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In the recent years, several interesting anomalies emerged in the experimental data of semi-leptonic B-meson decays. The ratios
RD(∗)≡B(B→D(∗)τˉν)/ B(B→D(∗)ℓˉν) withℓ=e,μ , obtained by latest averages of the measurements by BaBar [1, 2], Belle [3–6] and LHCb Collaboration [7–9], yield [10]RexpD=0.407±0.039(stat.)±0.024(syst.),RexpD∗=0.306±0.013(stat.)±0.007(syst.).
(1) In comparison to the branching fractions, these ratios have the advantage that, apart from the significant reduction of the experimental systematic uncertainties, the CKM matrix element
Vcb cancels out, and the sensitivity toB→D(∗) transition form factors becomes much weaker. The SM predictions read [10]RSMD=0.299±0.003,RSMD∗=0.258±0.005,
(2) which are obtained from the arithmetic averages of the most recent calculations performed by several groups [11-14]. The SM predictions for
RD andRD∗ have values below the experimental measurements by2.3σ and3.0σ , respectively. Taking into account the measurement correlation of −0.203 betweenRD andRD∗ , the combined experimental results exhibit about3.78σ deviation from the SM predictions [10]. For theBc→J/ψτˉν decay, which is mediated by the same quark-level process asB→D(∗)τˉν , the recently measured ratioRexpJ/ψ=0.71±0.17(stat.)± 0.18(syst.) at the LHCb [15] lies within about2σ above the SM predictionRSMJ/ψ=0.248±0.006 [16]. In addition, the LHCb measurements of the ratiosRK(∗)≡B(B→ K(∗)μ+μ−)/B(B→K(∗)e+e−) ,RexpK=0.745+0.090−0.074±0.036 forq2∈[1.0,6.0]GeV2 [17] andRexpK∗=0.69+0.11−0.07±0.05 forq2∈[1.1,6.0]GeV2 [18], are found to be about2.6σ and2.5σ lower than the SM expectation,RSMK(∗)≃1 [19, 20], respectively. These measurements, referred to as theRD(∗) ,RJ/ψ , andRK(∗) anomalies, may provide hints of the Lepton Flavor University (LFU) violation and have motivated numerous studies of new-physics (NP) both in the effective field theory (EFT) approach [21–34] and in specific NP models [35–60]. We refer to Refs. [61, 62] for recent reviews.The first measurement on the
D∗ longitudinal polarization fraction in theB→D∗τˉν decay has recently been reported by the Belle Collaboration [63, 64]PD∗L=0.60±0.08(stat.)±0.04(syst.),
which is consistent with the SM prediction of
PD∗L=0.46± 0.04 [65] at1.5σ . Previously, the Belle Collaboration also performed measurements onτ polarization in theB→D∗τˉν decay and obtained the resultPτL=−0.38± 0.51(stat.)+0.21−0.16(syst.) [5, 6]. Angular distributions can provide valuable information about the spin structure of the interaction inB→D(∗)τˉν decays, and they are good observables for the testing of various NP explanations [66–70]. Measurements of angular distributions are expected to significantly improve in the future. For example, Belle II with50ab−1 data can measurePτL with a precision of±0.07 [71].In this work, motivated by these recent experimental progresses, we study the
RD(∗) anomalies in the supersymmetry (SUSY) with R-parity violation (RPV). In this scenario, the down-type squarks interact with quarks and leptons via RPV couplings. Therefore, they contribute to theb→cτˉν transition at the tree level and could explain the currentRD(∗) anomalies [72–74]. BesidesB→D(∗)τˉν , we will also study theBc→J/ψτˉν ,Bc→ηcτˉν , andΛb→Λcτˉν decay. All of them depict theb→cτˉν transition at the quark level, whereas the latter two decays have not been measured yet. Using the latest experimental data of various low-energy flavor processes, we derive the constraints of the RPV couplings. Subsequently, predictions in the RPV SUSY are made for the fiveb→cτˉν decays, focusing on theq2 distributions of the branching fractions, LFU ratios, and various angular observables. We have also taken into account recent developments regarding the form factors [11, 14, 16, 75, 76]. Implications for future research at the high-luminosity LHC (HL-LHC) and SuperKEKB are briefly discussed.This paper is organized as follows: in Section 2, we briefly review the SUSY with RPV interactions. In Section 3, we recapitulate the theoretical formulae for various flavor processes, and discuss the SUSY effects. In Section 4, detailed numerical results and discussions are presented. We present the conclusions in Section 5. The relevant form factors are recapitulated in Appendix A.
