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

TL;DR

This paper investigates the rare lepton flavor violating decays of the Z boson within the Minimal R-symmetric Supersymmetric Standard Model, analyzing their potential observability at future colliders considering current experimental constraints.

Contribution

It provides a detailed analysis of Z boson LFV decays in MRSSM, incorporating experimental bounds and exploring parameter space for future detection prospects.

Findings

Predicted branching ratios are several orders below current experimental limits.

Z→eτ and Z→μτ decays could be observable in future experiments.

Constraints from l2→l1γ decays significantly limit flavor-violating parameters.

Abstract

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.

Tables

Table 1. Table 1: Current limits on LFV decays of Z 𝑍 Z boson.

Decay	Bound	Experiment	Bound	Experiment
$Z\to e\mu$	$1.7\times {10}^{-6}$	LEP (1995)ZPC9567	$7.5\times {10}^{-7}$	ATLAS (2014)AT14
$Z\to e\tau$	$9.8\times {10}^{-6}$	LEP (1995)ZPC9567	$5.8\times {10}^{-5}$	ATLAS (2018)AT18
$Z\to \mu \tau$	$1.2\times {10}^{-5}$	LEP (1995)ZPC9773	$1.3\times {10}^{-5}$	ATLAS (2018)AT18

Table 2. Table 2: The superfields with R-charge in MRSSM.

Field	Superfield		Boson		Fermion
Gauge vector	$\hat{g},\hat{W},\hat{B}$	0	$g,W,B$	0	$\tilde{g},\tilde{W}\tilde{B}$	+1
Matter	$\hat{l},{\hat{e}}^c$	+1	$\tilde{l},{\tilde{e}}_R^{\ast }$	+1	$l,e_R^{\ast }$	0
	$\hat{q},{\hat{d}}^c,{\hat{u}}^c$	+1	$\tilde{q},{\tilde{d}}_R^{\ast },{\tilde{u}}_R^{\ast }$	+1	$q,d_R^{\ast },u_R^{\ast }$	0
$H$-Higgs	${\hat{H}}_{d,u}$	0	$H_{d,u}$	0	${\tilde{H}}_{d,u}$	-1
R-Higgs	${\hat{R}}_{d,u}$	+2	$R_{d,u}$	+2	${\tilde{R}}_{d,u}$	+1
Adjoint chiral	$\widehat{𝒪},\hat{T},\hat{S}$	0	$O,T,S$	0	$\tilde{O},\tilde{T},\tilde{S}$	-1

Table 3. Table 3: Benchmark points.

Input	$tan\beta$	${\lambda }_d$,${\lambda }_u$	${\Lambda }_d$,${\Lambda }_u$	$v_S$	$v_T$	$M_D^B$	$M_D^W$	${\mu }_d$,${\mu }_u$	$m_T^2$	$m_l^2$,$m_r^2$
BMP1	3	1.0,-0.8	-1.0,-1.2	5.9	-0.33	600	500	400,400	${3000}^2$	${1000}^2$,${1000}^2$
BMP3	40	0.15,-0.15	-1.0,-1.15	-0.14	-0.34	250	500	400,400	${3000}^2$	${1000}^2$,${1000}^2$

Table 4. Table 4: Spectrum of the different BMPs in GeV.

Input	$H_1$	$H_2$	$A_1$	$H_1^{\pm }$	${\chi }_1^0$	${\chi }_2^0$	${\chi }_1^{\pm }$	$\tilde{\nu }$
BMP1	125.3	897	896	899	415	420	416	1002
BMP3	125.1	1245	1245	1248	251	408	408	1000

Table 5. Table 5: Current limits of LFV decays of l 2 → l 1 ​ γ → subscript 𝑙 2 subscript 𝑙 1 𝛾 l_{2}\rightarrow l_{1}\gamma .

Decay	Bound	Experiment	Decay	Bound	Experiment
$\mu \to e\gamma$	$4.2\times {10}^{-13}$	MEG(2016)MEG	$\tau \to e\gamma$	$3.3\times {10}^{-8}$	BABAR(2010)BABAR
$\tau \to \mu \gamma$	$4.4\times {10}^{-8}$	BABAR(2010)BABAR	-	-	-

Table 6. Table 6: The range of input parameters for the numerical scan with BMP3.

