Processing math: 79%

Phenomenology of bcτˉν decays in a scalar leptoquark model

  • During the past few years, signs of lepton flavor universality (LFU) violation have been observed in bcτˉν and bs+ transitions. Recently, the D and τ polarization fractions PDL and PτL in BDτˉν decay were likewise measured by the Belle collaboration. Motivated by these intriguing results, we revisit the RD() and RK() anomalies in a scalar leptoquark (LQ) model, where two scalar LQs, one of which is a SU(2)L singlet and the other a SU(2)L triplet, are introduced simultaneously. We consider five bcτˉν mediated decays, BD()τˉν, Bcηcτˉν, BcJ/ψτˉν, and ΛbΛcτˉν, and focus on the LQ effects on the q2 distributions of the branching fractions, LFU ratios, and various angular observables in these decays. Under the combined constraints of the available data on RD(), RJ/ψ, PτL(D), and PDL, we perform scans for the LQ couplings and make predictions for a number of observables. Numerically it is found that both the differential branching fractions and LFU ratios are largely enhanced by the LQ effects, with the latter expected to provide testable signatures at the SuperKEKB and High-Luminosity LHC experiments.
  • [1] Y. Li and C.-D. L, Sci. Bull., 63: 267-269 (2018), arXiv:1808.02990 doi: 10.1016/j.scib.2018.02.003
    [2] S. Bifani, S. Descotes-Genon, A. Romero Vidal et al, J. Phys. G, 46(2): 023001 (2019), arXiv:1809.06229
    [3] BaBar Collaboration, J. P. Lees et al, Phys. Rev. Lett., 109: 101802 (2012), arXiv:1205.5442 doi: 10.1103/PhysRevLett.109.101802
    [4] BaBar Collaboration, J. P. Lees et al, Phys. Rev. D, 88(7): 072012 (2013), arXiv:1303.0571
    [5] Belle Collaboration, M. Huschle et al, Phys. Rev. D, 92(7): 072014 (2015), arXiv:1507.03233
    [6] Belle Collaboration, Y. Sato et al, Phys. Rev. D, 94(7): 072007 (2016), arXiv:1607.07923
    [7] Belle Collaboration, S. Hirose et al, Phys. Rev. Lett., 118(21): 211801 (2017), arXiv:1612.00529 doi: 10.1103/PhysRevLett.118.211801
    [8] Belle Collaboration, S. Hirose et al, Phys. Rev. D, 97(10): 12004 (2018), arXiv:1709.00129
    [9] LHCb Collaboration, R. Aaij et al, Phys. Rev. Lett., 115(11): 111803 (2015), arXiv:1506.08614 doi: 10.1103/PhysRevLett.115.111803
    [10] LHCb Collaboration, R. Aaij et al, Phys. Rev. Lett., 120(17): 171802 (2018), arXiv:1708.08856 doi: 10.1103/PhysRevLett.120.171802
    [11] LHCb Collaboration, R. Aaij et al, Phys. Rev. D, 97(7): 072013 (2018), arXiv:1711.02505
    [12] Heavy Flavor Averaging Group Collaboration, Y. Amhis et al, Eur. Phys. J. C, 77: 895 (2017), arXiv:1612.07233
    [13] D. Bigi and P. Gambino, Phys. Rev. D, 94(9): 094008 (2016), arXiv:1606.08030
    [14] S. Jaiswal, S. Nandi, and S. K. Patra, JHEP, 12: 060 (2017), arXiv:1707.09977
    [15] F. U. Bernlochner, Z. Ligeti, M. Papucci et al, Phys. Rev. D, 95(11): 115008 (2017), arXiv:1703.05330
    [16] D. Bigi, P. Gambino, and S. Schacht, JHEP, 11: 061 (2017), arXiv:1707.09509
    [17] LHCb Collaboration, R. Aaij et al, Phys. Rev. Lett., 120(12): 121801 (2018), arXiv:1711.05623 doi: 10.1103/PhysRevLett.120.121801
    [18] W. Wang, Y.-L. Shen, and C.-D. Lu, Phys. Rev. D, 79: 054012 (2009), arXiv:0811.3748
    [19] LHCb Collaboration, R. Aaij et al, Phys. Rev. Lett., 113: 151601 (2014), arXiv:1406.6482 doi: 10.1103/PhysRevLett.113.151601
    [20] LHCb Collaboration, R. Aaij et al, JHEP, 08: 055 (2017), arXiv:1705.05802
    [21] G. Hiller and F. Kruger, Phys. Rev. D, 69: 074020 (2004), arXiv:hep-ph/0310219
    [22] M. Bordone, G. Isidori, and A. Pattori, Eur. Phys. J. C, 76(8): 440 (2016), arXiv:1605.07633
    [23] Y. Sakaki, M. Tanaka, A. Tayduganov et al, Phys. Rev. D, 91(11): 114028 (2015), arXiv:1412.3761
    [24] B. Bhattacharya, A. Datta, D. London et al, Phys. Lett. B, 742: 370-374 (2015), arXiv:1412.7164
    [25] L. Calibbi, A. Crivellin, and T. Ota, Phys. Rev. Lett., 115: 181801 (2015), arXiv:1506.02661 doi: 10.1103/PhysRevLett.115.181801
    [26] F. Feruglio, P. Paradisi, and A. Pattori, Phys. Rev. Lett., 118(1): 011801 (2017), arXiv:1606.00524 doi: 10.1103/PhysRevLett.118.011801
    [27] D. Choudhury, A. Kundu, R. Mandal et al, Phys. Rev. Lett., 119(15): 151801 (2017), arXiv:1706.08437 doi: 10.1103/PhysRevLett.119.151801
    [28] Q.-Y. Hu, X.-Q. Li, and Y.-D. Yang, bcτν Transitions in the Standard Model Effective Field Theory, arXiv: 1810.04939
    [29] A. Celis, M. Jung, X.-Q. Li et al, JHEP, 01: 054 (2013), arXiv:1210.8443
    [30] N. G. Deshpande and X.-G. He, Eur. Phys. J. C, 77(2): 134 (2017), arXiv:1608.04817
    [31] A. Celis, M. Jung, X.-Q. Li et al, Phys. Lett. B, 771: 168-179 (2017), arXiv:1612.07757
    [32] J. Zhu, B. Wei, J.-H. Sheng et al, Nucl. Phys. B, 934: 380-395 (2018), arXiv:1801.00917
    [33] S.-P. Li, X.-Q. Li, Y.-D. Yang et al, JHEP, 09: 149 (2018), arXiv:1807.08530
    [34] Q.-Y. Hu, X.-Q. Li, Y. Muramatsu et al, Phys. Rev. D, 99(1): 015008 (2019), arXiv:1808.01419
    [35] W. Altmannshofer, P. S. Bhupal Dev, and A. Soni, Phys. Rev. D, 96(9): 095010 (2017), arXiv:1704.06659
    [36] A. K. Alok, D. Kumar, S. Kumbhakar et al, arXiv: 1903.10486
    [37] T. Mandal, S. Mitra, and S. Raz, Phys. Rev. D, 99(5): 055028 (2019), arXiv:1811.03561
    [38] C.-T. Tran, M. A. Ivanov, J. G. Körner et al, Phys. Rev. D, 97(5): 054014 (2018), arXiv:1801.06927
    [39] D. Bečirević, S. Fajfer, N. Košnik et al, Phys. Rev. D, 94(11): 115021 (2016), arXiv:1608.08501
    [40] D. Das, C. Hati, G. Kumar et al, Phys. Rev. D, 96(9): 095033 (2017), arXiv:1705.09188
    [41] M. Blanke and A. Crivellin, Phys. Rev. Lett., 121(1): 011801 (2018), arXiv:1801.07256
    [42] N. Assad, B. Fornal, and B. Grinstein, Phys. Lett. B, 777: 324-331 (2018), arXiv:1708.06350
    [43] K. Azizi, Y. Sarac, and H. Sundu, arXiv: 1904.08267
    [44] R. Dutta, A. Bhol, and A. K. Giri, Phys. Rev. D, 88(11): 114023 (2013), arXiv:1307.6653
    [45] B. Fornal, S. A. Gadam, and B. Grinstein, Phys. Rev. D, 99(5): 055025 (2019), arXiv:1812.01603
    [46] Belle, Belle II Collaboration, K. Adamczyk, Semitauonic B decays at Belle/Belle II, in 10th International Workshop on the CKM Unitarity Triangle (CKM 2018) Heidelberg, Germany, September 17-21, 2018, 2019. arXiv: 1901.06380
    [47] Belle Collaboration, A. Abdesselam et al, Measurement of the D*− polarization in the decay B0D*−τ+ντ, arXiv: 1903.03102
    [48] A. K. Alok, D. Kumar, S. Kumbhakar et al, Phys. Rev. D, 95(11): 115038 (2017), arXiv:1606.03164
    [49] Z.-R. Huang, Y. Li, C.-D. Lu et al, Phys. Rev. D, 98(9): 095018 (2018), arXiv:1808.03565
    [50] J. Aebischer, J. Kumar, P. Stangl et al, A Global Likelihood for Precision Constraints and Flavour Anomalies, arXiv: 1810.07698
    [51] M. Blanke, A. Crivellin, S. de Boer et al, Impact of polarization observables and Bcτν on new physics explanations of the bcτν anomaly, arXiv: 1811.09603
    [52] S. Iguro, T. Kitahara, Y. Omura et al, JHEP, 02: 194 (2019), arXiv:1811.08899
    [53] M. Tanaka and R. Watanabe, Phys. Rev. D, 87(3): 034028 (2013), arXiv:1212.1878
    [54] Belle II Collaboration, W. Altmannshofer et al, The Belle II Physics Book, arXiv: 1808.10567
    [55] LHCb Collaboration, R. Aaij et al., Physics case for an LHCb Upgrade II - Opportunities in flavour physics, and beyond, in the HL-LHC era, arXiv: 1808.08865
    [56] A. Crivellin, D. Mller, and T. Ota, JHEP, 09: 040 (2017), arXiv:1703.09226
    [57] I. Dorner, S. Fajfer, A. Greljo et al, Phys. Rept., 641: 1-68 (2016), arXiv:1603.04993 doi: 10.1016/j.physrep.2016.06.001
    [58] M. Freytsis, Z. Ligeti, and J. T. Ruderman, Phys. Rev. D, 92(5): 054018 (2015), arXiv:1506.08896
    [59] M. Bauer and M. Neubert, Phys. Rev. Lett., 116(14): 141802 (2016), arXiv:1511.01900 doi: 10.1103/PhysRevLett.116.141802
    [60] X.-Q. Li, Y.-D. Yang, and X. Zhang, JHEP, 02: 068 (2017), arXiv:1611.01635
    [61] S. Fajfer and N. Konik, Phys. Lett. B, 755: 270-274 (2016), arXiv:1511.06024
    [62] F. F. Deppisch, S. Kulkarni, H. Ps et al, Phys. Rev. D, 94(1): 013003 (2016), arXiv:1603.07672
    [63] B. Dumont, K. Nishiwaki, and R. Watanabe, Phys. Rev. D, 94(3): 034001 (2016), arXiv:1603.05248
    [64] L. Di Luzio, A. Greljo, and M. Nardecchia, Phys. Rev. D, 96(11): 115011 (2017), arXiv:1708.08450
    [65] Y. Cai, J. Gargalionis, M. A. Schmidt et al, JHEP, 10: 047 (2017), arXiv:1704.05849
    [66] L. Calibbi, A. Crivellin, and T. Li, Phys. Rev. D, 98(11): 115002 (2018), arXiv:1709.00692
    [67] J. Kumar, D. London, and R. Watanabe, Phys. Rev. D, 99(1): 015007 (2019), arXiv:1806.07403
    [68] C. Hati, G. Kumar, J. Orloff et al, JHEP, 11: 011 (2018), arXiv:1806.10146
    [69] A. Crivellin, C. Greub, D. Mller et al, Phys. Rev. Lett., 122(1): 011805 (2019), arXiv:1807.02068 doi: 10.1103/PhysRevLett.122.011805
    [70] A. Angelescu, D. Beirevi, D. A. Faroughy et al, JHEP, 10: 183 (2018), arXiv:1808.08179
    [71] T. J. Kim, P. Ko, J. Li et al, Correlation between RD(*) and top quark FCNC decays in leptoquark models, arXiv: 1812.08484
    [72] D. Marzocca, JHEP, 07: 121 (2018), arXiv:1803.10972
    [73] D. Buttazzo, A. Greljo, G. Isidori et al, JHEP, 11: 044 (2017), arXiv:1706.07808
    [74] P. Arnan, D. Becirevic, F. Mescia et al, JHEP, 02: 109 (2019), arXiv:1901.06315
    [75] W. Detmold, C. Lehner, and S. Meinel, Phys. Rev. D, 92(3): 034503 (2015), arXiv:1503.01421
    [76] A. Datta, S. Kamali, S. Meinel et al, JHEP, 08: 131 (2017), arXiv:1702.02243
    [77] Flavour Lattice Averaging Group Collaboration, S. Aoki et al, FLAG Review 2019, arXiv: 1902.08191
    [78] A. Cerri et al, Opportunities in Flavour Physics at the HL-LHC and HE-LHC, arXiv: 1812.07638
    [79] K. Hagiwara, A. D. Martin, and M. F. Wade, Nucl. Phys. B, 327: 569-594 (1989)
    [80] F. U. Bernlochner, Z. Ligeti, and D. J. Robinson, N = 5, 6, 7, 8: Nested hypothesis tests and truncation dependence of |Vcb|, arXiv: 1902.09553
    [81] Y.-M. Wang, Y.-B. Wei, Y.-L. Shen et al, JHEP, 06: 062 (2017), arXiv:1701.06810
    [82] N. Gubernari, A. Kokulu, and D. van Dyk, JHEP, 01: 150 (2019), arXiv:1811.00983
    [83] C. W. Murphy and A. Soni, Phys. Rev. D, 98(9): 094026 (2018), arXiv:1808.05932
    [84] A. Berns and H. Lamm, JHEP, 12: 114 (2018), arXiv:1808.07360
    [85] W. Wang and R. Zhu, Model independent investigation of the RJ/ψ, ηc and ratios of decay widths of semileptonic Bc decays into a P-wave charmonium, arXiv: 1808.10830
    [86] D. Leljak, B. Melic, and M. Patra, On lepton flavour universality in semileptonic Bcηc, J/ψ decays, arXiv: 1901.08368
    [87] T. Gutsche, M. A. Ivanov, J. G. Krner et al, Phys. Rev. D, 91(7): 074001 (2015), arXiv:1502.04864
    [88] C. G. Boyd, B. Grinstein, and R. F. Lebed, Phys. Rev. D, 56: 6895-6911 (1997), arXiv:hep-ph/9705252
    [89] I. Caprini, L. Lellouch, and M. Neubert, Nucl. Phys. B, 530: 153-181 (1998), arXiv:hep-ph/9712417
    [90] G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Rev. Mod. Phys., 68: 1125-1144 (1996), arXiv:hep-ph/9512380 doi: 10.1103/RevModPhys.68.1125
    [91] W. Altmannshofer and D. M. Straub, Implications of b → s measurements, in Proceedings, 50th Rencontres de Moriond Electroweak Interactions and Unified Theories: La Thuile, Italy, March 14-21, 2015, pp. 333–338, 2015. arXiv: 1503.06199
    [92] S. Descotes-Genon, L. Hofer, J. Matias et al, JHEP, 06: 092 (2016), arXiv:1510.04239
    [93] T. Hurth, F. Mahmoudi, and S. Neshatpour, Nucl. Phys. B, 909: 737-777 (2016), arXiv:1603.00865
    [94] Muon g-2 Collaboration, G. W. Bennett et al, Phys. Rev. D, 73: 072003 (2006), arXiv:hep-ex/0602035
    [95] Particle Data Group Collaboration, M. Tanabashi et al, Phys. Rev. D, 98(3): 030001 (2018)
    [96] CMS Collaboration, A. M. Sirunyan et al, Phys. Rev. Lett., 121(24): 241802 (2018), arXiv:1809.05558 doi: 10.1103/PhysRevLett.121.241802
    [97] ATLAS Collaboration, M. Aaboud et al, Searches for third-generation scalar leptoquarks in \begin{document}$ \sqrt s $\end{document} = 13 TeV pp collisions with the ATLAS detector, arXiv: 1902.08103
    [98] CKMfitter Group Collaboration, J. Charles, A. Hocker, H. Lacker, S. Laplace, F. R. Le Diberder, J. Malcles, J. Ocariz, M. Pivk, and L. Roos, Eur. Phys. J. C, 41(1): 1-131 (2005)
    [99] M. Jung, X.-Q. Li, and A. Pich, JHEP, 10: 063 (2012), arXiv:1208.1251
    [100] C.-W. Chiang, X.-G. He, F. Ye et al, Phys. Rev. D, 96(3): 035032 (2017), arXiv:1703.06289
    [101] LHCb Collaboration, R. Aaij et al, Phys. Rev. Lett., 118(25): 251802 (2017), arXiv:1703.02508 doi: 10.1103/PhysRevLett.118.251802
    [102] C. Bobeth, M. Gorbahn, T. Hermann et al, Phys. Rev. Lett., 112: 101801 (2014), arXiv:1311.0903 doi: 10.1103/PhysRevLett.112.101801
    [103] J. Albrecht, F. Bernlochner, M. Kenzie et al, Future prospects for exploring present day anomalies in flavour physics measurements with Belle II and LHCb, arXiv: 1709.10308
    [104] J. F. Kamenik, S. Monteil, A. Semkiv et al, Eur. Phys. J. C, 77(10): 701 (2017), arXiv:1705.11106
    [105] Belle Collaboration, G. Caria, Measurement of R(D) and R(D*) with a semileptonic tag at Belle, Talk at ‘54th Rencontres de Moriond, Electroweak Interactions and Unified Theories, 2019’
    [106] elle Collaboration, A. Abdesselam et al., Measurement of \begin{document}$ \cal{R}$\end{document}(D) and \begin{document}$ \cal{R}$\end{document}(D*) with a semileptonic tagging method, arXiv: 1904.08794
