fu Feng, Shu-Min Zhao and Ke-Sheng Sun. $ { \bar{ B}\to{ {X_s \gamma }}}$ in BLMSSM[J]. Chinese Physics C.
Jian-Bin Chen, Meng Zhang, Li-Li Xing, Tai-Fu Feng, Shu-Min Zhao and Ke-Sheng Sun. $ { \bar{ B}\to{ {X_s \gamma }}}$ in BLMSSM[J]. Chinese Physics C. shu
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Received: 2020-02-02
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$ { \bar{ B}\to{ {X_s \gamma }}}$ in BLMSSM

    Corresponding author: Jian-Bin Chen, chenjianbin@tyut.edu.cn
    Corresponding author: Li-Li Xing, xinglili@tyut.edu.cn
    Corresponding author: Tai-Fu Feng, fengtf@hbu.edu.cn
    Corresponding author: Shu-Min Zhao, zhaosm@hbu.edu.cn
    Corresponding author: Ke-Sheng Sun, sunkesheng@bdu.edu.cn
  • 1. College of Physics and Optoelectronic Engineering, Taiyuan University of Technology, Taiyuan 030024, China
  • 2. Department of Physics, Hebei University, Baoding, 071002, China
  • 3. Department of Physics, Baoding University, Baoding, 071000, China

Abstract: Applying the effective Lagrangian method, we study the Flavor Changing Neutral Current $b\to s\gamma$ within the minimal supersymmetric extension of the standard model where baryon and lepton numbers are local gauge symmetries. Constraints on the parameters are investigated numerically with the experimental data on branching ratio of $\bar{B}\to X_s\gamma$. Additionally, we present the corrections to direct CP-violation in $\bar{B}\rightarrow X_s\gamma$ and time-dependent CP-asymmetry in $B\rightarrow K^*\gamma$. With appropriate assumptions on parameters, we find the direct CP-violation $A_{CP}$ is very small, while one-loop contributions to $S_{K^*\gamma}$ can be significant.

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    I.   INTRODUCTION
    • Since the Flavor Changing Neutral Current process(FCNC) $ b\to s\gamma $ originates only from loop diagrams, it is very sensitive to new physics beyond the Standard Model(SM). The updated average data of inclusive $ \bar{B}\rightarrow X_s\gamma $ is [1]

      $ \begin{array}{l} BR(\bar{B}\rightarrow X_s\gamma)_{exp} = (3.32\pm0.15)\times10^{-4}. \end{array} $

      (1)

      and the prediction of SM at next-next-to-leading order (NNLO) is[2-4]

      $ \begin{array}{l} BR(\bar{B}\rightarrow X_s\gamma)_{SM} = (3.40\pm0.17)\times10^{-4}. \end{array} $

      (2)

      Though the deviation of SM prediction from experimental results has been almost eliminated in the past few years, it is helpful to constrain parameters of new physics.

      The discovery of Higgs boson on Large Hadron Collider(LHC) makes SM the most successful theory in particle physics. Because of the hierarchy problem and missing of gravitational interaction, it is believed that SM is just an effective approximation of a more fundamental theory at higher scale. Among various extensions of SM, supersymmetric models have been studied for decades.

      As the simplest extension, the Minimal Supersymmetric Standard Model(MSSM)[5-7] solves the hierarchy problem as well as the instability of Higgs boson by introducing a superpartner for each SM particle. The Lightest Supersymmetric Particle (LSP) within this frmework also provides candidates of dark matter as Weakly Interacting Massive Particles (WIMPs). However the MSSM can not naturely generates tiny neutrino mass which is needed to explain the observation of neutrino oscillation. To acquire neutrino masses, heavy majorana neutrinos are introduced in the seesaw mechanism, which implies that the lepton numbers are broken. Besides, the baryon numbers are also expected to be broken because of the asymmetry of matter-antimatter in the universe. The authors of [8,9] present the so called BLMSSM model in which the baryon and lepton number are local gauged and spontaneously broken at TeV scale. The experimental bounds on proton decay lifetime is the main motivation of great desert hypothesis. In BLMSSM, the proton decay can be avoid with discrete symmetry called matter parity and R-parity[10]

      To describe the symmetries of baryon and lepton numbers, gauge group is enlarged to $ SU(3)_C \otimes SU(2)_L \otimes U(1)_Y \otimes U(1)_B \otimes U(1)_L $. Then corrections to various observations can be induced from new gauge boson and exotic fields within this scenario. In ref. [11], corrections to anomalous magnetic moment from one loop diagrams and two-loop Barr-Zee type diagrams are investigated with effective Lagrangian method. One-loop contributions to $ c(t) $ electric dipole moment in CP-Violating BLMSSM is presented in ref. [12]. To account for the experimental data on Higgs, the authors of [13] study the signals of $ h\to\gamma\gamma $ and $ h\to VV^*(V = Z,W) $ with a 125 GeV Higgs. In this work, we use the branching ratio to constrain the parameters. Furthermore, we present the corrections to CP-Violation of $ b\to s\gamma $ due to new parameters introduced in this model.

      Our presentation is organized as follows. In section II, we briefly introduce the construction of BLMSSM and the interactions we need for our caculation. After that, we present the one-loop corrections to branching ratio and CP-Violation with effective Lagrangian method in section III. Numerical results are discussed in section IV and the conclusions is given in section V.

    II.   INTRODUCTION TO BLMSSM
    • The BLMSSM is based on gauge symmetry $ SU(3)_C \otimes SU(2)_L \otimes U(1)_Y \otimes U(1)_B \otimes U(1)_L $. In order to cancel the anomalies of Baryon number(B), exotic quarks $ \hat{Q}_4 \sim (3 , 2 , 1/6 , B_4 , 0), \hat{U}^c_4 \sim (\bar{3} , 1 , -2/3, - B_4, 0 ), \hat{D}^c_4 \sim (\bar{3} , 1 , 1/3 , $$ -B_4 , 0 ), \hat{Q}_5^c \sim (\bar{3} , 2 , -1/6, -(1+B_4), 0 ), \hat{U}_5 \sim (3, 1 , 2/3 , 1 + B_4, 0 ),$$ \hat{D}_5 \sim (3, 1 , -1/3 , 1 + B_4, 0) $ are introduced. Baryon number are broken spontaneously after Higgs superfields $ \hat{\Phi}_B \sim ( 1 , 1 , 0 , 1 , 0 ), \hat{\varphi}_B \sim ( 1 , 1 , 0 , -1 , 0 ) $ acquire nonzero vacuum expectation values(VEVs). To deal with the anomalies of Lepton number(L), exotic leptons $ \hat{L}_4 \sim ( 1 , 2 , -1/2 , 0 , L_4), \hat{E}^c_4 \sim ( 1 , 1 , 1 , 0 , -L_4), \hat{N}^c_4 \sim ( 1 , 1 , 0 , 0 , -L_4), \hat{L}_5^c \sim ( 1 , 2 , 1/2 , 0 , -(3 + L_4)), \hat{E}_5 \sim ( 1 , 1 , -1 , 0 , 3 + L_4), \hat{N}_5 \sim ( 1 , 1 , 0 , 0 , 3 + L_4) $ are introduced, and $ \hat{\Phi}_L \sim ( 1 , 1 , 0 , 0 , -2 ), \hat{\varphi}_L \sim ( 1 , 1 , 0 , 0 , 2) $ are responsable for the breaking of lepton number[9]. The superfields $ \hat{X} \sim ( 1 , 1 , 0 ,2/3 + B_4, 0 ), \hat{X'} \sim ( 1 , 1 , 0 ,-(2/3 + B_4), 0) $ which mediate the decay of exotic quarks are added in this model to avoid their stability.