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The most general re-normalizable RPV terms in the superpotential are given by [77, 78]
WRPV=μiLiHu+12λijkLiLjEck+λ′ijkLiQjDck+12λ′′ijkUciDcjDck,
(3) where L and Q denote the
SU(2) doublet lepton and quark superfields, respectively. E and U (D) depict the singlet lepton and quark superfields, respectively. i, j and k indicate generation indices. To ensure the proton stability, we assume the couplingsλ′′ijk are zero. In semi-leptonic B meson decays, contribution from theλ term occurs through the exchange of sleptons, and it is much more suppressed than the one from theλ′ term, which occurs through the exchange of right-handed down-type squarks [72]. Therefore, we only consider theλ′ijkLiQjDck term in this work. For the SUSY scenario with theλ term, studies on theRD(∗) anomalies with slepton exchanges can be found in Refs. [79, 80].The interaction with
λ′ijk couplings can be expanded in terms of fermions and sfermions as [72]ΔLRPV=−λ′ijk[˜νiLˉdkRdjL+˜djLˉdkRνiL+˜dk∗RˉνciRdjL−Vjl(˜ℓiLˉdkRulL+˜ulLˉdkRℓiL+˜dk∗RˉℓciRulL)]+h.c.,
(4) where
Vij denotes the CKM matrix element. Here, all the SM fermionsdL,R ,ℓL,R , andνL are in their mass eigenstate. Since we neglect the tiny neutrino masses, the PMNS matrix is not needed for the lepton sector. For the sfermions, we assume that they are in the mass eigenstate. We refer to Ref. [77] for more details about the choice of basis. Finally, we adopt the assumption in Ref. [74] stating that only the third family is effectively supersymmetrized. This case is equivalent to the one where the first two generations are decoupled from the low-energy spectrum, as in Refs. [81, 82]. For the studies including the first two generation sfermions, we refer to Ref. [73], where both theRD(∗) andRK(∗) anomalies are discussed.The down-type squarks and the scalar leptoquark (LQ) discussed in Ref. [83] have similar interactions with the SM fermions. However, in the most general case, the LQ can couple to the right-handed
SU(2)L singlets, which is forbidden in the RPV SUSY. Such right-handed couplings are important to explain the(g−2)μ anomaly in the LQ scenario [83]. Moreover, these couplings can also affect semi-leptonic B decays. In particular, their contributions to theB→D(∗)τˉν decays are found to be small after considering other flavor constraints [52]. -
In this section, we introduce the theoretical framework of the relevant flavor processes and discuss the RPV SUSY effects in these processes.
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With the RPV SUSY contributions, the effective Hamiltonian responsible for
b→c(u)τˉντ transitions is given by [72]Heff=4GF√2∑i=u,cVib(1+CNPL,i)(ˉuiγμPLb)(ˉτγμPLντ),
(5) where tree-level sbottom exchange yields
CNPL,i=v24m2˜bRλ′333∑3j=1λ′∗3j3(VijVi3),
(6) with the Higgs vev v = 246 GeV. This Wilson coefficient is at the matching scale
μNP∼m˜bR . However, since the corresponding current is conserved, we can obtain the low-energy Wilson coefficient without considering the renormalization group evolution (RGE) effects, i.e.,CNPL,i(μb)=CNPL,i(μNP) .For
b→cℓˉν transitions, we consider five processes, includingB→D(∗)ℓˉν [84-86],Bc→ηcℓˉν [16, 87],Bc→J/ψℓˉν [88–96], andΛb→Λcℓˉν [97-100] decays. All these decays can be uniformly denoted asM(pM,λM)→N(pN,λN)+ℓ−(pℓ,λℓ)+ˉνℓ(pˉνℓ),