    [107] C. Murgui, A. Peuelas, M. Jung et al, Global fit to bcτν transitions, arXiv: 1904.09311
    [108] J. G. Korner and G. A. Schuler, Z. Phys. C, 46: 93 (1990)
    [109] H. E. Haber, Spin formalism and applications to new physics searches, in Spin structure in high-energy processes: Proceedings, 21st SLAC Summer Institute on Particle Physics, 26 Jul - 6 Aug 1993, Stanford, CA, pp. 231–272, 1994. hep-ph/9405376
    [110] Y. Sakaki, M. Tanaka, A. Tayduganov et al, Phys. Rev. D, 88(9): 094012 (2013), arXiv:1309.0301
    [111] D. Bardhan, P. Byakti, and D. Ghosh, JHEP, 01: 125 (2017), arXiv:1610.03038
  • [1] Y. Li and C.-D. L, Sci. Bull., 63: 267-269 (2018), arXiv:1808.02990 doi: 10.1016/j.scib.2018.02.003
    [2] S. Bifani, S. Descotes-Genon, A. Romero Vidal et al, J. Phys. G, 46(2): 023001 (2019), arXiv:1809.06229
    [3] BaBar Collaboration, J. P. Lees et al, Phys. Rev. Lett., 109: 101802 (2012), arXiv:1205.5442 doi: 10.1103/PhysRevLett.109.101802
    [4] BaBar Collaboration, J. P. Lees et al, Phys. Rev. D, 88(7): 072012 (2013), arXiv:1303.0571
    [5] Belle Collaboration, M. Huschle et al, Phys. Rev. D, 92(7): 072014 (2015), arXiv:1507.03233
    [6] Belle Collaboration, Y. Sato et al, Phys. Rev. D, 94(7): 072007 (2016), arXiv:1607.07923
    [7] Belle Collaboration, S. Hirose et al, Phys. Rev. Lett., 118(21): 211801 (2017), arXiv:1612.00529 doi: 10.1103/PhysRevLett.118.211801
    [8] Belle Collaboration, S. Hirose et al, Phys. Rev. D, 97(10): 12004 (2018), arXiv:1709.00129
    [9] LHCb Collaboration, R. Aaij et al, Phys. Rev. Lett., 115(11): 111803 (2015), arXiv:1506.08614 doi: 10.1103/PhysRevLett.115.111803
    [10] LHCb Collaboration, R. Aaij et al, Phys. Rev. Lett., 120(17): 171802 (2018), arXiv:1708.08856 doi: 10.1103/PhysRevLett.120.171802
    [11] LHCb Collaboration, R. Aaij et al, Phys. Rev. D, 97(7): 072013 (2018), arXiv:1711.02505
    [12] Heavy Flavor Averaging Group Collaboration, Y. Amhis et al, Eur. Phys. J. C, 77: 895 (2017), arXiv:1612.07233
    [13] D. Bigi and P. Gambino, Phys. Rev. D, 94(9): 094008 (2016), arXiv:1606.08030
    [14] S. Jaiswal, S. Nandi, and S. K. Patra, JHEP, 12: 060 (2017), arXiv:1707.09977
    [15] F. U. Bernlochner, Z. Ligeti, M. Papucci et al, Phys. Rev. D, 95(11): 115008 (2017), arXiv:1703.05330
    [16] D. Bigi, P. Gambino, and S. Schacht, JHEP, 11: 061 (2017), arXiv:1707.09509
    [17] LHCb Collaboration, R. Aaij et al, Phys. Rev. Lett., 120(12): 121801 (2018), arXiv:1711.05623 doi: 10.1103/PhysRevLett.120.121801
    [18] W. Wang, Y.-L. Shen, and C.-D. Lu, Phys. Rev. D, 79: 054012 (2009), arXiv:0811.3748
    [19] LHCb Collaboration, R. Aaij et al, Phys. Rev. Lett., 113: 151601 (2014), arXiv:1406.6482 doi: 10.1103/PhysRevLett.113.151601
    [20] LHCb Collaboration, R. Aaij et al, JHEP, 08: 055 (2017), arXiv:1705.05802
    [21] G. Hiller and F. Kruger, Phys. Rev. D, 69: 074020 (2004), arXiv:hep-ph/0310219
    [22] M. Bordone, G. Isidori, and A. Pattori, Eur. Phys. J. C, 76(8): 440 (2016), arXiv:1605.07633
    [23] Y. Sakaki, M. Tanaka, A. Tayduganov et al, Phys. Rev. D, 91(11): 114028 (2015), arXiv:1412.3761
    [24] B. Bhattacharya, A. Datta, D. London et al, Phys. Lett. B, 742: 370-374 (2015), arXiv:1412.7164
    [25] L. Calibbi, A. Crivellin, and T. Ota, Phys. Rev. Lett., 115: 181801 (2015), arXiv:1506.02661 doi: 10.1103/PhysRevLett.115.181801
    [26] F. Feruglio, P. Paradisi, and A. Pattori, Phys. Rev. Lett., 118(1): 011801 (2017), arXiv:1606.00524 doi: 10.1103/PhysRevLett.118.011801
    [27] D. Choudhury, A. Kundu, R. Mandal et al, Phys. Rev. Lett., 119(15): 151801 (2017), arXiv:1706.08437 doi: 10.1103/PhysRevLett.119.151801
    [28] Q.-Y. Hu, X.-Q. Li, and Y.-D. Yang, bcτν Transitions in the Standard Model Effective Field Theory, arXiv: 1810.04939
    [29] A. Celis, M. Jung, X.-Q. Li et al, JHEP, 01: 054 (2013), arXiv:1210.8443
    [30] N. G. Deshpande and X.-G. He, Eur. Phys. J. C, 77(2): 134 (2017), arXiv:1608.04817
    [31] A. Celis, M. Jung, X.-Q. Li et al, Phys. Lett. B, 771: 168-179 (2017), arXiv:1612.07757
    [32] J. Zhu, B. Wei, J.-H. Sheng et al, Nucl. Phys. B, 934: 380-395 (2018), arXiv:1801.00917
    [33] S.-P. Li, X.-Q. Li, Y.-D. Yang et al, JHEP, 09: 149 (2018), arXiv:1807.08530
    [34] Q.-Y. Hu, X.-Q. Li, Y. Muramatsu et al, Phys. Rev. D, 99(1): 015008 (2019), arXiv:1808.01419
    [35] W. Altmannshofer, P. S. Bhupal Dev, and A. Soni, Phys. Rev. D, 96(9): 095010 (2017), arXiv:1704.06659
    [36] A. K. Alok, D. Kumar, S. Kumbhakar et al, arXiv: 1903.10486
    [37] T. Mandal, S. Mitra, and S. Raz, Phys. Rev. D, 99(5): 055028 (2019), arXiv:1811.03561
    [38] C.-T. Tran, M. A. Ivanov, J. G. Körner et al, Phys. Rev. D, 97(5): 054014 (2018), arXiv:1801.06927
    [39] D. Bečirević, S. Fajfer, N. Košnik et al, Phys. Rev. D, 94(11): 115021 (2016), arXiv:1608.08501
    [40] D. Das, C. Hati, G. Kumar et al, Phys. Rev. D, 96(9): 095033 (2017), arXiv:1705.09188
    [41] M. Blanke and A. Crivellin, Phys. Rev. Lett., 121(1): 011801 (2018), arXiv:1801.07256
    [42] N. Assad, B. Fornal, and B. Grinstein, Phys. Lett. B, 777: 324-331 (2018), arXiv:1708.06350
    [43] K. Azizi, Y. Sarac, and H. Sundu, arXiv: 1904.08267
    [44] R. Dutta, A. Bhol, and A. K. Giri, Phys. Rev. D, 88(11): 114023 (2013), arXiv:1307.6653
    [45] B. Fornal, S. A. Gadam, and B. Grinstein, Phys. Rev. D, 99(5): 055025 (2019), arXiv:1812.01603
    [46] Belle, Belle II Collaboration, K. Adamczyk, Semitauonic B decays at Belle/Belle II, in 10th International Workshop on the CKM Unitarity Triangle (CKM 2018) Heidelberg, Germany, September 17-21, 2018, 2019. arXiv: 1901.06380
    [47] Belle Collaboration, A. Abdesselam et al, Measurement of the D*− polarization in the decay B0D*−τ+ντ, arXiv: 1903.03102
    [48] A. K. Alok, D. Kumar, S. Kumbhakar et al, Phys. Rev. D, 95(11): 115038 (2017), arXiv:1606.03164
    [49] Z.-R. Huang, Y. Li, C.-D. Lu et al, Phys. Rev. D, 98(9): 095018 (2018), arXiv:1808.03565
    [50] J. Aebischer, J. Kumar, P. Stangl et al, A Global Likelihood for Precision Constraints and Flavour Anomalies, arXiv: 1810.07698
    [51] M. Blanke, A. Crivellin, S. de Boer et al, Impact of polarization observables and Bcτν on new physics explanations of the bcτν anomaly, arXiv: 1811.09603
    [52] S. Iguro, T. Kitahara, Y. Omura et al, JHEP, 02: 194 (2019), arXiv:1811.08899
    [53] M. Tanaka and R. Watanabe, Phys. Rev. D, 87(3): 034028 (2013), arXiv:1212.1878
    [54] Belle II Collaboration, W. Altmannshofer et al, The Belle II Physics Book, arXiv: 1808.10567
    [55] LHCb Collaboration, R. Aaij et al., Physics case for an LHCb Upgrade II - Opportunities in flavour physics, and beyond, in the HL-LHC era, arXiv: 1808.08865
    [56] A. Crivellin, D. Mller, and T. Ota, JHEP, 09: 040 (2017), arXiv:1703.09226
    [57] I. Dorner, S. Fajfer, A. Greljo et al, Phys. Rept., 641: 1-68 (2016), arXiv:1603.04993 doi: 10.1016/j.physrep.2016.06.001
    [58] M. Freytsis, Z. Ligeti, and J. T. Ruderman, Phys. Rev. D, 92(5): 054018 (2015), arXiv:1506.08896
    [59] M. Bauer and M. Neubert, Phys. Rev. Lett., 116(14): 141802 (2016), arXiv:1511.01900 doi: 10.1103/PhysRevLett.116.141802
    [60] X.-Q. Li, Y.-D. Yang, and X. Zhang, JHEP, 02: 068 (2017), arXiv:1611.01635
    [61] S. Fajfer and N. Konik, Phys. Lett. B, 755: 270-274 (2016), arXiv:1511.06024
    [62] F. F. Deppisch, S. Kulkarni, H. Ps et al, Phys. Rev. D, 94(1): 013003 (2016), arXiv:1603.07672
    [63] B. Dumont, K. Nishiwaki, and R. Watanabe, Phys. Rev. D, 94(3): 034001 (2016), arXiv:1603.05248
    [64] L. Di Luzio, A. Greljo, and M. Nardecchia, Phys. Rev. D, 96(11): 115011 (2017), arXiv:1708.08450
    [65] Y. Cai, J. Gargalionis, M. A. Schmidt et al, JHEP, 10: 047 (2017), arXiv:1704.05849
    [66] L. Calibbi, A. Crivellin, and T. Li, Phys. Rev. D, 98(11): 115002 (2018), arXiv:1709.00692
    [67] J. Kumar, D. London, and R. Watanabe, Phys. Rev. D, 99(1): 015007 (2019), arXiv:1806.07403
    [68] C. Hati, G. Kumar, J. Orloff et al, JHEP, 11: 011 (2018), arXiv:1806.10146
    [69] A. Crivellin, C. Greub, D. Mller et al, Phys. Rev. Lett., 122(1): 011805 (2019), arXiv:1807.02068 doi: 10.1103/PhysRevLett.122.011805
    [70] A. Angelescu, D. Beirevi, D. A. Faroughy et al, JHEP, 10: 183 (2018), arXiv:1808.08179
    [71] T. J. Kim, P. Ko, J. Li et al, Correlation between RD(*) and top quark FCNC decays in leptoquark models, arXiv: 1812.08484
    [72] D. Marzocca, JHEP, 07: 121 (2018), arXiv:1803.10972
    [73] D. Buttazzo, A. Greljo, G. Isidori et al, JHEP, 11: 044 (2017), arXiv:1706.07808
    [74] P. Arnan, D. Becirevic, F. Mescia et al, JHEP, 02: 109 (2019), arXiv:1901.06315
    [75] W. Detmold, C. Lehner, and S. Meinel, Phys. Rev. D, 92(3): 034503 (2015), arXiv:1503.01421
    [76] A. Datta, S. Kamali, S. Meinel et al, JHEP, 08: 131 (2017), arXiv:1702.02243
    [77] Flavour Lattice Averaging Group Collaboration, S. Aoki et al, FLAG Review 2019, arXiv: 1902.08191
    [78] A. Cerri et al, Opportunities in Flavour Physics at the HL-LHC and HE-LHC, arXiv: 1812.07638
    [79] K. Hagiwara, A. D. Martin, and M. F. Wade, Nucl. Phys. B, 327: 569-594 (1989)
    [80] F. U. Bernlochner, Z. Ligeti, and D. J. Robinson, N = 5, 6, 7, 8: Nested hypothesis tests and truncation dependence of |Vcb|, arXiv: 1902.09553
    [81] Y.-M. Wang, Y.-B. Wei, Y.-L. Shen et al, JHEP, 06: 062 (2017), arXiv:1701.06810
    [82] N. Gubernari, A. Kokulu, and D. van Dyk, JHEP, 01: 150 (2019), arXiv:1811.00983
    [83] C. W. Murphy and A. Soni, Phys. Rev. D, 98(9): 094026 (2018), arXiv:1808.05932
    [84] A. Berns and H. Lamm, JHEP, 12: 114 (2018), arXiv:1808.07360
    [85] W. Wang and R. Zhu, Model independent investigation of the RJ/ψ, ηc and ratios of decay widths of semileptonic Bc decays into a P-wave charmonium, arXiv: 1808.10830
    [86] D. Leljak, B. Melic, and M. Patra, On lepton flavour universality in semileptonic Bcηc, J/ψ decays, arXiv: 1901.08368
    [87] T. Gutsche, M. A. Ivanov, J. G. Krner et al, Phys. Rev. D, 91(7): 074001 (2015), arXiv:1502.04864
    [88] C. G. Boyd, B. Grinstein, and R. F. Lebed, Phys. Rev. D, 56: 6895-6911 (1997), arXiv:hep-ph/9705252
    [89] I. Caprini, L. Lellouch, and M. Neubert, Nucl. Phys. B, 530: 153-181 (1998), arXiv:hep-ph/9712417
    [90] G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Rev. Mod. Phys., 68: 1125-1144 (1996), arXiv:hep-ph/9512380 doi: 10.1103/RevModPhys.68.1125
    [91] W. Altmannshofer and D. M. Straub, Implications of b → s measurements, in Proceedings, 50th Rencontres de Moriond Electroweak Interactions and Unified Theories: La Thuile, Italy, March 14-21, 2015, pp. 333–338, 2015. arXiv: 1503.06199
    [92] S. Descotes-Genon, L. Hofer, J. Matias et al, JHEP, 06: 092 (2016), arXiv:1510.04239
    [93] T. Hurth, F. Mahmoudi, and S. Neshatpour, Nucl. Phys. B, 909: 737-777 (2016), arXiv:1603.00865
    [94] Muon g-2 Collaboration, G. W. Bennett et al, Phys. Rev. D, 73: 072003 (2006), arXiv:hep-ex/0602035
    [95] Particle Data Group Collaboration, M. Tanabashi et al, Phys. Rev. D, 98(3): 030001 (2018)
    [96] CMS Collaboration, A. M. Sirunyan et al, Phys. Rev. Lett., 121(24): 241802 (2018), arXiv:1809.05558 doi: 10.1103/PhysRevLett.121.241802
    [97] ATLAS Collaboration, M. Aaboud et al, Searches for third-generation scalar leptoquarks in \begin{document}$ \sqrt s $\end{document} = 13 TeV pp collisions with the ATLAS detector, arXiv: 1902.08103
    [98] CKMfitter Group Collaboration, J. Charles, A. Hocker, H. Lacker, S. Laplace, F. R. Le Diberder, J. Malcles, J. Ocariz, M. Pivk, and L. Roos, Eur. Phys. J. C, 41(1): 1-131 (2005)
    [99] M. Jung, X.-Q. Li, and A. Pich, JHEP, 10: 063 (2012), arXiv:1208.1251
    [100] C.-W. Chiang, X.-G. He, F. Ye et al, Phys. Rev. D, 96(3): 035032 (2017), arXiv:1703.06289
    [101] LHCb Collaboration, R. Aaij et al, Phys. Rev. Lett., 118(25): 251802 (2017), arXiv:1703.02508 doi: 10.1103/PhysRevLett.118.251802
    [102] C. Bobeth, M. Gorbahn, T. Hermann et al, Phys. Rev. Lett., 112: 101801 (2014), arXiv:1311.0903 doi: 10.1103/PhysRevLett.112.101801
    [103] J. Albrecht, F. Bernlochner, M. Kenzie et al, Future prospects for exploring present day anomalies in flavour physics measurements with Belle II and LHCb, arXiv: 1709.10308
    [104] J. F. Kamenik, S. Monteil, A. Semkiv et al, Eur. Phys. J. C, 77(10): 701 (2017), arXiv:1705.11106
    [105] Belle Collaboration, G. Caria, Measurement of R(D) and R(D*) with a semileptonic tag at Belle, Talk at ‘54th Rencontres de Moriond, Electroweak Interactions and Unified Theories, 2019’
    [106] elle Collaboration, A. Abdesselam et al., Measurement of \begin{document}$ \cal{R}$\end{document}(D) and \begin{document}$ \cal{R}$\end{document}(D*) with a semileptonic tagging method, arXiv: 1904.08794
    [107] C. Murgui, A. Peuelas, M. Jung et al, Global fit to bcτν transitions, arXiv: 1904.09311
    [108] J. G. Korner and G. A. Schuler, Z. Phys. C, 46: 93 (1990)
    [109] H. E. Haber, Spin formalism and applications to new physics searches, in Spin structure in high-energy processes: Proceedings, 21st SLAC Summer Institute on Particle Physics, 26 Jul - 6 Aug 1993, Stanford, CA, pp. 231–272, 1994. hep-ph/9405376
    [110] Y. Sakaki, M. Tanaka, A. Tayduganov et al, Phys. Rev. D, 88(9): 094012 (2013), arXiv:1309.0301
    [111] D. Bardhan, P. Byakti, and D. Ghosh, JHEP, 01: 125 (2017), arXiv:1610.03038
  • 加载中