      Given the superfields above, one can construct the superpotential as

      $ \begin{array}{l} {\cal W}_{_{BLMSSM}} = {\cal W}_{_{MSSM}}+{\cal W}_{_B}+{\cal W}_{_L}+{\cal W}_{_X}, \end{array} $

      (3)

      where $ {\cal W}_{_{MSSM}} $ indicates the superpotential of MSSM, and

      $ \begin{aligned}[b] {\cal W}_{_B} =& \lambda_{_Q}\hat{Q}_{_4}\hat{Q}_{_5}^c\hat{\Phi}_{_B}+\lambda_{_U}\hat{U}_{_4}^c\hat{U}_{_5} \hat{\varphi}_{_B}+\lambda_{_D}\hat{D}_{_4}^c\hat{D}_{_5}\hat{\varphi}_{_B}+\mu_{_B}\hat{\Phi}_{_B}\hat{\varphi}_{_B} \\ & +Y_{_{u_4}}\hat{Q}_{_4}\hat{H}_{_u}\hat{U}_{_4}^c+Y_{_{d_4}}\hat{Q}_{_4}\hat{H}_{_d}\hat{D}_{_4}^c +Y_{_{u_5}}\hat{Q}_{_5}^c\hat{H}_{_d}\hat{U}_{_5}+Y_{_{d_5}}\hat{Q}_{_5}^c\hat{H}_{_u}\hat{D}_{_5}\;, \\ {\cal W}_{_L} =& Y_{_{e_4}}\hat{L}_{_4}\hat{H}_{_d}\hat{E}_{_4}^c+Y_{_{\nu_4}}\hat{L}_{_4}\hat{H}_{_u}\hat{N}_{_4}^c +Y_{_{e_5}}\hat{L}_{_5}^c\hat{H}_{_u}\hat{E}_{_5}+Y_{_{\nu_5}}\hat{L}_{_5}^c\hat{H}_{_d}\hat{N}_{_5} \\ & +Y_{_\nu}\hat{L}\hat{H}_{_u}\hat{N}^c+\lambda_{_{N^c}}\hat{N}^c\hat{N}^c\hat{\varphi}_{_L} +\mu_{_L}\hat{\Phi}_{_L}\hat{\varphi}_{_L}\;, \\ {\cal W}_{_X} =& \lambda_1\hat{Q}\hat{Q}_{_5}^c\hat{X}+\lambda_2\hat{U}^c\hat{U}_{_5}\hat{X}^\prime +\lambda_3\hat{D}^c\hat{D}_{_5}\hat{X}^\prime+\mu_{_X}\hat{X}\hat{X}^\prime\;. \end{aligned} $

      (4)

      The soft breaking terms are given by

      $ \begin{aligned}[b] {\cal L}_{_{soft}} =& {\cal L}_{_{soft}}^{MSSM}-(m_{_{\tilde{N}^c}}^2)_{_{IJ}}\tilde{N}_I^{c*}\tilde{N}_J^c -m_{_{\tilde{Q}_4}}^2\tilde{Q}_{_4}^\dagger ger\tilde{Q}_{_4}-m_{_{\tilde{U}_4}}^2\tilde{U}_{_4}^{c*}\tilde{U}_{_4}^c -m_{_{\tilde{D}_4}}^2\tilde{D}_{_4}^{c*}\tilde{D}_{_4}^c -m_{_{\tilde{Q}_5}}^2\tilde{Q}_{_5}^{c\dagger}\tilde{Q}_{_5}^c -m_{_{\tilde{U}_5}}^2\tilde{U}_{_5}^*\tilde{U}_{_5} -m_{_{\tilde{D}_5}}^2\tilde{D}_{_5}^*\tilde{D}_{_5}-m_{_{\tilde{L}_4}}^2\tilde{L}_{_4}^\dagger ger\tilde{L}_{_4} -M_{_{\tilde{\nu}_4}}^2\tilde{\nu}_{_4}^{c*}\tilde{\nu}_{_4}^c\\ &-m_{_{\tilde{E}_4}}^2\tilde{e}_{_4}^{c*}\tilde{e}_{_4}^c -m_{_{\tilde{L}_5}}^2\tilde{L}_{_5}^{c\dagger}\tilde{L}_{_5}^c -M_{_{\tilde{\nu}_5}}^2\tilde{\nu}_{_5}^*\tilde{\nu}_{_5}-m_{_{\tilde{E}_5}}^2\tilde{e}_{_5}^*\tilde{e}_{_5} -m_{_{\Phi_{_B}}}^2\Phi_{_B}^*\Phi_{_B}-m_{_{\varphi_{_B}}}^2\varphi_{_B}^*\varphi_{_B} -m_{_{\Phi_{_L}}}^2\Phi_{_L}^*\Phi_{_L} -m_{_{\varphi_{_L}}}^2\varphi_{_L}^*\varphi_{_L}-(m_{_B}\lambda_{_B}\lambda_{_B} +m_{_L}\lambda_{_L}\lambda_{_L}+h.c.) \\ & +\Big\{A_{_{u_4}}Y_{_{u_4}}\tilde{Q}_{_4}H_{_u}\tilde{U}_{_4}^c+A_{_{d_4}}Y_{_{d_4}}\tilde{Q}_{_4}H_{_d}\tilde{D}_{_4}^c +A_{_{u_5}}Y_{_{u_5}}\tilde{Q}_{_5}^cH_{_d}\tilde{U}_{_5}+A_{_{d_5}}Y_{_{d_5}}\tilde{Q}_{_5}^cH_{_u}\tilde{D}_{_5} +A_{_{BQ}}\lambda_{_Q}\tilde{Q}_{_4}\tilde{Q}_{_5}^c\Phi_{_B}+A_{_{BU}}\lambda_{_U}\tilde{U}_{_4}^c\tilde{U}_{_5}\varphi_{_B} +A_{_{BD}}\lambda_{_D}\tilde{D}_{_4}^c\tilde{D}_{_5}\varphi_{_B} \\ &+B_{_B}\mu_{_B}\Phi_{_B}\varphi_{_B} +h.c.\Big\} +\Big\{A_{_{e_4}}Y_{_{e_4}}\tilde{L}_{_4}H_{_d}\tilde{E}_{_4}^c+A_{_{N_4}}Y_{_{N_4}}\tilde{L}_{_4}H_{_u}\tilde{N}_{_4}^c +A_{_{e_5}}Y_{_{e_5}}\tilde{L}_{_5}^cH_{_u}\tilde{E}_{_5}+A_{_{N_5}}Y_{_{\nu_5}}\tilde{L}_{_5}^cH_{_d}\tilde{N}_{_5} +A_{_N}Y_{_N}\tilde{L}H_{_u}\tilde{N}^c\\ &+A_{_{N^c}}\lambda_{_{N^c}}\tilde{N}^c\tilde{N}^c\varphi_{_L} +B_{_L}\mu_{_L}\Phi_{_L}\varphi_{_L}+h.c.\Big\} +\Big\{A_1\lambda_1\tilde{Q}\tilde{Q}_{_5}^cX+A_2\lambda_2\tilde{U}^c\tilde{U}_{_5}X^\prime +A_3\lambda_3\tilde{D}^c\tilde{D}_{_5}X^\prime+B_{_X}\mu_{_X}XX^\prime+h.c.\Big\}. \end{aligned} $

      (5)

      The first term $ {\cal L}_{_{soft}}^{MSSM} $ denotes the soft breaking terms of MSSM. To break the gauge symmetry from $ SU(3)_C \otimes SU(2)_L \otimes U(1)_Y \otimes U(1)_B \otimes U(1)_L $ to electromagnetic symmetry $ U(1)_e $, nonzero VEVs $ v_u, v_d $ and $ v_B, \bar{v}_B, v_L,\bar{v}_L $ are allocated to $ SU(2)_L $ doublets $ H_u, H_d $ and $ SU(2)_L $ singlets $ \Phi_B, \varphi_B, \Phi_L, \varphi_L $.

      $ \begin{aligned}[b] H_u =& \left(\begin{array}{c} H_u^+\\ (v_u+H_u^0+iP_u^0)/\sqrt{2} \end{array}\right),\end{aligned} $

      $ \begin{aligned}[b] H_u = &\left(\begin{array}{c} (v_d+H_d^0+iP_d^0)/\sqrt{2}\\ H_d^- \end{array}\right),\\ \Phi_B =& (v_B+\Phi_B^0+iP_B^0)/\sqrt{2},\\ \varphi_B =& (\bar{v}_B+\varphi_B^0+i\bar{P}_B^0)/\sqrt{2},\\ \Phi_L = &(v_L+\Phi_L^0+iP_L^0)/\sqrt{2},\\ \varphi_L =& (\bar{v}_L+\varphi_L^0+i\bar{P}_L^0)/\sqrt{2}. \end{aligned} $

      (6)

      Here we take the notation $ \tan\beta = v_u/v_d, \tan\beta_B = \bar{v}_B/v_B $ and $ \tan\beta_L = \bar{v}_L/v_L $. After spontaneously breaking and unitary transformation from interactive eigenstate to mass eigenstate, one can extract the Feynman rules and mass spectrums in BLMSSM. The mass matrices of the particles that mediate the one-loop process $ b\to s\gamma $ can be found in ref. [14]. The Feynman rules that we need can be extracted from the following terms, where all the repeated index of generation should be summed over.

