Probing the R-parity violating supersymmetric effects in the exclusive $c\to d/s\ell\nu_\ell$ decays

A lot of branching ratios of the exclusive $c \to d/s\ell\nu_\ell$ ($\ell=e,\mu$) decays have been quite accurately measured by CLEO-c, BELLE, BABAR, BES(I,II,III), ALEPH and MARKIII collaborations. We probe the R-parity violating supersymmetric effects in the exclusive $c \to d/s\ell\nu_\ell$ decays. From the latest experimental measurements, we obtain new upper limits on the relevant R-parity violating coupling parameters within the decays, and many upper limits are obtained for the first time. Using the constrained new parameter spaces, we predict the R-parity violating effects on the observables, which have not been measured or have not been well measured yet. We find that the R-parity violating effects due to slepton exchange could be large on the branching ratios of $D_{d/s}\to e\nu_e$ decays and the normalized forward-backward asymmetries of $D_{u/d}\to \pi/K \ell\nu_\ell$ as well as $D_s\to K \ell\nu_\ell$ decays, and all branching ratios of the relevant semileptonic $D$ decays are sensitive to squark exchange couplings. Our results in this work could be used to probe new physics effects in the leptonic decays as well as the semileptonic decays, and will correlate with searches for direct supersymmetric signals at LHC and BESIII.

1 Introduction 2 The exclusive c → d/sℓν ℓ decays in SUSY without Rparity In the SM, the c → d/sℓν ℓ processes are mediated by a virtual W boson exchange, and the relevant four fermion effective Hamiltonian is In the most general superpotential of SUSY, the RPV superpotential is given by [34] whereL andQ are the SU (2) withλ irk ≡ n V * rn λ ink , and V rn is the SM CKM matrix element. Noted that (s)down-down-(s)neutrino vertices have the weak eigenbasis couplings λ ′ , while charged (s)lepton-(s)down-(s)up vertices have the up quark mass eigenbasis couplingsλ ′ . Very often in the literature (see e.g. [37][38][39][40][41]), one neglects the difference between λ ′ andλ ′ , based on the fact that diagonal elements of the CKM matrix dominate over nondiagonal ones.
In terms of Eq.(3), we can obtain the relevant four fermion effective Hamiltonian for thē c →d j ℓ + m ν ℓn processes with RPV couplings due to the squark and slepton exchange And the corresponding RPV feynman diagrams for thec →d k ℓ + m ν ℓn processes are displayed in Fig. 1.
Based on the effective Hamiltonian in Eq. (5), we will give the expressions of physical quantities for the RPV SUSY later in detail. In the following expressions and numerical analysis, we will keep the masses of the charged leptons, but ignore all neutrino masses.

D d/s → ℓν ℓ decays
Purely leptonic decays are the simplest and the cleanest decay modes of the pseudoscalar charged D + meson, and the decay amplitude of D + d k → ℓ + ν ℓ can be obtained in the terms of Eq.(5) After using the definitions of D meson decay constant [42] < 0|d k γ µ γ 5 c|D and we get the branching ratio for 2.2 D → P ℓν ℓ (P = π, K) decays In the terms of Eq.(5), D → P ℓ + ν ℓ decay amplitude can be written as Using the D → P transition form factors [43] with the factor c P accounts for the flavor content of particles (c P = √ 2 for π 0 , and c P = 1 for π − , K 0 , K − ) and s = q 2 (q = p D − p), the differential branching ratio for with where θ is the angle between the momentum of D meson and the charged lepton in the c.m.
system of ℓ − ν, and the kinematic factor λ P = m 4 Here, we give the definition of the normalized forward-backward (FB) asymmetry of charged lepton, which is more useful from the experimental point of view, Explicitly, for D → P ℓ + ν ℓ the normalized FB asymmetry is From Eq.(5), D → V ℓ + ν ℓ decay amplitude can be written as In terms of the D → V form factors [43] where with where The normalized FB asymmetry of D → V ℓ + ν ℓ can be written as

Numerical Results and Discussions
In this section, we summarize our numerical results and analysis of RPV couplings in the exclusivē c →d/sℓ + ν ℓ decays. When we study the effects due to SUSY without R-parity, we consider only one new coupling at one time, neglecting the interferences between different new couplings, but keeping their interferences with the SM amplitude. The input parameters are collected in the Appendix. To be conservative, the input parameters varied randomly within 1σ variance and the experimental bounds at 90% confidence level (CL) will be used to constrain parameters of the relevant new couplings.

