]>NUPHB14401S05503213(18)30197410.1016/j.nuclphysb.2018.07.011High Energy Physics – PhenomenologyFig. 1The allowed spaces due to slepton exchange couplings from the latest 95% CL experimental bounds of the exclusive b→cℓ−ν¯ℓ decays. RPVS1 and RPVS2 in the plot denote the allowed parameters of the RPV couplings λi33λ˜i23′⁎ from S1 and S2 experimental bounds, respectively.Fig. 1Fig. 2The effects of the constrained slepton exchange couplings shown in Fig. 1 on the branching ratios of Bc→J/ψℓ−ν¯ℓ,ηcℓ−ν¯ℓ and Λb→Λcℓ−ν¯ℓ decays.Fig. 2Fig. 3The effects of the constrained slepton exchange couplings shown in Fig. 1 on RJ/ψ, Rηc and RΛc.Fig. 3Fig. 4The effects of the constrained slepton exchange couplings shown in Fig. 1 on the differential branching ratios and their ratios of Bc→J/ψℓ−ν¯ℓ,ηcℓ−ν¯ℓ and Λb→Λcℓ−ν¯ℓ decays.Fig. 4Fig. 5The effects of the constrained slepton exchange couplings shown in Fig. 1 on the normalized forward–backward asymmetries of Bc→J/ψℓ−ν¯ℓ,ηcℓ−ν¯ℓ and Λb→Λcℓ−ν¯ℓ decays.Fig. 5Table 1Experimental ranges and our numerical predictions for the branching ratios (in units of 10−2) and their ratios. RPVS1 and RPVS2 in the table denote the NP predictions by considering the extra RPV couplings constrained from S1 and S2 experimental bounds, respectively. The same in Figs. 2–5.Table 1ObservableExp. rangesSM predictionsRPVS1RPVS2
B(Bc→J/ψℓ′−ν¯ℓ′)⋅⋅⋅[0.86 , 1.67][0.77 , 1.64][0.77 , 1.61]
B(Bc→J/ψτ−ν¯τ)⋅⋅⋅[0.26 , 0.46][0.22 , 0.47][0.26 , 0.47]

B(Bc→ηcℓ′−ν¯ℓ′)⋅⋅⋅[0.27 , 0.88][0.25 , 0.87][0.25 , 0.83]
B(Bc→ηcτ−ν¯τ)⋅⋅⋅[0.084, 0.263][0.087, 0.478][0.138, 0.464]

B(Λb→Λcℓ′−ν¯ℓ′)⋅⋅⋅[4.75 , 6.33][4.38 , 6.22][4.38 , 6.22]
B(Λb→Λcτ−ν¯τ)⋅⋅⋅[1.60 , 2.08][1.57 , 2.97][1.77 , 2.52]

RJ/ψ[0.225, 1.195][0.271, 0.314][0.252, 0.346][0.285, 0.346]

Rηc⋅⋅⋅[0.192, 0.613][0.204, 1.300][0.274, 1.300]

RΛc⋅⋅⋅[0.322, 0.356][0.328, 0.527][0.373, 0.443]
Probing the Rparity violating supersymmetric effects in Bc→J/ψℓ−ν¯ℓ,ηcℓ−ν¯ℓ and Λb→Λcℓ−ν¯ℓ decaysJieZhuabBinWeiaJinHuanShengacRuMinWanga⁎ruminwang@sina.comYangGaoaGongRuLubaInstitute of Theoretical Physics, Xinyang Normal University, Xinyang, Henan 464000, ChinaInstitute of Theoretical PhysicsXinyang Normal UniversityXinyangHenan464000ChinabInstitute of Particle and Nuclear Physics, Henan Normal University, Xinxiang, Henan 453007, ChinaInstitute of Particle and Nuclear PhysicsHenan Normal UniversityXinxiangHenan453007ChinacInstitute of Particle Physics, Central China Normal University, Wuhan, Hubei 430079, ChinaInstitute of Particle PhysicsCentral China Normal UniversityWuhanHubei430079China⁎Corresponding author.Editor: HongJian HeAbstractMotivated by recent RD, RD⁎ and RJ/ψ anomalies in B→Dℓ−ν¯ℓ, B→D⁎ℓ−ν¯ℓ and Bc→J/ψℓ−ν¯ℓ decays, respectively, we study possible Rparity violating supersymmetric effects in Bc→J/ψℓ−ν¯ℓ,ηcℓ−ν¯ℓ and Λb→Λcℓ−ν¯ℓ decays, which are also induced by b→cℓ−ν¯ℓ at quark level. We find that (I) the constrained slepton exchange couplings λi33λ˜i23′⁎ involving in b→cτ−ν¯τ transition from relevant latest experimental data still have quite large effects on all (differential) branching ratios and the normalized forward–backward asymmetries of the exclusive