${B^0} - {{\bar B}^0}$ mixing in supersymmetry with gauged baryon and lepton numbers

We perform an analysis on ${B^0} - {{\bar B}^0}$ mixing in the extension of the minimal supersymmetric standard model where baryon and lepton numbers are local gauge symmetries (BLMSSM) by using the effective Hamiltonian method. And the constraint of a 125 GeV Higgs to the parameter space has also been consid-ered. The numerical results indicate that the contributions of the extra particles can be sizeable in ${B^0} - {{\bar B}^0}$ mixing. For certain parameter sets, the theoretical prediction of mass differences $\Delta m_{B}$ agrees with the current experimental result. Furthermore, ${B^0} - {{\bar B}^0}$ mixing in the BLMSSM can preliminarily constrain the parameter space. With the development of more precise theoretical analysis and experimental determina-tions, the ${B^0} - {{\bar B}^0}$ mixing in the BLMSSM will have a clearer picture and the parameter space in this model will also be further constrained.


I. INTRODUCTION
The Minimal Supersymmetric Standard Model (MSSM) [1][2][3][4][5], as one of the most appealing options for the physics beyond the Standard Model (SM), has drawn the physicists' attention for a long time.As the simplest soft broken supersymmetry (SUSY) theory, the MSSM can solve hierarchy problem, ensure that the gauge couplings unify at high energies and provide a good dark matter candidate.To search for new particles predicted by SUSY, the Large Hadron Collider (LHC) has collected huge amounts of data, the CMS [7] and ATLAS [8] experiments now set strong limits on these parameter space [9][10][11][12].However, the present searches are largely based on the assumption of conserved R-parity [6].Some studies in the low-energy SUSY have been motivated by the results of the LHC [13][14][15][16][17][18][19][20][21][22][23], and R-parity violating scenarios of general MSSM have been proposed .
A model based on the gauge symmetry group SU(3)⊗SU( 2)⊗U(1) Y ⊗U(1) B ⊗U(1) L has been investigated at the TeV scale recently [49][50][51][52], where B stands for baryon number and L stands for lepton number.In this theory, the baryon and lepton numbers are local gauge symmetries spontaneously broken at the TeV scale.Breaking baryon number can explain the origin of the matter-antimatter asymmetry in the Universe.And breaking lepton number can explain the smallness of neutrino masses [53][54][55][56][57]. Two extensions of the SM where B and L are spontaneously broken gauge symmetries near the weak scale are constructed [58]: model I is a non-supersymmetric extension [59,60]; model II (BLMSSM) is a supersymmetric extension and is more favoured by the experiments [61].The BLMSSM has been studied in great detail and could avoid the current LHC bounds on the SUSY mass spectrum [62,63,65].Some further phenomenology analysis based on the BLMSSM coincide with the current experimental data well, the mass and decays of the lightest CP-even Higgs have been investigated in Refs.[65,66], and the neutron electric dipole moment in CP violating BLMSSM has also been studied [67].
The flavor changing neutral current (FCNC) processes are highly suppressed in the SM, therefore it is a fertile ground to search for physics beyond SM (BSM).FCNC processes such as b → sγ, K 0 − K0 and B 0 − B0 mixing have played an important role in particle physics over the last four decades.It is well known that CP violation was first observed in the decays of K 0 L meson in 1964 [68], and CP violation of the neutral B meson system was observed in 2001 [69].The first indication of a large top quark mass was also given by B 0 − B0 mixing [70,71].B-system decays have an advantage over the K-system to provide a direct test of the CP violating of SM and is free of corrections from strong interactions [72][73][74].The experiment results of B 0 − B0 mixing have been published by the ALEPH [75], DELPHI [76,77], L3 [78], OPAL [79,80] BaBar [81], Belle [82], CDF [83], DØ [84], and LHCb [85] collaborations.Current experimental result of mass difference is ∆m Exp B = 0.507 ±0.004 ps −1 = (3.337± 0.033) ×10 −13 GeV [86].Calculations for B 0 − B0 mixing have been done in the SM , the two-Higgs doublet model (2HDM), the MSSM and other models [87][88][89][90][91][92][93][94][95][96].The SM prediction for mass difference is ∆m SM B = 0.543 ± 0.091 ps −1 [97], which has a good agreement with the experiment.However, the theoretical error is around 17%, which is considerably larger than the experimental error.The running of LHC will resume in 2015 with higher energy and luminosity.Proposals for next-generation B-factories including SuperKEKB in Japan whose target luminosity is 8 × 10 35 cm −2 s −1 will start collecting data in the near future [98].This may also give some hints on physics beyond the SM.So it is important for experimental and theoretical physicist to search for new physics.As a candidate of new physics, the BLMSSM provides new FCNC at loop level in the B 0 − B0 mixing.We will carry out our calculations for B 0 − B0 mixing in this model.
Our presentation is organized as follows.In Section II, we briefly summarize the main features of the BLMSSM and introduce the superpotential as well as soft breaking terms, then we obtain the mass matrices and couplings needed for B 0 − B0 mixing.In Section III, we give the analytical formulae of the B 0 − B0 mixing in BLMSSM.The numerical analysis are shown in Section IV.Section V presents our conclusions.Finally, some related formulae are given in Appendix A-B.

