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bcτˉν decay in supersymmetry with R-parity violation

  • In past years, several hints of lepton flavor universality (LFU) violation have emerged from the bcτˉν and bs+ data. More recently, the Belle Collaboration has reported the first measurement of the D longitudinal polarization fraction in the BDτˉν decay. Motivated by this intriguing result, along with the recent measurements of RJ/ψ and τ polarization, we present the study of bcτˉν decays in supersymmetry (SUSY) with R-parity violation (RPV). We consider BD()τˉν, Bcηcτˉν, BcJ/ψτˉν and ΛbΛcτˉν modes and focus on the branching ratios, LFU ratios, forward-backward asymmetries, polarizations of daughter hadrons, and the τ lepton. The RPV SUSY was capable of explaining the RD() anomalies at the 2σ level, after taking into account various flavor constraints. In the allowed parameter space, the differential branching fractions and LFU ratios are largely enhanced by the SUSY effects, especially in the large dilepton invariant mass region. Moreover, a lower bound B(B+K+νˉν)>7.37× 10−6 is obtained. These observables could provide testable signatures at the high-luminosity LHC and SuperKEKB, and correlate with direct searches for SUSY.
  • In the recent years, several interesting anomalies emerged in the experimental data of semi-leptonic B-meson decays. The ratios RD()B(BD()τˉν)/B(BD()ˉν) with =e,μ, obtained by latest averages of the measurements by BaBar [1, 2], Belle [36] and LHCb Collaboration [79], 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 to BD() 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 and RD have values below the experimental measurements by 2.3σ and 3.0σ, respectively. Taking into account the measurement correlation of −0.203 between RD and RD, the combined experimental results exhibit about 3.78σ deviation from the SM predictions [10]. For the BcJ/ψτˉν decay, which is mediated by the same quark-level process as BD()τˉν, the recently measured ratio RexpJ/ψ=0.71±0.17(stat.)±0.18(syst.) at the LHCb [15] lies within about 2σ above the SM prediction RSMJ/ψ=0.248±0.006 [16]. In addition, the LHCb measurements of the ratios RK()B(BK()μ+μ)/B(BK()e+e), RexpK=0.745+0.0900.074±0.036 for q2[1.0,6.0]GeV2 [17] and RexpK=0.69+0.110.07±0.05 for q2[1.1,6.0]GeV2 [18], are found to be about 2.6σ and 2.5σ lower than the SM expectation, RSMK()1 [19, 20], respectively. These measurements, referred to as the RD(), RJ/ψ, and RK() 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 [2134] and in specific NP models [3560]. We refer to Refs. [61, 62] for recent reviews.

    The first measurement on the D longitudinal polarization fraction in the BDτˉν decay has recently been reported by the Belle Collaboration [63, 64]

    PDL=0.60±0.08(stat.)±0.04(syst.),

    which is consistent with the SM prediction of PDL=0.46±0.04 [65] at 1.5σ. Previously, the Belle Collaboration also performed measurements on τ polarization in the BDτˉν decay and obtained the result PτL=0.38±0.51(stat.)+0.210.16(syst.) [5, 6]. Angular distributions can provide valuable information about the spin structure of the interaction in BD()τˉν decays, and they are good observables for the testing of various NP explanations [6670]. Measurements of angular distributions are expected to significantly improve in the future. For example, Belle II with 50ab1 data can measure PτL 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 the bcτˉν transition at the tree level and could explain the current RD() anomalies [7274]. Besides BD()τˉν, we will also study the BcJ/ψτˉν, Bcηcτˉν, and ΛbΛcτˉν decay. All of them depict the bcτˉν 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 five bcτˉν decays, focusing on the q2 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.

    The most general re-normalizable RPV terms in the superpotential are given by [77, 78]

    WRPV=μiLiHu+12λijkLiLjEck+λijkLiQjDck+12λijkUciDcjDck,

    (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 couplings λijk are zero. In semi-leptonic B meson decays, contribution from the λ term occurs through the exchange of sleptons, and it is much more suppressed than the one from the λ term, which occurs through the exchange of right-handed down-type squarks [72]. Therefore, we only consider the λijkLiQjDck term in this work. For the SUSY scenario with the λ term, studies on the RD() anomalies with slepton exchanges can be found in Refs. [79, 80].

    The interaction with λijk couplings can be expanded in terms of fermions and sfermions as [72]

    ΔLRPV=λijk[˜νiLˉdkRdjL+˜djLˉdkRνiL+˜dkRˉνciRdjLVjl(˜iLˉdkRulL+˜ulLˉdkRiL+˜dkRˉciRulL)]+h.c.,

    (4)

    where Vij denotes the CKM matrix element. Here, all the SM fermions dL,R, L,R, and νL 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 the RD() and RK() 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 (g2)μ anomaly in the LQ scenario [83]. Moreover, these couplings can also affect semi-leptonic B decays. In particular, their contributions to the BD()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.

    With the RPV SUSY contributions, the effective Hamiltonian responsible for bc(u)τˉντ transitions is given by [72]

    Heff=4GF2i=u,cVib(1+CNPL,i)(ˉuiγμPLb)(ˉτγμPLντ),

    (5)

    where tree-level sbottom exchange yields

    CNPL,i=v24m2˜bRλ3333j=1λ3j3(VijVi3),

    (6)

    with the Higgs vev v = 246 GeV. This Wilson coefficient is at the matching scale μNPm˜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 bcˉν transitions, we consider five processes, including BD()ˉν [84-86], Bcηcˉν [16, 87], BcJ/ψˉν [8896], and ΛbΛcˉν [97-100] decays. All these decays can be uniformly denoted as

    M(pM,λM)N(pN,λN)+(p,λ)+ˉν(pˉν),

    (7)

    where (M,N)=(B,D),(Bc,ηc),(B,D),(Bc,J/ψ), and (Λb,Λc), and (,ˉν)=(e,ˉνe),(μ,ˉνμ), and (τ,ˉντ). For each particle i in the above decay, its momentum and helicity are denoted as pi and λi, 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,λ(MNˉν)=11+2|λM|λM|MλMλN,λ|2Q+Q512π3m3M×1m2q2dq2dcosθ,

    (8)

    where q=pMpN, m±=mM±mN, and Q±=m2±q2. The angle θ[0,π] denotes the angle between the three-momentum of and that of N in the -ˉν center-of-mass frame. The following observables can be derived with the differential decay width:

    • The decay width and branching ratio

    dBdq2=1ΓMdΓdq2=1ΓMλN,λdΓλN,λdq2,

    (9)

    where ΓM is the total width of the hadron M.

