B meson rare decays in the TNMSSM

We investigate the two loop electroweak corrections to B meson rare decays $\bar B\rightarrow X_s\gamma$ and $B_s^0\rightarrow \mu^+\mu^-$ in the minimal supersymmetry standard model (MSSM) extension with two triplets and one singlet (TNMSSM). The new particle contents and interactions in the TNMSSM can affect the theoretical predictions of the branching ratios ${\rm Br}(\bar B\rightarrow X_s\gamma)$ and ${\rm Br}(B_s^0\rightarrow \mu^+\mu^-)$, and the corrections from two loop diagrams to the process $\bar B\rightarrow X_s\gamma$ can reach around $4\%$. Considering the latest experimental measurements, the numerical results of ${\rm Br}(\bar B\rightarrow X_s\gamma)$ and ${\rm Br}(B_s^0\rightarrow \mu^+\mu^-)$ in the TNMSSM are presented and analyzed. It is found that the results in the TNMSSM can fit the updated experimental data well and the new parameters $T_{\lambda},\;\kappa,\;\lambda$ affect the theoretical predictions of ${\rm Br}(\bar B\rightarrow X_s\gamma)$ and ${\rm Br}(B_s^0\rightarrow \mu^+\mu^-)$ obviously.


I. INTRODUCTION
After the discovery of Higgs on the Large Hadron Collider (LHC) in 2012, all particles predicted by the SM have been found.However, there are still some problems that are difficult to be solved by the SM, such as the non-zero neutrino mass, resonable dark matter candidates and etc.It indicates new physics is needed to extend the SM.And since B meson rare decay processes B → X s γ and B 0 s → µ + µ − are not affected by the uncertainties of nonperturbative QCD, research on B physics is particularly sensitive to exploring new physical effects beyond the SM.In Refs.[1][2][3][4], the average experimental data on the branching ratios of B → X s γ and B 0 s → µ + µ − are provided as Br( B → X s γ) = (3.49± 0.19) × 10 −4 , Br(B 0 s → µ + µ − ) = (2.9 +0.7 −0.6 ) × 10 −9 . ( The branching ratios of B → X s γ and B 0 s → µ + µ − predicted by the SM are [5][6][7][8][9][10][11][12][13] Br( B → X s γ) SM = (3.36 ± 0.23) × 10 −4 , which coincides with the experimental data very well.Therefore, the new physics contributions to B → X s γ and B 0 s → µ + µ − are limited strictly by the accurate measurements on the processes B → X s γ and B 0 s → µ + µ − .As one of the most famous extensions of the SM, the B meson rare decay process B → X s γ is analyzed in the MSSM [14][15][16][17][18][19][20][21].In 1998, Ciuchini presented the QCD corrections to B → X s γ at Next-to-leading order (NLO) in the Two-Higgs doublet model (THDM) [22].Then, the two loop QCD corrections was proposed in Ref. [23].Not only the process B → X s γ, there are also many references researching on other B meson rare decay processes in the THDM [24][25][26][27][28][29].Recently, the authors of Refs.[30][31][32][33][34][35][36][37][38][39][40] have discussed the supersymmetric effects on the B meson rare decay processes.Meanwhile, Long et al present the computation of the flavor transition process b → sγ [41].The authors of Ref. [42] have investigated the two aspects of hadronic B decays, and then these processes in the case of CP violation have been discussed [43].Moreover, many possibilities for searching supersymmetry effects in different B meson rare decay processes have been proposed [44][45][46].When investigating the processes of rare B decay, the supersymmetry effects are very interesting, and B decay can be conducive for us to understand the characteristics of the supersymmetry model in detail while limiting the parameter space [47,48].