Cited by

1. Mahata, S., Mandal, M., Mahapatra, H. et al. Model dependent analysis of decays in beyond standard model[J]. Chinese Physics C, 2024, 48(9): 093106. doi: 10.1088/1674-1137/ad5426
2. Banik, S., Crivellin, A. Renormalization group evolution with scalar leptoquarks[J]. Journal of High Energy Physics, 2023, 2023(11): 121. doi: 10.1007/JHEP11(2023)121
3. Bolaños-Carrera, A., Sánchez-Vélez, R., Tavares-Velasco, G. Rare decay t →cγγ via scalar leptoquark doublets[J]. Physical Review D, 2023, 107(9): 095018. doi: 10.1103/PhysRevD.107.095018
4. Feng, J.L., Kling, F., Reno, M.H. et al. The Forward Physics Facility at the High-Luminosity LHC[J]. Journal of Physics G: Nuclear and Particle Physics, 2023, 50(3): 030501. doi: 10.1088/1361-6471/ac865e
5. Bhaskar, A., Madathil, A.A., Mandal, T. et al. Combined explanation of W -mass, muon g-2, RK (∗) and RD (∗) anomalies in a singlet-triplet scalar leptoquark model[J]. Physical Review D, 2022, 106(11): 115009. doi: 10.1103/PhysRevD.106.115009
6. Crivellin, A., Fuks, B., Schnell, L. Explaining the hints for lepton flavour universality violation with three S 2 leptoquark generations[J]. Journal of High Energy Physics, 2022, 2022(6): 169. doi: 10.1007/JHEP06(2022)169
7. Husek, T., Monsálvez-Pozo, K., Portolés, J. Constraints on leptoquarks from lepton-flavour-violating tau-lepton processes[J]. Journal of High Energy Physics, 2022, 2022(4): 165. doi: 10.1007/JHEP04(2022)165
8. Das, N., Dutta, R. New physics footprints in the angular distribution of Bs → Ds∗ (→Dsγ, Dsπ)τν decays[J]. Physical Review D, 2022, 105(5): 055027. doi: 10.1103/PhysRevD.105.055027
9. Crivellin, A., Eguren, J.F., Virto, J. Next-to-leading-order QCD matching for ∆F = 2 processes in scalar leptoquark models[J]. Journal of High Energy Physics, 2022, 2022(3): 185. doi: 10.1007/JHEP03(2022)185
10. Carvunis, A., Gangal, S., Crivellin, A. et al. Forward-backward asymmetry in B →d∗ℓν: One more hint for scalar leptoquarks?[J]. Physical Review D, 2022, 105(3): L031701. doi: 10.1103/PhysRevD.105.L031701
11. Crivellin, A., Schnell, L. Complete Lagrangian and set of Feynman rules for scalar leptoquarks[J]. Computer Physics Communications, 2022. doi: 10.1016/j.cpc.2021.108188
12. Sheng, J.-H., Hu, Q.-Y., Zhu, J. Investigation of the effects of some new physics models in semileptonic Ξb and Ωb decays[J]. Journal of Physics G: Nuclear and Particle Physics, 2022, 49(1): 015001. doi: 10.1088/1361-6471/ac0cc8
13. Hati, C., Kriewald, J., Orloff, J. et al. The fate of V1 vector leptoquarks: the impact of future flavour data[J]. European Physical Journal C, 2021, 81(12): 1066. doi: 10.1140/epjc/s10052-021-09824-z
14. Crivellin, A., Schnell, L. Addendum to "combined constraints on first generation leptoquarks"[J]. Physical Review D, 2021, 104(5): 055020. doi: 10.1103/PhysRevD.104.055020
15. Crivellin, A., Müller, D., Saturnino, F. Correlating h →μ+μ- To the Anomalous Magnetic Moment of the Muon via Leptoquarks[J]. Physical Review Letters, 2021, 127(2): 021801. doi: 10.1103/PhysRevLett.127.021801
16. Crivellin, A., Müller, D., Schnell, L. Combined constraints on first generation leptoquarks[J]. Physical Review D, 2021, 103(11): 115023. doi: 10.1103/PhysRevD.103.115023
17. Kowalska, K., Sessolo, E.M., Yamamoto, Y. Flavor anomalies from asymptotically safe gravity[J]. European Physical Journal C, 2021, 81(4): 272. doi: 10.1140/epjc/s10052-021-09072-1
18. Cheung, K., Huang, Z.-R., Li, H.-D. et al. Revisit to the b → cτν transition: In and beyond the SM[J]. Nuclear Physics B, 2021. doi: 10.1016/j.nuclphysb.2021.115354
19. Wang, S.-W.. Ξ b → Ξcτν̄ τ Decay in New Physics Models[J]. International Journal of Theoretical Physics, 2021, 60(3): 982-993. doi: 10.1007/s10773-021-04721-3
20. Crivellin, A., Greub, C., Müller, D. et al. Scalar leptoquarks in leptonic processes[J]. Journal of High Energy Physics, 2021, 2021(2): 182. doi: 10.1007/JHEP02(2021)182
21. Crivellin, A., Müller, D., Saturnino, F. Leptoquarks in oblique corrections and Higgs signal strength: status and prospects[J]. Journal of High Energy Physics, 2020, 2020(11): 94. doi: 10.1007/JHEP11(2020)094
22. Das, N., Dutta, R. Implication of b → cτ ν¯τ flavor anomalies on Bs → D∗sτ ν¯τ decay observables[J]. Journal of Physics G: Nuclear and Particle Physics, 2020, 47(11): aba422. doi: 10.1088/1361-6471/aba422
23. Sheng, J.-H., Zhu, J., Li, X.-N. et al. Probing new physics in semileptonic ςb and ωb decays[J]. Physical Review D, 2020, 102(5): 055023. doi: 10.1103/PhysRevD.102.055023
24. Bhattacharya, B., Datta, A., Kamali, S. et al. A measurable angular distribution for B¯ → D τ−v¯ τ decays[J]. Journal of High Energy Physics, 2020, 2020(7): 194. doi: 10.1007/JHEP07(2020)194
25. Gherardi, V., Marzocca, D., Venturini, E. Matching scalar leptoquarks to the SMEFT at one loop[J]. Journal of High Energy Physics, 2020, 2020(7): 225. doi: 10.1007/JHEP07(2020)225
26. Crivellin, A., Müller, D., Saturnino, F. Flavor phenomenology of the leptoquark singlet-triplet model[J]. Journal of High Energy Physics, 2020, 2020(6): 20. doi: 10.1007/JHEP06(2020)020
27. Ivanov, M.A., Körner, J.G., Santorelli, P. et al. D*Polarization as an Additional Constraint on New Physics in the(Formula Presented.) Transition[J]. Particles, 2020, 3(1): 193-207. doi: 10.3390/particles3010016
28. Chang, Q., Wang, X.-L., Zhu, J. et al. Study of b→c Induced b∗→Vℓνℓ Decays[J]. Advances in High Energy Physics, 2020. doi: 10.1155/2020/3079670
29. Mu, X.-L., Li, Y., Zou, Z.-T. et al. Investigation of effects of new physics in Λb → Λcτ ν τ decay INVESTIGATION of EFFECTS of NEW PHYSICS in ... XIAO-LONG MU et al.[J]. Physical Review D, 2019, 100(11): 113004. doi: 10.1103/PhysRevD.100.113004
30. Crivellin, A., Saturnino, F. Correlating tauonic B decays with the neutron electric dipole moment via a scalar leptoquark[J]. Physical Review D, 2019, 100(11): 115014. doi: 10.1103/PhysRevD.100.115014
31. Hati, C., Kriewald, J., Orloff, J. et al. A nonunitary interpretation for a single vector leptoquark combined explanation to the B-decay anomalies[J]. Journal of High Energy Physics, 2019, 2019(12): 6. doi: 10.1007/JHEP12(2019)006
32. Gómez, J.D., Quintero, N., Rojas, E. Charged current b →cτ ν τ anomalies in a general W′ boson scenario CHARGED CURRENT B →cτ ν τ ... GÓMEZ, QUINTERO, and ROJAS[J]. Physical Review D, 2019, 100(9): 093003. doi: 10.1103/PhysRevD.100.093003