(7) where
(M,N)=(B,D),(Bc,ηc),(B,D∗),(Bc,J/ψ) , and(Λb,Λc) , and(ℓ,ˉν)=(e,ˉνe),(μ,ˉνμ) , and(τ,ˉντ) . For each particle i in the above decay, its momentum and helicity are denoted aspi andλi , respectively. In particular, the helicity of pseudoscalar meson is zero, e.g.,λD=0 . After summation of the helicity of parent hadron M, the differential decay width for this process can be written as [67, 101]dΓλN,λℓ(M→Nℓ−ˉνℓ)=11+2|λM|∑λM|MλMλN,λℓ|2√Q+Q−512π3m3M×√1−m2ℓq2dq2dcosθℓ,
(8) where
q=pM−pN ,m±=mM±mN , andQ±=m2±−q2 . The angleθℓ∈[0,π] denotes the angle between the three-momentum ofℓ and that of N in theℓ -ˉν center-of-mass frame. The following observables can be derived with the differential decay width:• The decay width and branching ratio
dBdq2=1ΓMdΓdq2=1ΓM∑λN,λℓdΓλN,λℓdq2,
(9) where
ΓM is the total width of the hadron M.• The LFU ratio
RN(q2)=dΓ(M→Nτˉντ)/dq2dΓ(M→Nℓˉνℓ)/dq2,
(10) where
dΓ(M→Nℓˉνℓ)/dq2 in the denominator denotes the average of different decay widths of the electronic and muonic modes.• The lepton forward-backward asymmetry
AFB(q2)=∫10dcosθℓ(d2Γ/dq2dcosθℓ)−∫0−1dcosθℓ(d2Γ/dq2dcosθℓ)dΓ/dq2.
(11) • The polarization fractions
PτL(q2)=dΓλτ=+1/2/dq2−dΓλτ=−1/2/dq2dΓ/dq2,PNL(q2)=dΓλN=+1/2/dq2−dΓλN=−1/2/dq2dΓ/dq2,(for N=Λc)PNL(q2)=dΓλN=0/dq2dΓ/dq2,(for N=D∗,J/ψ)
(12) Explicit expressions of the helicity amplitudes
MλMλN,λℓ≡⟨Nℓˉνℓ|Heff|M⟩ and all the above observables can be found in Ref. [102] forB→D(∗)τˉν decays, and Ref. [76] for theΛb→Λcτˉν decay. The expressions forBc→ηcτˉν andBc→J/ψτˉν are analogical to the ones forB→Dτˉν andB→D∗τˉν , respectively. Since these angular observables are ratios of decay widths, they are largely free of hadronic uncertainties, and thus provide excellent tests of lepton flavor universality. The RPV SUSY effects generate the operator with the same chirality structure as in the SM, as shown in Eq. (5). Derivation of the following relation in all theb→cτˉν decays is straightforward:RNRSMN=|1+CNPL,2|2,
(13) for
N=D(∗),ηc,J/ψ , andΛc . Here, vanishing contributions to the electronic and muonic channels are assumed.The hadronic
M→N transition form factors are important inputs to calculate the observables introduced above. In recent years, notable progress has been achieved in this field [11-14, 75, 76, 87, 97, 103–110]. ForB→D(∗) transitions, it was already emphasized that the Caprini-Lellouch-Neubert (CLN) parameterization [111] does not account for uncertainties in the values of the subleading Isgur-Wise functions at zero recoil obtained with QCD sum rules [112–114], where the number of parameters is minimal [13]. In this work, we don’t use such simplified parameterization, but adopt the conservative approach in Refs. [11, 14], based on the Boyd-Grinstein-Lebed (BGL) parameterization [115]. Furthermore, we use theBc→ηc,J/ψ transition form factors obtained in the covariant light-front approach [16]. For theΛb→Λc transition form factor, we adopt the recent lattice QCD results from Refs. [75, 76]. Explicit expressions of all the form factors used in our work are recapitulated in Appendix A.For
b→uτˉν transitions, we considerB→τˉν ,B→πτˉν andB→ρτˉν decays. Similar to Eq. (13), we haveB(B→τˉν)B(B→τˉν)SM=B(B→πτˉν)B(B→πτˉν)SM=B(B→ρτˉν)B(B→ρτˉν)SM=|1+CNPL,1|2.