II. BLMSSM
In this section, we briefly review some main features of the BLMSSM.In the BLMSSM with gauged baryon (B) and lepton (L), by adding the new quarks with baryon number 2 and the new leptons with lepton number L 4 = 3 2 , one can cancel the baryonic and leptonic anomalies respectively [58].Compared with the MSSM, the BLMSSM includes many new fields.Tables I-IV list the superfields including the new quarks, new leptons, new Higgs, the exotic superfields X and X′ , respectively.As one can see, the left-handed superfields have the same absolute value of U(1) B as that of the right-handed superfields but with a contrary sign to cancel baryonic anomalies in the quark sector, similarly for the U(1) L in the leptonic sector to cancel leptonic anomalies.
TABLE II: Superfields including the new leptons in the BLMSSM.
In order to break baryon number spontaneously, we need to introduce the superfields ΦB and φB to acquire nonzero vacuum expectation values (VEVs), which also generate large TABLE IV: Superfields avoiding stability for the exotic quarks in the BLMSSM.
mass for the new quarks.Similarly, we introduce the superfields ΦL and φL to acquire VEVs spontaneously breaking lepton number.Finally, the exotic quarks should be unstable, so the model also includes the superfields X and X′ to avoid the stability for the exotic quarks.
The superpotential in BLMSSM is written as where W M SSM is the superpotential of MSSM, and In the superpotential above, the exotic quarks obtain TeV scale masses after Φ B , ϕ B acquiring nonzero VEVs, and the nonzero VEV of ϕ L implements the seesaw mechanism for the tiny neutrino masses.Correspondingly, the soft breaking terms are generally given as where is the soft breaking terms of MSSM, λ B , λ L are gauginos of U(1) B and U(1) L , respectively.After the SU(2) L doublets H u , H d and SU(2 the local gauge symmetry SU(2 L is broken down to the electromagnetic symmetry U(1) e .
After the symmetry breaking, we can obtain the physical spectrum of this model.The chargino mass matrix is as same as the chargino mass matrix in MSSM.Z + , Z − are the matrices to diagonalize the chargino mass mixing matrix M χ± which has some differences from that of MSSM, here m 2 is the mass squared of U(1) B gauge boson Z B , and the D-terms are In the basis ( Q2 * 4 , Dc 4 , Q1c 5 , D5 ), the mass term for the exotic bottom scalar quarks in the Lagrangian reads as where M 2 b′ is a 4 × 4 matrix, and the matrix elements are listed as follows The mass-squared matrix M 2 b′ is diagonalized by the unitary matrix Z b′ and the physical states are related to the gauge states by The mass squared matrix in the basis (X * , X ′ ) is Adopting the unitary transformation, the mass eigenstates are and the mass squared matrix M 2 X is diagonalized by In four-component Dirac spinors, the mass term for superfields X is given by here, we have defined So the parameter µ X is the mass of the particle X.
In mass basis, we obtain the couplings of quark-exotic quark and the superfields X We also obtain the couplings of quark-exotic scalar quark and the field X where λ 1 , λ 3 are the coupling coefficients, and δ, ǫ, ρ are the indices of the flavor.
Considering the radiative corrections, the mass squared matrix for the neutral CP-even Higgs in the basis (H 0 d , H 0 u ) is written as [99-110] where and the expressions of ∆ B,L 11 , ∆ B,L 12 , ∆ B,L 22 can be found in Refs.[65,66].A Higgs around 125 GeV has been observed at the LHC by ATLAS [111] and CMS [112] with the combined significances of 5.9 and 5.0 standard deviations, respectively.So after diagonalizing the mass squared matrix, the lightest neutral CP even Higgs m h 0 should satisfy this constraint.To obtain the Higgs h 0 with mass of 125 GeV gives a strong limit on the parameter space.
Considering this constraint, we can also obtain m 2 A 0 from the inverse solution of Eq. ( 21).We have where For the charged Higgs scalars, H ± 1,2 are related to the initial Higgs by the matrix Z H , and the charged Higgs mass m H ± 1 satisfy a relation with the pseudo-scalar Higgs mass m A 0 at tree-level: Using the Feynman-t'Hooft gauge, another charged Higgs boson H ± 2 has the same mass as the gauge boson W .
When external masses and momenta are neglected, the general form of the effective Hamiltonian for B 0 − B0 mixing at the weak scale can be expressed as [113] where G F denotes the Fermi constant, C α are the corresponding Wilson coefficients, O α are the effective operators, which read as where P R,L = (1 ± γ 5 ) /2 denote the chiral projectors, σ µν = [γ µ , γ ν ] /2, the SU(3) color indices here have omitted for simplicity.The box diagram contributions to B 0 − B0 mixing from the SM are displayed in Fig. 1, and the box diagrams contributing to B 0 − B0 mixing in the BLMSSM are shown in Fig. 2.
Note that the diagrams including the particles χ and X should make a Fierz rearrangement to ensure that the operators are color singlet states as follows The operators with a prime stand for the product of two color non-singlet quark current.
After this, the Wilson coefficients are given as follows For convenience, we have defined the ratio of mass square as: , and here Z λ iα , Z dλ iα ... have been defined as Here f 1 and f p 2 are the functions related to the one-loop integral functions.
The analytical expressions for the functions f p 2 (x 1 , x 2 , x 3 , x 4 ) and f 1 (x 1 , x 2 , x 3 , x 4 ) are listed in Appendix A. It should be noted that we need perform summation over the repeated indices in the calculations.
The matching scale is chosen as µ 0 = µ W in our calculations.Now we should evolve the coefficients from the scale µ W down to the B-meson scale By solving the remormalization group equation [114], we have with where γ (0) is the anomalous dimensions matrix (ADM) [114,115], and β 0 = 11Nc−2n f 3 with N c denoting the number of colors and n f denoting the number of active quark flavors.
The mass difference of B 0 − B0 mixing can be expressed as After substituting Eq. ( 26) into the above equation, at B-meson scale, the mass difference △m B can be written by  be noted that the value of the m Z B should not be too large, in order to avoid some tachyons appearing, as well as to coincide with the current experimental result on the mass of squarks.
Actually, the corrections of some other parameters to △m B are small, such as m D4 , m Q4 and B X , which we would not discuss in this paper.
In the following discussions, we choose λ 1 = 0.2 for simplicity.Now, we investigate the In Fig. 7, we study the dependence of △m B on the particle X mass µ X .The dotted line corresponds to the result when λ 3 = 0.2, the solid line corresponds to the result when λ 3 = 0.25, the dashed line corresponds to the result when λ 3 = 0.3.The light gray area  decays of squarks and gauginos without conflict with the current experiments.For instance, if the gluino is the lightest supersymmetric particle one could have signals with multitops and multibottoms such as pp → gg → ttbbjj (j stands for a light jet), which may be observed at the LHC [63,64].The projected sensitivity for future experiments that searching for the CLFV processes will be largely improved [122][123][124][125][126][127][128].And the running of LHC will resume in 2015 with higher energy and luminosity.So, it would be interesting to investigate this model.Any observation of BNV or CLFV whose branching fractions is large than that of SM prediction would be a clear sign for BSM physics.Investigating these BNV and CLFV processes can test the BLMSSM and provide constraints on the parameter space.