TNMSSM is an extension of Next-to minimal supersymmetric standard model (NMSSM) containing two SU(2) L triplets with hypercharge ±1, where the NMSSM introduces an additional scalar singlet compared with the MSSM [49].The new scalar singlet in the NMSSM is introduced to solve the µ problem in the MSSM [50,51].However, NMSSM fails to improve the little hierarchy problem [52][53][54][55][56][57].Fortunately, the author of Ref. [49] solved these problems in the TNMSSM by introducing two scalar triplets which are responsible for large correction to the lightest physical Higgs mass.The tiny neutrino mass measured at the neutrino oscillation experiments can be obtained by applying type II seesaw mechanism and a discrete flavor symmetry G F in TNMSSM (i.e. the flavored-TNMSSM) [58].In this work, we analyze the two loop electroweak corrections to B → X s γ and B 0 s → µ + µ − in the TNMSSM.Compared with the MSSM, new particle contents and interactions can make important contributions to the processes.
The paper is organized as follows.The superpotential, the soft breaking terms and the mass matrices of singly-charged Higgs and CP-even Higgs in the TNMSSM are reviewed briefly in Sec.II.Sec.III and Sec.IV give the corresponding Wilson coefficients and analytic expressions for Br( B → X s γ) and Br(B 0 s → µ + µ − ).Sec.V analyses the numerical results and Sec.VI gives a summary.The corresponding matrix elements and the concrete expressions of the Wilson coefficients are collected in the appendices.
The chiral superfields for quarks and leptons are given by where we ignore the index of generations and the quantum numbers of U(1) Y , SU(2) L and SU(3) C are indicated in the bracket, respectively.Additionally, the expressions and the quantum numbers of two Higgs triplets, two doublets and one singlet are assigned as In the previous expressions, T 0 and T 0 are two complex neutral superfields, while T + , T − are singly-charged Higgs and T ++ , T −− are doubly-charged Higgs.
The superpotential of the TNMSSM W TNMSSM contains two parts with where W MSSM is the superpotential of the MSSM, and W TS explains the extended scalar sector including two triplets and a SM gauge singlet, Here, we also neglect the index of generations.From Eq.( 6), we could find that there are only two MSSM Higgs doublets coupled with fermion multiplet via Yukawa coupling.Then, the general soft breaking terms are given by where When the Z 3 symmetry is imposed, the µ term only forms after the singlet S has obtained a vacuum expectation value (VEV) v S as µ = 1 √ 2 λv S .The coefficients in the Higgs sector are assumed to be real in the following calculations.
The SU(2) L U(1) Y electroweak symmetry breaking occurs when the neutral parts of Higgs fields obtain the VEVs Meanwhile, the VEVs of the triplets must to be small to avoid large ρ parameter correction [49] and the VEV of the singlet is required to be large for generating a large µ term like the case in the NMSSM [50].In the TNMSSM, the mass of Z gauge boson reads where g 1 and g 2 represent the gauge coupling constants of U(1) Y and SU(2) L respectively.
It will be seen from this that due to the triplets, the electroweak symmetry breaking VEV for the doublets compared to MSSM is changed as Minimizing the Higgs scalar potential we can deduce the squared mass matrices of the neutral Higgs and singly-charged Higgs.