(14) The SUSY contributions to both
b→uτˉν andb→cτˉν transitions depend on the same set of parameters,λ′313 ,λ′323 , andλ′333 . Therefore, the ratiosRD∗ are related to theB→τˉν decay. -
The Flavor-Changing Neutral Current (FCNC) decays
B+→K+νˉν andB+→π+νˉν are induced by theb→sνˉν andb→dνˉν transitions, respectively. In the SM, they are forbidden at the tree level and highly suppressed at the one-loop level due to the GIM mechanism. In the RPV SUSY, the sbottoms can contribute to these decays at the tree level, which results in strong constraints on the RPV couplings. Similar to theb→c(u)τˉν transitions, the RPV interactions do not generate new operators beyond the ones presented in the SM. Therefore, we have [73, 74]B(B+→K+νˉν)B(B+→K+νˉν)SM=23+13|1−v22m2˜bRπs2Wαemλ′333λ′∗323VtbV∗ts1Xt|2,B(B+→π+νˉν)B(B+→π+νˉν)SM=23+13|1−v22m2˜bRπs2Wαemλ′333λ′∗313VtbV∗td1Xt|2,
(15) where the gauge-invariant function
Xt=1.469±0.017 arises from the box and Z-penguin diagrams in the SM [116].The leptonic W and Z couplings are also important to probe the RPV SUSY effects [26, 117]. In particular, W and Z couplings involving left-handed
τ leptons can receive contributions from the loop diagrams mediated by top quark and sbottom. These effects modify the leptonic W and Z couplings as [74]gZτLτLgZℓLℓL=1−3|λ′333|216π211−2s2Wm2tm2˜bRfZ(m2tm2˜bR),gWτLντgWℓLνℓ=1−3|λ′333|216π214m2tm2˜bRfW(m2tm2˜bR),
(16) where
ℓ=e,μ andsW=sinθW withθW the weak mixing angle. The loop functionsfZ(x) andfW(x) have been calculated in Refs. [26, 74, 117] and are given byfZ(x)= 1/(x−1)−logx/(x−1)2 andfW(x)=1/(x−1)−(2−x)logx/ (x−1)2 . Experimental measurements on theZτLτL couplings have been performed at the LEP and SLD [118]. Their combined results yieldgZτLτL/gZℓLℓL=1.0013± 0.0019 [74]. TheWτLντ coupling can be extracted fromτ decay data. The measuredτ decay fractions compared to theμ decay fractions yieldgWτLντ/gWℓLνℓ=1.0007±0.0013 [74]. Both the leptonic W and Z couplings are measured at the few permille level. Therefore, they assert strong bounds on the RPV couplingλ′333 .RPV interactions can likewise affect K-meson decays, e.g.,
K→πνˉν , D-meson decays, e.g.,D→τˉν , andτ lepton decays, e.g.,τ→πν . However, as discussed in Ref. [74], their constraints are weaker than the ones from the processes discussed above. Moreover, the bound from theBc lifetime [119, 120] is not relevant, since the RPV SUSY contributions toBc→τˉν are not chirally enhanced compared to the SM.Other interesting anomalies arose in the recent LHCb measurements of
RK(∗)≡B(B→K(∗)μ+μ−)/ B(B→K(∗)e+e−) , which exhibit about2σ deviation from the SM prediction [17, 18] and are refered to asRK(∗) anomalies. TheRK(∗) anomalies imply hints of LFU violation inb→sℓ+ℓ− transition. In the RPV SUSY, the left-handed stop can affect this process at the tree level, and the right-handed sbottom can contribute at the one-loop level. However, as discussed in Ref. [73], once all other flavor constraints are taken into account, no parameter space in the RPV SUSY can explain the currentRK(∗) anomaly.Finally, we briefly comment on the direct searches for sbottoms at the LHC. Using data corresponding to
35.9 fb−1 at 13 TeV, the CMS collaboration has performed search for heavy scalar leptoquarks in thepp→tˉtτ+τ− channel. The results can be directly re-interpreted in the context of pair-produced sbottoms decaying into top quark andτ lepton pairs via the RPV couplingλ′333 . Then, the mass of the sbottom is excluded up to 810 GeV at 95% CL [121]. -
In this section, we proceed to present our numerical analysis for the RPV SUSY scenario introduced in Section 2. We derive the constraints of the RPV couplings and study their effects on various processes.
