Eur. Phys. J. C (2020) 80:1167 https://doi.org/10.1140/epjc/s10052-020-08750-w

Regular Article - Theoretical Physics

Searching for lepton flavor violating decays τ → Pl in the minimal R-symmetric supersymmetric standard model

Ke-Sheng Sun1,a,TaoGuo2,b,WeiLi3,4,c,Xiu-YiYang5,d, Shu-Min Zhao3,4,e 1 Department of Physics, , Baoding 071000, 2 School of Mathematics and , GEO University, 050031, China 3 Department of Physics, Hebei University, Baoding 071002, China 4 Key Laboratory of High-Precision Computation and Application of Quantum Field Theory of Hebei Province, Baoding 071002, China 5 School of Science, University of Science and Liaoning, Anshan 114051, China

Received: 8 October 2020 / Accepted: 10 December 2020 / Published online: 19 December 2020 © The Author(s) 2020

Abstract We analyze the lepton flavor violating decays stand for τ factories too, like BaBar or Belle, have joined τ → Pl (P = π, η, η; l = e,μ) in the scenario of in the pursuit of charged LFV decays coming from τ lep- the minimal R-symmetric supersymmetric standard model. ton [1]. All experiments have provided excellent bounds on The prediction on the branching ratios BR(τ → Pe) and the hadronic decays of τ, for the first time [2Ð4], such as BR(τ → Pμ) is affected by the mass insertion param- τ → μ(P, V, PP), where P(V ) stands for a pseudoscalar eters δ13 and δ23, respectively. These parameters are con- (vector) meson. The study of LFV decays of τ lepton are also strained by the experimental bounds on the branching ratios one of main goals of the future SuperKEKB/Belle II project BR(τ → e(μ)γ ) and BR(τ → 3e(μ)). The result shows under construction at KEK [5]. The present upper bounds on Z penguin dominates the prediction on BR(τ → Pl)ina the branching ratios of τ → Pl (P = π, η, η; l = e,μ)are large region of the parameter space. The branching ratios shown in Table 1 [6]. for BR(τ → Pl) are predicted to be, at least, five orders Assuming the integrated luminosity of 50 ab−1,thefuture of magnitude smaller than present experimental bounds and prospects of BR(τ → Pl) in Belle II will be extrapolated at three orders of magnitude smaller than future experimental the level of O(10−9Ð10−10) [9]. sensitivities. In various extensions of the SM, corrections to BR(τ → Pl) are enhanced by different LFV sources. There are a few studies within non-SUSY models, such as two Higgs dou- 1 Introduction blet models [10,11], 331 model [12], TC2 models [13], lit- tlest Higgs model with T parity [14], simplest little Higgs Searching for lepton flavor violating (LFV) decays is of great model [15], leptoquark models [16,17] and unparticle model importance in probing new physics (NP) beyond the standard [18]. Some models with heavy Dirac/Majorana neutrinos model (SM) since the theoretical prediction on these LFV can have BR(τ → Pl) close to the experimental sensitiv- decays is suppressed by small mass of neutrinos in the SM. ity [19Ð21]. In Type III seesaw model, there are tree level Much effort has been devoted to searching for LFV decays flavor changing neutral currents in the lepton sector which in experiment and the usually discussed decay channels are can enhance the prediction on BR(τ → Pl) [22,23]. There l2 → l1γ , l2 → 3l1, μÐe conversion in nuclei, semileptonic are also a few studies within SUSY models, such as MSSM τ decays, and so on. The experimental observation of LFV [24,25], unconstrained MSSM [26], supersymmetric seesaw decays of τ lepton is one goal of a bunch of excellent ded- mechanism model [27], R-parity violating SUSY [28], the icated experiments. The first generation of B-factories, that CMSSM-seesaw and NUHM-seesaw [29]. Within an effec- tive field theory framework, LFV decays τ → Pl are studied a e-mails: [email protected]; [email protected] (corre- to set constraints on the Wilson coefficients of the LFV oper- sponding author) ators [17,30Ð39]. b e-mail: [email protected] In this paper, we will study the LFV decays τ → Pl in c e-mail: [email protected] the minimal R-symmetric supersymmetric standard model d e-mail: [email protected] (MRSSM) [40]. The MRSSM has an unbroken global U(1)R e e-mail: [email protected] 123 1167 Page 2 of 10 Eur. Phys. J. C (2020) 80 :1167

Table 1 Current limits on LFV decays τ → Pl Decay Bound Experiment Decay Bound Experiment