Parameter	Min	Max	Step
$M_D^W$	600	800	10
$M_D^B$	600	1000	10

Table 7. Table 7: Upper limits on off-diagonal entries δ 12 superscript 𝛿 12 \delta^{12} , δ 13 superscript 𝛿 13 \delta^{13} and δ 23 superscript 𝛿 23 \delta^{23} .

Parameters	MRSSM	MSSM(${\Delta }_{LL}^{IJ}$/${\Delta }_{RR}^{IJ}$Rosiek1 )
${\delta }^{12}$	$1.0\times {10}^{-3}$	$8.4\times {10}^{-4}$/$5.0\times {10}^{-3}$
${\delta }^{13}$	$3.0\times {10}^{-1}$	$5.3\times {10}^{-1}$/$𝒪\left(1\right)$
${\delta }^{23}$	$3.0\times {10}^{-1}$	$4.6\times {10}^{-1}$/$𝒪\left(1\right)$

Table 8. Table 8: Upper predictions on LFV decays of Z 𝑍 Z boson in MRSSM.

Decay	Prediction	Future sensitivityZ1 ; Z2 ; Z3 ; Z4
Br($Z\to e\mu$)	$𝒪\left({10}^{-14}\right)$	$2.0\times {10}^{-9}$
Br($Z\to e\tau$)	$𝒪\left({10}^{-9}\right)$	$\left(1.3-6.5\right)\times {10}^{-8}$
Br($Z\to \mu \tau$)	$𝒪\left({10}^{-9}\right)$	$\left(0.44-2.2\right)\times {10}^{-8}$

Equations

$${\mathcal{W}}_{MRSSM}$$

$$\hat{T}=\left(\begin{array}[]{cc}\hat{T}^{0}/\sqrt{2}&\hat{T}^{+}\\ \hat{T}^{-}&-\hat{T}^{0}/\sqrt{2}\end{array}\right).$$

$$V_{SB,S}$$

$$V_{SB,DG}=M_D^B\tilde{B}\tilde{S}+M_D^W{\tilde{W}}^a{\tilde{T}}^a+M_D^O\tilde{g}\tilde{O}+h.c.,$$

$$m_{{\chi }^0}$$

$${\mu }_d^{eff,+}$$

$${\mu }_u^{eff,-}$$

$$(N^1{)}^{\ast }{m}_{{\chi }^0}(N^2{)}^{†}=diag(m_{{\chi }_1^0},...,m_{{\chi }_4^0}).$$

$$m_{\chi^{\pm}}=\left(\begin{array}[]{cc}g_{2}v_{T}+M^{W}_{D}&\frac{1}{\sqrt{2}}\Lambda_{d}v_{d}\\ \frac{1}{\sqrt{2}}g_{2}v_{d}&-\frac{1}{2}\Lambda_{d}v_{T}+\frac{1}{\sqrt{2}}\lambda_{d}v_{S}+\mu_{d}\end{array}\right),$$

$$(U^1{)}^{\ast }{m}_{{\chi }^{\pm }}(V^1{)}^{†}=diag(m_{{\chi }_1^{\pm }},m_{{\chi }_2^{\pm }}).$$

$$m^{2}_{\tilde{\nu}}=\begin{array}[]{c}m_{l}^{2}+\frac{1}{8}(g_{1}^{2}+g_{2}^{2})(v_{d}^{2}-v_{u}^{2})+g_{2}v_{T}M^{W}_{D}-g_{1}v_{S}M^{B}_{D},\end{array}$$

$$Z^V{m}_{\tilde{\nu }}^2(Z^V{)}^{†}=diag(m_{{\tilde{\nu }}_1}^2,m_{{\tilde{\nu }}_2}^2,m_{{\tilde{\nu }}_3}^2).$$

$$m^{2}_{\tilde{L}^{\pm}}=\left(\begin{array}[]{cc}(m^{2}_{\tilde{L}^{\pm}})_{LL}&0\\ 0&(m^{2}_{\tilde{L}^{\pm}})_{RR}\end{array}\right),$$