    • The LFU ratio

    RN(q2)=dΓ(MNτˉντ)/dq2dΓ(MNˉν)/dq2,

    (10)

    where dΓ(MNˉν)/dq2 in the denominator denotes the average of different decay widths of the electronic and muonic modes.

    • The lepton forward-backward asymmetry

    AFB(q2)=10dcosθ(d2Γ/dq2dcosθ)01dcosθ(d2Γ/dq2dcosθ)dΓ/dq2.

    (11)

    • The polarization fractions

    PτL(q2)=dΓλτ=+1/2/dq2dΓλτ=1/2/dq2dΓ/dq2,PNL(q2)=dΓλN=+1/2/dq2dΓλN=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,λNˉν|Heff|M and all the above observables can be found in Ref. [102] for BD()τˉν decays, and Ref. [76] for the ΛbΛcτˉν decay. The expressions for Bcηcτˉν and BcJ/ψτˉν are analogical to the ones for BDτˉν and BDτˉν, 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 the bcτˉν decays is straightforward:

    RNRSMN=|1+CNPL,2|2,

    (13)

    for N=D(),ηc,J/ψ, and Λc. Here, vanishing contributions to the electronic and muonic channels are assumed.

    The hadronic MN 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, 103110]. For BD() 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 [112114], 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 the Bcηc,J/ψ transition form factors obtained in the covariant light-front approach [16]. For the ΛbΛc 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 buτˉν transitions, we consider Bτˉν, Bπτˉν and Bρτˉν decays. Similar to Eq. (13), we have

    B(Bτˉν)B(Bτˉν)SM=B(Bπτˉν)B(Bπτˉν)SM=B(Bρτˉν)B(Bρτˉν)SM=|1+CNPL,1|2.

    (14)

    The SUSY contributions to both buτˉν and bcτˉν transitions depend on the same set of parameters, λ313, λ323, and λ333. Therefore, the ratios RD are related to the Bτˉν decay.

    The Flavor-Changing Neutral Current (FCNC) decays B+K+νˉν and B+π+νˉν are induced by the bsνˉν and bdνˉν 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 the bc(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|1v22m2˜bRπs2Wαemλ333λ323VtbVts1Xt|2,B(B+π+νˉν)B(B+π+νˉν)SM=23+13|1v22m2˜bRπs2Wαemλ333λ313VtbVtd1Xt|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]

    gZτLτLgZLL=13|λ333|216π2112s2Wm2tm2˜bRfZ(m2tm2˜bR),gWτLντgWLν=13|λ333|216π214m2tm2˜bRfW(m2tm2˜bR),

    (16)

    where =e,μ and sW=sinθW with θW the weak mixing angle. The loop functions fZ(x) and fW(x) have been calculated in Refs. [26, 74, 117] and are given by fZ(x)=1/(x1)logx/(x1)2 and fW(x)=1/(x1)(2x)logx/(x1)2. Experimental measurements on the ZτLτL couplings have been performed at the LEP and SLD [118]. Their combined results yield gZτLτL/gZLL=1.0013±0.0019 [74]. The WτLντ coupling can be extracted from τ decay data. The measured τ decay fractions compared to the μ decay fractions yield gWτLντ/gWLν=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 coupling λ333.

    RPV interactions can likewise affect K-meson decays, e.g., Kπνˉν, D-meson decays, e.g., Dτˉν, and τ lepton decays, e.g., τπν. However, as discussed in Ref. [74], their constraints are weaker than the ones from the processes discussed above. Moreover, the bound from the Bc lifetime [119, 120] is not relevant, since the RPV SUSY contributions to Bcτˉν are not chirally enhanced compared to the SM.

    Other interesting anomalies arose in the recent LHCb measurements of RK()B(BK()μ+μ)/ B(BK()e+e), which exhibit about 2σ deviation from the SM prediction [17, 18] and are refered to as RK() anomalies. The RK() anomalies imply hints of LFU violation in bs+ 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 current RK() anomaly.

    Finally, we briefly comment on the direct searches for sbottoms at the LHC. Using data corresponding to 35.9 fb1 at 13 TeV, the CMS collaboration has performed search for heavy scalar leptoquarks in the pptˉtτ+τ channel. The results can be directly re-interpreted in the context of pair-produced sbottoms decaying into top quark and τ lepton pairs via the RPV coupling λ333. 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 BD()τˉν, Bcηcτˉν, BcJ/ψτˉν, and ΛbΛcτˉν decays are given in Table 2. To obtain the theoretical uncertainties, we vary each input parameter within its 1σ 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 the ΛbΛcτˉν 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 ratios RD, RD and RJ/ψ deviate from the SM predictions by 2.33σ, 2.74σ and 1.87σ, respectively.

    Table 2

    Table 2.  Predictions for branching fractions and ratios R of five bcτˉν channels in SM and RPV SUSY. The sign "–" denotes no available measurements at present. Upper limits are all at 90% CL.
    observable unit SM RPV SUSY exp.
    B(Bτˉν) 104 0.947+0.1820.182 [0.760,1.546] 1.44±0.31 [10]
    B(B+π+νˉν) 106 0.146+0.0140.014 [0.091,14.00] < 14 [122]
    B(B+K+νˉν) 106 3.980+0.4700.470 [6.900,16.00] <16 [122]
    B(BDτˉν) 102 0.761+0.0210.055 [0.741,0.847] 0.90±0.24 [122]
    RD 0.300+0.0030.003 [0.314,0.330] 0.407±0.039±0.024 [10]
    B(Bcηcτˉν) 102 0.219+0.0230.029 [0.199,0.262]
    Rηc 0.280+0.0360.031 [0.262,0.342]
    B(BDτˉν) 102 1.331+0.1030.122 [1.270,1.554] 1.78±0.16 [122]
    RD 0.260+0.0080.008 [0.267,0.291] 0.306±0.013±0.007 [10]
    PτL 0.467+0.0670.061 [0.528,0.400] 0.38±0.51+0.210.16 [5, 6]
    PDL 0.413+0.0320.031 [0.382,0.445] 0.60±0.08±0.04 [63, 64]
    B(BcJ/ψτˉν) 102 0.426+0.0460.058 [0.387,0.512]
    RJ/ψ 0.248+0.0060.006 [0.254,0.275] 0.71±0.17±0.18 [15]
    B(ΛbΛcτˉν) 102 1.886+0.1070.165 [1.807,2.159]
    RΛc 0.332+0.0110.011 [0.337,0.372]
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    Table 1

