Searching for lepton flavor violating decays tau to Pl in Minimal R-symmetric Supersymmetric Standard Model

Considering the constraints from the experimental data on branching ratios of $\tau\rightarrow e \gamma$ and $\tau\rightarrow \mu \gamma$, we analyze the lepton flavor violating decays $\tau\rightarrow Pl$ ($P=\pi^0,\eta,\eta';\;l=e,\mu$) in the scenario of the minimal R-symmetric supersymmetric standard model. The numerical results show that the theoretical predictions on branching ratios of $\tau\rightarrow Pl$ can be enhanced close to the upper experimental bounds or future sensitivities, meanwhile the theoretical predictions on branching ratios of $\tau\rightarrow e \gamma$ and $\tau\rightarrow \mu \gamma$ satisfy the present experimental bounds.

In this paper, we will study the LFV decays τ → P l in the Minimal R-symmetric Supersymmetric Standard Model (MRSSM) [36]. The MRSSM has an unbroken global U (1) R symmetry and provides a new solution to the supersymmetric flavor problem in MSSM. In this model, R-symmetry forbids Majorana gaugino masses, µ term, A terms and all left-right squark and slepton mass mixings. The R-charged Higgs SU (2) L doubletsR u andR d are introduced in MRSSM to yield the Dirac mass terms of higgsinos. Additional superfieldsŜ, T andÔ are introduced to yield Dirac mass terms of gauginos. Studies on phenomenology in MRSSM can be found in literatures [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. 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 . Taking account of the constraints from radiative decays τ → lγ on the off-diagonal parameters, we explore τ → P l as a function of the off-diagonal parameters and other model parameters.
The paper is organized as follows. In Section II, we provide a brief introduction on MRSSM. In Section III, we present our notation and conventions for the operators and their corresponding Wilson coefficients. Then we present the Wilson coefficients for Feynman diagrams contributing to τ → P l in MRSSM in detail. The numerical results are presented in Section IV, and the conclusion is drawn in Section V.

II. MRSSM
In this section, we firstly provide a simple overview of MRSSM in order to fix the notations we use in this paper. The MRSSM has the same gauge symmetry SU (3) C × SU (2) L × U (1) Y as the SM and MSSM. The spectrum of fields in MRSSM contains the standard MSSM matter, Higgs and gauge superfields augmented by chiral adjointsÔ,T ,Ŝ and two R-Higgs iso-doublets. The superfields with R-charge in MRSSM are given in TABLE.II. The general   TABLE II: The superfields with R-charge in MRSSM.

Field
Superfield Boson Fermion Adjoint chiralÔ,T ,Ŝ 0 O, T, S 0Õ,T ,S -1 form of the superpotential of the MRSSM is given by [37] All trilinear scalar couplings involving Higgs bosons to squarks and sleptons are forbidden in Eq.(2) because the sfermions have an R-charge and these terms are non R-invariant, and this relaxes the flavor problem of the MSSM [36]. 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 as whereB,W andg are usually MSSM Weyl fermions. R-Higgs bosons do not develop vacuum expectation values(VEVs) since they carry R-charge 2. After electroweak symmetry breaking the singlet and triplet VEVs effectively modify the µ u and µ d , and the modified µ i parameters are given by The v T and v S are vacuum expectation values ofT andŜ which carry R-charge zero.
There are four complex neutral scalar fields and they can mix. Assuming the vacuum expectation values are real, the real and imaginary components in four complex neutral scalar fields do not mix, and the mass-square matrix breaks into two 4 × 4 sub-matrices.
In the scalar sector all fields mix and the SM-like Higgs boson is dominantly given by the up-type field. In the pseudo-scalar sector there is no mixing between MSSM-like states and singlet-triplet states, and the 4 × 4 mass-squared matrix breaks into two 2 × 2 submatrices.
The number of neutralino degrees of freedom in MRSSM is doubled compared to MSSM as the neutralinos are Dirac-type. The number of chargino degrees of freedom in MRSSM is also doubled compared to MSSM and these charginos can be grouped to two separated chargino sectors according to their R-charge. The χ ± -charginos sector has R-charge 1 electric charge; the ρ-charginos sector has R-charge -1 electric charge. Here, we don't discuss the ρ-charginos sector in detail since it doesn't contribute to the LFV decays. More information about the ρ-charginos can be found in Ref. [39,41,43,53]. For convenience, we present the tree-level mass matrices for scalar and pseudo-scalar Higgs bosons, neutralinos, charginos and squarks of the MRSSM in Appendix A.
In MRSSM the LFV decays mainly originate from the potential misalignment in sleptons mass matrices. In the gauge eigenstate basisν iL , the sneutrino mass matrix and the diagonalization procedure are where the last two terms are newly introduced by MRSSM. The slepton mass matrix and the diagonalization procedure are where The sources of LFV are the off-diagonal entries of the 3 × 3 soft supersymmetry breaking matrices m 2 l and m 2 r in Eqs. (4,5). From Eq.(5) we can see that the left-right slepton mass mixing is absent in MRSSM, whereas the A terms are present in MSSM.
Finally, the MRSSM has been implemented in the Mathematica package SARAH [55][56][57], and we use the Feynman rules generated with SARAH in our work.

