Searching for lepton flavor violating decays $\tau\rightarrow Pl$ in minimal R-symmetric supersymmetric standard model

In this section, we firstly provide a simple overview of the MRSSM in order to fix the notations used in this work. The MRSSM has the same gauge symmetry $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ as the SM and the MSSM. The spectrum of fields in the MRSSM contains the standard MSSM matter, the Higgs and gauge superfields augmented by the chiral adjoints $\hat{\cal O},\hat{T},\hat{S}$ and two $R$-Higgs iso-doublets. The superfields with R-charge in the MRSSM are given in TABLE.2.

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

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. The corresponding Dirac higgsino mass parameters are denoted as $\mu_{u}$ and $\mu_{d}$. Although R-symmetry forbids the $\mu$ terms of the MSSM, the bilinear combinations of the normal Higgs $SU(2)_{L}$ doublets $\hat{H}_{u}$ and $\hat{H}_{d}$ with the Higgs $SU(2)_{L}$ doublets $\hat{R}_{u}$ and $\hat{R}_{d}$ are allowed in Eq.(1). The parameters $\lambda_{u}$, $\lambda_{d}$, $\Lambda_{u}$ and $\Lambda_{d}$ are Yukawa-like trilinear terms involving the singlet $\hat{S}$ and the triplet $\hat{T}$.

For the phenomenological studies we take the soft-breaking scalar mass terms Die3

All the trilinear scalar couplings involving Higgs bosons to squark and slepton are forbidden in Eq.(2) because the sfermions have an R-charge and these terms are non R-invariant, and this has relaxed the flavor problem of the MSSM Kribs . The Dirac nature is a manifest feature of the MRSSM fermions. The soft-breaking Dirac mass terms of the singlet $\hat{S}$, triplet $\hat{T}$ and octet $\hat{O}$ take the form as

where $\tilde{B}$, $\tilde{W}$ and $\tilde{g}$ are the usually MSSM Weyl fermions. The 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 $\mu_{u}$ and $\mu_{d}$, and the modified $\mu_{i}$ parameters are given by

$v_{T}$ and $v_{S}$ are vacuum expectation values of $\hat{T}$ and $\hat{S}$.

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 $4\times 4$ 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 the MSSM-like states and the singlet-triplet states, and the $4\times 4$ mass-squared matrix breaks into two $2\times 2$ submatrices. The number of neutralino degrees of freedom in the MRSSM is doubled compared to the MSSM as the neutralinos are Dirac-type. 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 $\chi^{\pm}$-chargino sector has R-charge 1 electric charge; the $\rho$-chargino sector has R-charge -1 electric charge. Here, we do not discuss the $\rho$-chargino sector in detail since it does not contribute to the LFV decays. More information about the $\rho$-chargino can be found in Ref.Die3 ; Die5 ; sks1 ; KSS . 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, LFV decays mainly originate from the potential misalignment in slepton mass matrices. In the gauge eigenstate basis $\tilde{\nu}_{iL}$, the sneutrino mass matrix and the diagonalization procedure are

where the last two terms in mass matrix 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 $3\times 3$ soft supersymmetry breaking matrices $m_{l}^{2}$ and $m_{r}^{2}$ in Eqs.(4, 5). From Eq.(5) we can see that the left-right slepton mass mixing is absent in the MRSSM, whereas the $A$ terms are present in the MSSM. The relevant Feynman diagrams contributing to $\tau\rightarrow Pl$ are presented in FIG.1.

We now focus on the LFV processes $\tau\rightarrow Pl$. Using the effective Lagrangian method, we present the analytical expression for the decay width of $\tau\rightarrow Pl$. At the quark level, the interaction Lagrangian for $\tau\rightarrow Pl$ can be written as Flavor

where the index $\beta$(=$1,2$) denotes the generation of the emitted lepton and $l_{1}(l_{2})=e(\mu)$. Since only the axial-vector current contributes to $\tau\rightarrow Pl$, the coefficients in Eq.(6) do not include photon penguin contribution but they include $Z$ boson and scalar ones. The contribution to the Wilson coefficients $C^{I}_{XY}$ and $B^{I}_{XY}$ in Eq.(6) can be classified into self-energies, $Z$ penguins, Higgs penguins and box diagrams, as shown in FIG.1. Now consider the implication of virtual Higgs exchange for $\tau\rightarrow Pl$. Both the contributions from scalar and pseudo-scalar Higgs sector are considered in this work. However, all the Higgs penguin contribution is negligible since the couplings of scalar and pseudo-scalar Higgs to the light quarks are suppressed by their masses.