    Table 1.  Input parameters used in our numerical analysis.
    inputvalueunitRef.
    mpolet173.1±0.9GeV[122]
    mb(mb)4.18±0.03GeV[122]
    mc(mc)1.28±0.03GeV[122]
    A0.8396+0.00800.0298[123]
    λ0.224756+0.0001630.000065[123]
    ˉρ0.123+0.0230.023[123]
    ˉη0.375+0.0220.017[123]
    DownLoad: CSV
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    In the RPV SUSY scenario introduced in Section 2, the relevant parameters used to explain the RD() anomalies are (λ313,λ323,λ333) and m˜bR. In Section 3, we know only that the three products of the RPV couplings, (λ313λ333,λ323λ333,λ333λ333), 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 the 2σ (3σ) error bar, which is evaluated by adding the theoretical and experimental errors in quadrature, this point is regarded as allowed at the 2σ (3σ) level. From Section 3, it is known that the RPV couplings always appear in the form of λ3i3λ333/m2˜bR in all B decays. Therefore, we can assume m˜bR=1TeV without loss of generality, which is equivalent to absorbing m˜bR into λ3i3λ333. Furthermore, the choice of m˜bR=1TeV is compatible with the direct searches for the sbottoms at CMS [121]. In the SUSY contributions to the couplings gZτLτL and gWτLντ in Eq. (16), additional m˜bR dependence arises in the loop functions fZ(m2t/m2˜bR) and fW(m2t/m2˜bR), respectively. As described in the next subsection, our numerical results show that such m˜bR dependence is weak, and the choice of m˜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 and B+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<λ313λ333<0.090,(fromB+π+νˉν)0.098<λ323λ333<0.057,(fromB+K+νˉν)

    (17)

    at 2σ level. For the leptonic W and Z couplings, the current measurements on gWτLντ/gWLν and gZτLτL/gZLL have achieved the precision level of a few permille. We find that the latter can yield a stronger constraint, which reads

    λ333λ333<0.93,(fromgZτLτL/gZLL)

    (18)

    or |λ333|<0.96, at the 2σ level. This upper bound prevents the coupling λ333 from developing a Landau pole below the GUT scale [126].

    As discussed in Section 3, the RPV interactions affect bcτˉν transitions via the three products (λ313λ333,λ323λ333,λ333λ333). After considering the above individual constraints at 2σ level, the parameter space to explain the current measurements on RD(), RJ/ψ, PτL(D) and PDL is shown in Fig. 1 for m˜bR=1TeV. The BD()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 from B+K+νˉν and gZτLτL, the BD()τˉν decays are very sensitive to the product λ323λ333. Consequently, current RD() anomalies yield a lower bound on |λ323λ333|. Finally, the combined bounds in Fig. 1 read numerically,

    Figure 1

    Figure 1.  (color online) Allowed parameter space of (λ313λ333,λ323λ333,λ333λ333) by all flavor processes at 2σ level with m˜bR=1TeV, plotted in the (λ313λ333,λ323λ333) (a), (λ313λ333,λ333λ333) (b), and (λ323λ333,λ333λ333) (c) plane. Figure (d) shows the allowed region in (m˜bR,λ333λ333/m2˜bR) plane.

    0.082<λ313λ333<0.087,(fromcombinedconstraints)0.018<λ323λ333<0.057,0.033<λ333λ333<0.928.

    (19)

    As shown, a weak lower bound on λ333λ333 is also obtained. Although the constraints from the D polarization fraction PDL are much stronger than the ones from the τ polarization fraction PτL, this observable cannot provide further constraints on the RPV couplings. From previous discussions, we show the combined upper bound on λ333λ333/m2˜bR as a function of m˜bR in Fig. 1(d). The upper limit of λ333λ333/m2˜bR changes around 20% by varying m˜bR from 800 GeV to 2000 GeV. Therefore, the allowed parameter space for m˜bR1TeV 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 of RD and RD are strongly correlated, as expected from Eq. (13). The SUSY effects can only enhance the central value of RD() by about 8%, such that the ratios RD() approach, but still lie outside, the 2σ range of the HFLAV averages. Therefore, future refined measurements will provide a crucial test to the RPV SUSY explanation of RD() anomalies. At Belle II, precisions of RD() measurements are expected to be about 2%–4% [71] with a luminosity of 50ab1. Fig. 2(b), it can be seen that both RD and B(B+K+νˉν) deviate from their SM predictions. The lower bound for the latter is B(B+K+νˉν)>7.37×106, which is due to the lower bound of λ323λ333>0.018 obtained in the last section. Compared to the SM prediction B(B+K+νˉν)SM=(3.98±0.47)×106, such significant enhancement makes this decay an important probe of the RPV SUSY effects. In the future, Belle II with 50ab1 data can measure its branching ratio with a precision of 11% [71]. Another interesting correlation arises between B(B+K+νˉν) and gZτLτL/gZLL. As shown in Fig. 2(f), the RPV SUSY effects always enhance B(B+K+νˉν) and suppress gZτLτL/gZLL simultaneously. When gZτLτL/gZLL approaches the SM value 1, the branching ratio of B+K+νˉν maximally deviates from its SM prediction. In Fig. 2(d) and 2(e), we show the correlations involving Bτν decay. The SUSY prediction on B(Bτˉν) is almost in the SM 1σ range. Since the future Belle II sensitivity at 50ab1 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

    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 and RD are shown as the black region, and the 2σ (4σ) experimental region is depicted in gray (light gray). In other figures, the 1σ experimental region is shown in black. The 2σ regions for RD are also depicted in gray.

    Using the allowed parameter space at the 2σ level derived in the last subsection, we make predictions on the five bcτˉν decays, BD()τˉν, Bcηcτˉν, BcJ/ψτˉν, and ΛbΛcτˉν 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 the Bcηcτˉν and ΛbΛcτˉν decays. Table 2 shows that, although the SUSY predictions for the branching fractions and the LFU ratios in these two decays overlap with their 1σ SM range, they can be considerably enhanced by the RPV SUSY effects.

    Now we start to analyze the q2 distributions of the differential branching fraction B, LFU ratio R, lepton forward-backward asymmetry AFB, polarization fraction of τ lepton PτL, and the polarization fraction of daughter meson (PDL, PJ/ψL, PΛcL). For the two “BP” transitions BDτˉν and Bcηcτˉν, 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 in Bcηcτˉν suffer from larger theoretical uncertainties, which are due to the large uncertainties induced by the Bcηc form factors. In the RPV SUSY, the branching fraction of BDτˉν decay can be largely enhanced, while the LFU ratio is almost indistinguishable from the SM prediction. Therefore, it is difficult for the differential distribution of RD(q2) to provide testable signature of the RPV SUSY. Moreover, the RPV SUSY does not affect the forward-backward asymmetry AFB and τ polarization fraction PτL in these two decays, as shown in Fig. 3. The reason behind this is that the RPV couplings only modify the Wilson coefficient CL,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 the RD() anomaly, but involves scalar or tensor interactions [83, 127, 128].