semileptonic b→cτ−ν¯τ decays as well as the ratios of the (differential) branching ratios; (II) after satisfying the data of RD and RD⁎, the upper limit of RJ/ψ, Rηc and RΛc could be increased by 10%, 112% and 24%, respectively, from their upper limits of the Standard Model predictions by the λi33λ˜′⁎i23 couplings. The results in this work could be used to probe Rparity violating effects and will correlate with searches for direct supersymmetric signals at the running LHCb and the forthcoming BelleII.1IntroductionLepton flavor universality violation in the exclusive b→cℓ−ν¯ℓ and b→sℓ+ℓ− decays has attracted a lot of attention in the particle physics community and has significantly constrained many possible New Physics (NP) effects. For ratios RD(⁎)=B(B→D(⁎)τ−ν¯τ)B(B→D(⁎)ℓ′−ν¯ℓ′) with ℓ′=e or μ, the world averages of the BABAR [1,2], Belle [3–5] and LHCb [6,7] measurements are [8](1)RDExp.=0.407±0.039±0.024,RD⁎Exp.=0.304±0.013±0.007, which exceed the Standard Model (SM) predictions (RDSM=0.297±0.017 [9], RD⁎SM=0.252±0.003 [10]) by 1.9σ and 3.3σ, respectively. Nevertheless, it is worth noting that the last experimental reports from Belle [5] and LHCb [7] give RD⁎Exp.=0.270±0.035−0.025+0.028 and RD⁎Exp.=0.291±0.019±0.026±0.013, respectively, and their average (0.273±0.041) is consistent with the SM prediction within 1σ deviation. Very recently, LHCb reported a new measurement regarding b→cℓν in Bc decays [11](2)RJ/ψ=B(Bc+→J/ψτ+ντ)B(Bc+→J/ψμ+νμ)=0.71±0.17±0.18, which is about 2σ higher than its SM prediction [12,13].In the ratios RD, RD⁎ and RJ/ψ, the theoretical uncertainties, such as the relevant CKM matrix elements and form factors, are largely canceled, so any deviation from the SM prediction would clearly indicate the presence of NP. A lot of works of B→D(⁎)ℓ−ν¯ℓ decays have been studied, for examples, by using the heavy quark effective theory [14–16], the Lattice QCD techniques [17,18], the pQCD factorization approach with/without Lattice QCD input [19–21] and the relativistic quark model [22] in the framework of the SM, as well as by modelindependent approaches [23–32], charged Higgs [33–38], the W′ boson [39,40], lepton flavor violation [41–44], Rparity violation [45,46] and leptoquark [47–50] in the NP models. Otherwise, NP effects in Bc→J/ψℓ−ν¯ℓ,ηcℓ−ν¯ℓ and Λb→Λcℓ−ν¯ℓ decays have also been studied, for instance, in Refs. [13,51–54] and Refs. [34,55–61], respectively.In the supersymmetry without Rparity, the slepton exchange couplings could give large contributions to RD and RD⁎, which have been studied in Refs. [45,46]. Since Bc→J/ψℓ−ν¯ℓ,ηcℓ−ν¯ℓ and Λb→Λcℓ−ν¯ℓ decays are also induced at quark level by b→cℓ−ν¯ℓ, they involve the same set of Rparity violating (RPV) coupling constants as the B→Dℓ−ν¯ℓ and B→D⁎ℓ−ν¯ℓ decays. In this work, using the latest experimental data of all relevant exclusive b→cℓ−ν¯ℓ decays, we will analyze the constrained RPV contributions to the branching ratios and their ratios, differential branching ratios as well as normalized forward–backward asymmetries of the charged leptons in Bc→J/ψℓ−ν¯ℓ,ηcℓ−ν¯ℓ and Λb→Λcℓ−ν¯ℓ decays.The paper is organized as follows. In section 2, we