The most relevant input parameters used in our numerical analysis are presented in Table 1. Employing the theoretical framework described in Section 3, the SM predictions for the
B→D(∗)τˉν ,Bc→ηcτˉν ,Bc→J/ψτˉν , andΛb→Λcτˉν decays are given in Table 2. To obtain the theoretical uncertainties, we vary each input parameter within its1σ range and add each individual uncertainty in quadrature. For the uncertainties induced by form factors, we also include the correlations among the fit parameters. In particular, for theΛb→Λcτˉν decay, we follow the treatment of Ref. [75] to obtain the statistical and systematic uncertainties induced by the form factors. From Table 2, we can see that the experimental data on the ratiosRD ,RD∗ andRJ/ψ deviate from the SM predictions by2.33σ ,2.74σ and1.87σ , respectively.observable unit SM RPV SUSY exp. B(B→τˉν) 10−4 0.947+0.182−0.182 [0.760,1.546] 1.44±0.31 [10]B(B+→π+νˉν) 10−6 0.146+0.014−0.014 [0.091,14.00] < 14 [122] B(B+→K+νˉν) 10−6 3.980+0.470−0.470 [6.900,16.00] <16 [122] B(B→Dτˉν) 10−2 0.761+0.021−0.055 [0.741,0.847] 0.90±0.24 [122]RD 0.300+0.003−0.003 [0.314,0.330] 0.407±0.039±0.024 [10]B(Bc→ηcτˉν) 10−2 0.219+0.023−0.029 [0.199,0.262] – Rηc 0.280+0.036−0.031 [0.262,0.342] – B(B→D∗τˉν) 10−2 1.331+0.103−0.122 [1.270,1.554] 1.78±0.16 [122]RD∗ 0.260+0.008−0.008 [0.267,0.291] 0.306±0.013±0.007 [10]PτL −0.467+0.067−0.061 [−0.528,−0.400] −0.38±0.51+0.21−0.16 [5, 6]PD∗L 0.413+0.032−0.031 [0.382,0.445] 0.60±0.08±0.04 [63, 64]B(Bc→J/ψτˉν) 10−2 0.426+0.046−0.058 [0.387,0.512] – RJ/ψ 0.248+0.006−0.006 [0.254,0.275] 0.71±0.17±0.18 [15]B(Λb→Λcτˉν) 10−2 1.886+0.107−0.165 [1.807,2.159] – RΛc 0.332+0.011−0.011 [0.337,0.372] – Table 2. Predictions for branching fractions and ratios R of five
b→cτˉν channels in SM and RPV SUSY. The sign "–" denotes no available measurements at present. Upper limits are all at 90% CL. -
In the RPV SUSY scenario introduced in Section 2, the relevant parameters used to explain the
RD(∗) anomalies are(λ′313,λ′323,λ′333) andm˜bR . In Section 3, we know only that the three products of the RPV couplings,(λ′313λ′∗333, λ′323λ′∗333,λ′333λ′∗333) , appear in the various flavor processes. In the following analysis, we will assume that these products are real and derive bounds on them. We impose the experimental constraints in the same manner as in Refs. [124, 125], i.e., for each point in the parameter space, if the difference between the corresponding theoretical prediction and experimental data is less than the2σ (3σ) error bar, which is evaluated by adding the theoretical and experimental errors in quadrature, this point is regarded as allowed at the2σ (3σ) level. From Section 3, it is known that the RPV couplings always appear in the form ofλ′3i3λ′∗333/m2˜bR in all B decays. Therefore, we can assumem˜bR=1TeV without loss of generality, which is equivalent to absorbingm˜bR intoλ′3i3λ′∗333 . Furthermore, the choice ofm˜bR=1TeV is compatible with the direct searches for the sbottoms at CMS [121]. In the SUSY contributions to the couplingsgZτLτL andgWτLντ in Eq. (16), additionalm˜bR dependence arises in the loop functionsfZ(m2t/m2˜bR) andfW(m2t/m2˜bR) , respectively. As described in the next subsection, our numerical results show that suchm˜bR dependence is weak, and the choice ofm˜bR=1TeV does not lose much generality.As shown in Table 2, the current experimental upper bounds imposed on the branching ratio of
B+→K+νˉν andB+→π+νˉν are one order above their SM values. However, since the SUSY contributes to these decays at the tree level, the RPV couplings are strongly constrained as−0.082<λ′313λ′∗333<0.090,(fromB+→π+νˉν)−0.098<λ′323λ′∗333<0.057,(fromB+→K+νˉν)