τ → eπ 8.0 × 10−8 BELLE [7] τ → μπ 1.1 × 10−7 BABAR [8] τ → eη 9.2 × 10−8 BELLE [7] τ → μη 6.5 × 10−8 BELLE [7] τ → eη 1.6 × 10−7 BELLE [7] τ → μη 1.3 × 10−7 BELLE [7]

symmetry and provides a new solution to the supersymmet- U(1)Y as the SM and the MSSM. The spectrum of fields in the ric flavor problem that exists in the MSSM. In this model, MRSSM contains the standard MSSM matter, the Higgs and R-symmetry forbids the Majorana gaugino masses, μ term, gauge superfields augmented by the chiral adjoints Oˆ , Tˆ , Sˆ A terms and all left-right squark and slepton mass mixings. and two R-Higgs iso-doublets. The superfields with R-charge ˆ ˆ The R-charged Higgs SU(2)L doublets Ru and Rd are intro- in the MRSSM are given in Table 2. duced in the MRSSM to yield Dirac mass terms of higgsi- The general form of the superpotential of the MRSSM is nos. The additional superfields Sˆ, Tˆ and Oˆ are introduced given by [41] to yield Dirac mass terms of gauginos. The most unusual characteristic in the MRSSM is that large flavor violation is WMRSSM ˆ ˆ ˆ ˆ ˆ ˆ ˆ allowed in the squark and slepton mass matrices. The pres- = μd (Rd Hd ) + μu(Ru Hu) + d (Rd T )Hd ence of large flavor violation in the MRSSM means that it ˆ ˆ ˆ ˆ ˆ ˆ +u(Ru T )Hu + λd S(Rd Hd ) is no longer appropriate to discuss stops or selectrons nec- +λ ˆ( ˆ ˆ ) − ˆ( ˆ ˆ ) essarily. The large flavor violation opens the possibility for u S Ru Hu Yd d q Hd ˆ ˆ ˆ a wide variety of new signals at the LHC and is worthy of −Yeeˆ(l Hd ) + Yuuˆ(qˆ Hu), (1) significant study. Studies on phenomenology in the MRSSM ˆ ˆ can be found in Refs. [41Ð58]. It is interesting to explore where Hu and Hd are the MSSM-like Higgs weak iso- ˆ ˆ whether BR(τ → Pl) can be enhanced to be close to the doublets, Ru and Rd are the R-charged Higgs SU(2)L dou- current experiment limits or future experimental sensitivities blets. The corresponding Dirac higgsino mass parameters are while the predictions on other LFV processes do not exceed denoted as μu and μd . Although R-symmetry forbids the μ current experiment constraints. Thus, we choose decay chan- terms of the MSSM, the bilinear combinations of the normal ˆ ˆ nels τ → Pl as an object of the analysis. Similar to the case Higgs SU(2)L doublets Hu and Hd with the Higgs SU(2)L ˆ ˆ in the MSSM, LFV decays mainly originate from the off- doublets Ru and Rd areallowedinEq.(1). The parameters diagonal entries in slepton mass matrices m2 and m2. Taking λu, λd , u and d are Yukawa-like trilinear terms involving l r ˆ ˆ into account the experimental constraints from decay chan- the singlet S and the triplet T . nels τ → lγ and τ → 3l on the off-diagonal parameters, we For the phenomenological studies we take the soft- investigate the branching ratios BR(τ → Pl)asafunction breaking scalar mass terms [43] of the off-diagonal parameters and other model parameters. 2 0 2 − 2 2 0 2 + 2 V , = m (|H | +|H | ) + m (|H | +|H | ) The outline of this paper is organized as follows. In Sect. SB S Hd d d Hu u u +( ( − + − 0 0) + . .) 2, we provide a brief introduction on the MRSSM and present Bμ Hd Hu Hd Hu h c + − the definitions of the sneutrino mass matrix and slepton mass +m2 (|R0|2 +|R |2) + m2 (|R0|2 +|R |2) Rd d d Ru u u matrix. Then, we present conventions for the effective opera- − + +m2 (|T 0|2 +|T |2 +|T |2) tors and the corresponding Wilson coefficients. The existing T + 2 | |2 + 2 | 2|+ ˜∗ 2 ˜ + ˜∗ 2 ˜ constraints, benchmark points and the results of our calcula- mS S m O O dL,i mq,ijdL, j dR,i md,ijdR, j tion are shown in Sect. 3. In Sect. 4, the conclusion is drawn. +˜∗ 2 ˜ uL,i mq,ijuL, j The definitions of mass matrices of scalar and pseudo-scalar ∗ 2 ∗ 2 +˜u m u˜ , +˜e m e˜ , Higgs boson, neutralino, χ ±-chargino and squarks are listed R,i u,ij R j L,i l,ij L j +˜∗ 2 ˜ +˜ν∗ 2 ν˜ . in Appendix A. eR,i mr,ijeR, j L,i ml,ij L, j (2)