      $ \begin{aligned}[b] {\cal{L}}_{H^\pm du} =& \Bigg(-Y_d^IZ_H^{1i}P_L+Y_u^JZ_H^{2i}P_R\Bigg)K^{JI*}\bar{d}^Iu^JH_i^-,\\ {\cal{L}}_{\tilde{D} \chi^0d} =& \Bigg[\Bigg(\frac{-e}{\sqrt{2}s_Wc_W}Z_D^{Ii} (\frac{1}{3}Z_N^{1j}s_W-Z_N^{2j}c_W)+Y_d^IZ_D^{(I+3)i}Z_N^{3j}\Bigg)P_L\\ & +\Bigg(\frac{-e\sqrt{2}}{3c_W}Z_D^{(I+3)i}Z_N^{1j*}+Y_d^IZ_D^{Ii}Z_N^{3j*} \Bigg)P_R\Bigg] \bar{\chi}^0_jd^I\tilde{D}^+_i,\\ {\cal{L}}_{\tilde{D}\chi_B^0d} =& \frac{\sqrt{2}}{3}g_B\Bigg(Z_{N_B}^{1j}Z_D^{Ii}P_L +Z_{N_B}^{1j*}Z_D^{(I+3)i}P_R\Bigg)\bar{\chi}_{B_j}^0d^I\tilde{D}^+,\\ {\cal{L}}_{\tilde{U}\chi^-d} =& \Bigg[\Bigg(\frac{-e}{s_W}Z_U^{Ji*}Z_+^{1j} +Y_u^JZ_U^{(J+3)i*}Z_+^{2j}\Bigg)P_L-Y_d^IZ_U^{Ji*}Z_-^{2j*}P_R\Bigg]\\&\times K^{JI}\bar{\chi}^-d\tilde{U}^-,\\ {\cal{L}}_{Xb^\prime d} =& \Bigg[\lambda_{1}(W_b^{\dagger})_{j1}(Z_X)_{1k}P_L -\lambda_{3}^*(U_b^{\dagger})_{j2}(Z_X)_{2k}P_R\Bigg]\bar{b}^\prime_jd^IX_k,\\ {\cal {L}}_{\tilde{b}^\prime\tilde{X} d} =& -\Bigg[\lambda_1(W_{\tilde{b}^\prime}^*)_{3\rho}P_L +\lambda_3^*(W_{\tilde{b}^\prime})_{4\rho}P_R\Bigg] \bar{\tilde{X}}d^I\tilde{b}^\prime_\rho,\\ {\cal {L}}_{\tilde{D}\Lambda_G d} = &g_3\sqrt{2}Y_{\alpha\beta}^a \Bigg(-Z_D^{Ii}P_L+Z_D^{(I+3)i}P_R\Bigg)\bar{\Lambda}_G^a d_\beta^I\tilde{D}^+_{i\alpha}. \end{aligned} $

      (7)

      The interactions from MSSM are collected from ref. [7] for completeness, and the Feynman gauge are adopt in our derivation to keep consist with the MSSM sector.

    III.   ONE-LOOP CORRECTIONS TO $ b\rightarrow s\gamma $ FROM BLMSSM
    • The flavor transition process $ b\rightarrow s\gamma $ can be described by effective Hamiltonian at scale $ \mu = O(m_b) $ as follow [15-17]:

      $ \begin{aligned}[b] {\cal {H}}_{eff}(b\to s\gamma) =& -\frac{4G_F}{\sqrt{2}}V_{ts}^*V_{tb} \Bigg[C_1Q_1^c+C_2Q_2^c\\&+\sum_{i = 3}^6C_iQ_i+\sum_{i = 7}^8(C_iQ_i+\tilde{C}_i\tilde{Q}_i)\Bigg], \end{aligned} $

      (8)

      and the operators are given by ref. [18-22]:

      $ \begin{aligned}[b] {\cal{O}}_1^c =& (\bar{s}_L\gamma_\mu T^ab_L)(\bar{c}_L\gamma^\mu T^ab_L),\\ {\cal{O}}_2^c =& (\bar{s}_L\gamma_\mu b_L)(\bar{c}_L\gamma^\mu T^a b_L),\\ {\cal{O}}_3 =& (\bar{s}_L\gamma_\mu b_L)\sum_q(\bar{q}\gamma^\mu q), \end{aligned} $

      $ \begin{aligned}[b]{\cal{O}}_4 =& (\bar{s}_L\gamma_\mu T^ab_L)\sum_q(\bar{q}\gamma^\mu T^a q),\\ {\cal{O}}_5 =& (\bar{s}_L\gamma_\mu \gamma_\nu\gamma_\rho b_L)\sum_q(\bar{q}\gamma^\mu \gamma^\mu \gamma^\nu\gamma^\rho q),\\ {\cal{O}}_6 =& (\bar{s}_L\gamma_\mu \gamma_\nu\gamma_\rho T^a b_L)\sum_q(\bar{q}\gamma^\mu \gamma^\mu \gamma^\nu\gamma^\rho T^a q),\\ {\cal{O}}_7 = & e/g_s^2m_b(\bar{s}_L\sigma_{\mu\nu}b_R)F^{\mu\nu},\\ {\cal{O}}_8 =& 1/g_s^2m_b(\bar{s}_L\sigma_{\mu\nu}T^ab_R)G^{a,\mu\nu},\\ \tilde{{\cal{O}}}_7 =& e/g_s^2m_b(\bar{s}_R\sigma_{\mu\nu}b_L)F^{\mu\nu},\\ \tilde{{\cal{O}}}_8 =& 1/g_s^2m_b(\bar{s}_R\sigma_{\mu\nu}T^ab_L)G^{a,\mu\nu}. \end{aligned} $

      (9)

      Coefficients of these operators can be extracted from Feynman amplitudes that originate from considered diagrams. Actually only the Coefficients of $ {\cal{O}}_{7,8} $ and $ \tilde{{\cal{O}}}_{7,8} $ are needed if we adopt the branching ratio formula presented in ref. [15]:

      $ \begin{aligned}[textbf{}] BR(\bar{B}\rightarrow X_s\gamma)_{NP} =& 10^{-4}\times\Bigg\{(3.32\pm0.15)\\&+\frac{16\pi^2a_{77}}{\alpha_s^2(\mu_b)}\big[|C_{7,NP}(\mu_{EW})|^2+|\tilde{C}_{7,NP}(\mu_{EW})|^2\big]\\ & +\frac{16\pi^2a_{88}}{\alpha_s^2(\mu_b)}\big[|C_{8,NP}(\mu_{EW})|^2+|\tilde{C}_{8,NP}(\mu_{EW})|^2\big]\\& +\frac{ 4\pi}{\alpha_s(\mu_b)}\rm{Re}\big[a_7C_{7,NP}(\mu_{EW})\!+\!a_8C_{8,NP}(\mu_{EW})\big.\\& +\big.\frac{ 4\pi a_{78}}{\alpha_s(\mu_b)}\big(C_{7,NP}(\mu_{EW})C_{8,NP}(\mu_{EW})\\&+\tilde{C}_{7,NP}(\mu_{EW})\tilde{C}_{8,NP}(\mu_{EW})\big)\big]\Bigg\}, \end{aligned} $

      (10)

      where the first term is SM prediction. The others come from new physics in which $ C_{7,NP}(\mu_{EW}) $, $ C_{8,NP}(\mu_{EW}) $, $ \tilde{C}_{7,NP}(\mu_{EW}) $ and $ \tilde{C}_{8,NP}(\mu_{EW}) $ indicate Wilson coefficients at electroweak scale. It is an advantage of this expression that we don't have to evolve them down to hadronic scale $ \mu\sim m_b $ as the effect of evolution has already been involved in the coefficients $ a_{7,8,77,88,78} $. The numerical values of these coefficients are given in table 1.

      $a_7$ $a_8$ $a_{77}$ $a_{88}$ $a_{78}$
      $-7.184+0.612i$ $-2.225-0.557i$ 4.743 0.789 2.454−0.884i

      Table 1.  Numerical values for the coefficients $a_{7,8,77,88,78}$ at electroweak scale

      To obtain the New Physics corrections in BLMSSM, we investigate one-loop diagrams shown in Figure 1. Photons should be attached to all inner lines with electric charge to complete the diagrams of $ b\to s\gamma $ that contribute to $ {\cal{O}}_7 $ and $ \tilde{{\cal{O}}}_7 $. Similarly, diagrams of $ b\to sg $ can be completed with gluons attached to all the inner lines with color charge, and $ {\cal{O}}_8 $ and $ \tilde{{\cal{O}}}_8 $ originate from these process.

