B̄→ Xsγ in BLMSSM*

B̄→ Xsγ B̄→ Xsγ B→ K∗γ ACP S K∗γ Abstract: Applying the effective Lagrangian method, we study the flavor changing neutral current process 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 for the branching ratio of . Additionally, we present the corrections to direct CP-violation in and time-dependent CP-asymmetry in . With appropriate assumptions on the parameters, we find the direct CP-violation is very small, while one-loop contributions to can be significant.


I. INTRODUCTION
Since the Flavor Changing Neutral Current process(FCNC) b → sγ originates only from loop diagrams, it is very sensitive to new physics beyond the Standard Model(SM). The updated average data of inclusiveB → X s γ is [1] BR(B → X s γ) exp = (3.40 ± 0.21) × 10 −4 .
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) [10] 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 [11,12] 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 [13] To describe the symmetries of baryon and lepton numbers, gauge group is enlarged to SU(3) C ⊗ SU(2) L ⊗ U(1) Y ⊗ U(1) B ⊗ U(1) L . Then corrections to various observations can be induced from new gauge boson and exotic fields within this scenario. In ref. [14], 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. [15]. To account for the experimental data on Higgs, the authors of [16] study the signals of h → γγ and h → V V * (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 → sγ 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 are responsable for the breaking of lepton number [12]. The superfieldsX ∼ (1, 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 where W M SSM indicates the superpotential of MSSM, and Here we take the notation tan β = v u /v d , tan β B =v B /v B and tan β L =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 → sγ can be found in ref. [17]. The Feynman rules that we need can be extracted from the following terms, where all the repeated index of generation should be summed over.
III. ONE-LOOP CORRECTIONS TO b → sγ FROM BLMSSM The flavor transition process b → sγ can be described by effective Hamiltonian at scale µ = O(m b ) as follow [18]: and the operators are given by ref. [19][20][21]: Coefficients of these operators can be extracted from Feynman amplitudes that originate from considered diagrams. Actually only the Coefficients of O 7,8 andÕ 7,8 are needed if we adopt the branching ratio formula presented in ref. [18]: where the first term is SM prediction. The others come from new physics in which where The concrete expressions of relevant couplings are already given in previous section, and the form factors can be written as: where function ̺ m,n (x, y) is defined as: Corrections from all the other diagrams to C 7γ andC 7γ can be obtained similarly. In Figure 1.(b), the photon can only be attached to charged -1/3 squarkD. We present contributions from both neutralinos χ 0 i and baryon neutralinos χ 0 B at electroweak scale as With the photon attached to the charged +2/3 squarksŨ or chargino χ ± i in Figure 1.(c), the contributions to Wilson coefficients read The intermediate particles in Figure 1.(d) are the exotic quarks b ′ with charge -1/3 and superfield X introduced in BLMSSM. The contributions from this diagram are Correspondingly, the corrections of exotic squarksb ′ with charge -1/3 and fermionic particle X can be obtained from Figure 1.(e) From Figure 1.(f), we obtain the corrections from gluinos Λ G in MSSM, the Wilson coefficients at µ EW are with The corrections to C 8g andC 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 MSSMŨ ,D, exotic quarks b ′ with charge -1/3 and its supersymmetric partnersb ′ , as well as the gluinos Λ G . Wilson coefficients at electroweak scale can be formulated as: with the form factors listed below. As gluon can only be attached to intermediate fermion The Wilson coefficients obtained above can also be used to direct CP-violation inB → X s γ and the time-dependent CP-asymmetry in B → K * γ. The direct CP-violation A CP B→Xsγ and CP-asymmetry S K * γ are defined in hadronic scale [22][23][24][25][26] where the photon energy cut in A CP is taken as δ = 3, and φ d in S K * γ is phase of B d mixing amplititude. Here we use the experimental data sin φ d = 0.67 ± 0.02 given in ref. [27].
As the Wilson coefficients are calculated at electroweak scale µ EW , we need to evolve them down to hadronic scale µ ∼ m b with renormalization group equations.
where the Wilson coefficients are constructed as C T N P = (C 1,N P , · · · , C 6,N P , C ef f 7,N P , C ef f 8,N P ), C ′,T N P = (C ′ef f 7,N P , C ′ef f 8,N P ).
To be convenient, we take the numerrical results of effective coefficients C ef f from SM at next-to-next-to-leading logarithmic(NNLL) level, C ef f 7 (m b ) = −0.304, C ef f 8 (m b ) = −0.167. The evolving matrices involved in Eq. (27) are given aŝ with anomalous dimension matriceŝ