All the trilinear scalar couplings involving Higgs bosons to squark and slepton are forbidden in Eq. (2) because the 2 MRSSM sfermions have an R-charge and these terms are non R- invariant, and this has relaxed the flavor problem of the In this section, we firstly provide a simple overview of the MSSM [40]. The Dirac nature is a manifest feature of the MRSSM in order to fix the notations used in this work. The MRSSM fermions. The soft-breaking Dirac mass terms of ˆ ˆ ˆ MRSSM has the same gauge symmetry SU(3)C ×SU(2)L × the singlet S, triplet T and octet O take the form as 123 Eur. Phys. J. C (2020) 80 :1167 Page 3 of 10 1167

Table 2 The superfields with R-charge in MRSSM Field Superfield Boson Fermion

Gauge vector gˆ, Wˆ , Bˆ 0 g, W, B 0 g˜, W˜ B˜ +1 ˆ, ˆc ˜, ˜∗ , ∗ Matter l e +1 l eR +1 l eR 0 ˆ, ˆc, ˆc ˜, ˜∗ , ˜∗ , ∗ , ∗ q d u +1 q dR u R +1 q dR u R 0 ˆ ˜ H-Higgs Hd,u 0 Hd,u 0 Hd,u − 1 ˆ ˜ R-Higgs Rd,u +2 Rd,u +2 Rd,u +1 Adjoint chiral Oˆ , Tˆ , Sˆ 0 O, T, S 0 O˜ , T˜ , S˜ − 1

= B ˜ ˜ + W ˜ a ˜ a + O ˜ ˜ + . ., ν˜ VSB,DG MD BS MD W T MD gO h c (3) eigenstate basis iL, the sneutrino mass matrix and the diag- onalization procedure are where B˜ , W˜ and g˜ are the usually MSSM Weyl fermions. 1 m2 = m2 + (g2 + g2)(v2 − v2) The R-Higgs bosons do not develop vacuum expectation val- ν˜ l 8 1 2 d u ues (VEVs) since they carry R-charge 2. After electroweak + v W − v B, V 2( V )† = 2,diag, g2 T MD g1 S MD Z mν˜ Z mν˜ (4) symmetry breaking, the singlet and triplet VEVs effectively modify the μu and μd , and the modified μi parameters are where the last two terms in mass matrix are newly introduced given by by MRSSM. The slepton mass matrix and the diagonalization procedure are

ef f,+ 1 1   μ =  v + √ λ v + μ , ( 2) , d d T d S d 2 = me˜ LL 0 , E 2( E )† = 2 diag, 2 2 me˜ ( 2) Z me˜ Z me˜ (5) 0 me˜ RR ef f,− 1 1 μu =− uvT + √ λuvS + μu. 2 2 where

ˆ ˆ 2 2 1 2 2 1 2 2 2 v and v are vacuum expectation values of T and S. (m )LL = m + v |Ye| + (g − g )(v T S e˜ l 2 d 8 1 2 d There are four complex neutral scalar fields and they − v2) − g v M B − g v M W , can mix. Assuming the vacuum expectation values are real, u 1 S D 2 T D the real and imaginary components in four complex neutral ( 2) = 2 + 1v2| |2 + 1 2(v2 m ˜ RR mr d Ye g1 u scalar fields do not mix, and the mass-square matrix breaks e 2 4 − v2) + v B. into two 4 × 4 sub-matrices. In the scalar sector all fields d 2g1 S MD mix and the SM-like Higgs boson is dominantly given by the up-type field. In the pseudo-scalar sector there is no mixing The sources of LFV are the off-diagonal entries of the 3 × 3 2 2 between the MSSM-like states and the singlet-triplet states, soft supersymmetry breaking matrices ml and mr in Eqs. and the 4×4 mass-squared matrix breaks into two 2×2 sub- (4, 5). From Eq. (5) we can see that the left-right slepton matrices. The number of neutralino degrees of freedom in the mass mixing is absent in the MRSSM, whereas the A terms MRSSM is doubled compared to the MSSM as the neutrali- are present in the MSSM. The relevant Feynman diagrams nos are Dirac-type. The number of chargino degrees of free- contributing to τ → Pl are presented in Fig. 1. dom in the MRSSM is also doubled compared to the MSSM We now focus on the LFV processes τ → Pl.Using and these charginos can be grouped to two separated chargino the effective Lagrangian method, we present the analytical sectors according to their R-charge. The χ ±-chargino sec- expression for the decay width of τ → Pl. At the quark tor has R-charge 1 electric charge; the ρ-chargino sector level, the interaction Lagrangian for τ → Pl can be written has R-charge -1 electric charge. Here, we do not discuss as [62] ρ the -chargino sector in detail since it does not contribute X,Y=L,R to the LFV decays. More information about the ρ-chargino L = I (¯ τ)(¯ ) τ→Pl BXY lβ I PX d I PY d can be found in Refs. [43,45,47,57]. For convenience, we I =S,V present the tree-level mass matrices for scalar and pseudo- I ¯ +C (lβ P τ)(u¯ P u) + h.c., (6) scalar Higgs bosons, neutralinos, charginos and squarks of XY I X I Y the MRSSM in Appendix A. where the index β(=1, 2) denotes the generation of the emit- In MRSSM, LFV decays mainly originate from the poten- ted lepton and l1(l2) = e(μ). Since only the axial-vector tial misalignment in slepton mass matrices. In the gauge current contributes to τ → Pl, the coefficients in Eq. (6) 123 1167 Page 4 of 10 Eur. Phys. J. C (2020) 80 :1167