briefly review the theoretical expressions of the exclusive b→cℓ−ν¯ℓ decays. In section 3, using the constrained parameter spaces from relevant experimental measurements, we make a detailed classification research on the RPV effects on the quantities which have not been measured or not been well measured yet. Our conclusions are given in section 4.2Theoretical frameworkThe general effective Hamiltonian for b→cℓm−ν¯ℓn transitions can be written as [62,63](3)Heff(b→cℓm−ν¯ℓn)=GFVcb2{[GVc¯γμb−GAc¯γμγ5b]ℓ¯m−γμ(1−γ5)ν¯ℓn+[GSc¯b−GPc¯γ5b]ℓ¯m−(1−γ5)ν¯ℓn+[G˜Vc¯γμb−G˜Ac¯γμγ5b]ℓ¯m−γμ(1+γ5)ν¯ℓn+[G˜Sc¯b−G˜Pc¯γ5b]ℓ¯m−(1+γ5)ν¯ℓn}+h.c. In the SM, GV=GA=1 and all others are zero. If considering both the SM and the RPV contributions, we have [64](4)GV=GA=1−2GFVcb∑iλn3i′λ˜m2i′8md˜iR2,(5)GS=GP=2GFVcb∑iλinmλ˜i23′4mℓ˜iL2, and all others are zero.From general effective Hamiltonian given in Eq. (3), we obtain the differential branching ratios. The detail expressions can be found in A.1 and A.2 of Appendix. We only give the final ones in this subsection. For Bc→ηcℓ−ν¯ℓ and B→Dℓ−ν¯ℓ decays,(6)dB(Bq→Pℓ−ν¯ℓ)dq2=GF2Vcb2τBqp→Pq296π3mBq2(1−mℓ2q2)2{H02(GV2+G˜V2)(1+mℓ22q2)+3mℓ22q2[HtGV+q2mℓHSGS2+HtG˜V+q2mℓHSG˜S2]}, with(7)H0=2mBqp→Pq2F+(q2),Ht=mBq2−mP2q2F0(q2),HS=mBq2−mP2mb−mcF0(q2).For Bc→J/ψℓ−ν¯ℓ and B→D⁎ℓ−ν¯ℓ decays,(8)dB(Bq→Vℓ−ν¯ℓ)dq2=GF2Vcb2τBqp→Vq296π3mBq2(1−mℓ2q2)2{AAV2+mℓ22q2(AAV2+3AtP2)+A˜AV2+mℓ22q2(A˜AV2+3A˜tP2)}, where(9)AAV2=A02GA2+A‖2GA2+A⊥2GV2,A˜AV2=A02G˜A2+A‖2G˜A2+A⊥2G˜V2,AtP2=A0GA+q2mℓAPGP2,A˜tP2=A0G˜A+q2mℓAPG˜P2, with(10)A0=12mVq2[(mBq2−mV2−q2)(mBq+mV)A1(q2)−4mBq2p→V2mBq+mVA2(q2)],A‖=2(mBq+mV)A1(q2),A⊥=−4mBqV(q2)p→V2(mBq+mV),At=2mBqp→VA0(q2),AP=−2mBqp→VA0(q2)mb+mc.For baryonic Λb→Λcℓ−ν¯ℓ decays,(11)dB(Λb→Λcℓ−ν¯ℓ)dq2=GF2Vcb2τΛbp→Λcq2192π3mΛb2(1−mℓ2q2)2[B1+mℓ22q2B2+32B3+3mℓq2B4], where(12)B1=H1202+H−1202+H12122+H−12−12,B2=H1202+H−1202+H1212+H−12−12+3(H12t2+H−12t2),B3=H120SP2+H−120SP2,B4=Re[H12tH120SP⁎+H−12tH−120SP⁎], with(13)Hλ2λ1≡Hλ2λ1V−Hλ2λ1A,H120V=GVQ−q2[(mΛb+mΛc)f1q2−q2f2(q2)],H120A=GAQ+q2[(mΛb−mΛc)g1q2+q2g2(q2)],H121V=GV2Q−[−f1q2+(mΛb+mΛc)f2(q2)],H121A=GA2Q+[−g1q2−(mΛb−mΛc)g2(q2)],H12tV=GVQ+q2[(mΛb−mΛc)f1q2+q2f3(q2)],H12tA=GAQ−q2[(mΛb+mΛc)g1q2−q2g3(q2)],(14)H120SP≡H120S−H120P,H120S=GSQ+mb−mc[(mΛb−mΛc)f1(q2)+q2f3(q2)],H120P=GPQ−mb+mc[(mΛb+mΛc)g1(q2)−q2g3(q2)], where Q±=(mΛb±mΛc)2−q2. Either from parity or from explicit calculation, we have the relations H−λ2−λ1V=Hλ2λ1V, H−λ2−λ1A=−Hλ2λ1A, Hλ2λ1S=H−λ2−λ1S and Hλ2λ1P=−H−λ2−λ1P.In order to further study the RPV effects, we need calculate other two important physical quantities in M1→M2ℓ−ν¯ℓ decays to reduce the error. The ratio of differential branching ratio may be written as(15)dRM2dq2=dΓ(M1→M2τ−ν¯τ)/dsdΓ(M1→M2ℓ′−ν¯ℓ′)/ds. Noted that RM2 is obtained by separately integrating the numerators and denominators of above dRM2/dq2. The normalized forward–backward asymmetry is defined as(16)AFBM1→M2ℓ−ν¯ℓ(q2)=∫−10dcosθℓ[d2Γ(M1→M2ℓ−ν¯ℓ)dq2dcosθℓ]−∫01dcosθℓ[d2Γ(M1→M2ℓ−ν¯ℓ)dq2dcosθℓ]dΓ(M1→M2ℓ−ν¯ℓ)dq2.3Numerical results and discussionsThe main theoretical input parameters are the transition form factors, the CKM matrix element Vcb, the masses, the mean lives, etc. Relevant transition