    Figure 3

    Figure 3.  (color online) Differential observables in BDτˉν (left) and Bcηcτˉν (right) decays. The black curves (gray band) indicate the SM (SUSY) central values with 1σ theoretical uncertainty.

    The differential observables in the BDτˉν and BcJ/ψτˉν decays are shown in Fig. 4. As expected, these two “BV” 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 than 1σ. However, the LFU ratios RD(q2) and RJ/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 ratios RD(q2) and RJ/ψ(q2) in the RPV SUSY deviate from the SM predictions by about 2σ. Therefore, future measurements on these differential ratios could provide more information about the RD() anomaly and are important for the indirect searches for SUSY. In addition, as in the BDτˉν and Bcηcτˉν decays, the angular observables AFB, PτL and PD,J/ψL are not affected by the SUSY effects.

    Figure 4

    Figure 4.  (color online) Differential observables in BDτˉν (left) and BcJ/ψτˉν (right) decays. The black curves (gray band) indicate the SM (SUSY) central values with 1σ theoretical uncertainty.

    Figure 5 shows the differential observables in the ΛbΛcτν decay. The RPV SUSY effects significantly enhance the branching fraction and the LFU ratio. In particular, at the large dilepton invariant mass, the ratio RΛc(q2) in the SUSY exhibits a higher than 2σ discrepancy from the SM values. With large Λb 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

    Figure 5.  (color online) Differential observables in ΛbΛcτˉν decay. Other captions are the same as in Fig. 3.

    Recently, several hints of lepton flavor universality violation have been observed in the experimental data of semi-leptonic B decays. Motivated by the recent measurements of PDL, we have investigated the RPV SUSY effects in bcτˉν transitions. After considering various flavor processes, we obtain strong constraints of the RPV couplings, which are dominated by B(B+π+νˉν), B(B+K+νˉν), and gZτLτL. In the surviving parameter space, the RD() anomaly can be explained at the 2σ level, which results in bounds on the coupling products, 0.082<λ313λ333<0.087, 0.018<λ323λ333<0.057, and 0.033<λ333λ333<0.928. The upper bound on the coupling λ333 prevents this coupling from developing a Landau pole below the GUT scale.

    In the parameter space allowed by all the constraints, we make predictions for various flavor processes. For B+K+νˉν decay, a lower bound B(B+K+νˉν)>7.37×106 is obtained. Compared to the SM prediction (3.98±0.47)×106, this decay can provide an important probe of the RPV SUSY effects at Belle II. We also find interesting correlations among RD, RD, B(B+K+νˉν), B(Bτν), and gZτLτL/gZLL. For example, the RPV SUSY effects always enhance B(B+K+νˉν) and suppress gZτLτL/gZLL simultaneously, which makes one of them largely deviate from its SM value.

    Furthermore, we systematically investigated the RPV SUSY effects in five bcτˉν decays, including BD()τˉν, Bcηcτˉν, BcJ/ψτˉν, and ΛbΛcτˉν decays, while focusing on the q2 distributions of the branching fractions, the LFU ratios, and various angular observables. The differential ratios RD(q2), RJ/ψ(q2), and RΛc(q2) are significantly enhanced by the RPV SUSY effects in the large dilepton invariant mass region. Although the integrated ratios RD,J/ψ,Λc in the SUSY overlap with the 1σ range of the SM values, the differential ratios RD,J/ψ,Λc(q2) in this kinematical region exhibit a higher than 2σ discrepancy between the SM and SUSY predictions. In addition, the SM and RPV SUSY predictions of various angular observables are indistinguishable, since the RPV SUSY scenario does not generate new operators beyond the ones of SM.

    The decays B+K+νˉν and Bτˉν, as well as the differential observables in bcτˉν decays, have the potential to shed new light on the RD() anomalies and may serve as a test of the RPV SUSY. With the forthcoming SuperKEKB and the future HL-LHC, our results are expected to provide more information on the bcτˉν transitions and could correlate with the direct searches for SUSY in future high-energy colliders.

    We thank Jun-Kang He, Quan-Yi Hu, Xin-Qiang Li, Han Yan, Min-Di Zheng, and Xin Zhang for useful discussions.

    For the operator in Eq. (5), the hadronic matrix elements of BD transition can be parameterized in terms of form factors F+ and F0 [28, 102]. In the BGL parameterization, they can be written as expressions of a+n and a0n [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+w2N)/(1+w+2N), w=(m2B+m2Dq2)/(2mBmD), N=(1+r)/(2r), and r=mD/mB. Values of the fit parameters are taken from Ref. [11].

    For the BD transition, the relevant form factors are A0,1,2 and V. They can be written in terms of the BGL form factors as

    A0(q2)=mB+mD2mBmDP1(w),A1(q2)=f(w)mB+mD,A2(q2)=(mB+mD)[(m2Bm2Dq2)f(w)2mDF1(w)]λD(q2),V(q2)=mBmD(mB+mD)w21λD(q2)g(w),

    where w=(m2B+m2Dq2)/2mBmD and λD=[(mBmD)2q2][(mB+mD)2q2]. The four BGL form factors can be expanded as a series in z

    f(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+12)/(w+1+2) and r=mD/mB. Explicit expressions of the Blaschke factors P1± and B0, and the outer functions ϕi(z) can be found in Refs. [14, 129]. We also adopt the values of the fit parameters in Refs. [14, 129].

    The ΛbΛc hadronic matrix elements can be written in terms of the helicity form factors F0,+, and G0,+, [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 form

    f(q2)=11q2/(mfpole)2[af0+af1zf(q2)],

    while the form factor in the “higher-order fit” is given by

    fHO(q2)=11q2/(mfpole)2{af0,HO+af1,HOzf(q2)+af2,HO[zf(q2)]2},

    where zf(q2)=(tf+q2tf+t0)/(tf+q2+tf+t0), t0=(mΛbmΛc)2, and tf+=(mfpole)2. Values of the fit parameters are taken from Ref. [76].

    The form factors for BcJ/ψ and Bcηc transitions are taken from the results in the Covariant Light-Front Approach in Ref. [16].