      Figure 1.  One-loop Feynman diagrams of $b\to s$. The inner-line particles $\chi_B^0, X$ denote baryon neutralinos and new scalar particle introduced in BLMSSM. $b'$ and $\tilde{b}'$ are exotic quarks and squarks respectively. The photon and gluon can be attached in all possible ways

      The so-called flavor-changing self-energy diagrams in which photos or gluons are attached to external b or s quarks are not included during our calculations. As studied in Ref. [23-25], the contributions from those self-energy diagrams vanish when one of the external legs is on its mass shell. To preserve the Ward-Takahashi identity during the renormalization of $ \bar{s}bg $ and $ \bar{s}b\gamma $ vertices, one can always imposing the renormalized self-energies are zero as both b and s are on mass shell.

      In details, we attach a photon to SM quark $ u_i,(i = 1,2,3) $ or charged Higgs $ H^\pm $ in Figure 1.(a) to get a set of trigonal diagrams for $ b\to s\gamma $, while gluon can only be attached to up-type quarks $ u_i $ to form a specific diagram of $ b\to sg $. To give a complete correction originating from Figure 1.(a), contributions from all generations of $ u_i $ and Higgs should be summed over. From the amplitudes of these diagrams, one can extract Wilson coefficients of electric- and chromomagnetic-dipole operators $ {\cal{O}}_{7} $ and $ \tilde{{\cal{O}}}_{7} $ at electroweak scale,

      $ \begin{aligned}[b] \frac{G_F}{\sqrt{2}}C_{7\gamma}^a(\Lambda) =& -i\Lambda^{-2}(V^*_{ts}V_{tb})^{-1} \left\{ (\eta^L_{H^\pm})_{su_i}^\dagger (\eta^L_{H^\pm})_{u_ib}F_{1,\gamma}^{(a)}(x_{u_i},x_{H^\pm})\right.\\ & \left.+\frac{m_f}{m_b}(\eta^L_{H^\pm})_{su_i}^\dagger (\eta^R_{H^\pm})_{u_ib}F_{2,\gamma}^{(a)}(x_{u_i},x_{H^\pm})\right\},\\ \frac{G_F}{\sqrt{2}}\tilde{C}_{7\gamma}^a(\Lambda) = & \frac{G_F}{\sqrt{2}}C_{7\gamma}^a(\Lambda)\big(\eta^L_{H^\pm}\leftrightarrow\eta^R_{H^\pm}\big), \\[-5pt] \end{aligned} $

      (11)

      where $ x_i = m_i^2/\mu_{EW}^2 $. The concrete expressions of relevant couplings are already given in previous section. To be clear, the absence of divergences in the Wilson coefficients associated with one-loop triangle diagrams can be certified by expanding all the propagators in power of $ 1/(q^2-m^2_{f,S}) $, where $ q $ denotes the loop momentum. It can found that the order of q that appear in denominators are always higher than those in numerators. Thus, we do not have to deal with divergences during our evaluation. The form factors in Eq. (11) can be written as

      $ \begin{aligned}[b] F_{1,\gamma}^{(a)}(x,y) =& \Bigg[\frac{1}{72}\frac{\partial^3\varrho_{_{3,1}}}{\partial y^3} +\frac{1}{24}\frac{\partial^2\varrho_{_{2,1}}}{\partial y^2} -\frac{1}{ 6}\frac{\partial\varrho_{_{1,1}}}{\partial y}\Bigg](x,y),\\ F_{2,\gamma}^{(a)}(x,y) =& \Bigg[\frac{1}{12}\frac{\partial^2\varrho_{_{2,1}}}{\partial y^2} -\frac{1}{6}\frac{\partial\varrho_{_{1,1}}}{\partial y} -\frac{1}{3}\frac{\partial\varrho_{_{1,1}}}{\partial x}\Bigg](x,y), \end{aligned} $

      (12)

      where function $ \varrho_{m,n}(x,y) $ is defined as:

      $ \varrho_{_{m,n}}(x,y) = {x^m\ln^nx-y^m\ln^ny\over x-y}. $

      (13)

      Corrections from all the other diagrams to $ C_{7\gamma} $ and $ \tilde{C}_{7\gamma} $ can be obtained similarly. In Figure 1.(b), the photon can only be attached to charged −1/3 squark $ \tilde{D} $. We present contributions from both neutralinos $ \chi_i^0 $ and baryon neutralinos $ \chi_B^0 $ at electroweak scale as

      $ \begin{aligned}[b] \frac{G_F}{\sqrt{2}}C_{7\gamma}^b(\Lambda) =& -i\Lambda^{-2}(V^*_{ts}V_{tb})^{-1} \Bigg\{ (\xi^L_{\chi_i^0})_{s\tilde{D}}^\dagger (\xi^L_{\chi_i^0})_{\tilde{D}b}F_{1,\gamma}^{(b)}(x_{\chi_i^0},x_{\tilde{D}}) \\&+\frac{m_f}{m_b}(\xi^L_{\chi_i^0})_{s\tilde{D}}^\dagger (\xi^R_{\chi_i^0})_{\tilde{D}b}F_{2,\gamma}^{(b)}(x_{\chi_i^0},x_{\tilde{D}})\\ & +(\xi^L_{\chi_B^0})_{s\tilde{D}}^\dagger (\xi^L_{\chi_B^0})_{\tilde{D}b}F_{1,\gamma}^{(b)}(x_{\chi_B^0},x_{\tilde{D}}) \\&+\frac{m_f}{m_b}(\xi^L_{\chi_B^0})_{s\tilde{D}}^\dagger (\xi^R_{\chi_B^0})_{\tilde{D}b}F_{2,\gamma}^{(b)}(x_{\chi_B^0},x_{\tilde{D}})\Bigg\},\\ \frac{G_F}{\sqrt{2}}\tilde{C}_{7\gamma}^b(\Lambda) = & \frac{G_F}{\sqrt{2}}C_{7\gamma}^b(\Lambda)\big(\xi^L_{\chi_i^0}\leftrightarrow\xi^R_{\chi_i^0}\;,\;\xi^L_{\chi_B^0}\leftrightarrow\xi^R_{\chi_B^0}\big). \end{aligned} $

      (14)

      $ \begin{aligned}[b] F_{1,\gamma}^{(b)}(x,y) = & \Bigg[-\frac{1}{72}\frac{\partial^3\varrho_{_{3,1}}}{\partial^3y}+\frac{1}{24}\frac{\partial^2\varrho_{_{2,1}}}{\partial^2y}\Bigg](x,y),\\ F_{2,\gamma}^{(b)}(x,y) =& \Bigg[-\frac{1}{12}\frac{\partial^2\varrho_{_{2,1}}}{\partial^2y}+\frac{1}{6}\frac{\partial \varrho_{_{1,1}}}{\partial y}\Bigg](x,y). \end{aligned} $

      (15)

      With the photon attached to the charged +2/3 squarks $ \tilde{U} $ or chargino $ \chi^\pm_i $ in Figure 1.(c), the contributions to Wilson coefficients read

      $ \begin{aligned}[b] \frac{G_F}{\sqrt{2}}C_{7\gamma}^c(\Lambda) =& -i\Lambda^{-2}(V^*_{ts}V_{tb})^{-1} \Bigg\{ (\eta^L_{\tilde{U}})_{s\chi_i^{\pm}}^\dagger (\eta^L_{\tilde{U}})_{\chi_i^{\pm}b}F_{1,\gamma}^{(c)}(x_{\chi_i^{\pm}},x_{\tilde{U}}) \\&+\frac{m_f}{m_b}(\eta^L_{\tilde{U}})_{s\chi_i^{\pm}}^\dagger (\eta^R_{\tilde{U}})_{\chi_i^{\pm}b}F_{2,\gamma}^{(c)}(x_{\chi_i^{\pm}},x_{\tilde{U}})\Bigg\},\\ \frac{G_F}{\sqrt{2}}\tilde{C}_{7\gamma}^c(\Lambda) =& \frac{G_F}{\sqrt{2}}C_{7\gamma}^c(\Lambda)\big(\eta^L_{\tilde{U}}\leftrightarrow\eta^R_{\tilde{U}}\big). \end{aligned} $

      (16)