Figures(5) / Tables(2)

Get Citation
Han Yan, Ya-Dong Yang and Xing-Bo Yuan. Phenomenology of bcτˉν decays in a scalar leptoquark model[J]. Chinese Physics C, 2019, 43(8): 083105. doi: 10.1088/1674-1137/43/8/083105
Han Yan, Ya-Dong Yang and Xing-Bo Yuan. Phenomenology of bcτˉν decays in a scalar leptoquark model[J]. Chinese Physics C, 2019, 43(8): 083105.  doi: 10.1088/1674-1137/43/8/083105 shu
Milestone
Received: 2019-03-21
Article Metric

Article Views(3330)
PDF Downloads(27)
Cited by(32)
Policy on re-use
To reuse of subscription content published by CPC, the users need to request permission from CPC, unless the content was published under an Open Access license which automatically permits that type of reuse.
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Email This Article

Title:
Email:

Phenomenology of bcτˉν decays in a scalar leptoquark model

Abstract: During the past few years, signs of lepton flavor universality (LFU) violation have been observed in bcτˉν and bs+ transitions. Recently, the D and τ polarization fractions PDL and PτL in BDτˉν decay were likewise measured by the Belle collaboration. Motivated by these intriguing results, we revisit the RD() and RK() anomalies in a scalar leptoquark (LQ) model, where two scalar LQs, one of which is a SU(2)L singlet and the other a SU(2)L triplet, are introduced simultaneously. We consider five bcτˉν mediated decays, BD()τˉν, Bcηcτˉν, BcJ/ψτˉν, and ΛbΛcτˉν, and focus on the LQ effects on the q2 distributions of the branching fractions, LFU ratios, and various angular observables in these decays. Under the combined constraints of the available data on RD(), RJ/ψ, PτL(D), and PDL, we perform scans for the LQ couplings and make predictions for a number of observables. Numerically it is found that both the differential branching fractions and LFU ratios are largely enhanced by the LQ effects, with the latter expected to provide testable signatures at the SuperKEKB and High-Luminosity LHC experiments.

    HTML

    1.   Introduction
    • To date, the LHC has not provided any direct evidence for new physics (NP) particles beyond the standard model (SM). However, several hints referring to the lepton flavor university (LFU) violation emerge in the measurements of semileptonic b-hadron decays, which, if confirmed with more precise experimental data and theoretical predictions, depict unambiguous signs of NP [1, 2].

      The charged-current decays BD()lˉν, with =e, μ, or τ, have been measured by the BaBar [3, 4], Belle [58], and LHCb [911] collaborations. The ratios of the branching fractions, RD()B(BD()τˉν)/B(BD()ˉν), with =e and/or μ, obtained by the latest experimental averages by the heavy flavor averaging group read [12]

      RexpD=0.407±0.039(stat.)±0.024(syst.),

      (1)

      RexpD=0.306±0.013(stat.)±0.007(syst.),

      (2)

      both of which exceed their respective SM predictions [12]

      RSMD=0.299±0.003,RSMD=0.258±0.005,

      (3)

      by 2.3σ and 3.0σ, respectively. Considering the experimental correlation of −0.203 between RD and RD, the combined results exhibit a ~3.78σ deviation from the SM predictions [12]. This discrepancy, referred to as the RD() anomaly, may provide a hint of LFU violating NP [1, 2]. For the BcJ/ψˉν decay, a ratio RJ/ψ can be similarly defined. The recent LHCb measurement, RexpJ/ψ=0.71±0.17(stat.)±0.18(syst.) [17], lies about 2σ above the SM prediction, RSMJ/ψ=0.248±0.006 [18]. 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 1.0GeV2q26.0GeV2 [19] and RexpK=0.69+0.110.07±0.05 for 1.1GeV2q26.0GeV2 [20], are found to be about 2.6σ and ~2.5σ lower than the SM expectation, RSMK()1 [21, 22], respectively. These anomalies motivated numerous studies both in the effective field theory approach [2328] and in specific NP models [2945]. We refer to Refs. [1, 2] for recent reviews.

      Recently, the Belle collaboration reported the first preliminary result of the D longitudinal polarization fraction in the BDτˉν decay [46, 47]

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

      (4)

      which is consistent with the SM prediction PDL=0.46±0.04 [48] at 1.5σ. Together with the measurements of the τ polarization, PτL=0.38±0.51(stat.)+0.210.16(syst.) [7, 8], these results provide valuable information on the spin structure of the interaction involved in BD()τˉν decays and are good observables for testing of various NP scenarios [4853]. The measurement of angular observables in these decays will be considerably improved in the future [54, 55]. For example, the Belle II experiment with 50ab1 data can measure PτL with an expected precision of ±0.07 [54].

      In this work, motivated by these experimental progresses and future prospects, we study five bcτˉν decays, BD()τˉν, Bcηcτˉν, BcJ/ψτˉν, and ΛbΛcτˉν, in the leptoquark (LQ) model proposed in Ref. [56]. Models with one or more LQ states, which are colored bosons that couple to both quarks and leptons, depict some of the most popular scenarios employed to explain the RD() and RK() anomalies [5774]. In Ref. [56], the SM is extended with two scalar LQs, one of which is a SU(2)L singlet, whereas the other is a SU(2)L triplet. This model is also featured by the fact that these two LQs have the same mass and hypercharge, and their couplings to fermions are related by a discrete symmetry. In this manner, the anomalies in bcτˉν and bsμ+μ transitions can be explained simultaneously, while avoiding potentially dangerous contributions to bsνˉν decays. By taking into account recent developments on transition form factors [13, 14, 18, 7577], we derive constraints on LQ couplings in this model. Subsequently, predictions in the LQ model are made for the five bcτˉν decays, focusing on the q2 distributions of the branching fractions, LFU ratios, and various angular observables. Implications for future research at the High-Luminosity LHC (HL-LHC) [78] and SuperKEKB [54] are also briefly discussed.

      This paper is organized as follows: in Section 2, we provide a brief review of the LQ model proposed in Ref. [56]. In Section 3, we recapitulate the theoretical formulae for the various flavor processes and discuss the LQ effects on these decays. In Section 4, we present our detailed numerical analysis and discussions. Our conclusions are given in Section 5. The relevant transition form factors and helicity amplitudes are presented in the appendices.

    2.   Model
    • In this section, we recapitulate the LQ model proposed in Ref. [56], where a scalar LQ singlet Φ1 and a triplet Φ3 are added to the SM field content, to explain the observed flavor anomalies. Under the SM gauge group (SU(3)C,SU(2)L,U(1)Y), the LQ states Φ1 and Φ3 transform as (3,1,2/3) and (3,ˉ3,2/3), respectively. Their interactions with the SM fermions are described by the Lagrangian [56]

      L=λ1LjkˉQcjiτ2LkΦ1+λ3LjkˉQcjiτ2(τΦ3)Lk+h.c.,

      (5)

      where Qj and Lk denote the left-handed quark and lepton doublet with generation indices j and k, respectively. The couplings λ1Ljk and λ3Ljk are generally complex, however assumed to be real throughout this work. It is further assumed that these two scalar LQs have the same mass M, and their couplings to the SM fermions satisfy the following discrete symmetry [56]:

      λLjkλ1Ljk,λ3Ljk=eiπjλLjk.

      (6)

      With these two assumptions, the tree-level LQ contributions to the bsνˉν decays are canceled. After rotating to the mass eigenstate basis, the LQ couplings to the left-handed quarks involve the CKM elements as

      λLdjk=λLjk,λLujk=VjiλLik,

      (7)

      where Vij is the CKM matrix element.

    3.   Theoretical framework
    • In this section, we introduce the theoretical framework for the relevant flavor processes and discuss the LQ effects on these decays.

    • 3.1.   bcτˉν mediated processes

    • Including the LQ contributions, the effective Hamiltonian responsible for bciˉνj transitions is given by [56]

      Heff=4GF2VcbCijL(ˉcγμPLb)(ˉiγμPLνj),

      (8)

      with the Wilson coefficient CijL=CSM,ijL+CNP,ijL. The W-exchange contribution within the SM yields CSM,ijL=δij, and the LQ contributions result in

      CNP,ijL=28GFM2VckVcbλL3jλLki[1+(1)k].

      (9)

      This Wilson coefficient is given at the matching scale μNPM. However, as the corresponding current is conserved, we can obtain the low-energy Wilson coefficient, CNP,ijL(μb)=CNP,ijL, without considering the renormalization group evolution (RGE) effect.

      In this study, we consider five processes mediated by the quark-level bcˉν transition, including BD()ˉν, Bcηcˉν, BcJ/ψˉν, and ΛbΛcˉν decays. All these processes can be uniformly represented by

      M(pM,λM)N(pN,λN)+(p,λ)+ˉν(pˉν),

      (10)

      where (M,N)=(B,D()),(Bc,ηc),(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 by pi and λi, respectively. In particular, the helicity of a pseudoscalar meson is zero, i.e., λB(c),D,ηc=0. After averaging over the non-zero helicity of the hadron M, the differential decay rate of this process can be written as [53, 79]

      dΓλN,λ(MNˉν)=12mM12|λM|+1λM|MλMλN,λ|2dΦ3,

      (11)

      with the phase space

      dΦ3=Q+Q256π3m2M1m2q2dq2dcosθ,

      (12)

      where Q±=m2±q2, with m±=mM±mN and q2 depicts the dilepton invariant mass squared. θ[0,π] denotes the angle between the three-momentum of and that of N in the -ˉν center-of-mass frame. The helicity amplitudes MλMλN,λNˉν|Heff|M can be written as [76]

      MλMλN,λτ=GFVcb2(HSPλM,λNLSPλτ+λWηλWHVAλM,λN,λWLVAλτ,λW+λW1,λW2ηλW1ηλW2HTλM,λN,λW1,λW2LTλτ,λW1λW2),

      (13)

      where λWi denotes the helicity of the virtual vector bosons W, W1 and W2. The coefficient ηλWi=1 for λλWi=t, and ηλWi=1 for λλWi=0,±1. Explicit analytical expressions of the leptonic and hadronic helicity amplitudes H and L are given in appendices A and C.

      From Eq. (11), we can derive the following observables:

      • The differential decay width and branching fraction

      dB(MNˉν)dq2=1ΓMdΓ(MNˉν)dq2=1ΓMλN,λdΓλN,λ(MNˉν)dq2,

      (14)

      where ΓM=1/τM is the total width of the hadron M.

      • The q2-dependent LFU ratio

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

      (15)

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

      • The lepton forward-backward asymmetry

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

      (16)

      • The q2-dependent polarization fractions

      PτL(q2)=dΓλτ=+1/2/dq2dΓλτ=1/2/dq2dΓ/dq2,PNL(q2)=dΓλN=+1/2/dq2dΓλN=1/2/dq2dΓ/dq2,forN=Λc,PNL(q2)=dΓλN=0/dq2dΓ/dq2,forN=D,J/ψ,PNT(q2)=dΓλN=1/dq2dΓλN=1/dq2dΓ/dq2,forN=D,J/ψ.

      (17)

      Analytical expressions of all the above observables are given in Appendix C. As these angular observables depict ratios of the decay widths, they are largely free of hadronic uncertainties and thus provide excellent tests of the NP effects.

      As shown in Eq. (8), LQ effects generate an operator with the same chirality structure as in the SM. Therefore, it is straightforward to derive the following relation:

      RNRSMN=3i=1|δ3i+C3iL|2,

      (18)

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

      One of the main inputs in our calculations are the transition form factors. In this respect, notable progresses have been achieved in recent years [1316, 7577, 8087]. This study adopts the Boyd-Grinstein-Lebed (BGL) [13, 88] and Caprini-Lellouch-Neubert (CLN) [14, 89] parametrization for the BD and BD transition form factors, respectively. In these approaches, both the transition form factors and the CKM matrix element |Vcb| are simultaneously extracted from the experimental data. In addition, we use the Bcηc,J/ψ transition form factors obtained in the covariant light-front approach [18]. For the ΛbΛc transition form factor, we adopt the recent Lattice QCD results in Refs. [75, 76]. Explicit expressions of all the relevant transition form factors are recapitulated in Appendix B.

    • 3.2.   Other processes

    • With the LQ effects considered, the effective Hamiltonian for the bs+ij transition can be written as [90]

      Heff=4GF2VtbVtsaCijaOija+h.c.,

      (19)

      where the operators relevant to our study are

      Oij9=αe4π(ˉsγμPLb)(ˉiγμj),Oij10=αe4π(ˉsγμPLb)(ˉiγμγ5j).

      (20)

      The LQ contributions result in [56]

      CNP,ij9=CNP,ij10=22GFVtbVtsπαe1M2λL3jλL2i.

      (21)

      In the model-independent approach, the current bsμ+μ anomalies can be explained by a CNP,229=CNP,2210 -like contribution, with the permitted range given by [9193]

      0.91(0.71)CNP,229=CNP,22100.18(0.35),

      (22)

      at the 2σ (1σ) level, which in turn sets a constraint on λL22λL32. Furthermore, the LQ contributions to bsτ+τ and bcτˉντ transitions depend on the same product λL23λL33, therefore making a direct correlation between the branching fraction B(Bsτ+τ) and RD().