Then the decay width for $\tau\rightarrow Pl$ 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 $f_{\pi}$ is the pion decay constant. The expressions for coefficients $C(P)$,$D^{d,u}_{L}(P)$ are listed in TABLE.3 Arganda .

Here, $m_{\pi}$ and $m_{K}$ denote the masses of the neutral pion and Kaon, and $\theta_{\eta}$ denotes the $\eta-\eta^{\prime}$ mixing angle. In addition, $D^{d,u}_{R}(P)$ = - $(D^{d,u}_{L}(P))^{\ast}$.

Finally, the MRSSM has been implemented in the Mathematica package SARAH-4.14.3 SARAH ; SARAH1 ; SARAH2 ; Flavor . The masses of the MRSSM particles, mixing matrices and the Wilson coefficients of the corresponding operators in the effective lagrangian are computed by SPheno-4.0.4 SPheno1 ; SPheno2 modules written by SARAH.

Similarly to the most supersymmetry models, those LFV processes originate from the off-diagonal entries of the soft breaking terms $m_{l}^{2}$ and $m_{r}^{2}$ in the MRSSM, which are parametrized by the 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 $m_{l}^{2}$ and $m_{r}^{2}$ in Eq.(11) are equal, i.e., $\delta^{IJ}_{l}$ = $\delta^{IJ}_{r}$ = $\delta^{IJ}$. The experimental bounds 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 $l_{2}\rightarrow l_{1}\gamma$ and $l_{2}\rightarrow 3l_{1}$ to constrain the parameters $\delta^{IJ}$. It is noted that $\delta^{12}$ has been set zero in following discussion since it has no effect on the prediction of BR($\tau\rightarrow Pl$). Current limits on branching ratios of $l_{2}\rightarrow l_{1}\gamma$ and $l_{2}\rightarrow 3l_{1}$ are listed in TABLE.4 PDG .

Taking $\delta^{23}=0$ and the parameters in Eq.(9), we plot the predictions of BR$(\tau\rightarrow Pe)$ versus Log[$\delta^{13}$] in the left panel of FIG.2. Taking $\delta^{13}=0$ and the parameters in Eq.(9), we plot the predictions of BR$(\tau\rightarrow P\mu)$ versus Log[$\delta^{23}$] in the right panel of FIG.2. A linear relationship in logarithmic scale is displayed between BR$(\tau\rightarrow Pe(\mu))$ and the flavor violating parameter $\delta^{13}$($\delta^{23}$). The actual dependence on $\delta^{13}$ or $\delta^{23}$ 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, BR$(\tau\rightarrow\pi e)>$BR$(\tau\rightarrow\eta^{\prime}e)>$BR$(\tau\rightarrow\eta e)$ and BR$(\tau\rightarrow\pi\mu)>$BR$(\tau\rightarrow\eta^{\prime}\mu)>$BR$(\tau\rightarrow\eta\mu)$. The predictions on BR$(\tau\rightarrow\eta e(\mu))$ and BR$(\tau\rightarrow\eta^{\prime}e(\mu))$ are very close to each other. At $\delta^{13}(\delta^{23})$ = $10^{-0.25}\sim 0.56$, the prediction on BR$(\tau\rightarrow e(\mu)\gamma)$ is around $10^{-8}$ and this is very close to the current experimental bound. The prediction on BR$(\tau\rightarrow 3e(\mu))$ is around $10^{-10}$ and this is two orders of magnitude below the current experimental bound. The decay channels $\tau\rightarrow e(\mu)\gamma$ can set more strong constraint than channels $\tau\rightarrow 3e(\mu)$ on the flavor violating parameters. The predictions on BR$(\tau\rightarrow Pl)$ are far below the current experimental bounds. The prediction on BR$(\tau\rightarrow\pi e(\mu))$ is around $10^{-13}$ and this is five orders of magnitude below the current experimental bound and three orders of magnitude below the future experimental sensitivity.

Taking $\delta^{13}=0.5$, $\delta^{23}=0$ and the parameters in Eq.(9), we plot the predictions of BR$(\tau\rightarrow Pe)$ from various parts as a function of $\tan\beta$ in the left panel of FIG.3. Taking $\delta^{13}=0$, $\delta^{23}=0.5$ and the parameters in Eq.(9), we plot the theoretical predictions of BR$(\tau\rightarrow P\mu)$ from various parts as a function of $\tan\beta$ in the right panel of FIG.3. The lines corresponding to $Z$ penguin and box diagram indicate the values of BR$(\tau\rightarrow Pl)$ are given by only the listed contribution with all others set to zero. The total prediction for BR$(\tau\rightarrow Pl)$ is also indicated. We observe that $Z$ penguin dominates the prediction on BR$(\tau\rightarrow Pl)$ in a large region of the parameter space. For larger values of $\tan\beta$, the prediction from $Z$ penguin changes slowly, since it is directly proportional to $\frac{1}{1+\tan^{2}\beta}$ Fok . The contribution from box diagram is not sensitive to $\tan\beta$ and at least two orders of magnitude below $Z$ penguin. This is because of the small couplings of neutralino/chargino to quark and squark in box diagram.