      $ \begin{aligned}[b] F_{1,\gamma}^{(c)}(x,y) =& \Bigg[-\frac{1}{72}\frac{\partial^3\varrho_{_{3,1}}}{\partial^3y} +\frac{1}{ 6}\frac{\partial^2\varrho_{_{2,1}}}{\partial^2y} -\frac{1}{ 4}\frac{\partial \varrho_{_{1,1}}}{\partial y}\Bigg](x,y),\\ F_{2,\gamma}^{(c)}(x,y) = & \Bigg[-\frac{1}{12}\frac{\partial^2\varrho_{_{2,1}}}{\partial^2y} +\frac{1}{ 6}\frac{\partial\varrho_{_{1,1}}}{\partial y} -\frac{1}{ 2}\frac{\partial\varrho_{_{1,1}}}{\partial x}\Bigg](x,y). \end{aligned} $

      (17)

      The intermediate particles in Figure 1.(d) are the exotic quarks $ b^\prime $ with charge -1/3 and superfield $ X $ introduced in BLMSSM. The contributions from this diagram are

      $ \begin{aligned}[b] \frac{G_F}{\sqrt{2}}C_{7\gamma}^d(\Lambda) =& -i\Lambda^{-2}(V^*_{ts}V_{tb})^{-1} \Bigg\{ (\eta^L_{X^j})_{sb^\prime}^\dagger (\eta^L_{X^j})_{b^\prime b}F_{1,\gamma}^{(d)}(x_{b^\prime},x_{X^j}) \\&+\frac{m_f}{m_b}(\eta^L_{X^j})_{sb^\prime}^\dagger (\eta^R_{X^j})_{b^\prime b}F_{2,\gamma}^{(d)}(x_{b^\prime},x_{X^j})\Bigg\},\\ \frac{G_F}{\sqrt{2}}\tilde{C}_{7\gamma}^d(\Lambda) =& \frac{G_F}{\sqrt{2}}C_{7\gamma}^d(\Lambda)\big(\eta^L_{X^j}\leftrightarrow\eta^R_{X^j}\big). \end{aligned} $

      (18)

      Correspondingly, the corrections of exotic squarks $ \tilde{b}^\prime $ with charge -1/3 and fermionic particle $ X $ can be obtained from Figure 1.(e)

      $ \begin{aligned}[b] \frac{G_F}{\sqrt{2}}C_{7\gamma}^e(\Lambda) =& -i\Lambda^{-2}(V^*_{ts}V_{tb})^{-1} \\&\times \Bigg\{ (\eta^L_{\tilde{b}^\prime})_{s\tilde{X}^j}^\dagger (\eta^L_{\tilde{b}^\prime})_{\tilde{X}^jb}F_{1,\gamma}^{(e)} (x_{\tilde{X}^j},x_{\tilde{b}^\prime}) \\& +\frac{m_f}{m_b}(\eta^L_{\tilde{b}^\prime})_{s\tilde{X}^j}^\dagger (\eta^R_{\tilde{b}^\prime})_{\tilde{X}^jb}F_{2,\gamma}^{(e)} (x_{\tilde{X}^j},x_{\tilde{b}^\prime})\Bigg\},\\ \frac{G_F}{\sqrt{2}}\tilde{C}_{7\gamma}^e(\Lambda) =& \frac{G_F}{\sqrt{2}}C_{7\gamma}^e(\Lambda)\big(\eta^L_{\tilde{b}^\prime}\leftrightarrow\eta^R_{\tilde{b}^\prime}\big). \end{aligned} $

      (19)

      $ \begin{aligned}[b] F_{1,\gamma}^{(e)}(x,y) = & \Bigg[-\frac{1}{72}\frac{\partial^3\varrho_{_{3,1}}}{\partial^3y}+\frac{1}{24}\frac{\partial^2\varrho_{_{2,1}}}{\partial^2y}\Bigg](x,y),\\ F_{2,\gamma}^{(e)}(x,y) = & \Bigg[-\frac{1}{12}\frac{\partial^2\varrho_{_{2,1}}}{\partial^2y}+\frac{1}{ 6}\frac{\partial\varrho_{_{1,1}}}{\partial y}\Bigg](x,y). \end{aligned} $

      (20)

      From Figure 1.(f), we obtain the corrections from gluinos $ \Lambda_G $ in MSSM, the Wilson coefficients at $ \mu_{EW} $ are

      $ \begin{aligned}[b] \frac{G_F}{\sqrt{2}}C_{7\gamma}^f(\Lambda) =& -i\Lambda^{-2}(V^*_{ts}V_{tb})^{-1} \\&\times \Bigg\{ (\eta^L_{\tilde{D}})_{s\Lambda_G}^\dagger (\eta^L_{\tilde{D}})_{\Lambda_Gb}F_{1,\gamma}^{(f)}(x_{\Lambda_G},x_{\tilde{D}}) \\& +\frac{m_f}{m_b}(\eta^L_{\tilde{D}})_{s\Lambda_G}^\dagger (\eta^R_{\tilde{D}})_{\Lambda_Gb}F_{2,\gamma}^{(f)}(x_{\Lambda_G},x_{\tilde{D}})\Bigg\},\\ \frac{G_F}{\sqrt{2}}\tilde{C}_{7\gamma}^f(\Lambda) =& \frac{G_F}{\sqrt{2}}C_{7\gamma}^f(\Lambda)\big(\eta^L_{\tilde{D}}\leftrightarrow\eta^R_{\tilde{D}}\big). \end{aligned} $

      (21)

      with

      $ \begin{aligned}[b] F_{1,\gamma}^{(f)}(x,y) =& \Bigg[\frac{1}{24}\frac{\partial^3\varrho_{_{3,1}}}{\partial^3y}-\frac{1}{8}\frac{\partial^2\varrho_{_{2,1}}}{\partial^2y}\Bigg](x,y),\\ F_{2,\gamma}^{(f)}(x,y) = & \Bigg[\frac{1}{ 4}\frac{\partial^2\varrho_{_{2,1}}}{\partial^2y}-\frac{1}{2}\frac{\partial\varrho_{_{1,1}}}{\partial y}\Bigg](x,y). \end{aligned} $

      (22)

      The corrections to $ C_{8g} $ and $ \tilde{C}_{8g} $ at electroweak scale can be obtained by attaching the gluon to intermediate virtual particles with colors. For diagrams in Figure 1, the gluon can be attached to SM up-type quarks $ u_i $, squarks in MSSM $ \tilde{U}, \tilde{D} $, exotic quarks $ b^\prime $ with charge -1/3 and its supersymmetric partners $ \tilde{b}^\prime $, as well as the gluinos $ \Lambda_G $. Wilson coefficients at electroweak scale can be formulated as:

      $ \begin{aligned}[b] \frac{G_F}{\sqrt{2}}C_{8G}^a(\Lambda) =& -i\Lambda^{-2}(V^*_{ts}V_{tb})^{-1} \left\{ (\eta^L_{H^\pm})_{su_i}^\dagger (\eta^L_{H^\pm})_{u_ib}F_{1,g}^{(a)}(x_{u_i},x_{H^\pm}) +\frac{m_f}{m_b}(\eta^L_{H^\pm})_{su_i}^\dagger (\eta^R_{H^\pm})_{u_ib}F_{2,g}^{(a)}(x_{u_i},x_{H^\pm})\right\},\\ \frac{G_F}{\sqrt{2}}\tilde{C}_{8G}^a(\Lambda) = & \frac{G_F}{\sqrt{2}}C_{8G}^a(\Lambda) (\eta^L_{H^\pm} \leftrightarrow \eta^R_{H^\pm}\;,\; \eta^L_{G^\pm} \leftrightarrow \eta^R_{G^\pm} ),\\ \frac{G_F}{\sqrt{2}}C_{8G}^b(\Lambda) = & -i\Lambda^{-2}(V^*_{ts}V_{tb})^{-1} \left\{ (\xi^L_{\chi_i^0})_{s\tilde{D}}^\dagger (\xi^L_{\chi_i^0})_{\tilde{D}b}F_{1,g}^{(b)}(x_{\chi_i^0},x_{\tilde{D}}) +\frac{m_f}{m_b}(\xi^L_{\chi_i^0})_{s\tilde{D}}^\dagger (\xi^R_{\chi_i^0})_{\tilde{D}b}F_{2,g}^{(b)}(x_{\chi_i^0},x_{\tilde{D}})\right.\\ & +\left.(\xi^L_{\chi_B^0})_{s\tilde{D}}^\dagger (\xi^L_{\chi_B^0})_{\tilde{D}b}F_{1,g}^{(b)}(x_{\chi_B^0},x_{\tilde{D}}) +\frac{m_f}{m_b}(\xi^L_{\chi_B^0})_{s\tilde{D}}^\dagger (\xi^R_{\chi_B^0})_{\tilde{D}b}F_{2,g}^{(b)}(x_{\chi_B^0},x_{\tilde{D}})\right\},\\ \frac{G_F}{\sqrt{2}}\tilde{C}_{8G}^b(\Lambda) = & \frac{G_F}{\sqrt{2}}C_{8g}^b(\Lambda)\big(\xi^L_{\chi_i^0}\leftrightarrow\xi^R_{\chi_i^0}\;,\;\xi^L_{\chi_B^0}\leftrightarrow\xi^R_{\chi_B^0}\big),\\ \frac{G_F}{\sqrt{2}}C_{8G}^c(\Lambda) = & -i\Lambda^{-2}(V^*_{ts}V_{tb})^{-1} \left\{ (\xi^L_{\chi_i^\pm})_{s\tilde{D}}^\dagger (\xi^L_{\chi_i^\pm})_{\tilde{D}b}F_{1,g}^{(c)}(x_{\chi_i^\pm},x_{\tilde{U}}) +\frac{m_f}{m_b}(\xi^L_{\chi_i^\pm})_{s\tilde{D}}^\dagger (\xi^R_{\chi_i^\pm})_{\tilde{D}b}F_{2,g}^{(c)}(x_{\chi_i^\pm},x_{\tilde{U}})\right\},\\ \frac{G_F}{\sqrt{2}}\tilde{C}_{8G}^c(\Lambda) = & \frac{G_F}{\sqrt{2}}C_{8g}^c(\Lambda)\big(\xi^L_{\chi_i^\pm}\leftrightarrow\xi^R_{\chi_i^\pm}\big),\\ \frac{G_F}{\sqrt{2}}C_{8G}^d(\Lambda) =& -i\Lambda^{-2}(V^*_{ts}V_{tb})^{-1} \left\{ (\eta^L_{X^j})_{sb^\prime}^\dagger (\eta^L_{X^j})_{b^\prime b} F_{1,g}^{(d)}(x_{b^\prime},x_{X^j}) +\frac{m_f}{m_b}(\eta^L_{X^j})_{sb^\prime}^\dagger (\eta^R_{X^j})_{b^\prime b} F_{2,g}^{(d)}(x_{b^\prime},x_{X^j})\right\},\\ \frac{G_F}{\sqrt{2}}\tilde{C}_{8G}^d(\Lambda) =& \frac{G_F}{\sqrt{2}}C_{8G}^d(\Lambda)\big(\eta^L_{X^j}\leftrightarrow\eta^R_{X^j}\big),\\ \frac{G_F}{\sqrt{2}}C_{8G}^e(\Lambda) = & -i\Lambda^{-2}(V^*_{ts}V_{tb})^{-1} \left\{ (\xi^L_{\tilde{b}^\prime})_{s\tilde{X}^j}^\dagger (\xi^L_{\tilde{b}^\prime})_{\tilde{X}^jb}F_{1,g}^{(e)} (x_{\tilde{X}^j},x_{\tilde{b}^\prime}) +\frac{m_f}{m_b}(\xi^L_{\tilde{b}^\prime})_{s\tilde{X}^j}^\dagger (\xi^R_{\tilde{b}^\prime})_{\tilde{X}^jb}F_{2,g}^{(e)} (x_{\tilde{X}^j},x_{\tilde{b}^\prime})\right\},\\ \frac{G_F}{\sqrt{2}}\tilde{C}_{8G}^e(\Lambda) = & \frac{G_F}{\sqrt{2}}C_{8G}^e(\Lambda)\big(\xi^L_{\tilde{b}^\prime}\leftrightarrow\xi^R_{\tilde{b}^\prime})\big), \end{aligned} $

      (23)

      with the form factors listed below. As gluon can only be attached to intermediate fermion $ u_i $ and $ b^\prime $ in Figure 1.(a) and 1.(d), so the form factors have the same expressions. While in Figure 1.(b), 1.(c) and Figure 1.(e), the gluon can only be attached to scalar particles. Then form factors associated to these diagrams are the same. By summing over the contributions to Wilson coefficients when gluon attached to $ \Lambda_G $ and $ \tilde{D} $, we get form factors of Figure 1.(f).

      $ \begin{aligned}[b] F_{1,g}^{(a)}(x,y) =& F_{1,g}^{(d)}(x,y) = \Bigg[ -\frac{1}{24}\frac{\partial^3\varrho_{_{3,1}}}{\partial^3y} +\frac{1}{ 4}\frac{\partial^2\varrho_{_{2,1}}}{\partial^2y} -\frac{1}{ 4}\frac{\partial\varrho_{_{1,1}}}{\partial y}\Bigg](x,y),\\ F_{2,g}^{(a)}(x,y) =& F_{2,g}^{(d)}(x,y) = \Bigg[ -\frac{1}{4}\frac{\partial^2\varrho_{_{2,1}}}{\partial^2y} +\frac{1}{2}\frac{\partial \varrho_{_{1,1}}}{\partial y} -\frac{1}{2}\frac{\partial \varrho_{_{1,1}}}{\partial x}\Bigg](x,y),\\ F_{1,g}^{(b)}(x,y) = & F_{1,g}^{(c)}(x,y) = F_{1,g}^{(e)}(x,y) = \Bigg[ \frac{1}{24}\frac{\partial^3\varrho_{_{3,1}}}{\partial^3y} -\frac{1}{8}\frac{\partial^2\varrho_{_{2,1}}}{\partial^2y}\Bigg](x,y),\\ F_{2,g}^{(b)}(x,y) =& F_{2,g}^{(c)}(x,y) = F_{2,g}^{(e)}(x,y) = \Bigg[ \frac{1}{4}\frac{\partial^2\varrho_{_{2,1}}}{\partial^2y} -\frac{1}{2}\frac{\partial \varrho_{_{1,1}}}{\partial y}\Bigg](x,y),\\ F_{1,g}^{(f)}(x,y) = & \Bigg[\frac{1}{8}\frac{\partial^2\varrho_{_{2,1}}}{\partial^2y} -\frac{1}{4}\frac{\partial \varrho_{_{1,1}}}{\partial y}\Bigg](x,y),\\ F_{2,g}^{(f)}(x,y) =& \Bigg[-\frac{1}{2}\frac{\partial \varrho_{_{1,1}}}{\partial x}\Bigg](x,y). \end{aligned} $

      (24)

      The Wilson coefficients obtained above can also be used to direct CP-violation in $ \bar{B}\rightarrow X_s\gamma $ and the time-dependent CP-asymmetry in $ B\rightarrow K^*\gamma $. The direct CP-violation $ A^{CP}_{\bar{B}\rightarrow X_s\gamma} $ and CP-asymmetry $ S_{K^*\gamma} $ are defined in hadronic scale [26-30]

      $ \begin{aligned}[b] A^{CP}_{\bar{B}\rightarrow X_s\gamma} =& \left.\frac{\Gamma(\bar{B}\rightarrow X_s\gamma)-\Gamma(B\rightarrow X_{\bar{s}}\gamma)} {\Gamma(\bar{B}\rightarrow X_s\gamma)+\Gamma(B\rightarrow X_{\bar{s}}\gamma)}\right|_{E_\gamma > (1-\delta)E_\gamma^{max}}\\ \simeq & \frac{10^{-2}}{|C_7(\mu_b)|^2} \Big[1.23\mathfrak{I}\big(C_2(\mu_b)C_7^*(\mu_b)\big)\\ &-9.52\mathfrak{I}\big(C_8(\mu_b)C_7^*(\mu_b)\big)\\&+0.01\mathfrak{I}\big(C_2(\mu_b)C_8^*(\mu_b)\big)\Big], \end{aligned} $

      (25)

      $ S_{K^*\gamma} \simeq \frac{2\text{Im}(e^{-i\phi_d}C_7(\mu_b)C_7'(\mu_b))}{|C_7(\mu_b)|^2+|C_7'(\mu_b)|^2}, $

      (26)

      where the photon energy cut in $ A^{CP} $ is taken as $ \delta = 3 $, and $ \phi_d $ in $ S_{K^*\gamma} $ is phase of $ B_d $ mixing amplititude. Here we use the experimental data $ \sin\phi_d = 0.67\pm0.02 $ given in ref. [31].

      As the Wilson coefficients are calculated at electroweak scale $ \mu_{EW} $, we need to evolve them down to hadronic scale $ \mu\sim m_b $ with renormalization group equations.

