The LFV decays of Z boson in Minimal R-symmetric Supersymmetric Standard Model

A future $Z$-factory will offer the possibility to study rare $Z$ decays $Z\rightarrow l_1l_2$, as those leading to Lepton Flavor Violation final states. In this work, by taking account of the constraints from radiative two body decays $l_2\rightarrow l_1\gamma$, we investigate the Lepton Flavor Violation decays $Z\rightarrow l_1l_2$ in the framework of Minimal R-symmetric Supersymmetric Standard Model with two benchmark points from already existing literatures. The flavor violating off-diagonal entries $\delta^{12}$, $\delta^{13}$ and $\delta^{23}$ are constrained by the current experimental bounds of $l_2\rightarrow l_1\gamma$. Considering recent experimental constraints, we also investigate Br($Z\rightarrow l_1l_2$) as a function of $M_D^W$. The numerical results show that the theoretical prediction of Br($Z\rightarrow l_1l_2$) in MRSSM are several orders of magnitude below the current experimental bounds. The Lepton Flavor Violation decays $Z\rightarrow e\tau$ and $Z\rightarrow \mu\tau$ may be promising to be observed in future experiment.


I. INTRODUCTION
Rare decays are of great importance in searching for New Physics (NP) beyond the Standard Model (SM), and the Lepton Flavor Violating (LFV) decays are particularly appealing cause they are suppressed in SM, and their detection would be a manifest signal of NP.
In this paper, we have studied the LFV decays of Z boson in MRSSM. Similar to the case in MSSM, the LFV decays mainly originate from the off-diagonal entries in slepton mass matrices m 2 l and m 2 r [24]. Taking account of the constraint from radiative decay l 2 → l 1 γ on the off-diagonal parameters, we give the upper predictions on the LFV decays of Z boson with parameter spaces BMP1 and BMP3 [12]. Taking account of recent experimental limit on the masses of charginos and neutrilinos [25], we also explore the LFV decays of Z boson as a function of Dirac mass parameter M W D . A comparison on the upper bounds of off-diagonal parameters between MRSSM and MSSM is also displayed.
The paper is organized as follows. In Section II, we provide a brief introduction on MRSSM, and derive the analytic expressions for every Feynman diagram contributing to LFV decays of Z boson in MRSSM in detail. The numerical results are presented in Section III, and the conclusion is drawn in Section IV.

II. FORMALISM
In this section, we firstly provide a simple overview of MRSSM.
Adjoint chiralÔ,T ,Ŝ 0 O, T, S 0Õ,T ,S -1 given by [10] whereĤ u andĤ d are the MSSM-like Higgs weak iso-doublets,R u andR d are the R-charged Higgs SU(2) L doublets and the corresponding Dirac higgsino mass parameters are denoted as µ u and µ d . λ u , λ d , Λ u and Λ d are parameters of Yukawa-like trilinear terms involving the singletŜ and the tripletT , which is given bŷ Then, the soft-breaking terms involving scalar mass are It is noted that all trilinear scalar couplings involving Higgs bosons to squarks and sleptons are forbidden due to the R-symmetry. The Dirac nature is a manifest feature of MRSSM fermions and the soft-breaking Dirac mass terms of the singletŜ, tripletT and octetÔ take the form whereB,W andg are usually MSSM Weyl fermions. After EWSB, one can get the following 4 × 4 neutralino mass matrix where the modified µ i parameters are and the v T and v S are vacuum expectation values ofT andŜ which carry zero R-charge.
The neutralino mass matrix can be diagonalized by unitary matrices N 1 and N 2 The chargino mass matrix is given by and can be diagonalized by unitary matrices U 1 and V 1 The LFV interactions mainly originate from the potential misalignment between the leptons and sleptons mass matrices in the MRSSM. In the gauge eigenstate basisν iL , the sneutrino mass squared matrix is expressed as where the last two terms are newly introduced by MRSSM, and the mass matrix is diago- The slepton mass squared matrix takes the form where (m 2 The sources of LFV are the off-diagonal entries of the 3 × 3 soft supersymmetry breaking matrices m 2 l and m 2 r , where the A terms are absent. Note that, in the following, we replacẽ l withL ± to denote the sleptons. From Eq. (7), we can see that the left-right slepton mass mixing is also absent. The slepton mass matrix is diagonalized by ).
The interactions of charged sleptonsL ± and neutral sneutrinosν with neutralinos χ 0 and charginos χ ± are correspondingly given by the Lagrangian as [12,26] The interactions between Z boson and neutralinos χ 0 or charginos χ ± are given by the Lagrangian as The Zl 1 l 2 interaction Lagrangian can be written as [27] L Zl 1 l 2 =l 1 γ µ (C 1 The left-handed current coefficient where the charged lepton masses have been neglected and Γ Z is the total decay width of Z boson. The coefficients C 1 L/R and C 2 L/R are combinations of coefficients corresponding to each Feynman diagram in FIG.1 and take the form The coefficients in FIG.1 (a) and FIG.1 (b) take the same form where the couplings corresponding to FIG.1 (a) is and the couplings corresponding to FIG.1 where the couplings corresponding to FIG.1 (c) is and the couplings corresponding to FIG.1 (d) is Above loop integrals are given in term of Passarino-Veltman functions [28] and can be calculated by the Mathematica package Package-X [29] through a link to fortran library Collier [30][31][32][33], where the latter provides the numerical evaluation of one-loop scalar and tensor integrals in perturbative relativistic quantum field theories.