In the basis (H − d , H +, * u , T − , T +, * ) and (H −, * d , H + u , T −, * , T + ), the squared mass matrix for singly-charged Higgs can be expressed as The 4 × 4 squared mass matrix in Eq.( 16) can be diagonalized by the unitary matrix Then, we can get three mass eigenstates (H ± 1 , H ± 2 , H ± 3 ) for singly-charged Higgs and one state G ± for the massless Goldstone boson.Therefore, the TNMSSM has two more singlycharged Higgs due to the triplets, with respect to MSSM.These new defined singly-charged Higgs will bring contributions to loop corrections for B → X s γ and B 0 s → µ + µ − .At tree level, the squared mass matrix for CP-even Higgs is given in the basis (ℜH 0 d , ℜH 0 u , ℜS, ℜT 0 , ℜ T 0 ) as The corresponding matrix elements of the squared mass matrices for singly-charged Higgs and CP-even Higgs are collected in Appendix A.
Including the leading-log radiative corrections up to two loops for stop and top sector [67][68][69], the mass of the SM-like Higgs boson can be written as where m 0 h 1 is the lightest tree-level Higgs boson mass, α 3 is the running QCD coupling constant, M S = √ m t1 m t2 is the geometric mean of the stop masses m t1,2 , m t is the top quark pole mass, and Ãt = A t − µ cot β with A t = T u,33 being the trilinear Higgs stop coupling.
where g s represents the strong coupling, F µν is the electromagnetic field strength tensor, G µν is the gluon field strength tensor, and T a (a = 1, ..., 8) are the SU(3) generators.
As shown in Fig. 1, the main one loop Feynman diagrams contributing to the process B → X s γ in the TNMSSM are mediated by newly defined up-squarks, singly-charged Higgs and charginos.Compared to the MSSM, these new definitions of particles will affect the b t s FIG. 2: The relating two loop diagrams in which a closed heavy fermion loop is attached to virtual At two loop level, we consider the corrections from closed fermion loop in Fig. 2. In Fig. 2, new defined neutralinos, charginos and doubly-charged chargino in the TNMSSM bring new contributions to B → X s γ compared to the MSSM.Then, the two loop Wilson coefficients where Based on Wilson coefficients above, the branching ratio of B → X s γ in the TNMSSM can be given by where the overall factor R = 2.47 × 10 −3 , and the nonperturbative contribution where we choose the hadron scale µ b = 2.5 GeV and at NNLO level the SM contribution is [76][77][78][79].The Wilson coefficients for new physics at the bottom quark scale can be written as [80, 81] where IV. RARE DECAY B 0 s → µ + µ − Fig. 3 shows the main one loop vertex and box diagrams contributing to the process B 0 s → µ + µ − in the TNMSSM.In Fig. 3, newly defined up-squarks and pseudo-scalar Higgs bosons will make new contributions to the branching ratio Br(B 0 s → µ + µ − ) compared to the MSSM.Considering Fig. 2 and Fig. 3, the Wilson coefficients corresponding to the process B 0 s → µ + µ − at the EW scale can be written as S,NP (µ EW ) +C (6)  S,NP (µ EW ) + C (9)  S,NP (µ EW ) + C (11)  S,NP P,NP (µ EW ) +C (6)  P,NP (µ EW ) + C (9)  P,NP (µ EW ) + C (11)  P,NP (µ EW ) , 5)  9,NP (µ EW ) + C (6)  9,NP (µ EW ) + C (7)  9,NP (µ EW ) + C (8)  9,NP (µ EW ) +C (9)  9,NP (µ EW ) + C (10)  9,NP (µ FIG. 3: The one loop vertex and box diagrams contributing to B 0 s → µ + µ − in the TNMSSM.
10,NP (µ EW ) + C (6)  10,NP (µ EW ) + C (7)  10,NP (µ EW ) + C (8)  10,NP (µ EW ) +C (9)  10,NP (µ EW ) + C (10)  10,NP (µ The Wilson coefficients at hadronic energy scale from the SM to Next-to-Next-to-Logarithmic (NNLL) accuracy are shown in Table I.And the renormalization group equations are written as with and where Ref. [82] provided the anomalous dimension matrices as Based on the Wilson coefficients above, the branching ratio of B 0 s → µ + µ − can be given by where M B 0 s = 5.367GeV denotes the mass of neutral meson B 0 s and τ B 0 s = 1.466(31)ps denotes its life time.Moreover, the squared amplitude can be written as with where f B 0 s = (227 ± 8)MeV denote the decay constants.