As mentioned above, the contribution from Higgs penguin is negligible due to the small couplings of the scalar and pseudo-scalar Higgs to the light quarks. However, if the Higgs couplings to the strange components of the $\eta$ and $\eta^{\prime}$ mesons are large enough, which result in large $A^{0}$-$\eta$ and $A^{0}$-$\eta^{\prime}$ mixing, Higgs-mediated contribution could dominate the predictions on $\tau\rightarrow\mu\eta(\eta^{\prime})$ Arganda . Furthermore, it was pointed out in Ref.Celis1 that a one loop Higgs generated gluon operator does not suffer of the light-quark mass suppression and could give a sizeable contribution. The Wilson coefficients $C^{S}_{XY}$ and $B^{S}_{XY}$ corresponding to the Higgs penguin in Eq.(6) can also contribute to the LFV process $\mu$-e conversion. In simple formulas, the branching ratios BR($\tau\rightarrow Pl$) and conversion rate CR($\mu$-e, nucleus) are given by

Thus, the predicted CR($\mu$-e, nucleus) from Higgs penguin is much larger than BR($\tau\rightarrow Pl$) though both are negligible compared to the total contribution.

Taking $\delta^{13}=0.5$, $\delta^{23}=0$ and the parameters in Eq.(9), we plot the predictions of BR$(\tau\rightarrow Pe)$ as a function of the diagonal entries $mL$ in the left panel of FIG.4. Taking $\delta^{13}=0$, $\delta^{23}=0.5$ and parameters in Eq.(9), we plot the predictions of BR$(\tau\rightarrow P\mu)$ as a function of the diagonal entries $mL$ in the right panel of FIG.4. Here, $mL$$=\sqrt{(m^{2}_{l})_{11}}=\sqrt{(m^{2}_{l})_{22}}=\sqrt{(m^{2}_{l})_{33}}=\sqrt{(m^{2}_{r})_{11}}=\sqrt{(m^{2}_{r})_{22}}=\sqrt{(m^{2}_{r})_{33}}$. The prediction on BR$(\tau\rightarrow l\gamma)$ is at the level of $\mathcal{O}(10^{-9})$ and this is very close to the future experimental sensitivities Aushev . The prediction on BR$(\tau\rightarrow 3l)$ is at the level of $\mathcal{O}(10^{-11})$ and this is two orders of magnitude below the future experimental sensitivities Aushev . The predictions on BR$(\tau\rightarrow l\gamma)$, BR$(\tau\rightarrow 3l)$ and BR$(\tau\rightarrow Pl)$ in the MRSSM decreas as the slepton mass parameter $mL$ varies from 1 TeV to 5 TeV.

Taking $\delta^{13}=0.5$, $\delta^{23}=0$ and the parameters in Eq.(9), we plot the predictions of BR $(\tau\rightarrow Pe)$ as a function of the squark mass parameter $mQ$ in the left panel of FIG.5. Taking $\delta^{13}=0$, $\delta^{23}=0.5$ and the parameters in Eq.(9), we plot the predictions of BR $(\tau\rightarrow P\mu)$ as a function of the squark mass parameter $mQ$ in the right panel of FIG.5. Here, $mQ$=$\sqrt{(m^{2}_{\tilde{q}})_{11}}=\sqrt{(m^{2}_{\tilde{u}})_{11}}=\sqrt{(m^{2}_{\tilde{d}})_{11}}=\sqrt{(m^{2}_{\tilde{q}})_{22}}=\sqrt{(m^{2}_{\tilde{u}})_{22}}=\sqrt{(m^{2}_{\tilde{d}})_{22}}=\sqrt{(m^{2}_{\tilde{q}})_{33}}=\sqrt{(m^{2}_{\tilde{u}})_{33}}=\sqrt{(m^{2}_{\tilde{d}})_{33}}$. We clearly see that both the predictions on BR($\tau\rightarrow Pe$) and BR($\tau\rightarrow P\mu$), which increase slowly as $mQ$ varies from 1 TeV to 5 TeV, are not sensitive to $mQ$. The off-diagonal entries $\delta^{IJ}_{\tilde{q},\tilde{u},\tilde{d}}$ of the squark mass matrices $m^{2}_{\tilde{q}}$, $m^{2}_{\tilde{u}}$ and $m^{2}_{\tilde{d}}$ could give additional contributions to BR($\tau\rightarrow Pl$). Taking into account the experimental constraints on $\delta^{IJ}_{\tilde{q},\tilde{u},\tilde{d}}$ from low energy B meson physics observables, such as, BR($\bar{B}\rightarrow X_{s}\gamma$), BR($B^{0}_{s,d}\rightarrow\mu^{+}\mu^{-}$), the prediction of BR($\tau\rightarrow Pl$) takes values along a narrow band. Thus, the prediction of BR$(\tau\rightarrow Pl)$ is also not sensitive to the off-diagonal entries of the squark mass matrices.