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Dong-Yang Wang, Ya-Dong Yang and Xing-Bo Yuan. bcτˉν decay in supersymmetry with R-parity violation[J]. Chinese Physics C, 2019, 43(8): 083103. doi: 10.1088/1674-1137/43/8/083103
Dong-Yang Wang, Ya-Dong Yang and Xing-Bo Yuan. bcτˉν decay in supersymmetry with R-parity violation[J]. Chinese Physics C, 2019, 43(8): 083103.  doi: 10.1088/1674-1137/43/8/083103 shu
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bcτˉν decay in supersymmetry with R-parity violation

Abstract: In past years, several hints of lepton flavor universality (LFU) violation have emerged from the bcτˉν and bs+ data. More recently, the Belle Collaboration has reported the first measurement of the D longitudinal polarization fraction in the BDτˉν decay. Motivated by this intriguing result, along with the recent measurements of RJ/ψ and τ polarization, we present the study of bcτˉν decays in supersymmetry (SUSY) with R-parity violation (RPV). We consider BD()τˉν, Bcηcτˉν, BcJ/ψτˉν and ΛbΛcτˉν modes and focus on the branching ratios, LFU ratios, forward-backward asymmetries, polarizations of daughter hadrons, and the τ lepton. The RPV SUSY was capable of explaining the RD() anomalies at the 2σ level, after taking into account various flavor constraints. In the allowed parameter space, the differential branching fractions and LFU ratios are largely enhanced by the SUSY effects, especially in the large dilepton invariant mass region. Moreover, a lower bound B(B+K+νˉν)>7.37× 10−6 is obtained. These observables could provide testable signatures at the high-luminosity LHC and SuperKEKB, and correlate with direct searches for SUSY.

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    1.   Introduction
    • In the recent years, several interesting anomalies emerged in the experimental data of semi-leptonic B-meson decays. The ratios RD()B(BD()τˉν)/B(BD()ˉν) with =e,μ, obtained by latest averages of the measurements by BaBar [1, 2], Belle [36] and LHCb Collaboration [79], 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 to BD() 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 and RD have values below the experimental measurements by 2.3σ and 3.0σ, respectively. Taking into account the measurement correlation of −0.203 between RD and RD, the combined experimental results exhibit about 3.78σ deviation from the SM predictions [10]. For the BcJ/ψτˉν decay, which is mediated by the same quark-level process as BD()τˉν, the recently measured ratio RexpJ/ψ=0.71±0.17(stat.)±0.18(syst.) at the LHCb [15] lies within about 2σ above the SM prediction RSMJ/ψ=0.248±0.006 [16]. In addition, the LHCb measurements of the ratios RK()B(BK()μ+μ)/B(BK()e+e), RexpK=0.745+0.0900.074±0.036 for q2[1.0,6.0]GeV2 [17] and RexpK=0.69+0.110.07±0.05 for q2[1.1,6.0]GeV2 [18], are found to be about 2.6σ and 2.5σ lower than the SM expectation, RSMK()1 [19, 20], respectively. These measurements, referred to as the RD(), RJ/ψ, and RK() 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 [2134] and in specific NP models [3560]. We refer to Refs. [61, 62] for recent reviews.

      The first measurement on the D longitudinal polarization fraction in the BDτˉν decay has recently been reported by the Belle Collaboration [63, 64]

      PDL=0.60±0.08(stat.)±0.04(syst.),

      which is consistent with the SM prediction of PDL=0.46±0.04 [65] at 1.5σ. Previously, the Belle Collaboration also performed measurements on τ polarization in the BDτˉν decay and obtained the result PτL=0.38±0.51(stat.)+0.210.16(syst.) [5, 6]. Angular distributions can provide valuable information about the spin structure of the interaction in BD()τˉν decays, and they are good observables for the testing of various NP explanations [6670]. Measurements of angular distributions are expected to significantly improve in the future. For example, Belle II with 50ab1 data can measure PτL 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 the bcτˉν transition at the tree level and could explain the current RD() anomalies [7274]. Besides BD()τˉν, we will also study the BcJ/ψτˉν, Bcηcτˉν, and ΛbΛcτˉν decay. All of them depict the bcτˉν 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 five bcτˉν decays, focusing on the q2 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.

    2.   Supersymmetry with R-parity violation
    • The most general re-normalizable RPV terms in the superpotential are given by [77, 78]

      WRPV=μiLiHu+12λijkLiLjEck+λijkLiQjDck+12λijkUciDcjDck,

      (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 couplings λijk are zero. In semi-leptonic B meson decays, contribution from the λ term occurs through the exchange of sleptons, and it is much more suppressed than the one from the λ term, which occurs through the exchange of right-handed down-type squarks [72]. Therefore, we only consider the λijkLiQjDck term in this work. For the SUSY scenario with the λ term, studies on the RD() anomalies with slepton exchanges can be found in Refs. [79, 80].

      The interaction with λijk couplings can be expanded in terms of fermions and sfermions as [72]

      ΔLRPV=λijk[˜νiLˉdkRdjL+˜djLˉdkRνiL+˜dkRˉνciRdjLVjl(˜iLˉdkRulL+˜ulLˉdkRiL+˜dkRˉciRulL)]+h.c.,

      (4)

      where Vij denotes the CKM matrix element. Here, all the SM fermions dL,R, L,R, and νL 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 the RD() and RK() 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 (g2)μ anomaly in the LQ scenario [83]. Moreover, these couplings can also affect semi-leptonic B decays. In particular, their contributions to the BD()τˉν decays are found to be small after considering other flavor constraints [52].

    3.   Observables
    • In this section, we introduce the theoretical framework of the relevant flavor processes and discuss the RPV SUSY effects in these processes.

    • 3.1.   bc(u)τˉν transitions

    • With the RPV SUSY contributions, the effective Hamiltonian responsible for bc(u)τˉντ transitions is given by [72]

      Heff=4GF2i=u,cVib(1+CNPL,i)(ˉuiγμPLb)(ˉτγμPLντ),

      (5)

      where tree-level sbottom exchange yields

      CNPL,i=v24m2˜bRλ3333j=1λ3j3(VijVi3),

      (6)

      with the Higgs vev v = 246 GeV. This Wilson coefficient is at the matching scale μNPm˜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 bcˉν transitions, we consider five processes, including BD()ˉν [84-86], Bcηcˉν [16, 87], BcJ/ψˉν [8896], and ΛbΛcˉν [97-100] decays. All these decays can be uniformly denoted as

      M(pM,λM)N(pN,λN)+(p,λ)+ˉν(pˉν),

      (7)

      where (M,N)=(B,D),(Bc,ηc),(B,D),(Bc,J/ψ), and (Λb,Λc), and (,ˉν)=(e,ˉνe),(μ,ˉνμ), and (τ,ˉντ). For each particle i in the above decay, its momentum and helicity are denoted as pi and λi, 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,λ(MNˉν)=11+2|λM|λM|MλMλN,λ|2Q+Q512π3m3M×1m2q2dq2dcosθ,

      (8)

      where q=pMpN, m±=mM±mN, and Q±=m2±q2. The angle θ[0,π] denotes the angle between the three-momentum of and that of N in the -ˉν center-of-mass frame. The following observables can be derived with the differential decay width:

      • The decay width and branching ratio

      dBdq2=1ΓMdΓdq2=1ΓMλN,λdΓλN,λdq2,

      (9)

      where ΓM is the total width of the hadron M.