(17) at
2σ level. For the leptonic W and Z couplings, the current measurements ongWτLντ/gWℓLνℓ andgZτLτL/gZℓLℓL have achieved the precision level of a few permille. We find that the latter can yield a stronger constraint, which readsλ′333λ′∗333<0.93,(fromgZτLτL/gZℓLℓL)
(18) or
|λ′333|<0.96 , at the2σ level. This upper bound prevents the couplingλ′333 from developing a Landau pole below the GUT scale [126].As discussed in Section 3, the RPV interactions affect
b→cτˉν transitions via the three products(λ′313λ′∗333, λ′323λ′∗333,λ′333λ′∗333) . After considering the above individual constraints at2σ level, the parameter space to explain the current measurements onRD(∗) ,RJ/ψ ,PτL(D∗) andPD∗L is shown in Fig. 1 form˜bR=1TeV . TheB→D(∗)τˉν decays and other flavor observables are observed to put very stringent constraints on the RPV couplings. The combined constraints are slightly stronger than the individual ones in Eqs. (17) and (18). Moreover, after taking into account the bounds fromB+→K+νˉν andgZτLτL , theB→D(∗)τˉν decays are very sensitive to the productλ′323λ′∗333 . Consequently, currentRD(∗) anomalies yield a lower bound on|λ′323λ′∗333| . Finally, the combined bounds in Fig. 1 read numerically,Figure 1. (color online) Allowed parameter space of
(λ′313λ′∗333,λ′323λ′∗333,λ′333λ′∗333) by all flavor processes at2σ level withm˜bR=1TeV , plotted in the(λ′313λ′∗333,λ′323λ′∗333) (a),(λ′313λ′∗333,λ′333λ′∗333) (b), and(λ′323λ′∗333,λ′333λ′∗333) (c) plane. Figure (d) shows the allowed region in(m˜bR,λ′333λ′∗333/m2˜bR) plane.−0.082<λ′313λ′∗333<0.087,(fromcombinedconstraints)0.018<λ′323λ′∗333<0.057,0.033<λ′333λ′∗333<0.928.
(19) As shown, a weak lower bound on
λ′333λ′∗333 is also obtained. Although the constraints from theD∗ polarization fractionPD∗L are much stronger than the ones from theτ polarization fractionPτL , this observable cannot provide further constraints on the RPV couplings. From previous discussions, we show the combined upper bound onλ′333λ′∗333/m2˜bR as a function ofm˜bR in Fig. 1(d). The upper limit ofλ′333λ′∗333/m2˜bR changes around 20% by varyingm˜bR from800 GeV to2000 GeV . Therefore, the allowed parameter space form˜bR≠1TeV can approximately be obtained from Fig. 1(a)-1(c) by timing a factor of(m˜bR/1TeV)2 . -
In the parameter space allowed by all the constraints at the
2σ level, correlations among several observables are obtained, as shown in Fig. 2. In these figures, the SUSY predictions are central values without theoretical uncertainties. From Fig. 2(a), we can see that the central values ofRD andRD∗ are strongly correlated, as expected from Eq. (13). The SUSY effects can only enhance the central value ofRD(∗) by about 8%, such that the ratiosRD(∗) approach, but still lie outside, the2σ range of the HFLAV averages. Therefore, future refined measurements will provide a crucial test to the RPV SUSY explanation ofRD(∗) anomalies. At Belle II, precisions ofRD(∗) measurements are expected to be about 2%–4% [71] with a luminosity of50ab−1 . Fig. 2(b), it can be seen that bothRD∗ andB(B+→K+νˉν) deviate from their SM predictions. The lower bound for the latter isB(B+→K+νˉν)>7.37× 10−6 , which is due to the lower bound ofλ′323λ′∗333>0.018 obtained in the last section. Compared to the SM predictionB(B+→K+νˉν)SM= (3.98±0.47)×10−6 , such significant enhancement makes this decay an important probe of the RPV SUSY effects. In the future, Belle II with50ab−1 data can measure its branching ratio with a