The lepton flavor violating decays of vector mesons in the MRSSM

Recently several new bounds of the lepton flavor violating decays of the neutral vector mesons $J/\psi$ and $\Upsilon(nS)$ are reported from the experiments. In the work, we analyze these processes in the scenario of the minimal R-symmetric supersymmetric standard model by means of the effective Lagrangian method. The predicted branching ratios are affected by the mass insertion parameters $\delta^{ij}$ and the contributions from different parts are comparable. Taking account of the constraints on the mass insertion parameters from the experiments, the branching ratios for the most promising processes $\Upsilon(nS)\rightarrow l\tau$ are predicted to be ten orders of magnitude smaller than the present experimental bounds.

In this work, we will study the LFV decays V → l 1 l2 in the minimal R-symmetric supersymmetric standard model (MRSSM) [30].The MRSSM has the same gauge symmetry Y as the SM and the MSSM.It contains a continuous R-symmetry [31,32], Dirac gauginos and an N=2 sector.R-symmetry forbids Majorana gaugino masses, µ term, A terms and all the left-right squark and slepton mass mixings.There is a wellknown tanβ-enhancement in the MSSM for the anomalous magnetic dipole moment a µ [33,34] and a similar enhancement for the LFV processes µ → eγ and µ − e conversion [35].
In Section II, we provide a brief introduction to the MRSSM and present the notation and conventions for the operators and their corresponding Wilson coefficients.The numerical analysis is presented in Section III, and Section IV is devoted to a conclusion.

II. THE MRSSM
In this section, we first provide a simple overview of the MRSSM.The general form of the superpotential of the MRSSM is given by [37], where Ĥu and Ĥd are the MSSM-like Higgs weak iso-doublets, Ru and Rd are the R-charged For the phenomenological studies we take the soft breaking scalar mass terms All trilinear scalar couplings involving Higgs bosons to squarks and sleptons are forbidden due to the R-symmetry.The soft-breaking terms, which describe the Dirac mass terms for the gauginos and the interaction terms between the adjoint scalars and the auxiliary D-fields of the corresponding gauge multiplet, take the form where B, W and g are usually MSSM Weyl fermions, M B D , M W D and M O D are the mass of bino, wino and gluino, respctively.
The number of neutralino degrees of freedom in the MRSSM is doubled compared to the MSSM as the neutralinos are Dirac-type.The neutralino mass matrix and the diagonalization procedure are where the modified . g 1 and g 2 are coupling constants for the U (1) Y part and the SU (2) L part.v u and v d are the nonzero vacuum expectation values of two Higgs doublets and tan β= vu v d is defined.The number of chargino degrees of freedom in the MRSSM is also doubled compared to the MSSM and these charginos can be grouped to two separated chargino sectors according to their R-charge.The χ-chargino sector has R-charge 1 electric charge; the ρ-chargino sector has R-charge -1 electric charge.Here, we do not discuss the ρ-chargino sector in detail since it does not contribute to the LFV decays.The χ-chargino mass matrix and the diagonalization procedure are The slepton mass matrix and the diagonalization procedure are where One can see that the left-right slepton mass mixing is absent in the MRSSM, whereas the A terms are present in the MSSM.The sneutrino mass matrix and the diagonalization procedure are where the last two terms are newly introduced in the MRSSM.
The mass matrix for up squarks and down squarks, and the relevant diagonalization procedure are where We now focus on the LFV processes V → l 1 l2 .The relevant Feynman diagrams contributing to the LFV decays V → l 1 l2 in the MRSSM are presented in Fig. 1.Using the effective Lagrangian method, we present the analytical expression for the branching ratio of V → l 1 l2 .In the effective Lagrangian method, one can derive the effective Lagrangian relevant for V → l 1 l2 as [27] where ) is the left (right) chiral projection operator, G F is the Fermi constant and m q is the mass of the quark q.The first row in Eq.( 9) is the dipole part which are tightly constrained by the radiative LFV decays (e.g., l 1 → l 2 γ).The remains are the four-fermion dimension six lepton-quark Lagrangian where the coefficients do not include photonic contributions but Z boson and scalar ones.
The most general expression for the V → l 1 l2 decay amplitude can be written as where m V is the mass of the vector meson V , µ (p) is the polarization vector and p is its momentum.The coefficients are dimensionless constants which depend on the Wilson coefficients in Eq.( 9) as well as on hadronic effects associated with meson-to-vacuum matrix elements or decay constants.The coefficients in Eq.( 10) are given as where α is the fine structure constant, Q q is the charge of the quark q, y = m 2 m V and κ V =1/2 is a constant for pure q q states.From Eq. (11) we see that the contribution from the Higgs mediated self-energies and penguin diagrams is nonexistent since these diagrams only contribute to the Wilson coefficients corresponding to the scalar and the pseudo-scalar operators in Eq.( 9).The contribution from the tensor operators can be neglected in the MRSSM since the coefficients C ql 1 l 2 T L/T R calculated from the Feynman diagrams in Fig. 1 are zero.The decay constant f V and the transverse decay constant f T V are defined as Then the amplitude in Eq.( 10) leads to the branching ratio BR(V → l 1 l2 ), which is where the mass of the lighter of the two leptons is neglected.