      $ \begin{aligned}[b] \vec{C}_{NP}(\mu) =& \hat{U}(\mu,\mu_0)\vec{C}_{NP}(\mu_0),\\ \vec{C}'_{NP}(\mu) =& \hat{U}'(\mu,\mu_0)\vec{C}'_{NP}(\mu_0), \end{aligned} $

      (27)

      where the Wilson coefficients are constructed as

      $ \begin{aligned}[b] \vec{C}^\mathrm{T}_{NP} =& (C_{1,NP},\cdots,C_{6,NP},C^{eff}_{7,NP},C^{eff}_{8,NP}),\\ \vec{C}^{\prime,\mathrm{T}}_{NP} =& (C'^{eff}_{7,NP},C'^{eff}_{8,NP}). \end{aligned} $

      (28)

      The evolving matrices involved in Eq.(27) are given as

      $ \begin{aligned}[b] \hat{U}(\mu,\mu_0)\simeq & 1- \left[ \frac{1}{2\beta_0}\ln\frac{\alpha_s(\mu)}{\alpha_s(\mu_0)} \right]\hat{\gamma}^{(0)T},\\ \hat{U}'(\mu,\mu_0)\simeq & 1- \left[ \frac{1}{2\beta_0}\ln\frac{\alpha_s(\mu)}{\alpha_s(\mu_0)} \right]\hat{\gamma}'^{(0)T}, \end{aligned} $

      (29)

      with anomalous dimension matrices

      $ \begin{array}{l} \hat{\gamma}^{(0)} = \left(\begin{array}{cccccccc} -4 & \frac{8}{3} & 0 & -\frac{2}{9} & 0 & 0 & -\frac{208}{243} & \frac{173}{162}\\ 12 & 0 & 0 & \frac{4}{3} & 0 & 0 & \frac{416}{81} & \frac{70}{27}\\ 0 & 0 & 0 & -\frac{52}{3} & 0 & 2 & -\frac{176}{81} & \frac{14}{27}\\ 0 & 0 & -\frac{40}{9} & -\frac{100}{9} & \frac{4}{9} & \frac{5}{6} & -\frac{152}{243} & -\frac{587}{162}\\ 0 & 0 & 0 & -\frac{256}{3} & 0 & 20 & -\frac{6272}{81} & \frac{6596}{27}\\ 0 & 0 & -\frac{256}{9} & \frac{56}{9} & \frac{40}{9} & -\frac{2}{3} & \frac{4624}{243} & \frac{4772}{81}\\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{32}{3} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -\frac{32}{9} & \frac{28}{3} \end{array}\right), \end{array} $

      (30)

      and

      $ \begin{array}{l} \hat{\gamma}'^{(0)} = \left(\begin{array}{cc} \frac{32}{3} & 0\\ -\frac{32}{9} & \frac{28}{3} \end{array}\right). \end{array} $

      (31)

      As the renormalization group evolution of new physics contributions can be performed independently from the SM [32], we evolved the new physics contributions from electroweak scale to hadronic scale separately. Then the complete result of Wilson coefficients at hadronic scale are obtained by adding the SM parts denoted by $ C^{eff}_{7\gamma}, C^{eff}_{8g} $. To get a result as accurate as possible, we adopt the NNLO result from SM in our numerical analysis, $ C_7^{eff}(m_b) = -0.304, C_8^{eff}(m_b) = -0.167 $.

    IV.   NUMERICAL ANALYSIS
    • The consistency of SM prediction and experimental data on $ \bar{B}\to X_s\gamma $ sets stringent constraint on new physics parameters. In this section, we discuss the numerical results of branching ratio with some assumptions. The SM inputs are given in Table 2. All the parameters with mass dimension are given in the unit GeV. To be concise, we omit all the unit GeV in this section.

      $ \alpha $ $ 1/128 $ $ m_W $ $ 80.385 $ $ m_Z $ $ 91.188 $
      $ m_u $ $ 0.0023 $ $ m_c $ $ 1.275 $ $ m_t $ $ 173.5 $
      $ m_d $ $ 0.0048 $ $ m_s $ $ 0.095 $ $ m_b $ $ 4.18 $

      Table 2.  SM inputs in numerical analysis

      As we know, the heaver new particles appearing in inner lines are, the lower it contribute to Wilson Coefficient that we need. Besides, new particles with too heavy mass are not favorite because it is difficulty to reach on the nowadays colliders. On the other hand, no signal of new particles has been observed by now. Thus the masses of exotic particles have to be heaver than a few of TeV. Based on previous study on mass spectrum in BLMSSM in [14], the parameters introduced in BLMSSM are set as $ A_{BU} = A_{BD} = A_{BQ} = A'_{BU} = A'_{BD} = A'_{BQ} = A_{d_4} = A_{d_5} = A_{u_4} = A_{u_5} = $ $ A'_{d_4} = A'_{d_5} = A'_{u_4} = A'_{u_5} = 100, $ $ M^2_{\tilde{Q}_4} = M^2_{\tilde{Q}_5} = M^2_{\tilde{U}_4} = M^2_{\tilde{U}_5} = $$ M^2_{\tilde{D}_4} = M^2_{\tilde{D}_5} = 2500 $, $ m_1 = m_2 = 1200 $ and $ m_{Z_B} = 1000 $ to make sure the masses of new physics particles under experimental limitations. With the above setups, one can scan the other sensitive parameters with masses of new particles around a few TeV as a condition in the numerical program.

      As a new field introduced in BLMSSM, superfield $ X $ interacts with exotic quarks. The coplings between $ X $ and $ \hat{Q_5}, \hat{U_5} $ denoted by $ \lambda_i, (i = 1,2,3) $ are given in Eq.4. From the analytical expressions, one can find the Wilson coefficients are sensitive to these couplings as well as coefficients of mass term of $ X $, which turns up in $ {\cal{W}}_X $ as $ \mu_X $ and $ B_X $. We show the branching ratio varying with $ \lambda_1 $, $ \lambda_3 $, $ \mu_X $ and $ B_X $ in firgure 2. The dependency of $ \lambda_2 $ is not listed as it is similar to $ \lambda_1 $.

      In Figure 2.(a), one can find the branching ratio increases when $ \lambda_1 $ raises up. The experimental limitations are denoted by the gray area, and we have taken $ \lambda_3/(4\pi) = 0.07, \,\tan\beta = 10,\, \lambda_Q = 0.7,\, \lambda_U = 0.3,\, \lambda_D = 0.2, \, \mu = $ $ -600, \mu_B = m_{Z_B} = m_B = 1000, \mu_X = 1100, B_X = 400, v_{bt} = 6000 .$ It can be seen in this figure that branching ratio reachs the upper limitations of experimental limitations when $ \lambda_1/(4\pi) = 0.95 $, then we get the constrain $ \lambda_1/(4\pi)<0.95 $.

      Figure 2.  $ Br(\bar{B}\to X_s\gamma) $ varying with parameters relevant to superfield $ X $

      Similarly, we plot the branching ratio varying with $ \lambda_3 $ in Figure 2.(b). By taking $ \lambda_1/(4\pi) = 0.07 $, which satisfies the limitations obtained in Figure 2.(a), and $ \tan\beta = $ $ 5, \lambda_Q = 0.7, \lambda_U = 0.5, \lambda_D = 0.8, \mu = -800, \mu_B = \mu_X = 1100, m_{Z_B} = $ $ 1000, m_B = 500, B_X = 400, v_{bt} = 6000 $, we find the branching ratio rises very slowly when $ \lambda_3/(4\pi) $ runs from 0.01 to 1.5. The whole curve lies in the gray area, which means the branching ratio satisfies the experimental constraint under our assumptions.

      To investigate the trends of $ Br(\bar{B}\to X_s\gamma) $ varying with $ \mu_X $, we take $ \lambda_1/(4\pi) = 0.06, \lambda_3/(4\pi) = 0.08, \tan\beta = 5, \lambda_Q = $ $0.8, \lambda_U = 0.5, \lambda_D = 0.2, \mu = -600, \mu_B = 1000, B_X = 400, m_{z_B} = $ $1100, v_{bt} = 6000, m_B = 2500 $. We find from Figure 2.(c) that the branching ratio diminishes steeply with increasing of $ \mu_X $, and finally gets to the value of standard model. In Figure 2.(d), we plot the branching ratio varying with $ B_X $, where we take $ \lambda_1/(4\pi) = 0.15, \lambda_3/(4\pi) = 0.08, \tan\beta = $ $ 5,\, \lambda_Q = 0.7,\, \lambda_U = 0.2, \,\,\lambda_D = 0.3,\, \,\mu = -1000,\, \,\mu_B = 1100,\, \mu_X = $ $ 2500, m_{z_B} = 900, v_{bt} = 5500, m_B = 2000$. With the upper limitations of experimental result, we get the constraints $ B_X<1625 $.