$$(m_{{\tilde{L}}^{\pm }}^2{)}_{LL}$$

$$(m_{{\tilde{L}}^{\pm }}^2{)}_{RR}$$

$$Z^E{m}_{{\tilde{L}}^{\pm }}^2(Z^E{)}^{†}=diag(m_{{\tilde{L}}_1^{\pm }}^2,...,m_{{\tilde{L}}_6^{\pm }}^2).$$

$$-i\mathcal{L}$$

$$\displaystyle\mathcal{L}_{Zl_{1}l_{2}}=\bar{l}_{1}\Big{[}\gamma^{\mu}(C^{1}_{L}P_{L}+C^{1}_{R}P_{R})+p_{1}^{\mu}(C^{2}_{L}P_{L}+C^{2}_{R}P_{R})\Big{]}l_{2}Z_{\mu}.$$

$$Br(Z\to l_1{l}_2)$$

$$C_{L/R}^1$$

$$C_{L/R}^2$$

$$C_L^{1,a/b}$$

$$C_L^{2,a/b}$$

$$C_{1L}^{(a)}$$

$$C_2^{(a)}$$

$$C_{3L}^{(a)}$$

$$M_1$$

$$C_{1L}^{(b)}$$

$$C_{3L}^{(b)}$$

$$C_L^{1,c/d}$$

$$C_R^{1,c/d}$$

$$C_L^{2,c/d}$$

$$C_{1L}^{(c)}$$

$$C_{2L}^{(c)}$$

$$C_{2R}^{(c)}$$

Full text

R-symmetry; MRSSM; Lepton flavor violation

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. Search for such LFV decays has been pursued to date in a host of processes of leptons, $Z$ boson, Higgs boson and various hadrons. The present upper bounds on the various LFV decay channels of Z boson from both the LEP data and the LHC data is shown in TABLE.1 PDG . One can find a pedagogical introduction on the theoretical motivations for charged LFV and the experimental aspects in Ref.Calibbi .

As a new solution to the supersymmetric flavor problem in MSSM, the Minimal R-symmetric Supersymmetric Standard Model (MRSSM) is proposed in Ref.Kribs , where the R-symmetry is a fundamental symmetry proposed several decades ago and stronger than R-parity Fayet ; Salam . R-symmetry forbids Majorana gaugino masses, $\mu$ term, $A$ terms and all left-right squark and slepton mass mixings. The $R$-charged Higgs $SU(2)_{L}$ doublets $\hat{R}_{u}$ and $\hat{R}_{d}$ are introduced in MRSSM to yield the Dirac mass terms of higgsinos. Additional superfields $\hat{S}$, $\hat{T}$ and $\hat{O}$ are introduced to yield Dirac mass terms of gauginos. Studies on phenomenology in MRSSM can be found in literatures Die1 ; Die2 ; Die3 ; Die4 ; Die5 ; Kumar ; Blechman ; Kribs1 ; Frugiuele ; Jan ; Chakraborty ; Braathen ; Athron ; Alvarado .

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 $m_{l}^{2}$ and $m_{r}^{2}$ Kss . Taking account of the constraint from radiative decay $l_{2}\rightarrow l_{1}\gamma$ on the off-diagonal parameters, we give the upper predictions on the LFV decays of $Z$ boson with parameter spaces BMP1 and BMP3Die3 . Taking account of recent experimental limit on the masses of charginos and neutrilinos AT1806 , we also explore the LFV decays of $Z$ boson as a function of Dirac mass parameter $M_{D}^{W}$. 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. The spectrum of fields in MRSSM contains the standard MSSM matter, Higgs and gauge superfields augmented by chiral adjoints $\hat{\cal O},\hat{T},\hat{S}$ and two $R$-Higgs iso-doublets. The superfields with R-charge in MRSSM are given in TABLE.2.