      For the bsνˉν transitions, both the LQs Φ1 and Φ3 generate tree-level contributions. However, assuming that they have the same mass, their effects are canceled out due to the discrete symmetry in Eq. (6). In addition, this LQ scenario can accommodate the (g2)μ anomaly [94, 95], once the right-handed interaction term λRfiˉucfiΦ1 is introduced to Eq. (5) [56]. We do not consider such a term in this study. Further details can be found in Ref. [56], where various lepton flavor violating decays of leptons and B meson have also been discussed.

      Finally, we provide brief comments on direct searches for the LQs at high-energy colliders. Because the LQ contributions to bcτˉν transitions only involve the product λL23λL33, searches for the LQs with couplings to the second and third generations are more relevant to our work. At the LHC, both the CMS and ATLAS collaborations have performed searches for such LQs in several channels, e.g., LQtμ [96], LQtτ [97], LQbτ [97], etc. Current results from the LHC have excluded the LQs with masses below about 1TeV [95]. For example, searches for pair-produced scalar LQs decaying into t quark and μ lepton have been performed by the CMS Collaboration, in which a scalar LQ with mass below 1420GeV was excluded at 95% CL under the assumption of B(LQtμ)=1 [96]. All these collider constraints depend on the assumption of the total width of the LQ, which involves all the LQ couplings λLij. To apply the collider constraints to our scenario, one needs to perform a global fit on all the LQ couplings and derive bounds on the total width. Such analysis is beyond the scope of this study. Regarding the scenario with one singlet and one triplet LQ, we refer to Ref. [72] for a more detailed collider analysis. Furthermore, our analysis does not depend on the mass of the LQ, because LQ couplings always appear in the form of λL23λL33/M2 in bcτˉν transitions, as in Eq. (9).

    4.   Numerical analysis
    • In this section, we present our numerical analysis of the LQ effects on the decays considered. After deriving the constraints of the model parameters, we concentrate on the LQ effects on the five bcτˉν decays, i.e., BD()τˉν, Bcηcτˉν, BcJ/ψτˉν, and ΛbΛcτˉν.

    • 4.1.   SM predictions

    • In Table 1, we list the relevant input parameters used in our numerical analysis. Using the theoretical framework described in Section 3, the SM predictions for BD()τˉν, Bcηcτˉν, BcJ/ψτˉν, and ΛbΛcτˉν decays are given in Table 2. To obtain the theoretical uncertainties, we vary each input parameter within their respective 1σ range and add each individual uncertainty in quadrature. Correlations among fit parameters were considered to obtain uncertainties of the transition form factors. In particular, for the ΛbΛcτˉν decay, we follow the treatment of Ref. [75] to obtain the statistical and systematic uncertainties induced by the ΛbΛc transition form factors. From Table 2, the experimental data on the ratios RD, RD, and RJ/ψ are found to deviate from the SM predictions by 2.31σ, 2.85σ and 1.83σ, respectively.

      inputvalueunitRef.
      αs(mZ)0.1181±0.0011[95]
      mpolet173.1±0.9GeV[95]
      mb(mb)4.18±0.03GeV[95]
      mc(mc)1.275±0.025GeV[95]
      |Vcb|(semi-leptonic)41.00±0.33±0.74103[98]
      |Vub|(semi-leptonic)3.98±0.08±0.22103[98]

      Table 1.  Input parameters used in our numerical analysis.

      observableSMNPexpRef
      B(BDτˉν)0.711+0.0420.041[0.702,0.991]0.90±0.24[95]
      RD0.301+0.0030.003[0.313,0.400]0.407±0.039±0.024[12]
      B(Bcηcτˉν)0.204+0.0240.024[0.188,0.299]
      Rηc0.281+0.0350.031[0.263,0.416]
      B(BDτˉν)1.261+0.0870.085[1.234,1.788]1.78±0.16[95]
      RD0.258±0.008[0.263,0.351]0.306±0.013±0.007[12]
      PτL0.503±0.013[0.516,0.490]0.38±0.51+0.210.16[7, 8]
      PDL0.453±0.012[0.441,0.465]0.60±0.08±0.04[46, 47]
      B(BcJ/ψτˉν)0.398+0.0450.049[0.366,0.583]
      RJ/ψ0.248+0.0060.005[0.255,0.335]0.71±0.17±0.18[17]
      B(ΛbΛcτˉν)1.762+0.1050.104[1.737,2.457]
      RΛc0.333+0.0100.010[0.339,0.451]

      Table 2.  Predictions for branching fractions (in units of 102) and ratios RN of the five bcτˉν decay modes in the SM and the LQ scenario. The entry "––" indicates that no measurement is yet available for the corresponding observable.

    • 4.2.   Constraints

    • To obtain the permitted ranges of LQ parameters, we impose the experimental constraints in the same manner as in Refs. [99, 100]; i.e., for each point in the parameter space, if the difference between the corresponding theoretical prediction and experimental data is less than 2σ (3σ) error bar, which is calculated by adding the theoretical and experimental uncertainties in quadrature, this point is regarded as permitted at 2σ (3σ) level.

      In the LQ scenario introduced in Section 2, the LQ contributions to bcτˉν transitions are all controlled by the product λL23λL33. In the following analysis, the couplings λL23 and λL33 are assumed to be real. After considering the current experimental measurements of RD(), RJ/ψ, PτL(D), and PDL, we find that the constraints on λL23λL33 are dominated by RD and RD. The permitted ranges of λL23λL33 at 2σ level are obtained as follows

      2.90<λL23λL33<2.74,or0.03<λL23λL33<0.20,

      (23)

      where a common LQ mass M=1TeV is assumed. The solution with negative λL23λL33 corresponds to the case in which the LQ interactions dominate over the SM contributions. We do not pursue this possibility in the following analysis. For the solution with positive λL23λL33, the permitted regions of (λL23,λL33) at both 2σ and 3σ levels are shown in Fig. 1. In this figure, we also show the individual constraint from the D polarization fraction PDL, which remains weaker than the ones from RD(). In addition, the current measurement of the τ polarization fraction PτL in BDτν decay cannot provide any relevant constraint.

      Figure 1.  (color online) Combined constraints on (λL23,λL33) by all bcτˉν processes at 2σ (black) and 3σ (gray) levels. The dark (light) green area indicates the allowed region by PDL only at 2σ (3σ).

      As mentioned in Section 3, the LQ contributions to bsτ+τ and bcτˉντ depend on the same product λL23λL33. In the case of positive λL23λL33, we show in Fig. 2 the correlation between RD()/RSMD() and B(Bsτ+τ). The LQ effects enhance the branching fraction of Bsτ+τ in most of the parameter space. At present, the experimental upper limit 6.8×103 [101] is far above the SM prediction (7.73±0.49)×107 [102]. However, to obtain the 2σ experimental range of RD(), the LQ contributions enhance B(Bsτ+τ) by about 2–3 orders of magnitude compared to the SM prediction, which reaches the expected LHCb sensitivity 5×104 by the end of Upgrade II [55, 103]. The BK()τ+τ decay may also play an important role in probing the LQ effects. Although the Belle II experiment would improve the current upper limit 2.25×103 at a 90% confidence level by no more than two orders of magnitude, the proposed FCC-ee collider can yield a few thousand of B0K0τ+τ events from O(1013) Z decays [104].

      Figure 2.  (color online) Correlation between RD()/RSMD() and B(Bsτ+τ). The black (gray) region denotes the 2σ (3σ) experimental ranges of RD()/RSMD(). The horizontal dashed and dotted lines correspond to the current LHCb upper limit and the expected sensitivity by the end of LHCb Upgrade II, respectively. The black point indicates the SM prediction.

    • 4.3.   Predictions

    • Using the constrained parameter space at the 2σ level derived in the last subsection, we present predictions for the five bcτˉν processes. Table 2 shows the SM and LQ predictions for the branching fractions B and LFU ratios R of BD()τˉν, Bcηcτˉν, BcJ/ψτˉν, and ΛbΛcτˉν decays. The LQ predictions include the uncertainties induced by the transition form factors and CKM matrix elements. Considering that the polarization fractions PτL and PDL have already been measured, their SM and LQ predictions are also shown in Table 2. Although the LQ predictions for the branching fractions B and the LFU ratios R of the Bcηcτˉν and BcJ/ψτˉν decays lie within the 1σ range of their respective SM values, they can be significantly enhanced by LQ effects.

      We set out to analyze the q2 distributions of the branching fraction B, LFU ratio R, polarization fractions of the τ lepton (PτL), and daughter hadron (PDL,T, PJ/ψL,T, PΛcL), as well as the lepton forward-backward asymmetry AFB. The BDτν and Bcηcτˉν decays, both part of the "BP" transition, have differential observables in the SM and the LQ scenario as shown in Fig. 3. All differential observables of the BDτˉν and Bcηcτˉν decays are similar to each other, while the observables in the latter have larger theoretical uncertainties due to the less precise Bcηc transition form factors. Therefore, the BDτˉν decay is more sensitive to the LQ effects, with the differential branching fraction largely enhanced, especially near q27GeV2. The large difference between the SM and LQ predictions in this kinematic region could, therefore, provide a testable signature of the LQ effects. More interestingly, the q2 distribution of the ratio R in the LQ model is enhanced in the entire kinematic region and does not have overlap with the 1σ SM range. In the future, more precise measurements of these distributions are of importance to confirm the existence of a possible NP effect in the BDτˉν decay. With regard to the forward-backward asymmetry AFB and the τ-lepton polarization fraction PτL in both BDτˉν and Bcηcτˉν decays, the LQ predictions are indistinguishable from the ones in SM, because the LQ effects only modify the Wilson coefficient CνL, which is canceled out exactly in the definitions of these observables (see Eqs. (16) and (17),Fig. 3). This feature is different from the NP scenarios that use scalar or tensor operators to explain the RD() anomaly [5860].

      Figure 3.  (color online) q2 distributions of observables in BDτˉν (left) and Bcηcτˉν (right) decays. The black curves (gray band) indicate the SM (LQ) central values with 1σ theoretical uncertainty.

      The q2 distributions of the observables in BDτˉν and BcJ/ψτˉν decays are shown in Fig. 4. Because both of these two decays belong to "BV" transition, their differential observables are similar. While the differential branching fractions of these two decays are enhanced in the LQ model, their theoretical uncertainties are larger than the ones in the BDτˉν decay. For the q2 distributions of the ratios RD and RJ/ψ, they are largely enhanced in the entire kinematic region, especially in the large q2 region. More importantly, although the ranges of the q2-integrated ratio RD,J/ψ in the SM and the LQ scenario overlap at the 1σ level, the 1σ ranges of the differential ratio RD,J/ψ(q2) at large q2 in the SM and LQ exhiibt significant differences. The increases of RD and RJ/ψ in the large q2 region are larger than the one observed in RD. Measurements of the differential ratios in the large dilepton invariant mass region are, therefore, crucial to confirm the RD() anomaly and test the LQ model considered. Similarly to the ones in BDτˉν and Bcηcτˉν decays, the angular distributions AFB, PD,J/ψL,T, and PτL are likewise not affected by the LQ effects, as shown in Fig. 4.

      Figure 4.  (color online) q2 distributions of observables in BDτˉν (left) and BcJ/ψτˉν (right) decays. Other captions are the same as in Fig. 3.

      For the ΛbΛcτˉν decay, the q2 distributions of the observables are shown in Fig. 5. The situation is similar in the BDτˉν and BcJ/ψτˉν decays. The q2 distributions of the branching fraction B and the ratio RΛc are greatly enhanced by the LQ effects. In the large q2 region, the differential ratio RΛc exhibits a deviation between the 1σ permitted ranges of the SM and the LQ scenario. With the large numbers of Λb obtained at the HL-LHC [78], we expect that this prediction could provide helpful information on the LQ effects. For the angular distributions, the LQ effects vanish due to the same reason as in the mesonic decays.

      Figure 5.  (color online) q2 distributions of observables in ΛbΛcτˉν decay. Other captions are the same as in Fig. 3.

    5.   Conclusions
    • During the past few years, intriguing hints pointing towards an LFU violation have emerged in the BD()τˉν data. Motivated by the recent measurements of RJ/ψ, PτL, and PDL, we revisited the LQ model proposed in Ref. [56], where two scalar LQs, one of which is a SU(2)L singlet, whereas the other is a SU(2)L triplet, are introduced simultaneously. Taking into account the recent progress on the transition form factors and the most recent experimental data, we obtained constraints on the LQ couplings λL23 and λL33. Subsequently, we systematically investigated the LQ effects on the five bcτˉν decays, BD()τˉν, Bcηcτˉν, BcJ/ψτˉν, and ΛbΛcτˉν. In particular, we focused on the q2 distributions of the branching fractions, LFU ratios, and various angular observables. The main results of this study can be summarized as follows:

      • After considering the RD and RD data, we obtain the bound on the LQ couplings, 0.03<λL23λL33<0.20, at the 2σ level. The current measurements of RJ/ψ, PτL and PDL cannot provide further constraints on the LQ couplings.

      • The Bsτ+τ decay is strongly correlated with BD()τˉν. To reproduce the 2σ experimental range of RD(), the LQ effects enhance B(Bsτ+τ) by about 2–3 orders of magnitude compared to the SM prediction and hence reach the expected sensitivity of the LHCb Upgrade II.

      • The differential branching fractions and LFU ratios are largely enhanced by the LQ effects. Due to their small theoretical uncertainties, the latter provide testable signatures of the LQ model considered, especially in the large dilepton invariant mass squared region. Moreover, RΛc in the baryonic decay ΛbΛcτˉν has the potential to shed new light on the RD() anomalies.

      • Because no new operators are generated by the LQ effects, all angular distributions in the LQ model are the same as in the SM. We provide the most recent SM predictions for the τ-lepton forward-backward asymmetry, the τ, and meson polarization fractions of the five bcτˉν modes. Although precision measurements of these angular distributions are very challenging at the HL-LHC and SuperKEKB, they are crucial for the verification of the LQ scenario investigated in this work.

      The q2 distributions of the branching fractions, the LFU ratios, and the various angular observables in bcτˉν transitions can help to confirm possible NP resolutions of the RD() anomalies and distinguish among the various NP candidates. With the experimental progress expected from the SuperKEKB [54] and the future HL-LHC [78], our predictions for these observables can be further probed in the near future.