Fig. 1 Feynman diagrams τlν˜ (˜e)˜lττ ν (˜e)˜l τllν (˜e) contributing to τ → Pl in the MRSSM. Corrections from crossed diagrams of box ± 0 0c ± 0 0c ± 0 0c diagram are also considered χ (χ ,χ ) χ (χ ,χ ) χ (χ ,χ ) qqh, A0 qqh, A0 q Z0 q

τ τ ν˜ (˜e) l τ ν˜ (˜e) l τ χ± (χ0,χ0c) l

χ± χ± χ± (χ0,χ0c) ν˜ (˜e) ν˜ (˜e) (χ0,χ0c) (χ0,χ0c) qqZ0 q h, A0 q q h, A0 q

τ ν˜ (˜e) l τ χ± (χ0,χ0c) l τlν˜ (˜e)

χ± χ± χ± χ± ν˜ (˜e)˜ν (˜e) (χ0,χ0c)(χ0,χ0c) (χ0,χ0c) (χ0,χ0c) qqZ0 q Z0 q qqq˜

+ V − V + V − V − V + V ], do not include photon penguin contribution but they include BRL BRR CLL CLR CRL CRR Z boson and scalar ones. The contribution to the Wilson coefficients C I and B I in Eq. (6) can be classified into where fπ is the pion decay constant. The expressions for XY XY ( ) d,u( ) self-energies, Z penguins, Higgs penguins and box diagrams, coefficients C P ,DL P are listed in Table 3 [29]. as shown in Fig. 1. Now consider the implication of virtual Here, mπ and m K denote the masses of the neutral pion  Higgs exchange for τ → Pl. Both the contributions from and Kaon, and θη denotes the η−η mixing angle. In addition, d,u( ) =−( d,u( ))∗ scalar and pseudo-scalar Higgs sector are considered in this DR P DL P . work. However, all the Higgs penguin contribution is negli- Finally, the MRSSM has been implemented in the Math- gible since the couplings of scalar and pseudo-scalar Higgs ematica package SARAH-4.14.3 [59Ð62]. The masses of to the light quarks are suppressed by their masses. the MRSSM particles, mixing matrices and the Wilson Then the decay width for τ → Pl is given by coefficients of the corresponding operators in the effective lagrangian are computed by SPheno-4.0.4 [63,64] modules 1/2 2 2 2  λ (mτ , m , m ) written by SARAH. (τ → Pl) = l P |M|2, 3 (7) 16πmτ i, f where the averaged squared amplitude can be written as 3 Numerical analysis   |M|2 = [ ( I J∗ − I J∗) 2mτ ml aP aP bP bP The calculations of BR(τ → Pl) in the MRSSM are eval- , , = , i f I J S V uated within the framework of SARAH-4.14.3 [59Ð62] and +( 2 + 2 − 2 )( I J∗ + I J∗). mτ ml m P aP aP bP bP (8) SPheno-4.0.4 [63,64]. The experimental values of the Higgs , , mass and the W boson mass can impose stringent and non- The coefficients aS V and bS V are linear combinations of P P trivial constraints on the model parameters. The one loop and the Wilson coefficients in Eq. (6),   leading two loop corrections to the lightest (SM-like) Higgs  d u π D (P) D (P) S = f X ( S + S ) + X ( S + S ) , boson in the MRSSM have been computed in [43] and the aP BLX BRX CLX CRX 2 = , md mu new fields and couplings can give large contributions to the X L R    d u Higgs mass even for stop masses of order 1 TeV and no stop π D (P) D (P) S = f X ( S − S ) + X ( S − S ) , bP BRX BLX CRX CLX mixing. Meanwhile, the new fields and couplings can not give 2 md mu X=L,R too large contribution to the W boson mass and muon decay V fπ V V V a = C(P)(mτ − ml )[−B + B − B in the same regions of parameter space. A better agreement P 4 LL LR RL with the latest experimental value for the W boson mass has +BV + C V − C V + C V − C V ], RR LL LR RL RR been investigated in [46]. It combines all numerically rele- V fπ V V b = C(P)(mτ + ml )[−B + B vant contributions that are known in the SM in a consistent P 4 LL LR 123 Eur. Phys. J. C (2020) 80 :1167 Page 5 of 10 1167