The general form of the superpotential of the MRSSM is given by Die1

[TABLE]

where $\hat{H}_{u}$ and $\hat{H}_{d}$ are the MSSM-like Higgs weak iso-doublets, $\hat{R}_{u}$ and $\hat{R}_{d}$ are the $R$-charged Higgs $SU(2)_{L}$ doublets and the corresponding Dirac higgsino mass parameters are denoted as $\mu_{u}$ and $\mu_{d}$. $\lambda_{u}$, $\lambda_{d}$, $\Lambda_{u}$ and $\Lambda_{d}$ are parameters of Yukawa-like trilinear terms involving the singlet $\hat{S}$ and the triplet $\hat{T}$, which is given by

[TABLE]

Then, the soft-breaking terms involving scalar mass are

[TABLE]

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 singlet $\hat{S}$, triplet $\hat{T}$ and octet $\hat{O}$ take the form

[TABLE]

where $\tilde{B}$, $\tilde{W}$ and $\tilde{g}$ are usually MSSM Weyl fermions. After EWSB, one can get the following $4\times 4$ neutralino mass matrix

[TABLE]

where the modified $\mu_{i}$ parameters are

[TABLE]

and the $v_{T}$ and $v_{S}$ are vacuum expectation values of $\hat{T}$ and $\hat{S}$ which carry zero $R$-charge. The neutralino mass matrix can be diagonalized by unitary matrices $N^{1}$ and $N^{2}$

[TABLE]

The chargino mass matrix is given by

[TABLE]

and can be diagonalized by unitary matrices $U^{1}$ and $V^{1}$

[TABLE]

The LFV interactions mainly originate from the potential misalignment between the leptons and sleptons mass matrices in the MRSSM. In the gauge eigenstate basis $\tilde{\nu}_{iL}$, the sneutrino mass squared matrix is expressed as

[TABLE]

where the last two terms are newly introduced by MRSSM, and the mass matrix is diagonalized by unitary matrix $Z^{V}$

[TABLE]

The slepton mass squared matrix takes the form

[TABLE]

where

[TABLE]

The sources of LFV are the off-diagonal entries of the $3\times 3$ soft supersymmetry breaking matrices $m_{l}^{2}$ and $m_{r}^{2}$, where the A terms are absent. Note that, in the following, we replace $\tilde{l}$ with $\tilde{L}^{\pm}$ to denote the sleptons. From Eq.(11), we can see that the left-right slepton mass mixing is also absent. The slepton mass matrix is diagonalized by unitary matrix $Z^{E}$

[TABLE]

The interactions of charged sleptons $\tilde{L}^{\pm}$ and neutral sneutrinos $\tilde{\nu}$ with neutralinos $\chi^{0}$ and charginos $\chi^{\pm}$ are correspondingly given by the Lagrangian as Die3 ; SARAH

[TABLE]

The interactions between $Z$ boson and neutralinos $\chi^{0}$ or charginos $\chi^{\pm}$ are given by the Lagrangian as

[TABLE]

The relevant Feynman diagrams contributing to the LFV decays of $Z$ boson in MRSSM is presented in FIG.1, where FIG.1(a) and FIG.1(c) take more important role than others. The $Zl_{1}l_{2}$ interaction Lagrangian can be written as Flavor

[TABLE]

The left-handed current coefficient $C^{1}_{L}$ for FIG.1(a) and the right-handed current coefficient $C^{1}_{R}$ for FIG.1(c) are dominant in the final result. Then the branching ratios of LFV decays of $Z$ boson is calculated by

[TABLE]

where the charged lepton masses have been neglected and $\Gamma_{Z}$ 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

[TABLE]

The coefficients in FIG.1 (a) and FIG.1 (b) take the same form

[TABLE]

where the couplings corresponding to FIG.1 (a) is

[TABLE]

and the couplings corresponding to FIG.1 (b) is

[TABLE]

The coefficients in FIG.1 (c) and FIG.1 (d) take the same form

[TABLE]

where the couplings corresponding to FIG.1 (c) is

[TABLE]

and the couplings corresponding to FIG.1 (d) is

[TABLE]

Above loop integrals are given in term of Passarino-Veltman functions PVI

[TABLE]

and can be calculated by the Mathematica package Package-X X through a link to fortran library Collier collier ; collier1 ; collier2 ; collier3 , 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.Die3 as the default values for our parameter setup and display them in Table.3, 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.4. 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 Die1