2 3
where, the matrix elements B0 |O α | B 0 require non-perturbative QCD calculations by the lattice Monte Carlo estimates.The matrix element is parameterized as B0|O 1 | B 0 = B B (µ)f 2 B m 2 B, and the other hadronic matrix elements parameterized are listed in Appendix B.IV. THE NUMERICAL ANALYSISIn our calculations for the CKM matrix, we apply the Wolfenstein parametrization and set A = 0.81, λ = 0.22, ρ = 0.135, η = 0.349.For the hadronic matrix element, the recent average of the lattice results is f B d B B d = 216 ± 15 (MeV) [116], and we adopt the central value of the f B d B B d in our calculations.The other SM parameters are chosen as m W = 80.385 GeV, m u = 2.3 × 10 −3 GeV, m c = 1.275GeV, m t = 173.5 GeV, m b = 4.18 GeV, m d = 4.8 × 10 −3 GeV, m B = 5.279 GeV, G F = 1.166 × 10 −5 GeV −2 , α S (m W ) = 0.12 , α S (m b ) = 0.22 [86].Now we investigate the numerically behavior of these parameters to the B 0 − B0 mixing in BLMSSM.This model contains many parameters.In our following discussions, the parameters needed to study contain λ 1,3 , µ B , m Z B , m D 5 , µ X .The other parameters are adopted as Refs.[66, 67] which have been analyzed in the signals of decay channels h → γγ and h → V V * (V = Z, W ) with the Higgs mass around 125 GeV.m Q1,2,3 = m Ũ1,2,3 = m D1,2,3 = 1 TeV, A d,s,b = A u,c,t = −1 TeV, M 2 = 750 GeV,