Table 3 Coefficients for each pseudoscalar meson P P = π P = η P = η √ √ 1 1 C(P) 1 √ (sin θη + 2cosθη) √ ( 2sinθη − cos θη) 6 6 2 √ √ d mπ 1 2 2 2 1 2 2 2 D (P) − √ [(3mπ − 4m ) cos θη − 2 2m sin θη] √ [(3mπ − 4m ) sin θη + 2 2m cos θη] L 4 4 3 K K 4 3 K K 2 √ √ u mπ 1 2 1 2 D (P) √ mπ (cos θη − 2sinθη) √ mπ (sin θη + 2cosθη) L 4 4 3 4 3

Table 4 Current limits of l2 → l1γ and l2 → 3l1 Decay Bound Experiment Decay Bound Experiment

μ → eγ 4.2 × 10−13 SPEC (2016) [65] τ → eγ 3.3 × 10−8 BABAR (2010) [66] τ → μγ 4.4 × 10−8 BABAR (2010) [66] μ → 3e 1.0 × 10−12 SPEC (1988) [67] τ → 3e 2.7 × 10−8 BELL (2010) [68] τ → 3μ 2.1 × 10−8 BELL (2010) [68] way with all the MRSSM one loop corrections. A set of the where I, J = 1, 2, 3. To decrease the number of free param- updated benchmark point BMP1 is given in [46] and we dis- eters involved in our calculation, we assume that the off- 2 2 play them in Eq. (9) where all the mass parameters are in diagonal entries of ml and mr in Eq. (11) are equal, i.e., 2 δIJ δIJ δIJ GeV or GeV . l = r = . The experimental bounds on LFV decays, such as radiative two body decays l2 → l1γ , leptonic three 2 tan β = 3, Bμ = 500 ,λd = 1.0,λu =−0.8, body decays l2 → 3l1 and μÐe conversion in nuclei, can give δIJ d =−1.2,u =−1.1, strong constraints on the parameters . In the following, B = , W = ,μ = μ = ,v = . ,v we will use LFV decays l → l γ and l → 3l to constrain MD 550 M 600 d u 500 S 5 9 T 2 1 2 1 D IJ 12 =− . ,( 2) the parameters δ . It is noted that δ has been set zero in 0 38 ml 11 2 2 2 2 2 2 following discussion since it has no effect on the prediction of = (m )22 = (m )33 = (m )11 = (m )22 = (m )33 = 1000 , l l r r r BR(τ → Pl). Current limits on branching ratios of l → l γ (m2 ) = (m2) = (m2 ) = (m2 ) 2 1 q˜ 11 u˜ 11 d˜ 11 q˜ 22 and l2 → 3l1 are listed in Table 4 [6]. = (m2) = (m2 ) = 25002, 23 u˜ 22 d˜ 22 Taking δ = 0 and the parameters in Eq. (9), we plot ( 2 ) = ( 2) (τ → ) δ13 mq˜ 33 mu˜ 33 the predictions of BR Pe versus Log[ ]intheleft 13 2 2 panel of Fig. 2. Taking δ = 0 and the parameters in Eq. = (m ˜)33 = 1000 , mT = 3000, mS = 2000. (9) d (9), we plot the predictions of BR(τ → Pμ) versus Log[δ23] in the right panel of Fig. 2. A linear relationship in logarith- In the numerical analysis, the default values of the input mic scale is displayed between BR(τ → Pe(μ)) and the parameters are set same with those in Eq. (9). The off- flavor violating parameter δ13(δ23). The actual dependence diagonal entries of squark mass matrices m2 , m2, m2 and q˜ u˜ d˜ δ13 δ23 2 2 on or is quadratic. The mentioned linear dependence slepton mass matrices ml , mr in Eq. (9) are zero. The large is due to the fact that both x axis and y axis in Fig. 2 are |v | value of T is excluded by measurement of the W boson logarithmically scaled. In Fig. 2 the following hierarchy is v ( ) 0 mass because the VEV T of the SU 2 L triplet field T shown, BR(τ → πe)>BR(τ → ηe)>BR(τ → ηe) gives a correction to the W mass through [41] and BR(τ → πμ) > BR(τ → ημ) > BR(τ → ημ).The predictions on BR(τ → ηe(μ)) and BR(τ → ηe(μ)) are − . 2 = 1 2(v2 + v2) + 2v2 . very close to each other. At δ13(δ23) = 10 0 25 ∼ 0.56, the mW g2 u d g2 T (10) 4 prediction on BR(τ → e(μ)γ ) is around 10−8 and this is very close to the current experimental bound. The prediction Similarly to the most supersymmetry models, those LFV on BR(τ → 3e(μ)) is around 10−10 and this is two orders processes originate from the off-diagonal entries of the of magnitude below the current experimental bound. The 2 2 soft breaking terms ml and mr in the MRSSM, which are decay channels τ → e(μ)γ can set more strong constraint parametrized by the mass insertion than channels τ → 3e(μ) on the flavor violating parameters.  The predictions on BR(τ → Pl) are far below the current 2 IJ 2 2 experimental bounds. The prediction on BR(τ → πe(μ)) is (m )IJ = δ (m )II(m )JJ, l l  l l around 10−13 and this is five orders of magnitude below the ( 2) = δIJ ( 2) ( 2) , mr IJ r mr II mr JJ (11)