[TABLE]

To decrease the number of free parameters involved in our calculation, we assume that the diagonal entries of two $3\times 3$ matrices $m_{l}^{2}$ and $m_{r}^{2}$ are equal $(m_{l}^{2})_{II}=(m_{r}^{2})_{II}$, same to the values shown in TABLE.3, where $I=1,2,3$. Then, the only sources of LFV are off-diagonal entries of the soft breaking terms $m_{l}^{2}$, $m_{r}^{2}$, which are parameterized by mass insertion as in Rosiek3

[TABLE]

where $I,J=1,2,3$. We also assume $\delta^{IJ}_{l}$ = $\delta^{IJ}_{r}$ = $\delta^{IJ}$. The experimental limits on LFV decays, such as radiative two body decays $l_{2}\rightarrow l_{1}\gamma$, leptonic three body decays $l_{2}\rightarrow 3l_{1}$ and $\mu-e$ conversion in nuclei, can give strong constraints on the parameters $\delta^{IJ}$. In the following, we will use LFV decays $l_{2}\rightarrow l_{1}\gamma$ to constrain the parameters $\delta^{IJ}$. Current limits of LFV decays $l_{2}\rightarrow l_{1}\gamma$ listed in TABLE.5 PDG .

The sparticle mediated diagrams for $l_{2}\rightarrow l_{1}\gamma$ 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 $l_{2}\rightarrow l_{1}\gamma$ is given by DingY

[TABLE]

Then, in the limit $m_{1}\rightarrow 0$, the analytic expression of $Br(l_{2}\rightarrow l_{1}\gamma)$ is derived as

[TABLE]

where $\Gamma_{l_{2}}$ 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,

[TABLE]

In the limit $m_{1}\rightarrow 0$, the form factors $A^{(a)}$ and $B^{(a)}$ corresponding to FIG.2 (a) are given as

[TABLE]

and the form factors $A^{(b)}$ and $B^{(b)}$ corresponding to FIG.2 (b) are given as

[TABLE]

where the loop integrals $\mathcal{C}_{1,2,...}$ denote $\frac{i}{16\pi^{2}}\mathcal{C}_{1,2,...}(m_{2}^{2},0,0;M_{3},M_{2},M_{1})$. The couplings corresponding to FIG.2 (a) are given as

[TABLE]

and the couplings corresponding to FIG.2 (b) are given as

[TABLE]

Taking $\delta^{13}=0$, $\delta^{23}=0$, we plot the theoretical prediction of Br $(\mu\rightarrow e\gamma)$ versus $\delta^{12}$ and Br($Z\rightarrow e\mu$) versus $\delta^{12}$ in FIG.3(a) and FIG.3(b), where the horizontal dot line is the current experimental bounds of Br($\mu\rightarrow e\gamma$). The solid line stands for the result calculated with parameter setup BMP1 and the dash line stands for the result calculated with parameter setup BMP3. A linear relationship in logarithmic scale is displayed between Br$(\mu\rightarrow e\gamma)$ or Br($Z\rightarrow e\mu$) and the flavor violating parameter $\delta^{12}$. The prediction on Br$(\mu\rightarrow e\gamma)$ exceeds the current experiment limit at $\delta^{12}\sim 1.0\times 10^{-3}$. The parameter space of $\delta^{12}$ would been highly suppressed below to $\mathcal{O}(10^{-3})$ by taking account of the future sensitivity of experiment, which is Br$(\mu\rightarrow e\gamma)\sim 6\times 10^{-14}$ MEG1 . The current prediction of Br($Z\rightarrow e\mu$) in MRSSM is around $\mathcal{O}(10^{-13})$ and this prediction is six orders of magnitude below the current limit $\mathcal{O}(10^{-7})$. Based on the flavour expansion theorem, various LFV processes have been investigated in MSSM using a recently developed technique which performes a purely algebraic mass-insertion expansion of the amplitudes Rosiek1 . Taking account of the constraints from radiative charged lepton decays, upper bounds on the flavor violating parameters $\delta^{12}_{l}$ ($\Delta^{21}_{LL}$) and $\delta^{12}_{r}$ ($\Delta^{21}_{RR}$) are given in Ref.Rosiek1 with $\delta^{12}_{l}\sim 8.4\times 10^{-4}$ ($\tan\beta$=2) and $\delta^{12}_{r}\sim 5.0\times 10^{-3}$ ($\tan\beta$=2), and it shows flavor violating $\mu$ lepton decays still provide the most stringent bounds on supersymmetric effects.