We are also interested to the effect from other parameters on the prediction of BR($\tau\rightarrow Pl$) in the MRSSM. The predicted BR($\tau\rightarrow Pl$) decreases slowly along with the increase of the wino-triplino mass parameter $M_{D}^{W}$. However, the valid region of $M_{D}^{W}$ is constrained by the boundary conditions at the unification scale, and unphysical masses of neutral Higgs and charged Higgs are obtained when $M_{D}^{W}$ above 1 TeV. By scanning over these parameters, which are shown in Eq.(12),

the prediction is shown in relation to one input parameter (e.g. $\lambda_{d}$ or others). The parameters $\lambda_{d},\lambda_{u},\Lambda_{u}$ are constrained in a narrow region by the boundary conditions which is close to the benchmark points and $\Lambda_{d}$ can vary in the whole region. The results show that varying those parameters in Eq.(12) have very small effect on the prediction of BR($\tau\rightarrow Pl$) which takes values along a narrow band.

In this work, taking into account the constraints from LFV decays $\tau\rightarrow e(\mu)\gamma$ and $\tau\rightarrow 3e(\mu)$ on the flavor violating parameters, we analyze the LFV decays $\tau\rightarrow Pl$ in the framework of the minimal R-symmetric supersymmetric standard model.

We observe that $Z$ penguin dominates the prediction on BR$(\tau\rightarrow Pe(\mu))$, and other contributions are less dominant or negligible. As we all konw the LFV processes $l_{2}\rightarrow l_{1}\gamma$ can only obtain the dipole contribution from the gamma penguin. The LFV processes $l_{2}\rightarrow 3l_{1}$ and $\mu$-e conversion can obtain contributions from the gamma penguin (including dipole and non-dipole), $Z$ penguin, Higgs penguin and box diagram. While, for the LFV processes $\tau\rightarrow Pl$, contributions from $Z$ penguin, Higgs penguin and box diagram are included except for the gamma penguin. It is interesting to consider if it is possible to find a parameter region that LFV processes $\tau\rightarrow e(\mu)\gamma$ and $\tau$-$e(\mu)$ conversion are not excluded and which could still give large contribution to $\tau\rightarrow Pl$. The answer is negative at least in this work. The prediction of BR$(\tau\rightarrow Pl)$ is two or three orders of magnitude below BR$(\tau\rightarrow 3l)$.

In the MRSSM, the prediction on BR$(\tau\rightarrow Pe)$ and BR$(\tau\rightarrow P\mu)$ is affected by the mass insertions $\delta^{13}$ and $\delta^{23}$, respectively. The prediction on BR$(\tau\rightarrow Pe)$ would be zero if $\delta^{13}$=0 is assumed, and so are the prediction on BR$(\tau\rightarrow P\mu)$ if $\delta^{23}$=0 is assumed. Taking into account the experimental bounds on BR($\tau\rightarrow e(\mu)\gamma$) and BR($\tau\rightarrow 3e(\mu)$), the values of $\delta^{13}$ and $\delta^{23}$ are constrained around 0.5. The predictions on BR$(\tau\rightarrow Pe)$ and BR$(\tau\rightarrow P\mu)$ are found to be at the level of $\mathcal{O}(10^{-13}$-$10^{-14})$, which are five orders of magnitude below the present experimental upper limits. The processes $\tau\rightarrow\pi l$ may be the most competitive LFV semileptonic tau decay channels. The future prospects of BR$(\tau\rightarrow Pl)$ in Belle II are extrapolated at the level of $\mathcal{O}(10^{-9}$-$10^{-10})$ Altmannshofer and they are three orders of magnitude above the predictions in the MRSSM. Thus, LFV decays $\tau\rightarrow Pl$ may be out reach of the near future experiments.