form factors are taken from Refs. [12,15,16,58], the CKM matrix element is taken from the UTfit Collaboration [65], and others are gotten from PDG [66]. The 95% confidence level (CL) theoretical uncertainties of the input parameters are considered in our results.In our calculation, we consider only one NP coupling at one time and keep its interference with the SM amplitude to study the RPV effects. Due to the strong helicity suppression, the squark exchange couplings have no very obvious effects on the differential branching ratios and the normalized FB asymmetries of the semileptonic exclusive b→cℓ−ν¯ℓ decays. So we will only focus on the slepton exchange couplings in our following discussions. We assume the masses of the corresponding slepton are 500 GeV, for other values of the slepton masses, the bounds on the couplings in this paper can be easily obtained by scaling them by factor of f˜2≡(mℓ˜500GeV)2.The following experimental constraints at 95% CL will be considered in our analysis.(17)B(Bd→Dd⁎ℓ′−ν¯ℓ′)=(4.93±0.11)×10−2,B(Bd→Dd⁎τ−ν¯τ)=(1.67±0.13)×10−2,B(Bd→Ddℓ′−ν¯ℓ′)=(2.19±0.12)×10−2,B(Bd→Ddτ−ν¯τ)=(1.03±0.22)×10−2,B(Bu→D⁎0ℓ′−ν¯ℓ′)=(5.69±0.19)×10−2,B(Bu→D⁎0τ−ν¯τ)=(1.88±0.20)×10−2,B(Bu→Dℓ′−ν¯ℓ′)=(2.27±0.11)×10−2,B(Bu→Dτ−ν¯τ)=(7.7±2.5)×10−3,RD=0.407±0.046,RD⁎=0.304±0.015,RJ/ψ=0.71±0.25. Since the experimental measurements of B(B→D⁎τ−ν¯τ) and R(D⁎) obviously deviate from their SM predictions, two schemes of 95% CL experimental bounds will be used in this work. S1:All relevant experimental bounds except for B(B→D⁎τ−ν¯τ) and R(D⁎) at 95% CL.S2:All relevant experimental bounds except for B(B→D⁎τ−ν¯τ) at 95% CL.Slepton exchange couplings λi11λ˜i23′⁎, λi22λ˜i23′⁎ and λi33λ˜i23′⁎ contribute to the exclusive b→ce−ν¯e, b→cμ−ν¯μ and b→cτ−ν¯τ decays, respectively. For λi11λ˜i23′⁎ and λi22λ˜′⁎i23, which contribute to both b→cℓ′−ν¯ℓ′ and b→sℓ′+ℓ′− transitions, the stronger constraints come from the exclusive b→sℓ′+ℓ′− decays (λi11λ˜i23′⁎<5.75×10−4, λi22λ˜i23′⁎<1.63×10−5) [67,68], which will be used in our numerical results. Fig. 1 shows the allowed coupling spaces of λi33λ˜i23′⁎ from the latest 95% CL experimental measurements of the exclusive b→cτ−ν¯τ in the cases of S1 and S2. Fig. 1 shows us that both moduli and RPV weak phases of λi33λ˜i23′⁎ are constrained in the cases of both S1 and S2. In S1 case, we obtain λi33λ˜i23′⁎≤0.67, and the slight difference between this constrained space within S1 case and one in Fig. 3 of Ref. [46] comes from the updated input parameters and experimental measurements. If considering the RD⁎ bound, i.e., in the cases of S2, very strong bounds on λi33λ˜i23′⁎ are obtained, 0.11≤λi33λ˜i23′⁎≤0.30 and ϕRPV≤82∘.Noted that the moduli of the RPV coupling productions, λi33λ˜i23′⁎, were only obtained from the exclusive b→cℓν in this work and Ref. [46]. If neglecting the slight difference between λ′ and λ˜′, these upper limits also could be obtained from the products on previous individual relevant couplings. The previous strongest (weakest) bounds are λ133λ123′⁎≤0.13(4.1) and λ233λ223′⁎≤0.16(3.3) with 500 GeV sfermion masses from the charged currents and neutral currents [69,70]. If considering