Taking $\delta^{12}=0$, $\delta^{23}=0$, we plot the theoretical prediction of Br $(\tau\rightarrow e\gamma)$ versus $\delta^{13}$ and Br($Z\rightarrow e\tau$) versus $\delta^{13}$ in FIG.3(d) and FIG.3(d), where the horizontal dot line is the current experimental bounds of Br($\tau\rightarrow e\gamma$). The solid line stands for the result calculated with parameter setup BMP1 and the dash line stands for the result calculated with parameter setup BMP3. Both predictions on Br$(\tau\rightarrow e\gamma)$ and Br($Z\rightarrow e\tau$) decrease as the flavor violating parameter $\delta^{13}$ varies from 0.9 to 0.1. The prediction on Br$(\tau\rightarrow e\gamma)$ exceeds the current experiment limit at $\delta^{13}\sim 0.75$. Taking account of the future experimental expectation on Br$(\tau\rightarrow e\gamma)$, which is around $2.3\times 10^{-9}$ SuperB , the parameter space of $\delta^{13}$ would been suppressed below to $0.3$. The upper theoretical prediction on Br($Z\rightarrow e\tau$) in MRSSM is around $\mathcal{O}(10^{-8})$ and this prediction is two orders of magnitude below the current limit $\mathcal{O}(10^{-6})$. Recent upper bounds on the flavor violating parameters $\delta^{13}_{l}$ ($\Delta^{31}_{LL}$) and $\delta^{13}_{r}$ ($\Delta^{31}_{RR}$) in MSSM are given by $\delta^{13}_{l}\sim 4.6\times 10^{-1}$ ($\tan\beta$=2) and $\delta^{13}_{r}\sim\mathcal{O}(1)$ ($\tan\beta$=2) Rosiek1 .

Taking $\delta^{12}=0$, $\delta^{13}=0$, we plot the theoretical prediction of Br $(\tau\rightarrow\mu\gamma)$ versus $\delta^{23}$ and Br($Z\rightarrow\mu\tau$) versus $\delta^{23}$ in FIG.3(e) and FIG.3(f), where the horizontal dot line is the current experimental bounds of Br($\tau\rightarrow\mu\gamma$). The solid line stands for the result calculated with parameter setup BMP1 and the dash line stands for the result calculated with parameter setup BMP3. Both predictions on Br$(\tau\rightarrow\mu\gamma)$ and Br($Z\rightarrow\mu\tau$) decrease sharply as the flavor violating parameter $\delta^{13}$ is close to 0.1. The prediction on Br$(\tau\rightarrow\mu\gamma)$ exceeds the current experiment limit at $\delta^{23}\sim 0.8$. Taking account of the future experimental expectation on Br$(\tau\rightarrow\mu\gamma)$, which is around ($1.8\times 10^{-9}$) SuperB , the parameter space of $\delta^{23}$ would been also suppressed below to $0.3$. The upper theoretical prediction on Br($Z\rightarrow\mu\tau$) in MRSSM is also around $\mathcal{O}(10^{-8})$ and this prediction is three orders of magnitude below the current limit $\mathcal{O}(10^{-5})$. Recent upper bounds on the flavor violating parameters $\delta^{23}_{l}$ ($\Delta^{32}_{LL}$) and $\delta^{23}_{r}$ ($\Delta^{32}_{RR}$) in MSSM are given by $\delta^{23}_{l}\sim 5.3\times 10^{-1}$ ($\tan\beta$=2) and $\delta^{23}_{r}\sim\mathcal{O}(1)$ ($\tan\beta$=2) Rosiek1 .