Observable
Exp. data SM predictions SUSY w/λ *  from c → ue + e − transition [45] and |λ ′ * [36]. If neglecting the difference between λ ′ andλ ′ , our bound on λ ′ * 11iλ ′ 12i fromc →de + ν e is more than two orders weaker than one from c → ue + e − transition or D 0 −D 0 mixing. Now we will analyze the constrained RPV effects in the exclusivec →de + ν e decays. Using the constrained parameter spaces shown in Fig. 2, we can predict the constrained RPV effects on the branching ratios, the differential branching ratios and the normalized FB asymmetries of charged leptons. The numerical results for the branching ratios are listed in the last two columns of Table   1, and the constrained RPV effects of λ * i11λ ′ i21 and λ ′ * 11iλ ′ 12i in the exclusivec →de + ν e decays are displayed in Fig. 3 and Fig. 4, respectively. Comparing the RPV SUSY predictions to the SM ones or experimental bounds given in Table 1 as well as shown in Fig. 3 and Fig. 4, we give some remarks as follows.
For the slepton exchange coupling λ * i11λ ′ i21 , since its contribution to B(D + d → e + ν e ) is increased by m D /m e , as shown in Fig. 3 (a-b) u → π − e + ν e ) and B(D + d → π 0 e + ν e ) give very strong bounds on the semileptonic decay branching ratio predictions with λ * i11λ ′ i21 coupling. As for the differential branching ratios and the normalized FB asymmetries of relevant semileptonic D decays, slepton exchange RPV contributions to D 0 u → π − e + ν e , D + d → π 0 e + ν e and D + s → K 0 e + ν e (D 0 u → ρ − e + ν e , D + d → ρ 0 e + ν e and D + s → K * 0 e + ν e ) are very similar to each other. We would take D 0 u → π − e + ν e and Figure 3: The constrained effects of RPV coupling λ * i11λ ′ i21 due to the slepton exchange in the exclusivec →de + ν e decays. D 0 u → ρ − e + ν e as examples (the similar in the subsections of the exclusivec →dµ + ν µ ,se + ν e ,sµ + ν µ decays), which are shown by Fig. 3 (c,e) and Fig. 3 (d,f), respectively. We can see that present accurate experimental measurements of B(D 0 u → π − e + ν e ) and B(D + d → π 0 e + ν e ) also give very strong bounds on their differential branching ratios, nevertheless, other differential branching ratios (including dB(D + s → K 0 e + ν e )/ds) are not constrained so much by present experimental measurements given in Table 1. The RPV predictions of the four differential branching ratios of D + s → K 0 e + ν e , D 0 u → ρ − e + ν e , D + d → ρ 0 e + ν e and D + s → K * 0 e + ν e decays can not be distinguished from their SM ones at all s range. As displayed in Fig. 3 (e), the constrained slepton exchange coupling has quite large effects on the normalized FB asymmetries of D 0 u → π − e + ν e , D + d → π 0 e + ν e and D + s → K 0 e + ν e decays, but these values are very tiny.  it has obvious effects in the six semileptonic D decays. From the last column of Table 1, one can see that the experimental measurements of all relevant semileptonic D decays except D + s → K * 0 e + ν e give bounds on λ ′ * 11iλ ′ 12i . Except B(D 0 u → π − e + ν e ) and B(D + d → π 0 e + ν e ), which are strongly constrained by their experimental measurements, as shown in Fig. 4 (a-f), all other branching ratios are sensitive to both modulus and weak phase of is much less than its experimental upper limit, therefore we do not show the experimental upper limit in Fig. 4 (a-b). The constrained squark exchange contributions to the differential branching ratios and the normalized FB asymmetries of D 0 u → π − e + ν e , D + d → π 0 e + ν e and D + s → K 0 e + ν e (D 0 u → ρ − e + ν e , D + d → ρ 0 e + ν e and D + s → K * 0 e + ν e ) are very similar to each other. We would also take D 0 u → π − e + ν e and D 0 u → ρ − e + ν e as examples (the similar in the subsections of the exclusivē c →dµ + ν µ ,se + ν e ,sµ + ν µ decays), which are displayed in Fig. 4 (g-j). Fig. 4 (h-i) show us that our constrained λ ′ * 11iλ ′ 12i coupling could enlarge the allowed ranges of dB(D 0 u → ρ − e + ν e ), but it could shrink the allowed ranges ofĀ F B (D 0 u → π − e + ν e ). Noted that, if considering the further constraints from D 0 −D 0 mixing and c → ue + e − transition, i.e., |λ ′ * 12i coupling has small effects in the exclusivec →de + ν e decays.
The numerical results for the branching ratios are listed in the last two columns of Table 3. The constrained RPV effects due to the slepton exchange and squark exchange are displayed in Fig. 9 and Fig. 10, respectively. One can see that the RPV effects in the exclusivec →se + ν e decays are similar to ones in the exclusivec →de + ν e decays. Table 3: Branching ratios of the exclusivec →se + ν e decays (in units of 10 −2 ) except for B(D + s → e + ν e ) (in units of 10 −7 ). "a" denotes the experimental data and "b" denotes the corresponding experimental bounds at 90% CL.