III. NUMERICAL ANALYSIS
We carried out the calculation by using the spectrum generator SPheno-4.0.5 [54,55], where the model implementations are generated by the public code SARAH-4.14.5 [56][57][58][59][60].The generic expressions for the Wilson coefficients in Eq.( 11) is derived with the help of the package PreSARAH-1.0.3.The explicit expressions for the Wilson coefficients in the MRSSM is obtained by adapting the generic expressions to the specific details of the MRSSM by SARAH.The Fortran code of the branching ratio in Eq.( 13) is written by authors and this code is used by SARAH to generate the Fortran modules for SPheno.The numerical calculation of the branching ratio is done by SPheno.Note that the conventions of Ref. [27] are different from those presented in Ref. [59].The Wilson coefficients are related by More details on how to implement new observables in SPheno can be found in Ref. [60].We present the numerical values of the vector masses, the decay constants and BR(V → ee) in Table II, which are taken from Ref. [61] and Ref. [27], and all the mass parameters are in GeV.Following the suggestion in Ref. [62], the transverse decay constants are set f T V =f V except for J/ψ, which has f T J/ψ = 0.410GeV.Note that the branching ratio in Eq.( 13) is largely independent of the values of the decay constants [27].
The most accurate prediction for the mass of the W boson as well as the SM-like Higgs boson in the MRSSM is studied in Ref. [41].A set of the benchmark point is given by [41] where all the mass parameters are in GeV or GeV 2 .The predicted W boson mass in the MRSSM is comparable with the result from the combination of Large Electron-Positron collider and Fermilab Tevatron collider measurements [63] and the result from the ATLAS collaboration [64].By changing the values of some parameters, e.g.m SU SY , v T , Λ u and Λ d , the recent result on W boson mass from CDF collaboration [65] can also be accommodated in the MRSSM.It is noted that these parameters have very small effect on the prediction of BR(V → l 1 l2 ) which take values along a narrow band.In the numerical analysis, the default values of the input parameters are set same with those in Eq.( 15).The off-diagonal entries of the slepton mass matrices m 2 l , m 2 r and the squark mass matrices m 2 q , m 2 ũ, m 2 d in Eq.( 15) are zero.
The LFV decays mainly originate from the off-diagonal entries of the soft breaking terms m 2 l and m 2 r .These off-diagonal entries of 3 × 3 matrices m 2 l and m 2 r are parameterized by the mass insertions where I, J = 1, 2, 3. To decrease the number of free parameters involved in our calculation, we assume δ IJ l = δ IJ r = δ IJ and then δ IJ = δ 12 , δ 13 or δ 23 .The parameters δ IJ are constrained by the experimental bounds on the LFV decays, such as the radiative two body decays l 2 → l 1 γ, the leptonic three body decays l 2 → 3l 1 and µ − e conversion in nuclei.Current bounds of these LFV decays are listed in Table.III [61].In the following we will use these bounds to constrain the parameters δ IJ .The predictions for BR(Υ(nS) → l 1 l2 ) in each plot are very close to each other.The prediction on BR(V → e − µ + ), BR(V → e − τ + ) and BR(V → µ − τ + ) is affected by the mass insertions δ 12 , δ 13 and δ 23 , respectively.The predictions on BR(V → l 1 l2 ) in the MRSSM are found to be below 10 −14 , which are at least seven orders of magnitude below the present experimental upper limits.A linear relationship in logarithmic scale is displayed between the branching ratios and the flavor violating parameters δ IJ .The actual dependence on δ IJ is quadratic.The mentioned linear dependence is due to the fact that both x axis and y axis in Fig. 2 are logarithmically scaled.In Fig. 2 the following hierarchy is shown, The same hierarchy appears in several new physics [9,22].The most challenging experimental prospects for δ 12 arise for µ → eγ.Considering the new sensitivity for BR(µ → eγ) in the future projects will be about 6 × 10 −14 from MEG II [66], δ 12 is constrained to around 10 −3 .The constraints on δ 13 from τ → eγ and τ → 3e are comparable and δ 13 is constrained to around 10 −0.2 .The case for δ 23 is same with δ 13 .
We observe that the contributions from different parts are comparable and the following . It shows that varying tanβ has very small effect on the prediction of BR(Υ(3S) → l 1 l2 ), which takes values along a narrow band, and indicates the tanβ-enhancement for BR(Υ(3S) → l 1 l2 ) does not exist in the MRSSM.
We present the contour plots of BR(Υ(3S) → l 1 l2 ) in the mQ∼mL plane in Fig. 4, where d) ii (i = 1, 2, 3) and mL= (m 2 l ) ii = (m 2 r ) ii (i = 1, 2, 3) are assumed.The predictions for BR(Υ(3S) → l 1 l2 ) are sensitive to mQ and mL.The predictions on BR(Υ(3S) → l 1 l2 ) increase slowly as mQ varies from 1 to 5 TeV and decrease slowly as mL varies from 1 to 5 TeV.In a wide region of mQ and mL, the predicted BR(Υ(3S) → l 1 l2 ) can change about one order of magnitude.The off-diagonal entries δ IJ q,ũ, d of the squark mass matrices m 2 q , m 2 ũ and m 2 d have very small effect on the prediction of BR(Υ(3S) → l 1 l2 ) which take values along a narrow band.The effect from the other parameters in Eq.( 15) is same with that in Ref. [52,53].
The final results on the upper bounds of BR(V → l 1 l2 ) in the MRSSM are given in Table IV, which are obtained by assuming δ 12 =10 −3 , δ 13 =10 −0.2 (≈ 0.63) and δ 23 =10 −0.2 , respectively, where the results in the literature are also included for comparison.By means of an effective field theory, the data in Ref. [28] are obtained from the recast of high-p T dilepton tails at the LHC for the left-handed scenario, where the dipole operators are not considered.The expressions for BR(V → l 1 l2 ) in Ref. [28] are same except a few adjustments (e.g., mass, decay constant and full width).Since the mass, decay constant and full width for ψ(2S) are very close to J/ψ, the bounds for ψ(2S) → l 1 l2 are at the same level with J/ψ → l 1 l2 .For the same reason, the bounds for Υ(1S) → l 1 l2 and Υ(2S) → l 1 l2 are at the same level with Υ(3S) → l 1 l2 .The predicted values for BR(V → lτ ) in the MRSSM lie a range between Ref. [9] and Ref. [22], and the predicted values for BR(V → eµ) are below Ref. [9] and Ref. [22].All the direct bounds in new physics are smaller than the indirect bounds in Ref. [28].IV that the bound on BR(J/ψ → e − τ + ) in the MRSSM is about 5.3 × 10 −18 and this is ten orders of magnitude below the recently reported bound from the BESIII collaboration [4].The bound on BR(J/ψ → µ − τ + ) in the MRSSM is about 5.3 × 10 −18 and this is twelve orders of magnitude below the current bound and ten orders of magnitude below the future experimental sensitivity (1.5 × 10 −8 ) [67].The bound on BR(J/ψ → e − µ + ) in the MRSSM is about 1.5×10 −23 and this is far below the current bound and the future experimental sensitivities (6.0 × 10 −9 ) [67].The bound on BR(Υ(nS) → lτ + ) in the MRSSM is about O(10 −16 ) and this is ten orders of magnitude below the recently reported bound from the Belle Collaboration [1].The bound on BR(Υ(nS) → e − µ + ) in the MRSSM is about O(10 −21 ) and this is fourteen orders of magnitude below the recently reported bound from the Belle Collaboration [1] and the BaBar collaboration [3].