      • The LFU ratio

      RN(q2)=dΓ(MNτˉντ)/dq2dΓ(MNˉν)/dq2,

      (10)

      where dΓ(MNˉν)/dq2 in the denominator denotes the average of different decay widths of the electronic and muonic modes.

      • The lepton forward-backward asymmetry

      AFB(q2)=10dcosθ(d2Γ/dq2dcosθ)01dcosθ(d2Γ/dq2dcosθ)dΓ/dq2.

      (11)

      • The polarization fractions

      PτL(q2)=dΓλτ=+1/2/dq2dΓλτ=1/2/dq2dΓ/dq2,PNL(q2)=dΓλN=+1/2/dq2dΓλN=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,λNˉν|Heff|M and all the above observables can be found in Ref. [102] for BD()τˉν decays, and Ref. [76] for the ΛbΛcτˉν decay. The expressions for Bcηcτˉν and BcJ/ψτˉν are analogical to the ones for BDτˉν and BDτˉν, 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 the bcτˉν decays is straightforward:

      RNRSMN=|1+CNPL,2|2,

      (13)

      for N=D(),ηc,J/ψ, and Λc. Here, vanishing contributions to the electronic and muonic channels are assumed.

      The hadronic MN 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, 103110]. For BD() 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 [112114], 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 the Bcηc,J/ψ transition form factors obtained in the covariant light-front approach [16]. For the ΛbΛc 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 buτˉν transitions, we consider Bτˉν, Bπτˉν and Bρτˉν decays. Similar to Eq. (13), we have

      B(Bτˉν)B(Bτˉν)SM=B(Bπτˉν)B(Bπτˉν)SM=B(Bρτˉν)B(Bρτˉν)SM=|1+CNPL,1|2.

      (14)

      The SUSY contributions to both buτˉν and bcτˉν transitions depend on the same set of parameters, λ313, λ323, and λ333. Therefore, the ratios RD are related to the Bτˉν decay.

    • 3.2.   Other processes

    • The Flavor-Changing Neutral Current (FCNC) decays B+K+νˉν and B+π+νˉν are induced by the bsνˉν and bdνˉν 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 the bc(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|1v22m2˜bRπs2Wαemλ333λ323VtbVts1Xt|2,B(B+π+νˉν)B(B+π+νˉν)SM=23+13|1v22m2˜bRπs2Wαemλ333λ313VtbVtd1Xt|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]

      gZτLτLgZLL=13|λ333|216π2112s2Wm2tm2˜bRfZ(m2tm2˜bR),gWτLντgWLν=13|λ333|216π214m2tm2˜bRfW(m2tm2˜bR),

      (16)

      where =e,μ and sW=sinθW with θW the weak mixing angle. The loop functions fZ(x) and fW(x) have been calculated in Refs. [26, 74, 117] and are given by fZ(x)=1/(x1)logx/(x1)2 and fW(x)=1/(x1)(2x)logx/(x1)2. Experimental measurements on the ZτLτL couplings have been performed at the LEP and SLD [118]. Their combined results yield gZτLτL/gZLL=1.0013±0.0019 [74]. The WτLντ coupling can be extracted from τ decay data. The measured τ decay fractions compared to the μ decay fractions yield gWτLντ/gWLν=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 coupling λ333.

      RPV interactions can likewise affect K-meson decays, e.g., Kπνˉν, D-meson decays, e.g., Dτˉν, and τ lepton decays, e.g., τπν. However, as discussed in Ref. [74], their constraints are weaker than the ones from the processes discussed above. Moreover, the bound from the Bc lifetime [119, 120] is not relevant, since the RPV SUSY contributions to Bcτˉν are not chirally enhanced compared to the SM.

      Other interesting anomalies arose in the recent LHCb measurements of RK()B(BK()μ+μ)/ B(BK()e+e), which exhibit about 2σ deviation from the SM prediction [17, 18] and are refered to as RK() anomalies. The RK() anomalies imply hints of LFU violation in bs+ 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 current RK() anomaly.

      Finally, we briefly comment on the direct searches for sbottoms at the LHC. Using data corresponding to 35.9 fb1 at 13 TeV, the CMS collaboration has performed search for heavy scalar leptoquarks in the pptˉtτ+τ channel. The results can be directly re-interpreted in the context of pair-produced sbottoms decaying into top quark and τ lepton pairs via the RPV coupling λ333. Then, the mass of the sbottom is excluded up to 810 GeV at 95% CL [121].

    4.   Numerical results and discussions
    • 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 BD()τˉν, Bcηcτˉν, BcJ/ψτˉν, and ΛbΛcτˉν decays are given in Table 2. To obtain the theoretical uncertainties, we vary each input parameter within its 1σ 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 the ΛbΛcτˉν 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 ratios RD, RD and RJ/ψ deviate from the SM predictions by 2.33σ, 2.74σ and 1.87σ, respectively.

      observable unit SM RPV SUSY exp.
      B(Bτˉν) 104 0.947+0.1820.182 [0.760,1.546] 1.44±0.31 [10]
      B(B+π+νˉν) 106 0.146+0.0140.014 [0.091,14.00] < 14 [122]
      B(B+K+νˉν) 106 3.980+0.4700.470 [6.900,16.00] <16 [122]
      B(BDτˉν) 102 0.761+0.0210.055 [0.741,0.847] 0.90±0.24 [122]
      RD 0.300+0.0030.003 [0.314,0.330] 0.407±0.039±0.024 [10]
      B(Bcηcτˉν) 102 0.219+0.0230.029 [0.199,0.262]
      Rηc 0.280+0.0360.031 [0.262,0.342]
      B(BDτˉν) 102 1.331+0.1030.122 [1.270,1.554] 1.78±0.16 [122]
      RD 0.260+0.0080.008 [0.267,0.291] 0.306±0.013±0.007 [10]
      PτL 0.467+0.0670.061 [0.528,0.400] 0.38±0.51+0.210.16 [5, 6]
      PDL 0.413+0.0320.031 [0.382,0.445] 0.60±0.08±0.04 [63, 64]
      B(BcJ/ψτˉν) 102 0.426+0.0460.058 [0.387,0.512]
      RJ/ψ 0.248+0.0060.006 [0.254,0.275] 0.71±0.17±0.18 [15]
      B(ΛbΛcτˉν) 102 1.886+0.1070.165 [1.807,2.159]
      RΛc 0.332+0.0110.011 [0.337,0.372]

      Table 2.  Predictions for branching fractions and ratios R of five bcτˉν channels in SM and RPV SUSY. The sign "–" denotes no available measurements at present. Upper limits are all at 90% CL.

      inputvalueunitRef.
      mpolet173.1±0.9GeV[122]
      mb(mb)4.18±0.03GeV[122]
      mc(mc)1.28±0.03GeV[122]
      A0.8396+0.00800.0298[123]
      λ0.224756+0.0001630.000065[123]
      ˉρ0.123+0.0230.023[123]
      ˉη0.375+0.0220.017[123]

      Table 1.  Input parameters used in our numerical analysis.