      Note Added. After the completion of this work, the Belle Collaboration announced their results of RD and RD with a semileptonic tagging method [105,106]. The measured values are RexpD=0.307±0.037(stat.)±0.016(syst.) and RexpD=0.283±0.018(stat.)±0.014(syst.). After including this new measurement, the world averages become Ravg,2019D=0.337±0.030 and Ravg,2019D=0.299±0.013 [107]. The deviation of the current world averages from the SM predictions decreases from 3.8σ to 3.1σ [105]. Because the difference between the new and previous averages is small, our numerical results are expected to remain qualitatively unchanged. For example, the updated bounds on λL23λL33 in Eq. (23) becomes 2.88<λL23λL33<2.73 and 0.02<λL23λL33<0.17.

      We thank Xin-Qiang Li for useful discussions.

    A.   Appendix A: Helicity amplitudes in bcτˉν decays
    • In the presence of NP, the most general effective Hamiltonian for the bcτˉν transition can be written as [23, 76]

      Heff=22GFVcb[(1+gL)(ˉcγμPLb)(ˉτγμPLντ)+gR(ˉcγμPRb)(ˉτγμPLντ)+12gS(ˉcb)(ˉτPLντ)+12gP(ˉcγ5b)(ˉτPLντ)+gT(ˉcσμνPLb)(ˉτσμνPLντ)]+h.c..

      (24)

      In this appendix, for completeness, we consider the most general case of NP and provide the helicity amplitudes in the five bcτˉν decays, BD()τˉν, Bcηcτˉν, BcJ/ψτˉν, and ΛbΛcτˉν. Explicit expressions of the spinors and polarization vectors used to calculate the helicity amplitudes are also presented.

    • A.1.   Kinematic conventions

    • To calculate the hadronic helicity amplitudes of MNτˉν in Eq. (13), we work in the M rest frame and follow the notation of Ref. [79]:

      pμM=(mM,0,0,0),pμN=(EN,0,0,|pN|),qμ=(q0,0,0,|q|),

      (25)

      where qμ is the four-momentum of the virtual vector boson in the M rest frame, and

      q0=12mM(m2Mm2N+q2),EN=12mM(m2M+m2Nq2),|q|=|pN|=12mMQ+Q,Q±=(mM±mN)2q2.

      (26)

      Subsequently, substituting the momentum into Eq. (A12), the Dirac spinors in the ΛbΛcτντ decay can be written as

      uΛb(pΛb,λΛb)=2mΛb(χ(pΛb,λΛb)0),uΛc(pΛc,λΛc)=(E+mΛcχ(pΛc,λΛc)2λΛcEmΛcχ(pΛc,λΛc)),

      (27)

      where χ(pΛb,1/2)=χ(pΛc,1/2)=(1,0)T,χ(pΛb,1/2)=χ(pΛc,1/2)=(0,1)T..

      In the BDτˉν decay, the polarization vectors of the D meson are given by

      ϵμ(pD,0)=1mD(|pD|,0,0,ED),ϵμ(pD,±)=12(0,±1,i,0).

      (28)

      In all the five bcτˉν decays, the polarization vectors for the virtual vector boson W can be written as

      ϵμ(t)=1q2(q0,0,0,|q|),ϵμ(0)=1q2(|q|,0,0,q0),ϵμ(±)=12(0,1,i,0),

      (29)

      and the orthonormality and completeness relation [108]

      μϵμ(m)ϵμ(n)=gmn,m,nϵμ(m)ϵν(n)gmn=gμν,m,n{t,±,0},

      (30)

      where gmn=diag(+1,1,1,1).

      In the calculation of the leptonic helicity amplitudes, we work in the rest frame of the virtual vector boson W, which is equivalent to the rest frame of the τ-ˉντ system. Following Ref. [79], we have

      qμ=(q2,0,0,0),pμτ=(Eτ,|pτ|sinθτ,0,|pτ|cosθτ),pμˉν=|pτ|(1,sinθτ,0,cosθτ),

      (31)

      where |pτ|=q2v2/2, Eτ=|pτ|+m2τ/q2, v=1m2τ/q2, and θτ denotes the angle between the three-momenta of the τ and the N.

      The Dirac spinors for τ and ˉντ read

      uτ(pτ,λτ)=(Eτ+mτχ(pτ,λτ)2λτEτmτχ(pτ,λτ)),vˉντ(pτ,12)=Eν(ξ(pτ,12)ξ(pτ,12)),

      (32)

      respectively. Further details are given in Appendix A.2

      The polarization vectors of the virtual vector boson in the W rest frame are written as

      \tag{A10} \bar{\epsilon}^{\mu}(t) = (1,0,0,0), \quad \bar{\epsilon}^{\mu}(0) = (0,0,0,-1), \quad \bar{\epsilon}^{\mu}(\pm) = \frac{1}{\sqrt{2}}(0,\mp 1,i,0),

      (33)

      which can also be obtained from Eq. (A6) by a Lorentz transformation and satisfy the orthonormality and completeness relation in Eq. (A7).

    • A.2.   Dirac spinor

    • The definitions of the helicity operator h_{\vec{p}} and its eigenstates are given as follows [109]

      h_{\vec{p}}\equiv\frac{1}{2}\hat{\vec{p}}\cdot\vec{\sigma},\quad\hat{\vec{p}}\equiv\frac{\vec{p}}{|\vec{p}|},\quad h_{\vec{p}}\; \chi(\vec{p},s) = s\; \chi(\vec{p},s),

      where \vec{p} denotes the momentum of the particle, and \vec{\sigma} = \{\sigma^1,\sigma^2,\sigma^3\} are the Pauli matrices. Eigenstates of the helicity operator h_{\vec{p}} read

      \tag{A11} \begin{split} \chi \left(\vec p,\frac{1}{2}\right) =& {\left( \begin{array}{l} \cos \frac{\theta }{2}\\ {{\rm e}^{{\rm i}\phi }}\sin \frac{\theta }{2} \end{array} \right),}\quad{\chi \left(\vec p, - \frac{1}{2}\right) = \left( \begin{array}{l} - {{\rm e}^{ - {\rm i}\phi }}\sin \frac{\theta }{2}\\ \;\;\;\;\cos \frac{\theta }{2} \end{array} \right),}\\ \chi \left( - \vec p,\frac{1}{2}\right) =& {\left( \begin{array}{l} \;\;\;\sin \frac{\theta }{2}\\ - {{\rm e}^{{\rm i}\phi }}\cos \frac{\theta }{2} \end{array} \right),}\quad{\chi \left( - \vec p, - \frac{1}{2}\right) = \left( \begin{array}{l} {{\rm e}^{ - {\rm i}\phi }}\cos \frac{\theta }{2}\\ \;\;\;\;\sin \frac{\theta }{2} \end{array} \right),} \end{split}

      (34)

      for the normalized momentum \hat{\vec{p}} = \{\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta\} .

      Using these eigenstates, the solution of Dirac equation (\gamma^{\mu} p_{\mu}-m)u(\vec{p},s) = 0 in Dirac representation can be written as

      \tag{A12} u(\vec p,s) = \left( \begin{array}{l} \;\sqrt {E + m} \;\chi (\vec p,s)\\ 2s\sqrt {E - m} \;\chi (\vec p,s) \end{array} \right).

      (35)

      Further, the spinor for the antiparticle can be obtained by v(\vec{p},s)\equiv C\bar{u}(\vec{p},s)^{T} = i\gamma^0\gamma^2 \bar{u}(\vec{p},s)^{T} , whose explicit expression reads

      \tag{A13}v(\vec p,s) = \left( \begin{array}{l} \; \sqrt {E - m} \;\xi (\vec p,s)\\ - 2s\sqrt {E + m} \;\xi (\vec p,s) \end{array} \right),

      (36)

      where \xi(\vec{p},s) = \chi(\vec{p},-s) and \xi(\vec{p},s) satisfies h_{\vec{p}}\; \xi(\vec{p},s) = -s\; \xi(\vec{p},s) .

      The spinors in Weyl representation read

      \tag{A14} \begin{split} u_W(\vec{p},s) =& \left( \begin{array}{l} \sqrt{E-2s\left\vert{{\vec{p}}}\right\vert}\; \chi(\vec{p},s)\\ \sqrt{E+2s\left\vert{{\vec{p}}}\right\vert} \; \chi(\vec{p},s) \end{array} \right), \\ v_W(\vec{p},s) =& \left( \begin{array}{l} -2s\sqrt{E+2s\left\vert{{\vec{p}}}\right\vert}\; \xi(\vec{p},s)\\ \hphantom{-} 2s\sqrt{E-2s\left\vert{{\vec{p}}}\right\vert} \; \xi(\vec{p},s) \end{array} \right). \end{split}

      (37)

      They can also be obtained from the Dirac representation by the relation u_{W}(\vec{p},s) = X u(\vec{p},s) with the transformation matrix

      X = \frac{1}{{\sqrt 2 }}\left( {\begin{array}{*{20}{l}} 1&{ - 1}\\ 1&1 \end{array}} \right).

      In the \tau - \bar\nu_\tau center-of-mass frame, we emphasize that if the \tau spinor is specified as u(\vec{p},s) in the leptonic helicity amplitude, then the \bar\nu_\tau spinor has the form v(-\vec{p},s) , as in Eq. (A9). All calculations in our work are depictec in Dirac representation.

    • A.3.   Leptonic helicity amplitudes

    • The leptonic helicity amplitudes in Eq. (13) are defined as [79]

      \tag{A15} \begin{split} L^{SP}_{\lambda_\tau} = &\left\langle \tau\bar{\nu}_\tau\right|\bar{\tau} (1-\gamma_5)\nu_\tau\left| 0\right\rangle = \bar{u}_\tau(\vec{p}_\tau,\lambda_{\tau})(1-\gamma_5)v_{\bar{\nu}_\tau}(-\vec{p}_\tau,1/2), \\ L^{VA}_{\lambda_\tau,\lambda_W} = &\bar{\epsilon}^{\mu} (\lambda_W)\left\langle \tau\bar{\nu}_\tau\right|\bar{\tau}\gamma_\mu (1-\gamma_5)\nu_\tau\left| 0\right\rangle \\=& \bar{\epsilon}^{\mu}(\lambda_W)\bar{u}_\tau(\vec{p}_\tau,\lambda_{\tau})\gamma_\mu (1-\gamma_5)v_{\bar{\nu}_\tau}(-\vec{p}_\tau, 1/2) , \\ L^{T}_{\lambda_\tau,\lambda_{W_1} ,\lambda_{W_2}} = &-i\bar{\epsilon}^{\mu} (\lambda_{W_1})\bar{\epsilon}^{\nu} (\lambda_{W_2})\left\langle \tau\bar{\nu}_\tau\right|\bar{\tau}\sigma_{\mu \nu} (1-\gamma_5)\nu_\tau\left| 0\right\rangle \\ = &-i\bar{\epsilon}^{\mu} (\lambda_{W_1})\bar{\epsilon}^{\nu} (\lambda_{W_2})\bar{u}_\tau(\vec{p}_\tau,\lambda_{\tau})\sigma_{\mu \nu} (1-\gamma_5)v_{\bar{\nu}_\tau}(-\vec{p}_\tau, 1/2), \end{split}

      (38)

      Obtaining L^T_{\lambda_\tau,\lambda_{W_1} ,\lambda_{W_2}} = -L^T_{\lambda_\tau ,\lambda_{W_2},\lambda_{W_1}} is straightforward. The non-zero leptonic helicity amplitudes read

      \tag{A16} \begin{split} L^{SP}_{1/2}& = 2\sqrt{q^2}v,\\ L^{VA}_{1/2,t}& = 2m_\tau v,\\ L^{VA}_{1/2,0}& = -2m_\tau v \cos\theta_\tau ,\\ L^{VA}_{-1/2,0}& = 2\sqrt{q^2}v \sin\theta_\tau,\\ L^{VA}_{1/2,\pm}& = \mp\sqrt{2} m_\tau v \sin\theta_\tau,\\ L^{VA}_{-1/2,\pm}& = \sqrt{2 q^2}v (-1 \mp \cos\theta_\tau),\\ L^{T}_{1/2,0,\pm} & = \pm L^{T}_{1/2,\pm,t} = \sqrt{2 q^2}v\sin\theta_\tau, \\ L^{T}_{1/2,t,0}& = L^{T}_{1/2,+,-} = -2\sqrt{q^2}v \cos\theta_\tau,\\ L^{T}_{-1/2,0,\pm}& = \pm L^{T}_{-1/2,\pm,t} = \sqrt{2}m_\tau v (\pm 1+ \cos\theta_\tau), \\ L^{T}_{-1/2,t,0}& = L^{T}_{-1/2,+,-} = 2m_\tau v \sin\theta_\tau. \end{split}

      (39)
    • A.4.   Hadronic helicity amplitudes

    • The hadronic helicity amplitudes M \to N are defined as

      \tag{A17} \begin{split} H^S_{\lambda_M,\lambda_N}& = \left\langle N(\lambda_N)\right|\bar{c} b\left| M(\lambda_M)\right\rangle,\\ H^P_{\lambda_M,\lambda_N}& = \left\langle N(\lambda_N)\right|\bar{c}\gamma_5 b\left| M(\lambda_M)\right\rangle,\\ H^V_{\lambda_M,\lambda_N,\lambda_{W}}& = \,\epsilon_\mu^{*}(\lambda_{W})\left\langle N(\lambda_N)\right|\bar{c}\gamma^\mu b\left| M(\lambda_M)\right\rangle, \\ H^A_{\lambda_M,\lambda_N,\lambda_{W}}& = \,\epsilon_\mu^{*}(\lambda_{W})\left\langle N(\lambda_N)\right|\bar{c}\gamma^\mu\gamma_5 b\left| M(\lambda_M)\right\rangle,\\ H^{T_1,\lambda_M}_{\lambda_N,\lambda_{W_1} ,\lambda_{W_2}}& = i\epsilon_\mu^{*}(\lambda_{W_1})\epsilon_\nu^{*}(\lambda_{W_2})\left\langle N(\lambda_N)\right|\bar{c}\sigma^{\mu \nu} b\left| M(\lambda_M)\right\rangle, \\ H^{T_2,\lambda_M}_{\lambda_N,\lambda_{W_1} ,\lambda_{W_2}}& = i\epsilon_\mu^{*}(\lambda_{W_1})\epsilon_\nu^{*}(\lambda_{W_2})\left\langle N(\lambda_N)\right|\bar{c}\sigma_{\mu \nu}\gamma_5 b\left| M(\lambda_M)\right\rangle, \end{split}

      (40)

      and

      \tag{A18} \begin{split} H^{SP}_{\lambda_M,\lambda_N}& = g_S H^S_{\lambda_M,\lambda_N}+g_P H^P_{\lambda_M,\lambda_N}, \\ H^{VA}_{\lambda_M,\lambda_N,\lambda_{W}}& = (1+g_L+g_R)H^V_{\lambda_N,\lambda_{W}}-(1+g_L-g_R)H^A_{\lambda_M,\lambda_N,\lambda_{W}}, \\ H^{T,\lambda_M}_{\lambda_N,\lambda_{W_1} ,\lambda_{W_2}}& = g_TH^{T_1,\lambda_M}_{\lambda_N,\lambda_{W_1} ,\lambda_{W_2}}-g_T H^{T_2,\lambda_M}_{\lambda_N,\lambda_{W_1} ,\lambda_{W_2}} , \end{split}

      (41)

      H^{T,\lambda_M}_{\lambda_N,\lambda_{W_1} ,\lambda_{W_2}} = -H^{T,\lambda_M}_{\lambda_N,\lambda_{W_2} ,\lambda_{W_1}} is easily obtained. The amplitudes H_{\lambda_N,\lambda_{W_1},\lambda_{W_2}}^{T_1,\lambda_M} and H_{\lambda_N,\lambda_{W_1},\lambda_{W_2}}^{T_2,\lambda_M} are connected by the relation \sigma_{\mu \nu}\gamma_{5} = -(i/2)\epsilon^{\mu\nu\alpha\beta}\sigma_{\alpha\beta} , where \epsilon^{0123} = -1 .