III. NUMERICAL ANALYSIS
In the numerical analysis, we use the benchmark points in Ref. [12] as the default values for our parameter setup and display them in Table.III, where the slepton mass matrices are diagonal and all mass parameters are in GeV or GeV 2 and the mass spectra for the BMPs are shown in Table.IV. Note that large value of |v T | is excluded by measurement of W mass cause the vev v T of the SU(2) L triplet field T 0 gives a correction to W mass through [10]   off-diagonal entries of the soft breaking terms m 2 l , m 2 r , which are parameterized by mass insertion as in [34] where I, J = 1, 2, 3. We also assume δ IJ l = δ IJ r = δ IJ . The experimental limits on LFV decays, such as radiative two body decays l 2 → l 1 γ, leptonic three body decays l 2 → 3l 1 and µ − e conversion in nuclei, can give strong constraints on the parameters δ IJ . In the following, we will use LFV decays l 2 → l 1 γ to constrain the parameters δ IJ . Current limits of LFV decays l 2 → l 1 γ listed in TABLE.V [1].  The sparticle mediated diagrams for l 2 → l 1 γ in MRSSM are shown in FIG.2. Taking account of the gauge invariance, and assuming the photon is on shell and transverse, the amplitude for l 2 → l 1 γ is given by [37] M(l 2 → l 1 γ) = ǫ µ * ū l 1 (p l 1 )[iq ν σ µν (A + Bγ 5 )]u l 2 (p l 2 ). (11) Then, in the limit m 1 → 0, the analytic expression of Br(l 2 → l 1 γ) is derived as where Γ l 2 is the total decay width of l 2 and the form factors A and B is a combination of form factors for every Feynman diagram in FIG.2, In the limit m 1 → 0, the form factors A (a) and B (a) corresponding to FIG.2 (a) are given as The couplings corresponding to FIG.2 (a) are given as and the couplings corresponding to FIG.2 (b) are given as Taking δ 13 = 0, δ 23 = 0, we plot the theoretical prediction of Br (µ → eγ) versus δ 12 and Br(Z → eµ) versus δ 12 in FIG.3(a) and FIG.3(b  Recently, the ATLAS collaboration has released a search for chargino-neutralino production in two and three lepton final states employing RJR techniques that target specific event topologies [25], which state that charginos and neutralinos must be heavier than 600 GeV at 95% CL. To be compatible with the experimental limit, the parameters M W D , M B D , µ u and µ d , which dominant the masses of charginos and neutralinos, should be enlarged. The selection of BMP1 and BMP3 is shown in figure 8.2 in reference [41], where the parameters are set to BMP1 and BMP3 for the top and bottom row respectively and the benchmark point is marked by a star in each plot. It is shown that the valid region is µ u (µ d ) < 500 GeV for BMP1, and this leads to at least two sparticle mass are lighter than 600 GeV. For