    • 4.1.   Constraints

    • In the RPV SUSY scenario introduced in Section 2, the relevant parameters used to explain the RD() anomalies are (λ313,λ323,λ333) and m˜bR. In Section 3, we know only that the three products of the RPV couplings, (λ313λ333,λ323λ333,λ333λ333), 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 the 2σ (3σ) error bar, which is evaluated by adding the theoretical and experimental errors in quadrature, this point is regarded as allowed at the 2σ (3σ) level. From Section 3, it is known that the RPV couplings always appear in the form of λ3i3λ333/m2˜bR in all B decays. Therefore, we can assume m˜bR=1TeV without loss of generality, which is equivalent to absorbing m˜bR into λ3i3λ333. Furthermore, the choice of m˜bR=1TeV is compatible with the direct searches for the sbottoms at CMS [121]. In the SUSY contributions to the couplings gZτLτL and gWτLντ in Eq. (16), additional m˜bR dependence arises in the loop functions fZ(m2t/m2˜bR) and fW(m2t/m2˜bR), respectively. As described in the next subsection, our numerical results show that such m˜bR dependence is weak, and the choice of m˜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 and B+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<λ313λ333<0.090,(fromB+π+νˉν)0.098<λ323λ333<0.057,(fromB+K+νˉν)

      (17)

      at 2σ level. For the leptonic W and Z couplings, the current measurements on gWτLντ/gWLν and gZτLτL/gZLL have achieved the precision level of a few permille. We find that the latter can yield a stronger constraint, which reads

      λ333λ333<0.93,(fromgZτLτL/gZLL)

      (18)

      or |λ333|<0.96, at the 2σ level. This upper bound prevents the coupling λ333 from developing a Landau pole below the GUT scale [126].

      As discussed in Section 3, the RPV interactions affect bcτˉν transitions via the three products (λ313λ333,λ323λ333,λ333λ333). After considering the above individual constraints at 2σ level, the parameter space to explain the current measurements on RD(), RJ/ψ, PτL(D) and PDL is shown in Fig. 1 for m˜bR=1TeV. The BD()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 from B+K+νˉν and gZτLτL, the BD()τˉν decays are very sensitive to the product λ323λ333. Consequently, current RD() anomalies yield a lower bound on |λ323λ333|. Finally, the combined bounds in Fig. 1 read numerically,

      Figure 1.  (color online) Allowed parameter space of (λ313λ333,λ323λ333,λ333λ333) by all flavor processes at 2σ level with m˜bR=1TeV, plotted in the (λ313λ333,λ323λ333) (a), (λ313λ333,λ333λ333) (b), and (λ323λ333,λ333λ333) (c) plane. Figure (d) shows the allowed region in (m˜bR,λ333λ333/m2˜bR) plane.

      0.082<λ313λ333<0.087,(fromcombinedconstraints)0.018<λ323λ333<0.057,0.033<λ333λ333<0.928.

      (19)

      As shown, a weak lower bound on λ333λ333 is also obtained. Although the constraints from the D polarization fraction PDL are much stronger than the ones from the τ polarization fraction PτL, this observable cannot provide further constraints on the RPV couplings. From previous discussions, we show the combined upper bound on λ333λ333/m2˜bR as a function of m˜bR in Fig. 1(d). The upper limit of λ333λ333/m2˜bR changes around 20% by varying m˜bR from 800 GeV to 2000 GeV. Therefore, the allowed parameter space for m˜bR1TeV can approximately be obtained from Fig. 1(a)-1(c) by timing a factor of (m˜bR/1TeV)2.

    • 4.2.   Predictions

    • 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 of RD and RD are strongly correlated, as expected from Eq. (13). The SUSY effects can only enhance the central value of RD() by about 8%, such that the ratios RD() approach, but still lie outside, the 2σ range of the HFLAV averages. Therefore, future refined measurements will provide a crucial test to the RPV SUSY explanation of RD() anomalies. At Belle II, precisions of RD() measurements are expected to be about 2%–4% [71] with a luminosity of 50ab1. Fig. 2(b), it can be seen that both RD and B(B+K+νˉν) deviate from their SM predictions. The lower bound for the latter is B(B+K+νˉν)>7.37×106, which is due to the lower bound of λ323λ333>0.018 obtained in the last section. Compared to the SM prediction B(B+K+νˉν)SM=(3.98±0.47)×106, such significant enhancement makes this decay an important probe of the RPV SUSY effects. In the future, Belle II with 50ab1 data can measure its branching ratio with a precision of 11% [71]. Another interesting correlation arises between B(B+K+νˉν) and gZτLτL/gZLL. As shown in Fig. 2(f), the RPV SUSY effects always enhance B(B+K+νˉν) and suppress gZτLτL/gZLL simultaneously. When gZτLτL/gZLL approaches the SM value 1, the branching ratio of B+K+νˉν maximally deviates from its SM prediction. In Fig. 2(d) and 2(e), we show the correlations involving Bτν decay. The SUSY prediction on B(Bτˉν) is almost in the SM 1σ range. Since the future Belle II sensitivity at 50ab1 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 and RD are shown as the black region, and the 2σ (4σ) experimental region is depicted in gray (light gray). In other figures, the 1σ experimental region is shown in black. The 2σ regions for RD are also depicted in gray.

      Using the allowed parameter space at the 2σ level derived in the last subsection, we make predictions on the five bcτˉν decays, BD()τˉν, Bcηcτˉν, BcJ/ψτˉν, and ΛbΛcτˉν 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 the Bcηcτˉν and ΛbΛcτˉν decays. Table 2 shows that, although the SUSY predictions for the branching fractions and the LFU ratios in these two decays overlap with their 1σ SM range, they can be considerably enhanced by the RPV SUSY effects.