Conclusion
In this paper, we have studied RPV effects in the 26 semileptonic and leptonic D meson decays, Considering the theoretical uncertainties and the experimental errors, we have constrained fairly parameter spaces of RPV coupling constants from the present experimental data, and many obounds are obtained for the first time. Furthermore, we have predicted the RPV effects on the branching ratios, the differential branching ratios and the normalized FB asymmetries of charged leptons, which have not been measured or have not been well measured yet.
We have found that the constrained RPV effects due to slepton exchange could be large on the branching ratios of D d/s → eν e decays and the normalized FB asymmetries of D u/d → π/Kℓν ℓ as well as D s → Kℓν ℓ decays. The RPV contributions due to squark exchange couplings could enhance the predictions of all semileptonic branching ratios, which are very sensitive to both moduli and weak phases of the relevent RPV coupling products. Such correlated signals would provide strong evidence for RPV interactions. The results in this paper could be useful for probing the RPV SUSY effects, and will correlate strongly with searches for the direct SUSY signals at future experiments.  [44] τ Du = 0.4101 ± 0.0015 ps, τ Ds = 0.500 ± 0.007 ps, τ D d = 1.040 ± 0.007 ps. [44] f D d = 0.201 ± 0.017 GeV, f Ds = 0.249 ± 0.017 GeV. [46] |V cd | = 0.22520 ± 0.0065, |V cs | = 0.97344 ± 0.00016. [44] in the whole kinematically accessible region, we use the following parametrization where F (q 2 ) can be any of the form factors f + , f 0 , A 1 , A 2 , A ′ 3 and V . For D → π(K) decays, we may use the single pole approximations for the form factor f + in the large q 2 region [43].
with f D * g D * D π = 2.7± 0.8GeV, f D * s g D * s D K = 3.1± 0.6GeV . With the above considerations, we obtain the form factors in the whole kinematically accessible region shown in numerical results are presented in Table 6. Table 6: Fit for form factors involving the D → π(K, K * , ρ) and D s → φ(K, K * ) transitions valid for general s [43].