    B.   Appendix B: Form factors
    • The hadronic matrix elements for the B\to D transition can be parameterized in terms of form factors F_{+,0,T} [110, 111]. In the BGL parametrization, the form factors F_{+,0} can be written as expressions of a_n^+ and a_n^0 [13],

      \tag{B1} \begin{split}F_{+}(z) =& \frac{1}{P_{+}(z)\phi_{+}(z,{\cal N})}\sum\limits_{n = 0}^{\infty} a_{n}^{+}z^n(w,{\cal N}), \\ F_{0}(z) =& \frac{1}{P_{0}(z)\phi_{0}(z,{\cal N})}\sum\limits_{n = 0}^{\infty} a_{n}^{0}z^n(w,{\cal N}), \end{split}

      (42)

      where r = m_D / m_B , {\cal N} = (1+r)/(2\sqrt{r}) , w = (m_{B}^2+m_{D}^2-q^{2})/(2m_{B}m_{D}) , z(w,{\cal N}) = (\sqrt{1+w}-\sqrt{2{\cal N}})/(\sqrt{1+w}+\sqrt{2{\cal N}}) , and F_+(0) = F_0(0) . The values of the fit parameters are taken from Ref. [13]. Expressions of the tensor form factor F_T can be found in Ref. [110].

      For B\to D^* transition, the relevant form factors \{V,A_{0,1,2}\} can be written in terms of the form factors \{h_V,h_{A_{1,2,3}}\} in the heavy quark effective theory (HQET) [110],

      \tag{B2} \begin{split}V(q^2) =& { m_+ \over 2\sqrt{m_B m_{D^{*}}} } \, h_V(w), \\ A_0(q^2) =& { 1 \over 2\sqrt{m_B m_{D^{*}}} } \left[ { m_+^2 \!-\! q^2 \over 2m_{D^{*}} } \, h_{A_1}(w) -\! { m_+m_- + q^2 \over 2m_B } \, h_{A_2}(w) -\! {m_+m_– q^2 \over 2m_{D^{*}} } \, h_{A_3}(w) \right] , \\ A_1(q^2) =& { m_+^2 - q^2 \over 2\sqrt{m_B m_{D^{*}}} m_+ } \, h_{A_1}(w), \\ A_2(q^2) =& {m_+ \over 2\sqrt{m_B m_{D^{*}}} } \left[ h_{A_3}(w) + { m_{D^{*}} \over m_B } h_{A_2}(w) \right] , \end{split}

      (43)

      where m_\pm = m_B \pm m_{D^*} and w = (m_B^2+m_{D^*}^2-q^2)/2m_Bm_{D^*} . In the CLN parametrization, the HQET form factors can be expressed as [89]

      \begin{split} \frac{h_V(w)}{h_{A_1}(w) } = R_1(w) , \quad \frac{h_{A_2}(w)}{h_{A_1}(w)} = { R_2(w)-R_3(w) \over 2\,r_{D^{*}} }, \end{split}

      \tag{B3}\begin{split} \frac{h_{A_3}(w)}{h_{A_1}(w)} = { R_2(w)+R_3(w) \over 2 } , \end{split}

      (44)

      with r = m_{D^*}/m_B . Numerically we obtain,

      \tag{B4} \begin{split} h_{A_1}(w) =& h_{A_1}(1)[1-8\rho_{D^*}^2 z + (53 \rho_{D^*}^2 -15)z^2 - (231 \rho_{D^*}^2 -91)z^3],\\ R_1(w) =& R_1(1)-0.12(w-1)+0.05(w-1)^2,\\ R_2(w) =& R_2(1)+0.11(w-1)-0.06(w-1)^2,\\ R_3(w) =& 1.22 -0.052(w-1) +0.026(w-1)^2, \end{split}

      (45)

      with z = (\sqrt{w+1}-\sqrt{2})/(\sqrt{w+1}+\sqrt{2}) . The fit parameters R_{1}(1) , R_{2}(1) , h_{A_1}(1) and \rho_{D^{*}}^2 are taken from Ref. [14]. Expressions of the tensor form factors T_{1,2,3} can be found in Ref. [110].

      The \Lambda_b\rightarrow\Lambda_c hadronic matrix elements can be written in terms of ten helicity form factors \{F_{0,+,\perp},G_{0,+,\perp},h_{+,\perp},\widetilde{h}_{+,\perp}\} [75, 76]. Following Ref. [75], the lattice calculations are fitted to two (Bourrely-Caprini-Lellouch) BCL z-parametrizations. In the so called "nominal" fit, a form factor f reduces to the form

      \tag{B5} f(q^2) = \frac{1}{1-q^2/(m_{\rm pole}^f)^2} \big[ a_0^f + a_1^f\:z^f(q^2) \big],

      (46)

      while a form factor f in the higher-order fit is given by

      \tag{B6} \begin{split} f_{\rm HO}(q^2) =& \frac{1}{1-q^2/(m_{\rm pole}^f)^2} \left\{ a_{0,{\rm HO}}^f + a_{1,{\rm HO}}^f\:z^f(q^2) \right.\\&\left.+ a_{2,{\rm HO}}^f\:[z^f(q^2)]^2 \right\},\end{split}

      (47)

      where t_0 = (m_{\Lambda_b} - m_{\Lambda_c})^2 , t_+^f = (m_{\rm pole}^f)^2 , and z^f(q^2) = \left.\left(\sqrt{t_+^f-q^2}-\sqrt{t_+^f-t_0}\right)\right/ \left(\sqrt{t_+^f-q^2}+\sqrt{t_+^f-t_0}\right) .The values of the fit parameters and all the pole masses are taken from Ref. [76].

      In addition, the form factors for B_c \to J/\psi\ell\bar{\nu_\ell} and B_c \to \eta_c\ell\bar{\nu_\ell} decays are taken form the results in the Covariant Light-Front Approach in Ref. [18].

    C.   Appendix C: Observables in { { b \to c \tau \bar\nu}} decays

      C.1.   { B \to D \tau \bar\nu} and { B_c \to \eta_c \tau \bar\nu} decays

    • Because similar expressions hold for the B \to D \tau \bar\nu and B_c \to \eta_c \tau \bar\nu decays, we only provide the theoretical formulae of the former. Using the form factors in Appendix B, the non-zero helicity amplitudes for the B \to D \tau \bar\nu decay in Eq. (A18) can be written as

      \begin{split} H_{0}^{VA}(q^2) & = (1+g_{\rm L}+g_{\rm R})\sqrt{\frac{Q_+Q_-}{q^2}} F_{+}(q^2) , \end{split}

      \tag{C1} \begin{split} H_{t}^{VA}(q^2) & = (1+g_{\rm L}+g_{\rm R})\frac{m_B^2-m_D^2}{\sqrt{q^2}} F_0(q^2) ,\\ H^{SP}(q^2) & = g_S{m_B^2-m_D^2 \over m_b-m_c} F_0(q^2) , \\ H_{-,+}^T(q^2) & = H_{t,0}^T(q^2) = g_T{\sqrt{Q_+Q_-} \over m_B+m_D} F_T(q^2) . \end{split}

      (48)

      Subsequently, the differential decay width in Eq. (11) and angular observables in Eq. (16) and (17) are obtained

      \tag{C2} \begin{split} \frac{{\rm d}\Gamma}{{\rm d}q^2} = &\frac{N_D}{2} \biggl[\frac{3m_{\tau}^2}{q^2}|H^{VA}_{t}|^2+ \Bigl(2+\frac{m_\tau^2}{q^2}\Bigr) |H^{VA}_{0}|^2+3|H^{SP}|^2 +16\Big(1+\frac{2m_\tau^2}{q^2}\Big) \\ &\times|H^{T}_{t,0}|^2 +\frac{6m_\tau}{\sqrt{q^2}} \Re[H^{SP}H^{VA*}_{t}]+\frac{24m_\tau}{\sqrt{q^2}} \Re[H^{T}_{t,0}H^{VA*}_{0}] \biggr], \end{split}

      (49)

      \tag{C3} \frac{{\rm d}A_{\rm FB}}{{\rm d} q^2} = \frac{3N_D}{2}\Re \biggl[\biggl(4H^{T*}_{t,0}+ \frac{m_\tau}{\sqrt{q^2}}H_0^{VA*}\biggr) \biggl(H^{SP} + \frac{m_\tau}{\sqrt{q^2}}H_t^{VA} \biggr) \biggr],

      (50)

      \tag{C4} \begin{split} \frac{{\rm d}P_{L}^{\tau}}{{\rm d} q^2} = &\frac{1}{\rm{d}\Gamma/{\rm d}q^2}\frac{N_D}{2} \biggl[\frac{3m_\tau^2}{q^2}|H^{VA}_{t}|^2+ \left(\frac{m_\tau^2}{q^2}-2 \right)|H^{VA}_{0}|^2 +3|H^{SP}|^2\\&+16 \left(1-\frac{2m_\tau^2}{q^2} \right)|H^{T}_{t,0}|^2 +\frac{6 m_\tau}{\sqrt{q^2}} \Re[H^{SP}H^{VA*}_{t}] \\ &-\frac{8 m_\tau}{\sqrt{q^2}}\Re[H^{T}_{t,0}H^{VA*}_{0}] \biggr] , \end{split}

      (51)

      with

      \tag{C5} N_D = \frac{G_{F}^{2}|V_{cb}|^2}{192\pi^{3}}\frac{q^2\sqrt{Q_+ Q_-}}{m_{B}^3}\Big(1-\frac{m_\tau^2}{q^2}\Big)^2.

      (52)
    • C.2.   { B \to D^* \tau \bar\nu} and { B_c \to J/\psi \tau \bar\nu} decays

    • Because similar expressions hold for the B \to D^* \tau \bar\nu and B_c \to J/\psi \tau \bar\nu decays, only theoretical formulae of the former are provided in this subsection. Using the form factors in Appendix B, the non-zero helicity amplitudes for the B \to D^* \tau \bar\nu decay in Eq. (A18) can be written as

      \tag{C6} \begin{split} H_0^{SP}(q^2) & = -g_P{ \sqrt{Q_+Q_-} \over m_b+m_c } A_0(q^2) , \\ H^{VA}_{\pm,\pm}(q^2) & = -(1+g_L-g_R) m_+ A_1(q^2) \pm (1+g_L+g_R){ \sqrt{Q_+Q_-} \over m_+ } V(q^2) , \\ H^{VA}_{0,t}(q^2) & = -(1+g_L-g_R){ \sqrt{Q_+Q_-} \over \sqrt{q^2} } A_0(q^2) , \\ H^{VA}_{0,0}(q^2) & = \frac{(1+g_L-g_R)}{2m_{D^*}\sqrt{q^2}}\biggl[- m_+(m_+ m_- -q^2) A_1(q^2) +{Q_+Q_- \over m_+ } A_2(q^2)\biggr] , \\ H^{T}_{\pm,\pm,t}(q^2) = &\pm H^{T}_{\pm,\pm,0}(q^2) = \frac{g_{T}}{\sqrt{q^{2}}} \biggl[\mp\sqrt{Q_+Q_-} T_1(q^2)-m_+m_- T_2(q^2) \biggr] , \\ H^{T}_{0,t,0}(q^2) = &\; H^{T}_{0,-,+}(q^2) = \frac{g_T}{2m_{D^{*}}}\biggl[-(m_B^2+3m_{D^{*}}^2-q^2) T_2(q^2)\\&+\frac{Q_+Q_-}{m_+ m_-}T_3(q^2) \biggr] , \end{split}

      (53)

      with m_\pm = m_B \pm m_{D^*} . Then, the differential decay width in Eq. (11) and the angular observables in Eq. (16) and (17) are obtained, respectively, as

      \tag{C7} \begin{split} \frac{{\rm d}\Gamma}{{\rm d} q^2} = & N_{D^*} \biggl[\frac{3m_\tau^2}{2q^2} |H^{VA}_{0,t}|^2+ \Bigl( 1+\frac{m_\tau^2}{2q^2} \Bigr)(|H^{VA}_{-,-}|^2+|H^{VA}_{0,0}|^2+|H^{VA}_{+,+}|^2) \\ &+\frac{3}{2}|H^{SP}_{0}|^2 +8 \Bigl( 1+\frac{2 m_\tau^2}{q^2} \Bigr)(|H^{T}_{0,t,0}|^2+|H^{T}_{+,+,t}|^2+|H^{T}_{-,-,t}|^2) \\ &+ \frac{3m_\tau}{\sqrt{q^2}}\Re[H^{SP}_{0} H^{VA*}_{0,t}]+\frac{12 m_\tau}{\sqrt{q^2}}(\Re[H^{T}_{0,t,0}H^{VA*}_{0,0}-H^{T}_{+,+,t}H^{VA*}_{+,+}\\&-H^{T}_{-,-,t}H^{VA*}_{-,-}])\biggr], \end{split}