III. DECAY WIDTH AND WILSON COEFFICIENTS
Using the effective Lagrangian method, we present analytical expressions for decay width of τ → P l. At the quark level, the interaction Lagrangian for τ → P l can be written as [58] where the index β(=1, 2) denotes the generation of emitted lepton and l 1 (l 2 ) = e(µ). Since only the axial-vector current contributes to τ → P l, the coefficients in Eq.(6) do not include photonic contributions but they include Z boson and scalar ones. Then the decay width for τ → P l is given by where the averaged squared amplitude can be written as The coefficients a S,V P and b S,V P are linear combinations of the Wilson coefficients in Eq.(6) where f π is the pion decay constant. The expressions for coefficients C(P ),D d,u L (P ) are listed in TABLE.III [24]. Here, m π and m K denote the masses of the neutral pion and Kaon, and θ η denote the η − η mixing angle. In addition, D d,u R (P )=−(D d,u L (P )) * . The contributions to Wilson coefficients C I XY and B I XY can be classified into Z penguins, Higgs penguins and box diagrams, shown in FIG.1, FIG.2 and FIG.3. Photon penguins are not included since only the axial-vector current contributes to τ → P l. In the following, we will calculate the Wilson coefficients separately.
where the subscript X is defined as and so does Y . The symbol κ equals −1 if the loop lines contain χ 0c , and 1 otherwise.
The symbols M 1 , M 2 and M 3 stand for masses of particles in loop lines and the explicit expressions for FIG.1(a-d) are The symbols C 1 X and C 3 X in Eqs.(9,10) stand for the left-handed or right-handed couplings of the interaction between leptons and sleptons. The symbol C 2 X in Eq.(9) stands for the left-handed or right-handed coupling of the interaction between Z boson and neutralinos or charginos. The couplings C 1 X , C 2 X and C 3 X for FIG.1(a,b) are given by By an interchange of sum indexes (i ↔ k, j ↔ k) of the couplings C 1 X and C 3 X for FIG.1  (a) and FIG.1 (b), one can get the expressions of the couplings C 1 X and C 3 X for FIG.1 (c) and FIG.1 (d) respectively. The symbol C 2 in Eq.(10) stands for the coupling of the interaction between Z boson and two sneutrinos or two sleptons, and it is noted worthwhile that the relevant Feynman rules in MRSSM are same with those in MSSM. The couplings C 2 for FIG.1 (c,d) are given by The symbol C 4 Y in Eqs.(9,10) stands for the left-handed or right-handed coupling of the interaction between Z boson and two u quarks, for which the relevant Feynman rules in MRSSM are same with those in SM. The couplings C 4 Y are given by The symbols B 0 , C 0 , C 1 and C 00 denote the Passarino-Veltman integrals which take the form of These loop integrals are calculated by Mathematica package Package-X through a link to a fortran library Collier which is developed for the numerical evaluation of one-loop scalar and tensor integrals in perturbative relativistic quantum field theory [61]. The explicit expressions of these loop integrals are given in Refs. [62][63][64] and M S scheme is used to delete the infinite terms.
The Wilson coefficients B V XY corresponding to FIG.1 (a-d) can be formulated by replacing the couplings of u quark in Eq. (14) with the couplings of d quark in Eq. (15).