IV. NUMERICAL ANALYSIS
The consistency of SM prediction and experimental data onB → X s γ 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 II. All the parameters with mass dimension are given in the unit GeV. To be concise, we omit all the unit GeV in this section. Other free parameters introduced in BLMSSM are set to be As a new field introduced in BLMSSM, superfield X interacts with exotic quarks. The coplings between X andQ 5 ,Û 5 denoted by λ 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 W X as µ X and B X . We show the branching ratio varying with λ 1 , λ 3 , µ X and B X in firgure 2. The dependency of λ 2 is not listed as it is similar to λ 1 .   Additionally, we plot the direct CP-violation ofB → X s γ and time-dependent CPasymmetry of B → K * γ varying with λ 1 , λ 3 , µ X , B X , λ Q , λ D and v bt . Within the framework of SM, we have −0.6% < A SM CP < +2.8% [28], and the average value of this observable is A exp CP = −0.009 ± 0.018 [1]. Within some uncertainty, the theoretical value is consistent with the experimental result. Compared with direct CP-violation ofB → X s γ, there is significant deviation between SM prediction and experimental result of S K * γ . The SM prediction of time-dependent CP-asymmetry in B → K * γ at LO level is given as S SM K * γ ≃ (−2.3 ± 1.6)% [29] and the experimental result is S K * γ ≃ −0.15 ± 0.22 [1,19].
To investigate A CP B→Xsγ and S K * γ 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 B→Xsγ (solid line) are hardly affected by the change of λ 1 , λ 3 , µ X , B X . Though corrections from one-loop level are almost zero, the numerical results are consistent with experimental data.
As shown in Figure 4.(a), one-loop corrections to S K * γ (dashed line) in BLMSSM can reach −0.25 with appropriate inputs. By changing the free parameters, one finds S K * γ can be as small as zero in Figure 4.(b). In Figure 4.(c), it can be seen that S K * γ raise obviously with increasing of µ X , and finally gets stable around zero. The S K * γ varying with B X are given in Figure 4.(d). When B X raises up, we can see that S K * γ decreases. Within the range of parameters λ 1 , λ 3 , µ X and B X , we find S K * γ is consistent with experimental data.
In Figure 5, we take into account the parameters λ Q and λ D . When λ 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 λ D , S K * γ raises from −0.28 to −0.02. Under our assumptions, we conclude that λ Q and λ D affect S K * γ apparently, and the numerical results of new physics correction are consistent with experimental data. However, the direct CP-violation ofB → X s γ depends on λ Q and λ D weakly, and the one-loop contributions from BLMSSM are very small. The last Figure 6 illustrates the trend of S K * γ and A CP B→Xsγ varying with v bt . By taking λ 1 /(4π) = 0.8, λ 3 /(4π) = 0.9, B X = 400 and λ Q = 0.4e 0.625π , we find that S K * γ increases from −0.26 to −0.06. The A CP B→Xsγ stays around zero within the range 100 < v bt < 10000. on the analytical expressions, constraints on parameters are given in the numerical section with the experimental data of branching ratio ofB → X s γ. The direct CP-violation of B → X s γ in BLMSSM is very small, and depend on the free parameters weakly. However, the time-dependent CP-asymmetry S K * γ in B → K * γ varies with µ X , B X , λ Q , λ D and v bt obviously. The contributions from new physics can reach −0.28 under appropriate setup of