V. NUMERICAL ANALYSES
This section provides the numerical discussion of the branching ratios of B meson rare decays B → X s γ and B 0 s → µ + µ − by considering the latest multiple experimental constraints of particles.It includes that the SM-like Higgs mass m h is keeped around 125. 25 GeV, the neutralino mass is limited to more than 116 GeV, the chargino mass is limited to more than 1100 GeV, the slepton mass is limited to more than 700 GeV and the squark mass is maintained at the TeV order of magnitude [1,[83][84][85][86][87][88][89][90].Additionally, the first two generations of squarks are strongly constrained by direct searches at the LHC [91,92].
However, compared to the previous two generations, the mass of the third generation squark which can affect the mass of SM-like Higgs do not suffer strong constraints.Therefore we take m q = m ũ = diag(2, 2, m t) TeV, and the discussion about the observed Higgs signal in Ref. [93] limits m t > ∼ 1.5 TeV.For simplicity, we also choose As a key parameter, T u 3 = A t affects the SM-like Higgs mass and the numerical calculation obviously.In order to obtain reasonable numerical results, we need to find some sensitive parameters for discussion.Similarly to the MSSM [94], the new physics contributions to the branching ratios Br( B → X s γ) and Br(B 0 s → µ + µ − ) are also depended on tan β, A t and singly-charged Higgs mass.Moreover, according to their great impacts on singly-charged Higgs mass, neutralino mass and chargino mass, we found three other new parameters κ, λ and T λ that can affect Br( B → X s γ) and Br(B 0 s → µ + µ − ) in the TNMSSM.We will plot the relational and scatter diagrams and explore the effects of these parameters on the branching ratios and the allowed ranges between λ, κ and T λ .
By considering the experimental constraints described above, we adopt the parameters in the following numerical calculation as To illustrate the effects of A t and tan β on the branching ratios, we take λ = 0.4, κ = decreases with the increasing of A t and the slope of evolution is steeper as tan β is bigger.
In Fig. 4(b), we can see that the variation relationship between A t and Br(B 0 s → µ + µ − ) is almost the shape of parabolic.Moreover, when tan β is bigger, A t is limited strongly in the smaller range by the experimental data on Br(B 0 s → µ + µ − ).It will be seen from this that the new physics can provide the considerable contributions to Br(B 0 s → µ + µ − ) for large tan β and A t .
To see the effects of λ, κ, and T λ on the branching ratios, we first take λ = 0.4, tan β = 5, A t = 0.5 TeV and plot the graph of Br( B → X s γ) and Br(B 0 s → µ + µ − ) varying with T λ in Fig. 5, for κ = 0.4 (solid line), κ = 0.5 (dashed line), κ = 0.6 (dotted line), respectively.Then, we take T λ = 100 GeV, tan β = 5, A t = 0.5 TeV and plot the graph of Br( B → X s γ) and Br(B 0 s → µ + µ − ) varying with κ in Fig. 6, for λ = 0.9 (solid line), λ = 0.7 (dashed line), λ = 0.5 (dotted line), respectively.The gray area denotes the experimental 1σ bounds.Fig. 5 shows that Br( B → X s γ) and Br(B 0 s → µ + µ − ) increase with the decreasing of T λ when T λ is negative and the three lines mix together for large T λ .Thus it can be seen that the smaller κ is taken, the smaller range of T λ is limited and large T λ can get rid of the influence of κ.In Fig. 6(a), Br( B → X s γ) decreases gradually with the increasing of κ and κ is less affected by λ.However, from Fig. 6(b), Br(B 0 s → µ + µ − ) decreases dramatically near κ = 0.4 for λ = 0.5, κ = 0.6 for λ = 0.7 and κ = 0.9 for λ = 0.9 respectively.It indicates that λ has strong limitations on κ and the limitations will be discussed in the following by drawing scatter plots.
Meanwhile, for comparing and reflecting the specific differences between one loop and two loop corrections to B → X s γ and B 0 s → µ + µ − , we take T λ = 100 GeV, λ = 0.7 and plot the graph of Br( B → X s γ) and Br(B 0 s → µ + µ − ) varying with κ in Fig. 7.The solid and dashed line denote two loop and one loop predictions respectively.Fig. 7(a) shows that, the relative corrections from two loop diagrams to one loop corrections of Br( B → X s γ) can reach around 4%, which can produce a more precise prediction on the process B → X s γ.
In Fig. 7(b), we can see that the two lines almost overlap which shows that the two loop corrections are negligible compared with one loop corrections.Therefore, in the analysis of the numerical calculations above, we always use the more precise two loop predictions.Now, for revealing how λ, κ, and T λ be constrained by the experimental measurements of B meson rare decays, we scan the sensitive parameters under the consideration of the experimental constraints above and Br( B → X s γ), Br(B 0 s → µ + µ − ) within one standard deviation.The random ranges of input parameters are as follows: tan β = (1, 40), λ = (0.01, 0.99), κ = (0.01, 0.99), µ = (1.1,1.5) TeV, T λ = (−1, 1) TeV, A t = (−4, 4) TeV. ( Then, we plot the allowed ranges of κ versus λ, T λ versus λ and κ versus T λ in Fig. 8. Fig. 8(a) shows that the vast majority of points are concentrated in the areas κ > λ and the number of points gradually decreases as κ < λ.In Fig. 8(b), we can find that the density of points decreases with the increasing of λ and this phenomenon is more obvious when λ > 0.6.As shown in Fig. 8(c), the negative range of T λ is gradually limited when κ < 0.6.