the strongest bounds in the charged currents and neutral currents, only very narrow spaces are left in S2 case. In addition, very strong bounds on λi33 had been obtained from neutrino masses, λi33<10−3 with 500 GeV sfermion masses [69]. So combining their constraints from neutrino masses and the individual bounds from the charged currents and neutral currents, the bounds on λi33λi23′⁎ will be very strong. However, we note that the constraints on λi33 from neutrino masses would depend on the explicit neutrino masses models with trilinear couplings only, bilinear couplings only, or both [69]. In the following analysis, we will use our bounds on λi33λ˜i23′⁎, and will give the discussion for previous bounds.Now we discuss the constrained slepton exchange effects in Bc→J/ψℓ−ν¯ℓ,ηcℓ−ν¯ℓ and Λb→Λcℓ−ν¯ℓ decays. Our numerical predictions for the branching ratios and their ratios are summarized in last columns of Table 1. We also show their sensitivities to the moduli and weak phases of the slepton exchange couplings λi33λ˜i23′⁎ in Figs. 2–3. For convenient analysis and comparison, we also give all SM predictions of Bc→J/ψℓ−ν¯ℓ,ηcℓ−ν¯ℓ and Λb→Λcℓ−ν¯ℓ decays. We have the following remarks for the branching ratios and their ratios:•Branching ratios with ℓ=ℓ′: The constrained slepton couplings have but not large effects on B(Bc→J/ψℓ′−ν¯ℓ′,ηcℓ′−ν¯ℓ′) and B(Λb→Λcℓ′−ν¯ℓ′). And these branching ratios are not very sensitive to relevant slepton exchange couplings, so we will not display their sensitivities to RPV couplings as similar as Fig. 2.•Branching ratios with ℓ=τ: As displayed in Fig. 2, the constrained slepton couplings have very obvious effects on all branching ratios with ℓ=τ, and they are very sensitive to both moduli and weak phases of λi33λ˜i23′⁎. If also considering the experimental bounds of RD⁎, the lower limits of B(Bc→J/ψτ−ν¯τ,ηcτ−ν¯τ) as well as both upper and lower limits of B(Λb→Λcτ−ν¯τ) are further constrained.•Ratios of the branching ratios: As shown in Fig. 3, the ratios of the branching ratios are also very sensitive to λi33λ˜i23′⁎ couplings, the further experimental constraints of RD⁎ give obvious lower limits of RJ/ψ and RΛc. Present experimental measurement of RJ/ψ with the large uncertainty could not give any further constraint on the slepton exchange couplings. The upper limit of RJ/ψ, Rηc and RΛc could be increased by 10%, 112% and 24%, respectively, from their upper limits of the SM predictions. The upper limit of RPV prediction of RJ/ψ is about half of the central value of the experimental measurement, but is within 1.5σ. Noted that, combining the strongest bounds from the charged currents and neutral currents, the constrained RPV couplings still have large effects on the branching ratios with ℓ=τ and the ratios of the branching ratios. But if considering the bounds on λi33 from neutrino masses, the constrained RPV couplings have no obvious effects on them.Now we discuss the constrained slepton exchange coupling effects on the differential branching ratios and their ratios of Bc→J/ψℓ−ν¯ℓ,ηcℓ−ν¯ℓ and Λb→Λcℓ−ν¯ℓ decays, which are shown in Fig. 4. From the first and last columns of Fig. 