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 AT1806 , which state that charginos and neutralinos must be heavier than 600 GeV at 95% CL. To be compatible with the experimental limit, the parameters $M_{D}^{W}$, $M_{D}^{B}$, $\mu_{u}$ and $\mu_{d}$, 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 PhD , 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 $\mu_{u}(\mu_{d})<$ 500 GeV for BMP1, and this leads to at least two sparticle mass are lighter than 600 GeV. For BMP3, the valid region for $\mu_{u}(\mu_{d})$ can enlarged above 600 GeV together with $M_{D}^{W}$ and $M_{D}^{B}$. Then, to quantitatively study the LFV decays of Z boson, we perform a scan under BMP3 over the parameters $M_{D}^{W}$ and $M_{D}^{B}$, with $\mu_{u}(\mu_{d})=$ 600 GeV. The ranges of variation over the MRSSM parameters are displayed in TABLE.6, all mass parameters are in GeV. Moreover, the three flavor violating parameters $\delta^{12}$=$10^{-3}$, $\delta^{13}$ =0.3 and $\delta^{23}$ =0.3 are assumed.

Over a general scan of thousands of points in parameter spaces according to TABLE.6, we display the predictions on Br($Z\rightarrow e\mu$), Br($Z\rightarrow e\tau$) and Br($Z\rightarrow\mu\tau$) as a function of $M_{D}^{W}$ in FIG.4(a), (b) and (c) respectively. It shows that the predictions on Br($Z\rightarrow e\mu$), Br($Z\rightarrow e\tau$) and Br($Z\rightarrow\mu\tau$) decrease as $M_{D}^{W}$ varies from 600 GeV to 800 GeV. For Br($Z\rightarrow e\mu$), the range of the prediction is narrowed above $\mathcal{O}(10^{-14})$ at $M_{D}^{W}$=600 GeV and narrowed to $\mathcal{O}(10^{-15}-10^{-14})$ at $M_{D}^{W}$=800 GeV. For Br($Z\rightarrow e\tau$) and Br($Z\rightarrow\mu\tau$), the range of the prediction is narrowed above $\mathcal{O}(10^{-9})$ at $M_{D}^{W}$=600 GeV and $\mathcal{O}(10^{-10}-10^{-9})$ at $M_{D}^{W}$=800 GeV.

IV Conclusions

In this work, taking account of the constraints on the parameter space from radiative charged lepton decays Br($l_{2}\rightarrow l_{1}\gamma$), we analyze the LFV decays of $Z\rightarrow l_{1}l_{2}$ as a function of the flavor violating off-diagonal entries $\delta^{12}$, $\delta^{13}$ and $\delta^{23}$ in the framework with R-symmetric Supersymmetric Standard Model. A summary on the upper limits on flavor violating parameters in MRSSM is given in TABLE.7, where the results in MSSM are also included for supplement. It displays the upper limits in two different models are very close to each other.

The LFV decays of $Z\rightarrow l_{i}l_{j}$ depend strongly on the three flavor violating parameters $\delta^{ij}$ in soft breaking terms $m_{l}$ and $m_{r}$, i.e., if set $\delta_{ij}$=0, then the branching ratios of $Z\rightarrow l_{i}l_{j}$ equal zero. Taking account of constraints from $l_{2}\rightarrow l_{1}\gamma$ we summarize the theoretical predictions of $Z\rightarrow l_{1}l_{2}$ in MRSSM in TABLE.8, and the value is taken from FIG.4 with $M_{D}^{W}$=600 GeV. The upper prediction on Br($Z\rightarrow e\mu$) is seven orders of magnitude below current experimental bound and we may make more efforts to observe it in future experiment. The upper prediction on Br($Z\rightarrow e\tau$) and Br($Z\rightarrow\mu\tau$) are at the same order and close to the current experimental bound. Thus, the LFV decays $Z\rightarrow e\tau$ and $Z\rightarrow\mu\tau$ may be more promising to be observed in future experiment.

Acknowledgements.

The work has been supported by the National Natural Science Foundation of China (NNSFC) with Grants No.11747064, No.11805140, the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (Grant No. 2017113), the Foundation of Department of Education of Liaoning province with Grant No. 2016TSPY10 and the Youth Foundation of the University of Science and Technology Liaoning with Grant No. 2016QN11.