123 1167 Page 6 of 10 Eur. Phys. J. C (2020) 80 :1167

Fig. 2 Left panel: dependence on mass insertion δ13 of BR(τ → Pe). Right panel: dependence on mass insertion δ23 of BR(τ → Pμ)

(a) (b)

Fig. 3 Contributions to BR(τ → Pe) and BR(τ → Pμ) from Z penguins, box diagrams and total diagrams. BR(τ → ηe(μ)) are not shown in plots cause they are very close to BR(τ → ηe(μ))

(a) (b) current experimental bound and three orders of magnitude Higgs couplings to the strange components of the η and η below the future experimental sensitivity. mesons are large enough, which result in large A0Ðη and A0Ð Taking δ13 = 0.5, δ23 = 0 and the parameters in Eq. η mixing, Higgs-mediated contribution could dominate the (9), we plot the predictions of BR(τ → Pe) from various predictions on τ → μη(η) [29]. Furthermore, it was pointed parts as a function of tan β in the left panel of Fig. 3. Taking out in Ref. [70] that a one loop Higgs generated gluon oper- δ13 = 0, δ23 = 0.5 and the parameters in Eq. (9), we plot ator does not suffer of the light-quark mass suppression and the theoretical predictions of BR(τ → Pμ) from various could give a sizeable contribution. The Wilson coefficients β S S parts as a function of tan in the right panel of Fig. 3.The CXY and BXY corresponding to the Higgs penguin in Eq. lines corresponding to Z penguin and box diagram indicate (6) can also contribute to the LFV process μÐe conversion. the values of BR(τ → Pl) are given by only the listed con- In simple formulas, the branching ratios BR(τ → Pl) and tribution with all others set to zero. The total prediction for conversion rate CR(μÐe, nucleus) are given by BR(τ → Pl) is also indicated. We observe that Z penguin dominates the prediction on BR(τ → Pl) in a large region of 5 2 mτ f − the parameter space. For larger values of tan β, the prediction BR(τ → Pl) ∼ P |C S |2 ∼ 10 4|C S |2, 64π XY XY from Z penguin changes slowly, since it is directly propor- 3Zm5 α3 Z 4 F2 1 μ ef f p S 2 tional to + 2 β [69]. The contribution from box diagram is CR(μ − e, nucleus) ∼ |C | 1 tan π 2 XY not sensitive to tan β and at least two orders of magnitude 8 capt ∼ 10| S |2( ). below Z penguin. This is because of the small couplings of 10 CXY nucleus = Ti neutralino/chargino to quark and squark in box diagram. As mentioned above, the contribution from Higgs pen- Thus, the predicted CR(μÐe, nucleus) from Higgs penguin guin is negligible due to the small couplings of the scalar is much larger than BR(τ → Pl) though both are negligible and pseudo-scalar Higgs to the light quarks. However, if the compared to the total contribution. 123 Eur. Phys. J. C (2020) 80 :1167 Page 7 of 10 1167