4, one can see that the constrained slepton exchange couplings have very large effects on all differential branching ratios with ℓ=τ and all ratios of the differential branching ratios. In S2 case, the 95% CL experimental measurement of RD⁎ given obviously further constraints on the lower limits of dB(Bc→J/ψτντ)/dq2, dB(Bc→ηcτντ)/dq2, dRJ/ψ/dq2, dRηc/dq2, dRΛc/dq2 as well as both upper and lower limits of dB(Λb→Λcτντ)/dq2. From the second column of Fig. 4, one can see that the constrained slepton exchange couplings still have some effects on dB(Bc→J/ψℓ′νℓ′)/dq2, dB(Bc→ηcℓ′νℓ′)/dq2 and dB(Λb→Λcℓ′νℓ′)/dq2. In addition, RD⁎ could not give obviously further constraints on all differential branching ratios with ℓ=ℓ′.Fig. 5 displays the constrained slepton exchange coupling effects on the normalized forward–backward asymmetries of Bc→J/ψℓ−ν¯ℓ,ηcℓ−ν¯ℓ and Λb→Λcℓ−ν¯ℓ decays. Since the SM and RPV predictions of the normalized forward–backward asymmetries in cases of ℓ=μ and ℓ=e are quite different, we show them all in Fig. 5. For ℓ=τ case, as shown in the first column of Fig. 5, the significant effects on all normalized forward–backward asymmetries are allowed in case of S1, the experimental measurement of RD⁎ could give very strongly further bounds on these normalized forward–backward asymmetries. So the measurement of these normalized forward–backward asymmetries could test our RPV predictions and further shrink or reveal the parameter spaces of the slepton exchange couplings. In addition, as shown in Fig. 5 (b3), the constrained slepton exchange couplings still provide quite obvious effects on AFBBc→ηce−ν¯e(q2), its sign could be changed, nevertheless, this quantity is tiny.4ConclusionMotivated by RD, RD⁎ and RJ/ψ anomalies reported by LHCb, BABAR and Belle Collaborations, we have studied RPV supersymmetric effects in Bc→J/ψℓ−ν¯ℓ,ηcℓ−ν¯ℓ and Λb→Λcℓ−ν¯ℓ decays, which are also induced by b→cℓ−ν¯ℓ at quark level. Since the squark exchange couplings have tiny effects in these decays, we have only focused on the slepton exchange couplings in this work.The slepton exchange couplings λi33λ˜′⁎i23 involve in the exclusive b→cτ−ν¯τ decays. The latest relevant experimental measurements at 95% CL give obvious bounds on the both moduli and weak phases of λi33λ˜i23′⁎, and these couplings could explain the recent RD, RD⁎ and RJ/ψ anomalies at the same time. We have found that, if considering all relevant 95% CL experimental bounds except for B(Bd→Dd⁎τ−ν¯τ) and R(D⁎), which are obviously deviate from their SM predictions, the constrained slepton couplings have great effects on all (differential) branching ratios with ℓ=τ and the normalized forward–backward asymmetries with ℓ=τ as well as the ratios of the (differential) branching ratios. And the most of branching ratios with ℓ=τ and ratios of the branching ratios are very sensitive to the both moduli and weak phases of λi33λ˜i23′⁎. The upper limits of RJ/ψ, Rηc and RΛc could be increased by 10%, 112% and 24% from their upper limits of the SM predictions, respectively. The upper limit of RJ/ψ RPV prediction is 1.5σ away from its experimental measurement. If also considering