Higgs boson contribution
The Higgs penguin diagrams contributing to τ → P l at one loop level in MRSSM are presented in FIG.2. The Wilson coefficients C S XY for FIG.2 (a-d) can be expressed as The explicit expressions for M 1 , M 2 , M 3 , C 1 X and C 3 X for FIG.2 (a), (b), (c) and (d) are same with those in FIG.1 (a), (b), (c) and (d) respectively. The symbol C 2 X in Eq.(16) stands for the left-handed or right-handed coupling of the interaction between Higgs boson and two neutralinos or two charginos, and can be expressed by The relevant couplings C 2 X for FIG.2 (b) with χ 0c can be available by an interchange of the sum indexes i ↔ j in C 2 X for FIG.2 (b) with χ 0 . The symbol C 2 in Eq.(17) stands for coupling of interaction between Higgs boson and two sneutrinos or two sleptons, and can be expressed by It is noted worthwhile that, assuming both M W D and M B D are real numbers, the couplings C 2 for FIG.2 (c,d) with A 0 are zero and the relevant contribution can be neglected.
The symbol C 4 Y in Eqs. (16,17) stands for the left-handed or right-handed coupling of the interaction between Higgs boson and two u quarks, for which the relevant Feynman rules in MRSSM are same with those in MSSM. The couplings C 4 Y for FIG.2 (a-d) are given by The Wilson coefficients B S XY corresponding to FIG.2 (a-d) can be formulated by replacing the couplings of u quark in Eq. (20) with the couplings of d quark in Eq. (21).
Box diagrams contribution  3 (a,b) can be expressed as The Using the identities in Eq. (24), which are deduced from a generalized Fierz identities in chirality-diagonal and chirality-flipped cases [65], one can obtain the relations between C S(V ) The symbols C 1 X and C 4 X in Eqs. (22,23) stand for the couplings of interaction between leptons and sleptons, and the explicit expressions are same with those in FIG.1 (a) and (b) respectively. The symbol C 2 X in Eqs. (22,23) stands for the couplings of interaction between anti-u quark and squarks, and the symbol C 3 X in Eqs. (22,23) stands for the couplings of interaction between u quark and squarks. The expressions of C 2 X and C 3 X are given by The Wilson coefficients B S XY and B V XY corresponding to FIG.3 (a,b) can be formulated by replacing the couplings of u quark in Eq. (26) with the couplings of d quark in Eq. (27).
In the numerical analysis, the default values of the input parameters are set same with those in Eq. (28). The off-diagonal entries of squark mass matrices m 2 q , m 2 u , m 2 d and slepton mass matrices m 2 l , m 2 r in Eq.(28) are zero. The large value of |v T | is excluded by measurement of W boson mass because the VEV v T of the SU (2) L triplet field T 0 gives a correction to W mass through [37] Similarly to most supersymmetry models, the LFV processes originate from the offdiagonal entries of the soft breaking terms m 2 l and m 2 r in MRSSM, which are parameterized by mass insertion where I, J = 1, 2, 3. To decrease the number of free parameters involved in our calculation, we assume that the off-diagonal entries of m 2 l and m 2 r in Eq.(30) are equal, i.e., 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 which are discussed in Ref. [54]. It is noted that δ 12 has been set zero in following discussion since it has no effect on the predictions of BR(τ → P l). Current limits of LFV decays l 2 → l 1 γ are listed in   given by only the listed contribution with all others set to zero. The total prediction for BR(τ → P e(µ)) is also indicated. It shows Z penguins dominate the predictions on BR(τ → P e(µ)), and the Higgs penguins contribution and box diagrams contribution are negligible, which is different from some SUSY models (e.g., [24], where the Higgs penguins contribution is dominant and Z penguins contribution is subdominant). The predictions on BR(τ → P e, P µ) from Higgs penguins increase as tan β varies from 3 to 40 while the total predictions and the predictions from Z penguins or box diagrams take a narrow band. The total predictions on BR(τ → P e, P µ) are one order or two orders of magnitude lower than  We clearly see that both the predictions for BR(τ → P e) and BR(τ → P µ) show a weak dependence on M W D , and the predictions on BR(τ → P e, P µ) in MRSSM decrease slowly as M W D varies from 100 GeV to 1000 GeV. We are also interested to the effects from other parameters on the predictions of BR(τ → P l) in MRSSM. By scanning over these parameters, which are shown in Eq.(31), −1.5 < λ d , λ u , Λ d , Λ u < 1.5, 300 GeV < µ d , µ u , m S , m T , m A < 3000 GeV, 300 GeV < (mq) II , (mũ) II , (md) II < 3000 GeV, (31) the predictions are shown in relation to one input parameter (e.g. m T or others). The results show that varying those parameters in Eq. (31) have almost no effect on the predictions of BR(τ → P l) which take values along a narrow band.

V. CONCLUSIONS
In this work, taking account of the constraints from τ → eγ and τ → µγ on the parameter space, we analyze the LFV decays of τ → P l in the framework of the Minimal R-symmetric Supersymmetric Standard Model.
In MRSSM, the theoretical predictions on BR(τ → P e) and BR(τ → P µ) affected by the mass insertion δ 13 and δ 23 , respectively. The predictions on BR(τ → P e) would be zero if δ 13 =0 is assumed, and so are the predictions on BR(τ → P µ) if δ 23 =0 is assumed. Z penguins dominate the predictions on BR(τ → P e(µ)), and other contribution are negligible. where the submatrices (c β = cosβ, s β = sinβ) are In the weak basis (σ d , σ u , σ S , σ T ), the pseudo-scalar Higgs boson mass matrix and the diagonalization procedure are The mass eigenstates λ ± i and physical four-component Dirac charginos are The mass matrix for up squarks and down squarks, and the relevant diagonalization procedure are where