Fig. 4 Left panel: dependence on mL of BR(τ → Pe). Right panel: dependence on mL of BR(τ → Pμ). The mass parameter mL is in TeV

(a) (b)

Taking δ13 = 0.5, δ23 = 0 and the parameters in Eq. We are also interested to the effect from other parameters (9), we plot the predictions of BR(τ → Pe) as a function on the prediction of BR(τ → Pl) in the MRSSM. The pre- of the diagonal entries mL in the left panel of Fig. 4. Taking dicted BR(τ → Pl) decreases slowly along with the increase δ13 = δ23 = . W 0, 0 5 and parameters in Eq. (9), we plot the pre- of the wino-triplino mass parameter MD . However, the valid (τ → μ) W dictions of BR P as a function of the diagonal entries region of MD is constrained by the boundary conditions at 2 the unification scale, and unphysical masses of neutral Higgs mL in the right panel of Fig. 4. Here, mL = (m )11 =   l W    and charged Higgs are obtained when MD above 1 TeV. By ( 2) = ( 2) = ( 2) = ( 2) = ( 2) ml 22 ml 33 mr 11 mr 22 mr 33. scanning over these parameters, which are shown in Eq. (12), ThepredictiononBR(τ → lγ)is at the level of O(10−9) and − 1.5 <λd ,λu,d ,u < 1.5, this is very close to the future experimental sensitivities [71]. (12) <μ ,μ , < , The prediction on BR(τ → 3l) is at the level of O(10−11) 300 GeV d u m A 1000 GeV and this is two orders of magnitude below the future exper- the prediction is shown in relation to one input parameter (e.g. (τ → γ) imental sensitivities [71]. The predictions on BR l , λd or others). The parameters λd ,λu,u are constrained in BR(τ → 3l) and BR(τ → Pl) in the MRSSM decreas as a narrow region by the boundary conditions which is close to the slepton mass parameter mL varies from 1 to 5 TeV. the benchmark points and d can vary in the whole region. Taking δ13 = 0.5, δ23 = 0 and the parameters in Eq. (9), The results show that varying those parameters in Eq. (12) we plot the predictions of BR (τ → Pe) as a function of have very small effect on the prediction of BR(τ → Pl) the squark mass parameter mQ in the left panel of Fig. 5. which takes values along a narrow band. Taking δ13 = 0, δ23 = 0.5 and the parameters in Eq. (9), we plot the predictions of BR (τ → Pμ) as a function of the squark mass parametermQin the right panel of Fig. 5. Here, 4 Conclusions mQ = (m2 ) = (m2) = (m2 ) = (m2 ) =  q˜ 11 u˜ 11 d˜ 11 q˜ 22    In this work, taking into account the constraints from LFV (m2) = (m2 ) = (m2 ) = (m2) = (m2 ) . u˜ 22 d˜ 22 q˜ 33 u˜ 33 d˜ 33 decays τ → e(μ)γ and τ → 3e(μ) on the flavor violat- We clearly see that both the predictions on BR(τ → Pe) and ing parameters, we analyze the LFV decays τ → Pl in BR(τ → Pμ), which increase slowly as mQvaries from 1 to the framework of the minimal R-symmetric supersymmet- 5 TeV,are not sensitive to mQ. The off-diagonal entries δIJ q˜,u˜,d˜ ric standard model. of the squark mass matrices m2 , m2 and m2 could give addi- We observe that Z penguin dominates the prediction on q˜ u˜ d˜ tional contributions to BR(τ → Pl). Taking into account the BR(τ → Pe(μ)), and other contributions are less dominant experimental constraints on δIJ from low energy B meson or negligible. As we all konw the LFV processes l2 → l1γ q˜,u˜,d˜ ¯ → γ 0 → can only obtain the dipole contribution from the gamma pen- physics observables, such as, BR(B Xs ), BR(Bs,d + − guin. The LFV processes l → 3l and μÐe conversion μ μ ), the prediction of BR(τ → Pl) takes values along 2 1 can obtain contributions from the gamma penguin (includ- a narrow band. Thus, the prediction of BR(τ → Pl) is also ing dipole and non-dipole), Z penguin, Higgs penguin and not sensitive to the off-diagonal entries of the squark mass box diagram. While, for the LFV processes τ → Pl, con- matrices. tributions from Z penguin, Higgs penguin and box diagram are included except for the gamma penguin. It is interest- 123 1167 Page 8 of 10 Eur. Phys. J. C (2020) 80 :1167

Fig. 5 Left panel: dependence on mQ of BR(τ → Pe). Right panel: dependence on mQ of BR(τ → Pμ). Mass parameter mQ is in TeV