IV. CONCLUSIONS
Although the higher order LFV processes in the SM are permitted, these are extremely suppressed by powers of the small neutrino masses and are not observable in current or planned experiments.Therefore, observation of the LFV decays would be a clear signature of new physics.In this work, we analyze the LFV decays of vector mesons V → l 1 l2 in the MRSSM, by taking account of the constraints on the parameter space from the radiative charged lepton decays l 1 → l 2 γ, leptonic three body decays l 1 → 3l 2 and µ − e conversion in nuclei.The prediction on BR(V → e − µ + ), BR(V → e − τ + ) and BR(V → µ − τ + ) is affected by the mass insertions δ 12 , δ 13 and δ 23 , respectively.The final results on the upper bounds of BR(V → l 1 l2 ) in the MRSSM are given in Table IV, which are obtained by assuming δ 12 =10 −3 , δ 13 =10 −0.2 and δ 23 =10 −0.2 , respectively, where the results in the literature are also included for comparison.The predictions on BR(V → l 1 l2 ) in the MRSSM are far below the current upper limits.Thus, the LFV decays V → l 1 l2 may be out reach of the near future experiments.
The studies of the radiative lepton flavor violating (RLFV) decays of vector mesons V → γl 1 l2 could provide important complementary access to search of new physics [27].
Besides the dipole, vector and tensor operators, the RLFV decays V → γl 1 l2 could receive contributions from the axial, scalar and pseudoscalar operators which are not accessible in V → l 1 l2 , e.g., the Higgs mediated self-energies and penguin diagrams.It might be possible that the RLFV processes V → γl 1 l2 can be enhanced close to the current or future experimental sensitivities while the LFV processes V → l 1 l2 are still out reach of the current experiments.

Higgs SU ( 2 )
L doublets and the corresponding Dirac higgsino mass parameters are denoted as µ u and µ d .Y e , Y u and Y d are the Yukawa couplings of charged lepton, up type quark and down type quark, respectively.λ u , λ d , Λ u and Λ d are parameters of Yukawa-like trilinear terms involving the singlet Ŝ and the triplet T , which is given by T

TABLE I :
Current experimental bounds of the LFV decays of the neutral vector mesons.

TABLE IV :
The upper bounds on BR(V → l 1 l2 ) in the literature and in the MRSSM.