precision of 11% [71]. Another interesting correlation arises betweenB(B+→K+νˉν) andgZτLτL/gZℓLℓL . As shown in Fig. 2(f), the RPV SUSY effects always enhanceB(B+→K+νˉν) and suppressgZτLτL/gZℓLℓL simultaneously. WhengZτLτL/gZℓLℓL approaches the SM value 1, the branching ratio ofB+→K+νˉν maximally deviates from its SM prediction. In Fig. 2(d) and 2(e), we show the correlations involvingB→τν decay. The SUSY prediction onB(B→τˉν) is almost in the SM1σ range. Since the future Belle II sensitivity at50ab−1 is comparable to the current theoretical uncertainties [71], significantly more precise theoretical predictions are required in the future to probe the SUSY effects.Figure 2. (color online) Correlations among various observables. SM predictions correspond to the green cross, while the correlations in the RPV SUSY are depicted by red points. In Fig. 2(a), the current HFLAV averages for
RD andRD∗ are shown as the black region, and the2σ (4σ ) experimental region is depicted in gray (light gray). In other figures, the1σ experimental region is shown in black. The2σ regions forRD∗ are also depicted in gray.Using the allowed parameter space at the
2σ level derived in the last subsection, we make predictions on the fiveb→cτˉν decays,B→D(∗)τˉν ,Bc→ηcτˉν ,Bc→J/ψτˉν , andΛb→Λcτˉν decays. In Table 2, the SM and SUSY predictions of the various observables in these decays are presented. The SUSY predictions have included the uncertainties induced by the form factors and CKM matrix elements. At present, there are no available measurements on theBc→ηcτˉν andΛb→Λcτˉν decays. Table 2 shows that, although the SUSY predictions for the branching fractions and the LFU ratios in these two decays overlap with their1σ SM range, they can be considerably enhanced by the RPV SUSY effects.Now we start to analyze the
q2 distributions of the differential branching fractionB , LFU ratio R, lepton forward-backward asymmetryAFB , polarization fraction ofτ leptonPτL , and the polarization fraction of daughter meson (PD∗L ,PJ/ψL ,PΛcL ). For the two “B→P ” transitionsB→Dτˉν andBc→ηcτˉν , their differential observables in the SM and RPV SUSY are shown in Fig. 3. All the differential distributions of these two decays are very similar, whereas the observables inBc→ηcτˉν suffer from larger theoretical uncertainties, which are due to the large uncertainties induced by theBc→ηc form factors. In the RPV SUSY, the branching fraction ofB→Dτˉν decay can be largely enhanced, while the LFU ratio is almost indistinguishable from the SM prediction. Therefore, it is difficult for the differential distribution ofRD(q2) to provide testable signature of the RPV SUSY. Moreover, the RPV SUSY does not affect the forward-backward asymmetryAFB andτ polarization fractionPτL in these two decays, as shown in Fig. 3. The reason behind this is that the RPV couplings only modify the Wilson coefficientCL,2 , and its effects in the numerator and denominator in Eqs. (11) and (12) cancel out exactly. This feature could be used to distinguish from the NP candidates, which can explain theRD(∗) anomaly, but involves scalar or tensor interactions [83, 127, 128].Figure 3. (color online) Differential observables in
B→Dτˉν (left) andBc→ηcτˉν (right) decays. The black curves (gray band) indicate the SM (SUSY) central values with1σ theoretical uncertainty.The differential observables in the