      (54)

      \tag{C8} \begin{split} \frac{{\rm d}A_{\rm FB}}{{\rm d} q^2} = & \frac{3N_{D^*}}{4} \biggl[\frac{2m_\tau^2}{q^2} \Re[H^{VA}_{0,0}H^{VA*}_{0,t}]-|H^{VA}_{-,-}|^2+|H^{VA}_{+,+}|^2+8\Re[H_0^{SP} H_{0,t,0}^{T*}] \\ & +\frac{16m_\tau^2}{q^2}(|H^{T}_{+,+,t}|^2-|H^{T}_{-,-,t}|^2)+\frac{2m_\tau}{\sqrt{q^2}}\Re[H^{SP}_{0}H^{VA*}_{0,0}] \\ &+\frac{8m_\tau}{\sqrt{q^2}}\Re[H^{T}_{0,t,0}H^{VA*}_{0,t}+H^{T}_{-,-,t}H^{VA*}_{-,-}-H^{T}_{+,+,t}H^{VA*}_{+,+}] \biggr], \end{split}

      (55)

      \tag{C9} \begin{split} \frac{{\rm d}P_{L}^{{D^*}}}{{\rm d} q^2} =& \frac{1}{{\rm d}\Gamma/{\rm d}q^2}\frac{N_{D^*}}{2}\left[\frac{3m_\tau^2}{q^2}|H^{VA}_{0,t}|^2+ \left( 2 + \frac{m_\tau^2}{q^2} \right) |H^{VA}_{0,0}|^2 +3|H^{SP}_{0}|^2\right. \\&+16 \left(1+\frac{2m_\tau^2}{q^2} \right)|H^{T}_{0,t,0}|^2 +\frac{6 m_\tau}{\sqrt{q^2}}\Re[H^{SP}_{0}H^{VA*}_{0,t}]\\ &\left.+\frac{24 m_\tau}{\sqrt{q^2}}\Re[H^{T}_{0,t,0}H^{VA*}_{0,0}] \right], \end{split}

      (56)

      \tag{C10} \begin{split} \frac{{\rm d} P_L^\tau}{{\rm d}q^2} =& \frac{1}{{\rm d}\Gamma/{\rm d}q^2}\frac{N_{D^*}}{2}\biggl[\frac{3m_\tau^2}{q^2}|H^{VA}_{0,t}|^2+\Bigl(\frac{m_\tau^2}{q^2}-2\Bigr)(|H^{VA}_{+,+}|^2+|H^{VA}_{0,0}|^2+|H^{VA}_{-,-}|^2 ) \\ &+3|H^{SP}_{0}|^2+\frac{6 m_\tau}{\sqrt{q^2}}\Re[H^{SP}_{0}H^{VA*}_{0,t}]+16 \Bigl(1-\frac{2m_\tau^2}{q^2} \Bigr)(|H^{T}_{0,t,0}|^2\\ &+|H^{T}_{-,-,t}|^2+|H^{T}_{+,+,t}|^2) +\frac{8m_\tau}{\sqrt{q^2}}\Re[H^{T}_{-,-,t}H^{VA*}_{-,-}\\ &+H^{T}_{+,+,t}H^{VA*}_{+,+}-H^{T}_{0,t,0}H^{VA*}_{0,0}] \biggr], \end{split}

      (57)

      \tag{C11} \begin{split} \frac{{\rm d} P_{T}^{{D^*}}}{ {\rm d} q^2} =& \frac{1}{{\rm d}\Gamma/{\rm d} q^2}\frac{N_{D^*}}{2}\biggl[\Bigl(2 + \frac{m_\tau^2}{q^2} \Bigr)(|H^{VA}_{+,+}|^2-|H^{VA}_{-,-}|^2) \\ &+16(1+\frac{2m_\tau^2}{q^2})(|H^{T}_{+,+,t}|^2-|H^{T}_{-,-,t}|^2)\\ &+\frac{24m_\tau}{\sqrt{q^2}}\Re[H^{T}_{-,-,t}H^{VA*}_{-,-}-H^{T}_{+,+,t}H^{VA*}_{+,+}] \biggr], \end{split}

      (58)

      with

      \tag{C12} N_{D^*} = \frac{G_{F}^{2}|V_{cb}|^2}{192\pi^{3}}\frac{q^2\sqrt{Q_+ Q_-}}{m_{B}^3}\Big(1-\frac{m_\tau^2}{q^2}\Big)^2.

      (59)
    • C.3.   { \Lambda_b \to \Lambda_c \tau \bar\nu} decay

    • Using the transition form factors in Appendix B, the helicity amplitudes for the \Lambda_b \to \Lambda_c decay in Eq. (A18) can be written as

      \tag{C13} \begin{split} H^{SP}_{\pm 1/2, \pm 1/2} = &F_0g_S \frac{\sqrt{Q_+}}{m_b-m_c} m_- \mp G_0g_P\frac{\sqrt{Q_-}}{m_b+m_c} m_+, \\ H^{VA}_{\pm 1/2, \pm 1/2,t} = &F_0(1+g_L+g_R)\frac{\sqrt{Q_+}}{\sqrt{q^2}} m_- \mp G_0(1+g_L-g_R)\frac{\sqrt{Q_-}}{\sqrt{q^2}} m_+ , \\ H^{VA}_{\pm 1/2, \pm 1/2,0} = &F_+ (1+g_L+g_R)\frac{\sqrt{Q_-}}{\sqrt{q^2}} m_+ \mp G_+ (1+g_L-g_R)\frac{\sqrt{Q_+}}{\sqrt{q^2}} m_-, \\ H^{VA}_{\mp 1/2,\pm 1/2,\pm} = &F_\perp (1+g_L+g_R)\sqrt{2Q_-} \mp G_\perp (1+g_L-g_R)\sqrt{2Q_+} , \\ H^{T,\pm 1/2}_{\pm 1/2,t,0} = &H^{T,\pm 1/2}_{\pm 1/2,-,+} = g_T\Big[h_+\sqrt{Q_-}\pm \widetilde{h}_+\sqrt{Q_+}\Big] , \\ H^{T,\pm 1/2}_{\mp 1/2,t,\mp} = &\mp H^{T,\pm1/2}_{\mp 1/2,0,\mp} = g_T\frac{\sqrt{2}}{\sqrt{q^2}}\Big[h_\perp m_+\sqrt{Q_-}\mp\widetilde{h}_\perp m_- \sqrt{Q_+}\Big] , \end{split}

      (60)

      with m_\pm = m_{\Lambda_b}\pm m_{\Lambda_c} . Thus, the differential decay width in Eq. (11) can be written as

      \tag{C14} \begin{split} \frac{{ \rm d}\Gamma}{{\rm d} q^2} = & N_{\Lambda_c} \biggl[ A_1^{VA}+\frac{m_\tau^2}{2q^2}A_2^{VA} +\frac{3}{2}A_3^{SP}+8\Big(1+\frac{2m_\tau^2}{q^2}\Big)A_4^{T}\\&+\frac{3m_\tau}{\sqrt{q^2}} (A_5^{VA-SP}+ 4 A_6^{VA-T}) \biggr], \end{split}

      (61)

      with

      \tag{C15} \begin{split} N_{\Lambda_c} & = \frac{G_{F}^{2}|V_{cb}|^2}{384\pi^{3}}\frac{q^2\sqrt{Q_+ Q_-}}{m_{\Lambda_b}^3}\Big(1-\frac{m_\tau^2}{q^2}\Big)^2, \\ A_1^{VA} = &|H^{VA}_{-1/2,1/2,+}|^2+ \sum |H^{VA}_{s,s,0}|^2+|H^{VA}_{1/2,-1/2,-}|^2 , \\ A_2^{VA} = & A_1^{VA}+3 \sum |H^{VA}_{s,s,t}|^2, \\ A_3^{SP} = &\sum |H^{SP}_{s,s}|^2 , \\ A_4^{T} = &\sum |H^{T,s}_{s,t,0}|^2+|H^{T,1/2}_{-1/2,t,-}|^2+|H^{T,-1/2}_{1/2,t,+}|^2, \\ A_5^{VA-SP} = &\sum \Re[H^{SP*}_{s,s} H^{VA}_{s,s,t}] , \\ A_6^{VA-T} = &\sum \Re[H^{VA*}_{s,s,0}H^{T,s}_{s,t,0}] + \Re[H^{VA*}_{-1/2,1/2,+}H^{T,-1/2}_{1/2,t,+}]\\&+ \Re[H^{VA*}_{1/2,-1/2,-}H^{T,1/2}_{-1/2,t,-}], \end{split}

      (62)

      where \sum depicts the summation over s = \pm 1/2 . For the forward-backward asymmetry in Eq. (16), we obtain

      \tag{C16} \begin{split} \frac{ {\rm d} A_{\rm FB}} { {\rm d} q^2} =& \frac{ N_{\Lambda_c} }{{\rm d} \Gamma / {\rm d} q^2}\frac{3}{4} \biggl[ B_1^{VA}+\frac{2m_\tau^2}{q^2} \bigl(B_2^{VA}+8 B_3^T \bigr)\\&+ \frac{2m_\tau}{\sqrt{q^2}} \bigl(B_4^{VA-SP} +4 B_5^{VA-T} \bigr) + 8B_6^{SP-T} \biggr], \end{split}

      (63)

      where

      \tag{C17} \begin{split} B_1^{VA} = &\; |H^{VA}_{-1/2,1/2,+}|^2-|H^{VA}_{1/2,-1/2,-}|^2 , \\ B_2^{VA} = &\; \sum \Re[H^{VA*}_{s,s,t}H^{VA}_{s,s,0}] , \\ B_3^{T} = &\; |H^{T,-1/2}_{1/2,t,+}|^2-|H^{T,1/2}_{-1/2,t,-}|^2, \\ B_4^{VA-SP} = & \sum \Re[H^{SP*}_{s,s}H^{VA}_{s,s,0}], \\ B_5^{VA-T} = & \sum\Re[H^{VA*}_{s,s,t} H^{T,s}_{s,t,0}]+\Re[H^{VA*}_{-1/2,1/2,+} H^{T,-1/2}_{1/2,t,+}]\\&-\Re[H^{VA*}_{1/2,-1/2,-} H^{T,1/2}_{-1/2,t,-}], \\ B_6^{SP-T} = &\; \sum\Re[H^{SP*}_{s,s}H^{T,s}_{s,t,0}]. \end{split}

      (64)

      For the \Lambda_c longitudinal polarization fraction in Eq. (17), we obtain

      \tag{C18} \begin{split} \frac{{\rm d}P_{L}^{\Lambda_c}}{{\rm d} q^2} = &\frac{N_{\Lambda_c}}{{\rm d}\Gamma/{\rm d}q^2}\frac{1}{2}\biggl[2 C_1^{VA}+\frac{m_\tau^2}{q^2}C_2^{VA}+3C_3^{SP} \\ &+16 \Bigl(1 + \frac{2m_\tau^2}{q^2} \Bigr)C_4^{T} +6\frac{m_\tau}{\sqrt{q^2}} \bigl(C_5^{VA-SP}+ 4C_6^{VA-T} \bigr) \biggr], \end{split}

      (65)

      where

      \tag{C19} \begin{split} C_1^{VA} = &|H^{VA}_{1/2,1/2,0}|^2-|H^{VA}_{-1/2,-1/2,0}|^2+|H^{VA}_{-1/2,1/2,+}|^2-|H^{VA}_{1/2,-1/2,-}|^2, \\ C_2^{VA} = &C_1^{VA} -3|H^{VA}_{-1/2,-1/2,t}|^2 +3|H^{VA}_{1/2,1/2,t}|^2, \\ C_3^{SP} = &|H^{SP}_{1/2,1/2}|^2-|H^{SP}_{-1/2,-1/2}|^2, \\ C_4^{T} = &\sum 2s|H^{T,s}_{s,t,0}|^2+|H^{T,-1/2}_{1/2,t,+}|^2 -|H^{T,1/2}_{-1/2,t,-}|^2, \\ C_5^{VA-SP} = & \sum 2s\Re[H^{SP*}_{s,s}H^{VA}_{s,s,t}], \\ C_6^{VA-T} = &\sum 2s\Re\big[H^{T,s}_{s,t,0}H^{VA*}_{s,s,0}\big]+\Re\big[H^{T,-1/2}_{1/2,t,+}H^{VA*}_{-1/2,1/2,+}\big] \\&-\Re\big[H^{T,1/2}_{-1/2,t,-}H^{VA*}_{1/2,-1/2,-}\big]. \end{split}

      (66)

      For the \tau -lepton longitudinal polarization fraction, we obtain

      \tag{C20} \begin{split} \frac{{\rm d} P_L^\tau}{{\rm d} q^2} = &\frac{N_{\Lambda_c}}{{\rm d}\Gamma/{\rm d}q^2}\frac{1}{2}\biggl[-2 D_1^{VA}+\frac{m_\tau^2}{q^2}D_2^{VA}+3D_3^{SP} \\ &+16 \Bigl(1-\frac{2m_\tau^2}{q^2} \Bigr)D_4^{T}+\frac{m_\tau}{\sqrt{q^2}} \bigl( 6D_5^{VA-SP}-8D_6^{VA-T} \bigr) \biggr], \end{split}

      (67)

      where

      \tag{C21}\begin{split} D_1^{VA} = & \sum |H^{VA}_{s,s,0}|^2 + |H^{VA}_{-1/2,1/2,+}|^2 + |H^{VA}_{1/2,-1/2,-}|^2, \\ D_2^{VA} = & D_1^{VA}+3\sum |H^{VA}_{s,s,t}|^2, \\ D_3^{SP} = & \sum |H^{SP}_{s,s}|^2, \\ D_4^{T} = & \sum |H^{T,s}_{s,t,0}|^2+|H^{T,-1/2}_{1/2,t,+}|^2 + |H^{T,1/2}_{-1/2,t,-}|^2 , \\ D_5^{VA-SP} = & \sum \Re[H^{SP*}_{s, s}H^{VA}_{ s, s,t}], \\ D_6^{VA-T} = &\sum \Re[H^{T,s}_{s,t,0}H^{VA*}_{s,s,0}] +\Re[H^{T,-1/2}_{1/2,t,+}H^{VA*}_{-1/2,1/2,+}]\\&+\Re[H^{T,1/2}_{-1/2,t,-}H^{VA*}_{1/2,-1/2,-}]. \end{split}

      (68)
Reference (111)

目录

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return