      In Figure 3.(a), we present the branching ratio varying with $ \lambda_Q $, which is the coupling truns up in superpotential term $ \lambda_{_Q}\hat{Q}_{_4}\hat{Q}_{_5}^c\hat{\Phi}_{_B} $. With $ \lambda_1/(4\pi) = 0.08, \lambda_3/(4\pi) = 0.06, $$ \tan\beta = 20, \lambda_U = 0.3, \lambda_D = 0.6, \mu = -800, \mu_B = 1000, \mu_X = 1200, $$B_X = 400, m_{z_B} = 1000, v_{bt} = 5000, m_B = 1500 $, we find branching ratio decreases when $ \lambda_Q $ gets larger. To consist with the experimental data, one has $ \lambda_Q>0.62 $. Another interesting parameter is $ v_{bt} $, which is defined as $ v_{bt} = \sqrt{\bar{v}_B^2+v_B^2} $, where $ v_B $ and $ \bar{v}_B $ are VEVs of $ \Phi_B $ and $ \varphi_B $ respectively. We plot branching ratio varying with $ v_{bt} $ in Figure 3.(b) with $ \lambda_1/(4\pi) = \lambda_3/(4\pi) = 0.1, \tan\beta = 5, \lambda_Q = 0.7, $$ \lambda_U = 0.2, \lambda_D = 0.7, \mu = -1000, \mu_B = 1100, \mu_X = 1400, B_X = 400, $$ m_{z_B} = 1000, m_B = 1500 $. To satisfy the experimental constraints, we have $ v_{bt}>4200 $.

      Figure 3.  $ Br(\bar{B}\to X_s\gamma) $ varying with $ \lambda_Q $ and $ v_{bt} $

      Additionally, we plot the direct CP-violation of $ \bar{B}\to X_s\gamma $ and time-dependent CP-asymmetry of $ B\rightarrow K^*\gamma $ varying with $ \lambda_1, \lambda_3, \mu_X, B_X, \lambda_Q, \lambda_D $ and $ v_{bt} $. Within the framework of SM, we have $ -0.6 \%<A_{CP}^{SM}<+2.8 \% $ [33], and the average value of this observable is $ A_{CP}^{exp} = -0.009\pm 0.018 $[35,34]. Within some uncertainty, the theoretical value is consistent with the experimental result. Compared with direct CP-violation of $ \bar{B}\to X_s\gamma $, there is significant deviation between SM prediction and experimental result of $ S_{K^*\gamma} $. The SM prediction of time-dependent CP-asymmetry in $ B\rightarrow K^*\gamma $ at LO level is given as $ S_{K^*\gamma}^{SM}\simeq (-2.3\pm1.6) \% $ [36] and the experimental result is $ S_{K^*\gamma}\simeq -0.15\pm 0.22 $ [34].

      To investigate $ A^{CP}_{\bar{B}\to X_s\gamma} $ and $ S_{K^*\gamma} $ numerically, some parameters are taken to be complex, and the area within experimental boundaries are filled to be gray in the presented figures. In Figure 4, we plot the dependency of parameters relevant to superfield $ X $. Under our assumptions of free parameters introduced in BLMSSM, we find that $ A^{CP}_{\bar{B}\to X_s\gamma} $ (solid line) are hardly affected by the change of $ \lambda_1, \lambda_3, \mu_X, B_X $. Though corrections from one-loop level are almost zero, the numerical results are consistent with experimental data.

      Figure 4.  $ A^{CP}_{\bar{B}\rightarrow X_s\gamma} $ and $ S_{K^*\gamma} $ varying with paremeters relevant to superfield $ X $

      As shown in Figure 4.(a), one-loop corrections to $ S_{K^*\gamma} $ (dashed line) in BLMSSM can reach $ -0.25 $ with appropriate inputs. By changing the free parameters, one finds $ S_{K^*\gamma} $ can be as small as zero in Figure 4.(b). In Figure 4.(c), it can be seen that $ S_{K^*\gamma} $ raise obviously with increasing of $ \mu_X $, and finally gets stable around zero. The $ S_{K^*\gamma} $ varying with $ B_X $ are given in Figure 4.(d). When $ B_X $ raises up, we can see that $ S_{K^*\gamma} $ decreases. Within the range of parameters $ \lambda_1, \lambda_3, \mu_X $ and $ B_X $, we find $ S_{K^*\gamma} $ is consistent with experimental data.

      In Figure 5, we take into account the parameters $ \lambda_Q $ and $ \lambda_D $. When $ \lambda_Q $ runs from 0.01 to 2.0, the time-dependent CP-asymmetry decrease from $ 0.02 $ to $ -0.22 $. While for the increasing of $ \lambda_D $, $ S_{K^*\gamma} $ raises from $ -0.28 $ to $ -0.02 $. Under our assumptions, we conclude that $ \lambda_Q $ and $ \lambda_D $ affect $ S_{K^*\gamma} $ apparently, and the numerical results of new physics correction are consistent with experimental data. However, the direct CP-violation of $ \bar{B}\to X_s\gamma $ depends on $ \lambda_Q $ and $ \lambda_D $ weakly, and the one-loop contributions from BLMSSM are very small.

      Figure 5.  $ A^{CP}_{\bar{B}\rightarrow X_s\gamma} $ and $ S_{K^*\gamma} $ varying with $ \lambda_Q $ and $ \lambda_D $

      The last Figure 6 illustrates the trend of $ S_{K^*\gamma} $ and $ A^{CP}_{\bar{B}\to X_s\gamma} $ varying with $ v_{bt} $. By taking $ \lambda_1/(4\pi) = 0.8, \lambda_3/(4\pi) = 0.9, B_X = 400 $ and $ \lambda_Q = 0.4e^{0.625\pi} $, we find that $ S_{K^*\gamma} $ increases from $ -0.26 $ to $ -0.06 $. The $ A^{CP}_{\bar{B}\to X_s\gamma} $ stays around zero within the range $ 100<v_{bt}<10000 $.

      Figure 6.  $ A^{CP}_{\bar{B}\rightarrow X_s\gamma} $ and $ S_{K^*\gamma} $ varying with $ v_{bt} $

      To analyze the dependence of Wilson coefficients and CP-asymmetry on the scale $ \mu_b $, numerical results of $ C_{7,8}^{NP} $, $ \tilde{C}_{7,8}^{NP} $, $ A_{CP} $ and $ S_{K^*\gamma} $ are given in Table 3. The input parameters are the same as in Figure 5.(b) with $ \lambda_D = 0.1 $. It can be seen that the CP violation/asymmetry get more significant when $ \mu_b $ become lager. By printing the coefficients from different diagrams seperately, one can find the dominant corrections come from Figure 1.(e), which contains exotic squark charged -1/3 and superfield $ \tilde X $.

      $ \mu_b $ $ |C_7^{NP}| $ $ |C_8^{NP}| $ $ |\tilde{C}_7^{NP}| $ $ |\tilde{C}_8^{NP}| $ $ A_{CP} $ $ S_{K^*\gamma} $
      $ m_b/2=2.09 $ GeV $ 0.078 $ $ 0.013 $ $ 0.053 $ $ 0.014 $ $ -0.019 $ $ -0.200 $
      $ m_b=4.18 $ GeV $ 0.100 $ $ 0.016 $ $ 0.067 $ $ 0.017 $ $ -0.025 $ $ -0.220 $
      $ 2m_b=8.36 $ GeV $ 0.118 $ $ 0.018 $ $ 0.078 $ $ 0.020 $ $ -0.028 $ $ -0.228 $

      Table 3.  Dependence of $ C_{7,8}^{NP} $, $ \tilde{C}_{7,8}^{NP} $, $ A_{CP} $ and $ S_{K^*\gamma} $ on typical scales

    V.   CONCLUSIONS
    • As an interesting process of FCNC, we investigate the transition $ b\to s\gamma $ within the framework of BLMSSM. With effective Hamiltonian method, we present the Wilson coefficients extracted from amplitudes corresponding to the concerned one-loop diagrams. Based on the analytical expressions, constraints on parameters are given in the numerical section with the experimental data of branching ratio of $ \bar{B}\to X_s\gamma $. The direct CP-violation of $ \bar{B}\to X_s\gamma $ in BLMSSM is very small, and depend on the free parameters weakly. However, the time-dependent CP-asymmetry $ S_{K^*\gamma} $ in $ B\rightarrow K^*\gamma $ varies with $ \mu_X, B_X, \lambda_Q, \lambda_D $ and $ v_{bt} $ obviously. The contributions from new physics can reach $ -0.28 $ under appropriate setup of the parameters.

Reference (36)

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