5 , m ν 4 =
m ν 5 = 90 GeV, m e 4 = m e 5 = B X = 100 GeV, m Q4 = 790 GeV.(38) In order to see the dependence of the mass difference △m B on the parameters space in the BLMSSM, we fix m Q5 = 1 TeV, m D5 = 1 TeV, µ B = 500 GeV, µ X = 2.4 TeV, m Z B = 1 TeV.From the Wilson coefficients listed in Section III, one can see that the mass difference △m B is the continuous function of the parameters λ 1 and λ 3 , and because of the fourth power of λ 1,3 the △m B should remarkably increase with the increasing of |λ 1 | and |λ 3 | .So λ 1 and λ 3 play an important role to the theoretical prediction on △m B .Next, the influence of the parameters λ 1,3 to △m B will be discussed in detail.We plot the contours corresponding to the mass difference ∆m B in the parameter space of λ 1 and λ 3 in Fig. 3.We can see that △m B increases as |λ 1,3 | increases, and sensitively depends on |λ 1,3 | when |λ 1 | and |λ 3 | are both larger than 0.2.As one can see, the values of |λ 1 | and |λ 3 | that all is larger than 0.25 are disfavored by experiment results under this given assumption.Next, we investigate the dependence of △m B on the parameter m Z B .In Fig. 4, we plot △m B varying with the mass of neutral U(1) B gauge boson Z B , when λ 1 = 0.25 and λ 3 = 0.2.The figure shows that △m B decreases as the m Z B increases.However, it should

Λ 3 0. 3 Λ 3 FIG. 6 :
FIG. 6: The mass difference ∆m B varies with the parameter m D5 for three values of λ 3 .The light gray area denotes the ∆m SM B at 1σ, and the gray area denotes the ∆m Exp B at 1σ.

Λ 3 0. 3 Λ 3 FIG. 7 :
FIG. 7: The mass difference ∆m B as a function of X mass µ X for three values of λ 3 .The light gray area denotes the ∆m SM B at 1σ, and the gray area denotes the ∆m Exp B at 1σ.

TABLE I :
Superfields including the new quarks in the BLMSSM.

TABLE III :
Superfields including the new Higgs in the BLMSSM.