(a) (b) ing to consider if it is possible to find a parameter region give appropriate credit to the original author(s) and the source, pro- that LFV processes τ → e(μ)γ and τÐe(μ) conversion are vide a link to the Creative Commons licence, and indicate if changes not excluded and which could still give large contribution were made. The images or other third party material in this article τ → are included in the article’s Creative Commons licence, unless indi- to Pl. The answer is negative at least in this work. cated otherwise in a credit line to the material. If material is not The prediction of BR(τ → Pl) is two or three orders of included in the article’s Creative Commons licence and your intended magnitude below BR(τ → 3l). use is not permitted by statutory regulation or exceeds the permit- In the MRSSM, the prediction on BR(τ → Pe) and ted use, you will need to obtain permission directly from the copy- right holder. To view a copy of this licence, visit http://creativecomm (τ → μ) δ13 BR P is affected by the mass insertions and ons.org/licenses/by/4.0/. δ23, respectively. The prediction on BR(τ → Pe) would Funded by SCOAP3. be zero if δ13 = 0 is assumed, and so are the predic- tiononBR(τ → Pμ) if δ23 = 0 is assumed. Taking into account the experimental bounds on BR(τ → e(μ)γ ) Appendix A: Mass matrices at tree level in the MRSSM and BR(τ → 3e(μ)), the values of δ13 and δ23 are con- strained around 0.5. The predictions on BR(τ → Pe) and In the weak basis (φd ,φu,φS,φT ), the scalar Higgs boson − BR(τ → Pμ) are found to be at the level of O(10 13Ð mass matrix and the diagonalization procedure are −14)   10 , which are five orders of magnitude below the present M MT experimental upper limits. The processes τ → πl may be the M = 11 21 , hM ( h)† = Mdiag, h M M Z h Z h (A1) most competitive LFV semileptonic tau decay channels. The 21 22 future prospects of BR(τ → Pl) in Belle II are extrapolated where the submatrices (cβ = cosβ, sβ = sinβ)are at the level of O(10−9Ð10−10) [9] and they are three orders 2 2 + 2 2 −( 2 + 2 ) m Z cβ m Asβ m Z m A sβ cβ of magnitude above the predictions in the MRSSM. Thus, M11 = , −(m2 + m2 )sβ cβ m2 s2 + m2 c2 τ → Z A Z β A β LFV decays Pl may be out reach of the near future √ √ v ( λ μef f,+ − D)v( λ μef f,− + D) experiments. M = d 2 d d g1 MB u 2 u u g1 MB , 21 ef f,+ D ef f,1 D vd (d μ + g2 M ) −vu (u μu + g2 M ) ⎛ d W W ⎞ Acknowledgements This work has been supported partly by the λ2 v2+λ2 v2 λ  v2−λ  v2 4(M D)2 + m2 + d d u u d d d√ u u u National Natural Science Foundation of China (NNSFC) under Grants M = ⎝ B S 2 2 2 ⎠ . 22 λ  v2−λ  v2 2 v2+2 v2 nos. 11905002 and 11805140, the Scientific Research Foundation of d d d√ u u u 4(M D )2 + m2 + d d u u the Higher Institutions of Hebei Province under Grant no. 2 2 W T 4 BJ2019210, the Foundation of Baoding University under Grant no. In the weak basis (σd ,σu,σS,σT ), the pseudo-scalar 2018Z01, the Foundation of Department of Education of Liaoning province under Grant no. 2020LNQN14, and the Natural Science Foun- Higgs boson mass matrix and the diagonalization procedure dation of Hebei province under Grant no. A2020201002. are ⎛ v ⎞ Bμ u Bμ 00 Data Availability Statement This manuscript has no associated data or vd ⎜ vd ⎟ the data will not be deposited. [Authors’ comment: This is a theoretical Bμ Bμ 00 ⎜ vu ⎟ study and there is no experimental data associated with it.] M =⎜ λ2 v2+λ2 v2 λ  v2−λ  v2 ⎟, A0 ⎜ 00m2 + d d u u d d d√ u u u ⎟ ⎝ S 2 2 2 ⎠ Open Access This article is licensed under a Creative Commons Attri- λ  v2−λ  v2 2 v2+2 v2 00d d d√ u u u m2 + d d u u bution 4.0 International License, which permits use, sharing, adaptation, 2 2 T 4 distribution and reproduction in any medium or format, as long as you A A † diag Z M 0 (Z ) = M . A A0 (A2) 123 Eur. Phys. J. C (2020) 80 :1167 Page 9 of 10 1167

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