B→D∗τˉν andBc→J/ψτˉν decays are shown in Fig. 4. As expected, these two “B→V ” processes have very similar distributions. In these two decays, the enhancement by the RPV SUSY effects is not large enough to make the branching ratios deviate from the SM values by more than1σ . However, the LFU ratiosRD∗(q2) andRJ/ψ(q2) are significantly enhanced in the entire kinematical region, especially in the large dilepton invariant mass region. In this end-point region, the theoretical predictions suffer from very small uncertainties compared to the other kinematical region. By this virtue, the LFU ratiosRD∗(q2) andRJ/ψ(q2) in the RPV SUSY deviate from the SM predictions by about2σ . Therefore, future measurements on these differential ratios could provide more information about theRD(∗) anomaly and are important for the indirect searches for SUSY. In addition, as in theB→Dτˉν andBc→ηcτˉν decays, the angular observablesAFB ,PτL andPD∗,J/ψL are not affected by the SUSY effects.Figure 4. (color online) Differential observables in
B→D∗τˉν (left) andBc→J/ψτˉν (right) decays. The black curves (gray band) indicate the SM (SUSY) central values with1σ theoretical uncertainty.Figure 5 shows the differential observables in the
Λb→Λcτν decay. The RPV SUSY effects significantly enhance the branching fraction and the LFU ratio. In particular, at the large dilepton invariant mass, the ratioRΛc(q2) in the SUSY exhibits a higher than2σ discrepancy from the SM values. With largeΛb samples at the future HL-LHC, this decay is expected to provide complementary information to the direct SUSY searches. In addition, as in the other decays, the RPV SUSY effects vanish in various angular observables.Figure 5. (color online) Differential observables in
Λb→Λcτˉν decay. Other captions are the same as in Fig. 3. -
For the operator in Eq. (5), the hadronic matrix elements of
B→D transition can be parameterized in terms of form factorsF+ andF0 [28, 102]. In the BGL parameterization, they can be written as expressions ofa+n anda0n [11],F+(z)=1P+(z)ϕ+(z,N)∞∑n=0a+nzn(w,N),F0(z)=1P0(z)ϕ0(z,N)∞∑n=0a0nzn(w,N),
where
z(w,N)=(√1+w−√2N)/(√1+w+√2N) ,w=(m2B+m2D−q2)/ (2mBmD) ,N=(1+r)/(2√r) , andr=mD/mB . Values of the fit parameters are taken from Ref. [11].For the
B→D∗ transition, the relevant form factors areA0,1,2 and V. They can be written in terms of the BGL form factors asA0(q2)=mB+mD∗2√mBmD∗P1(w),A1(q2)=f(w)mB+mD∗,A2(q2)=(mB+mD∗)[(m2B−m2D∗−q2)f(w)−2mD∗F1(w)]λD∗(q2),V(q2)=mBmD∗(mB+mD∗)√w2−1√λD∗(q2)g(w),
where
w=(m2B+m2D∗−q2)/2mBmD∗ andλD∗=[(mB−mD∗)2−q2] [(mB+mD∗)2−q2] . The four BGL form factors can be expanded as a series in zf(z)=1P1+(z)ϕf(z)∞∑n=0afnzn,F1(z)=1P1+(z)ϕF1(z)∞∑n=0zF1nzn,g(z)=1P1−(z)ϕg(z)∞∑n=0agnzn,P1(z)=√r(1+r)B0−(z)ϕP1(z)∞∑n=0aP1nzn,
where
z=(√w+1−√2)/(√w+1+√2) andr=mD∗/mB . Explicit expressions of the Blaschke factorsP1± andB0− , and the outer functionsϕi(z) can be found in Refs. [14, 129]. We also adopt the values of the fit parameters in Refs. [14, 129].The
Λb→Λc hadronic matrix elements can be written in terms of the helicity form factorsF0,+,⊥ andG0,+,⊥ [75, 76]. Following Ref. [75], the lattice calculations are fitted to two Bourrely-Caprini-Lellouch z-parameterization [130]. In the so-called “nominal fit”, a form factor has the following formf(q2)=11−q2/(mfpole)2[af0+af1zf(q2)],
while the form factor in the “higher-order fit” is given by
fHO(q2)=11−q2/(mfpole)2{af0,HO+af1,HOzf(q2)+af2,HO[zf(q2)]2},
where
zf(q2)=(√tf+−q2−√tf+−t0)/(√tf+−q2+√tf+−t0) ,t0=(mΛb− mΛc)2 , andtf+=(mfpole)2 . Values of the fit parameters are taken from Ref. [76].The form factors for
Bc→J/ψ andBc→ηc transitions are taken from the results in the Covariant Light-Front Approach in Ref. [16].
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