the strongest bounds from the charged currents and neutral currents, the constrained RPV couplings still have large effects on the branching ratios with ℓ=τ and the ratios of the branching ratios.The slepton exchange couplings λi11λ˜′⁎i23 and λi22λ˜i23′⁎ involve in the exclusive b→ce−ν¯e,se+e− and b→cμ−ν¯μ,sμ+μ− decays, respectively. The constrained couplings of λi11λ˜i23′⁎ and λi22λ˜i23′⁎ from the exclusive b→se+e−,sμ+μ− decays have quite small effects on the branching ratios and their ratios of the exclusive semileptonic b→ce−ν¯e and b→cμ−ν¯μ decays, nevertheless, the constrained λi11λ˜′⁎i23 couplings still have obviously effects on the normalized forward–backward asymmetries of Bc→ηce−ν¯e, but AFBBc→ηce−ν¯e(q2) is tiny.The large amount of data is expected in the near future from LHCb and BELLE II, and the precise measurements of the ratios of the branching ratios and the normalized forward–backward asymmetries of the exclusive semileptonic b→cτντ decays would enable us to test our RPV predictions and further shrink or reveal the parameter spaces of the slepton exchange couplings.AcknowledgementsThis work is supported by the National Natural Science Foundation of China under Contract No. 11047145, Nanhu Scholars Program and the High Performance Computing Lab of Xinyang Normal University.Appendix AA.1Formulae of the Bq→Mℓ−ν¯ℓ decaysThe hadronic matrix elements for Bq→P/V transition can be parameterized by the form factors as(18)<P(p′)q′¯γμbBq(p)>=F+(q2)[(p+p′)μ−mBq2−mP2q2qμ]+F0(q2)mBq2−mP2q2qμ,<V(p′,ϵ⁎)q′¯γμbBq(p)>=2iV(q2)mBq+mVεμνρσϵ⁎νp′ρpσ,<V(p′,ϵ⁎)q′¯γμγ5bBq(p)>=2mVA0(q2)ϵ⁎⋅qq2qμ+(mBq+mV)A1(q2)[ϵμ⁎−ϵ⁎⋅qq2qμ]−A2(q2)ϵ⁎⋅q(mBq+mV)[(p+p′)μ−mBq2−mV2q2qμ], where q=p−p′ is the momentum transfer, F+,F0 and V,A0,A1,A2 are the form factors of Bq→P and Bq→V transitions, respectively. Noted that, in our numerical results, we take the B→D/D⁎ form factors from Refs. [15,16] and the Bc→ηc,J/ψ form factors from Refs. [12].The double differential branching ratios of Bq→Pℓ−ν¯ℓ decays can be represented as(19)dB(Bq→Pℓ−ν¯ℓ)dq2dcosθℓ=GF2Vcb2τBqp→Pq2128π3mBq2(1−mℓ2q2)2{H02sin2θℓ(GV2+G˜V2)+mℓ2q2H0GVcosθl−(HtGV+q2mℓHSGS)2+mℓ2q2H0G˜Vcosθl−(HtG˜V+q2mℓHSG˜S)2}, where p→M≡λ(mBq2,mM2,q2)/2mBq with λ(a,b,c)≡a2+b2+c2−2(ab+bc+ca).The double differential branching ratios of Bq→Vℓ−ν¯ℓ decays are(20)dB(Bq→Vℓ−ν¯ℓ)dq2dcosθl=GF2Vcb2τBqp→Vq2256π3mBq2(1−mℓ2q2)2{2A02(GA2+G˜A2)sin2θℓ+(1+cos2θℓ)[A‖2(GA2+G˜A2)+A⊥2(GV2+G˜V2)]−4cosθℓRe[A‖A⊥(GAGV⁎−G˜AG˜V⁎)]+mℓ2q2sin2θℓ[A‖2(GA2+G˜A2)+A⊥2(GV2+G˜V2)]+2mℓ2q2A0GAcosθl−(AtGA+q2mlAPGP)2+2mℓ2q2A0G˜Acosθl−(AtG˜A+q2mlAPG˜P)2}.A.2Formulae of the Λb→Λcℓ−ν¯ℓ decaysThe hadronic matrix elements for Λb→Λcℓ−ν¯ℓ transition can be parameterized as [60]<Λc(p2,λ2)c¯γμbΛb(p1,λ1)>=u¯2(p2,λ2)[f1(q2)γμ+if2(q2)σμνqν+f3(q2)qμ]u1(p1,λ1),<Λc(p2,λ2)c¯γμγ5bΛb(p1,λ1)>=u¯2(p2,λ2)[g1(q2)γμ+ig2(q2)σμνqν+g3(q2)qμ]γ5u1(p1,λ1),<Λc(p2,λ2)c¯bΛb(p1,λ1)>=u¯2(p2,λ2)[f1(q2)q̸mb−mc+f3(q2)q2mb−mc]u1(p1,λ1),<Λc(p2,λ2)c¯γ5bΛb(p1,λ1)>=u¯2(p2,λ2)[−g1(q2)q̸mb+mc−g3(q2)q2mb+mc]γ5u1(p1,λ1), where q=(p1−p2), σμν=i[γμ,γν]/2, λi is the helicity of baryons, and fi(q2), gi(q2) are Λb→Λc form factors. 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