VI. SUMMARY
B meson rare decays are sensitive to the searching on new physics beyond the SM.In this paper, we investigate the two loop electroweak corrections to the processes B → X s γ and B 0 s → µ + µ − in the TNMSSM which extends the MSSM with two triplets and one singlet.Under the consideration of the constraints from the observed Higgs signal and the updated experimental data of the branching ratios, the numerical results indicate that the corrections from two loop diagrams to the process B → X s γ in the TNMSSM can reach around 4%, which can produce a more precise theoretical prediction.Moreover, the new physics effects in the TNMSSM can fit the experimental data for the rare decays B → X s γ and B 0 s → µ + µ − , and the corresponding parameter space is limited strictly by considering Br( B → X s γ) and Br(B 0 s → µ + µ − ) in the experimental 1σ intervals.The new parameters λ, κ, T λ in the TNMSSM have great impacts on the theoretical predictions of Br( B → X s γ), Br(B 0 s → µ + µ − ) and maintaining Br( B → X s γ) and Br(B 0 s → µ + µ − ) within the experimental 1σ interval prefers λ, κ in the range λ < 0.6, κ > 0.6 and positive T λ .

Appendix A: The corresponding matrix elements
The matrix elements of the squared mass matrix for singly-charged Higgs can are given by where tan β = vu v d and tan β ′ = v T v T .The matrix elements of the squared mass matrix for CP-even Higgs are written as where

Appendix B: The corresponding Wilson coefficients
The one loop Wilson coefficients for the process B → X s γ are written as where C L,R XY Z denote the constant parts of the interaction vertices about particles XY Z, L and R represent the left and right-handed parts respectively.
, the concrete expressions for I k (k = 1, ..., 4) can be given as: Assuming m χ ± i , m χ 0 j ≫ m W , the two loop Wilson coefficients for the process B → X s γ can be given by , 1, The concrete expressions for P and J are given by: where m F runs all m χ ± i , m χ 0 j .The one loop Wilson coefficients for the process B 0 s → µ + µ − are written as C (10)  10,NP (µ EW ) = C (11)  S,NP (µ EW ) = − u i ,H ± j ,ν l C (11)  P,NP (µ EW ) = − u i ,H ± j ,ν l where V denotes photon γ, Z boson and Ct V t , C χ0 j V χ 0 j , C χ+ i V χ + i denote the vector parts of the corresponding interaction vertex.
The two loop Wilson coefficients for the process B 0 s → µ + µ − are written as The concrete expressions for G k (k = 1, ..., 4) are written as x 2  4 lnx 4 (x 1 − x 4 )(x 2 − x 4 )(x 3 − x 4 ) . (B25) where a real photon or gluon is attached in all possible ways.prediction of the process B → X s γ.In the TNMSSM, the one loop Wilson coefficients corresponding to Fig. 1 are C (a) 7,N P (µ EW ), C (b) 7,N P (µ EW ), C (c) 7,N P (µ EW ) and C (d) 7,N P (µ EW ) and their concrete expressions can be obtained from Appendix B.

Fig. 2
are also collected in Appendix B. In addition, C 8g,N P (µ EW ), C ′ 8g,N P (µ EW ), C W W 8,N P (µ EW ), C W H 8,N P (µ EW ) are Wilson coefficients of the process b → sg in the TNMSSM, which can make contributions to the process b → sγ through the QCD running.Similarly, the Wilson coefficients of the process b → sg at EW scale can be written as

9 FIG. 7 :
FIG. 7:The comparison graph of two loop (solid lines) result and one loop (dashed lines) result when λ = 0.7, T λ = 100 GeV, where the gray area denotes the experimental 1σ interval.

TABLE I :
At hadronic scale µ = m b ≃ 4.65GeV, Wilson coefficients from the SM to NNLL accuracy.All of the Wilson coefficients calculated above are gauge invariant and the concrete expressions on the right side of Eq.(30) are collected in Appendix B. In addition, the Wilson coefficients can also be evolved from EW scale µ EW down to hadronic scale µ ∼ m [72] the renormalization group equations.For obtaining hadronic matrix elements conveniently, we define effective coefficients as[72]