      Now we start to analyze the q2 distributions of the differential branching fraction B, LFU ratio R, lepton forward-backward asymmetry AFB, polarization fraction of τ lepton PτL, and the polarization fraction of daughter meson (PDL, PJ/ψL, PΛcL). For the two “BP” transitions BDτˉν and Bcηcτˉν, 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 in Bcηcτˉν suffer from larger theoretical uncertainties, which are due to the large uncertainties induced by the Bcηc form factors. In the RPV SUSY, the branching fraction of BDτˉν decay can be largely enhanced, while the LFU ratio is almost indistinguishable from the SM prediction. Therefore, it is difficult for the differential distribution of RD(q2) to provide testable signature of the RPV SUSY. Moreover, the RPV SUSY does not affect the forward-backward asymmetry AFB and τ polarization fraction PτL in these two decays, as shown in Fig. 3. The reason behind this is that the RPV couplings only modify the Wilson coefficient CL,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 the RD() anomaly, but involves scalar or tensor interactions [83, 127, 128].

      Figure 3.  (color online) Differential observables in BDτˉν (left) and Bcηcτˉν (right) decays. The black curves (gray band) indicate the SM (SUSY) central values with 1σ theoretical uncertainty.

      The differential observables in the BDτˉν and BcJ/ψτˉν decays are shown in Fig. 4. As expected, these two “BV” 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 than 1σ. However, the LFU ratios RD(q2) and RJ/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 ratios RD(q2) and RJ/ψ(q2) in the RPV SUSY deviate from the SM predictions by about 2σ. Therefore, future measurements on these differential ratios could provide more information about the RD() anomaly and are important for the indirect searches for SUSY. In addition, as in the BDτˉν and Bcηcτˉν decays, the angular observables AFB, PτL and PD,J/ψL are not affected by the SUSY effects.

      Figure 4.  (color online) Differential observables in BDτˉν (left) and BcJ/ψτˉν (right) decays. The black curves (gray band) indicate the SM (SUSY) central values with 1σ theoretical uncertainty.

      Figure 5 shows the differential observables in the ΛbΛcτν decay. The RPV SUSY effects significantly enhance the branching fraction and the LFU ratio. In particular, at the large dilepton invariant mass, the ratio RΛc(q2) in the SUSY exhibits a higher than 2σ discrepancy from the SM values. With large Λb 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 ΛbΛcτˉν decay. Other captions are the same as in Fig. 3.

    5.   Conclusions
    • Recently, several hints of lepton flavor universality violation have been observed in the experimental data of semi-leptonic B decays. Motivated by the recent measurements of PDL, we have investigated the RPV SUSY effects in bcτˉν transitions. After considering various flavor processes, we obtain strong constraints of the RPV couplings, which are dominated by B(B+π+νˉν), B(B+K+νˉν), and gZτLτL. In the surviving parameter space, the RD() anomaly can be explained at the 2σ level, which results in bounds on the coupling products, 0.082<λ313λ333<0.087, 0.018<λ323λ333<0.057, and 0.033<λ333λ333<0.928. The upper bound on the coupling λ333 prevents this coupling from developing a Landau pole below the GUT scale.

      In the parameter space allowed by all the constraints, we make predictions for various flavor processes. For B+K+νˉν decay, a lower bound B(B+K+νˉν)>7.37×106 is obtained. Compared to the SM prediction (3.98±0.47)×106, this decay can provide an important probe of the RPV SUSY effects at Belle II. We also find interesting correlations among RD, RD, B(B+K+νˉν), B(Bτν), and gZτLτL/gZLL. For example, the RPV SUSY effects always enhance B(B+K+νˉν) and suppress gZτLτL/gZLL simultaneously, which makes one of them largely deviate from its SM value.

      Furthermore, we systematically investigated the RPV SUSY effects in five bcτˉν decays, including BD()τˉν, Bcηcτˉν, BcJ/ψτˉν, and ΛbΛcτˉν decays, while focusing on the q2 distributions of the branching fractions, the LFU ratios, and various angular observables. The differential ratios RD(q2), RJ/ψ(q2), and RΛc(q2) are significantly enhanced by the RPV SUSY effects in the large dilepton invariant mass region. Although the integrated ratios RD,J/ψ,Λc in the SUSY overlap with the 1σ range of the SM values, the differential ratios RD,J/ψ,Λc(q2) in this kinematical region exhibit a higher than 2σ discrepancy between the SM and SUSY predictions. In addition, the SM and RPV SUSY predictions of various angular observables are indistinguishable, since the RPV SUSY scenario does not generate new operators beyond the ones of SM.

      The decays B+K+νˉν and Bτˉν, as well as the differential observables in bcτˉν decays, have the potential to shed new light on the RD() anomalies and may serve as a test of the RPV SUSY. With the forthcoming SuperKEKB and the future HL-LHC, our results are expected to provide more information on the bcτˉν transitions and could correlate with the direct searches for SUSY in future high-energy colliders.

      We thank Jun-Kang He, Quan-Yi Hu, Xin-Qiang Li, Han Yan, Min-Di Zheng, and Xin Zhang for useful discussions.

    Appendix A: Form factors
    • For the operator in Eq. (5), the hadronic matrix elements of BD transition can be parameterized in terms of form factors F+ and F0 [28, 102]. In the BGL parameterization, they can be written as expressions of a+n and a0n [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+w2N)/(1+w+2N), w=(m2B+m2Dq2)/(2mBmD), N=(1+r)/(2r), and r=mD/mB. Values of the fit parameters are taken from Ref. [11].

      For the BD transition, the relevant form factors are A0,1,2 and V. They can be written in terms of the BGL form factors as

      A0(q2)=mB+mD2mBmDP1(w),A1(q2)=f(w)mB+mD,A2(q2)=(mB+mD)[(m2Bm2Dq2)f(w)2mDF1(w)]λD(q2),V(q2)=mBmD(mB+mD)w21λD(q2)g(w),

      where w=(m2B+m2Dq2)/2mBmD and λD=[(mBmD)2q2][(mB+mD)2q2]. The four BGL form factors can be expanded as a series in z

      f(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+12)/(w+1+2) and r=mD/mB. Explicit expressions of the Blaschke factors P1± and B0, and the outer functions ϕi(z) can be found in Refs. [14, 129]. We also adopt the values of the fit parameters in Refs. [14, 129].

      The ΛbΛc hadronic matrix elements can be written in terms of the helicity form factors F0,+, and G0,+, [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 form

      f(q2)=11q2/(mfpole)2[af0+af1zf(q2)],

      while the form factor in the “higher-order fit” is given by

      fHO(q2)=11q2/(mfpole)2{af0,HO+af1,HOzf(q2)+af2,HO[zf(q2)]2},

      where zf(q2)=(tf+q2tf+t0)/(tf+q2+tf+t0), t0=(mΛbmΛc)2, and tf+=(mfpole)2. Values of the fit parameters are taken from Ref. [76].

      The form factors for BcJ/ψ and Bcηc transitions are taken from the results in the Covariant Light-Front Approach in Ref. [16].

Reference (130)

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