QCD calculations of Bπ, K form factors with higher-twist corrections

  • Cai-Dian Lü
  • Yue-Long Shen
  • Yu-Ming WangEmail author
  • Yan-Bing WeiEmail author
Open Access
Regular Article - Theoretical Physics


We update QCD calculations of Bπ, K form factors at large hadronic recoil by including the subleading-power corrections from the higher-twist B-meson light-cone distribution amplitudes (LCDAs) up to the twist-six accuracy and the strange-quark mass effects at leading-power in Λ/mb from the twist-two B-meson LCDA ϕ B + (ω, μ). The higher-twist corrections from both the two-particle and three-particle B-meson LCDAs are computed from the light-cone QCD sum rules (LCSR) at tree level. In particular, we construct the local duality model for the twist-five and -six B-meson LCDAs, in agreement with the corresponding asymptotic behaviours at small quark and gluon momenta, employing the method of QCD sum rules in heavy quark effective theory at leading order in αs. The strange quark mass effects in semileptonic BK form factors yield the leading-power contribution in the heavy quark expansion, consistent with the power-counting analysis in soft-collinear effective theory, and they are also computed from the LCSR approach due to the appearance of the rapidity singularities. We demonstrate explicitly that the SU(3)-flavour symmetry breaking effects between Bπ and BK form factors, free of the power suppression in Λ/mb, are suppressed by a factor of \( {\alpha}_s\left(\sqrt{m_b\Lambda}\right) \) in perturbative expansion, and they also respect the large-recoil symmetry relations of the heavy-to-light form factors at least at one-loop accuracy. An exploratory analysis of the obtained sum rules for Bπ, K form factors with two distinct models for the B-meson LCDAs indicates that the dominant higher-twist corrections are from the Wandzura-Wilczek part of the two-particle LCDA of twist five g B (ω, μ) instead of the three-particle B-meson LCDAs. The resulting SU(3)-flavour symmetry violation effects of Bπ, K form factors turn out to be insensitive to the non-perturbative models of B-meson LCDAs. We further explore the phenomenological aspects of the semileptonic Bπℓν decays and the rare exclusive processes BKνν, including the determination of the CKM matrix element |Vub|, the normalized differential q2 distributions and precision observables defined by the ratios of branching fractions for the above-mentioned two channels in the same intervals of q2.


NLO Computations QCD Phenomenology 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    Fermilab Lattice and MILC collaborations, |V ub| from Bπℓν decays and (2 + 1)-flavor lattice QCD, Phys. Rev. D 92 (2015) 014024 [arXiv:1503.07839] [INSPIRE].
  2. [2]
    Fermilab Lattice and MILC collaboration, Bπℓℓ form factors for new-physics searches from lattice QCD, Phys. Rev. Lett. 115 (2015) 152002 [arXiv:1507.01618] [INSPIRE].
  3. [3]
    J.A. Bailey et al., BKℓ + decay form factors from three-flavor lattice QCD, Phys. Rev. D 93 (2016) 025026 [arXiv:1509.06235] [INSPIRE].ADSGoogle Scholar
  4. [4]
    C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, An effective field theory for collinear and soft gluons: heavy to light decays, Phys. Rev. D 63 (2001) 114020 [hep-ph/0011336] [INSPIRE].
  5. [5]
    C.W. Bauer, D. Pirjol and I.W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D 65 (2002) 054022 [hep-ph/0109045] [INSPIRE].
  6. [6]
    M. Beneke, A.P. Chapovsky, M. Diehl and T. Feldmann, Soft collinear effective theory and heavy to light currents beyond leading power, Nucl. Phys. B 643 (2002) 431 [hep-ph/0206152] [INSPIRE].
  7. [7]
    M. Beneke and T. Feldmann, Multipole expanded soft collinear effective theory with non-Abelian gauge symmetry, Phys. Lett. B 553 (2003) 267 [hep-ph/0211358] [INSPIRE].
  8. [8]
    M. Beneke, Y. Kiyo and D.s. Yang, Loop corrections to subleading heavy quark currents in SCET, Nucl. Phys. B 692 (2004) 232 [hep-ph/0402241] [INSPIRE].
  9. [9]
    T. Becher and R.J. Hill, Loop corrections to heavy-to-light form-factors and evanescent operators in SCET, JHEP 10 (2004) 055 [hep-ph/0408344] [INSPIRE].
  10. [10]
    R. Bonciani and A. Ferroglia, Two-loop QCD corrections to the heavy-to-light quark decay, JHEP 11 (2008) 065 [arXiv:0809.4687] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    H.M. Asatrian, C. Greub and B.D. Pecjak, NNLO corrections to \( \overline{B}\to {X}_u\ell \overline{\nu} \) in the shape-function region, Phys. Rev. D 78 (2008) 114028 [arXiv:0810.0987] [INSPIRE].ADSGoogle Scholar
  12. [12]
    M. Beneke, T. Huber and X.-Q. Li, Two-loop QCD correction to differential semi-leptonic bu decays in the shape-function region, Nucl. Phys. B 811 (2009) 77 [arXiv:0810.1230] [INSPIRE].
  13. [13]
    G. Bell, NNLO corrections to inclusive semileptonic B decays in the shape-function region, Nucl. Phys. B 812 (2009) 264 [arXiv:0810.5695] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  14. [14]
    G. Bell, M. Beneke, T. Huber and X.-Q. Li, Heavy-to-light currents at NNLO in SCET and semi-inclusive \( \overline{B}\to {X}_s{\ell}^{+}{\ell}^{-} \) decay, Nucl. Phys. B 843 (2011) 143 [arXiv:1007.3758] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    R.J. Hill, T. Becher, S.J. Lee and M. Neubert, Sudakov resummation for subleading SCET currents and heavy-to-light form-factors, JHEP 07 (2004) 081 [hep-ph/0404217] [INSPIRE].
  16. [16]
    M. Beneke and D. Yang, Heavy-to-light B meson form-factors at large recoil energy: spectator-scattering corrections, Nucl. Phys. B 736 (2006) 34 [hep-ph/0508250] [INSPIRE].
  17. [17]
    M. Beneke and T. Feldmann, Factorization of heavy to light form-factors in soft collinear effective theory, Nucl. Phys. B 685 (2004) 249 [hep-ph/0311335] [INSPIRE].
  18. [18]
    P. Ball and R. Zwicky, Improved analysis of Bπeν from QCD sum rules on the light cone, JHEP 10 (2001) 019 [hep-ph/0110115] [INSPIRE].
  19. [19]
    P. Ball and R. Zwicky, New results on Bπ, K, η decay formfactors from light-cone sum rules, Phys. Rev. D 71 (2005) 014015 [hep-ph/0406232] [INSPIRE].
  20. [20]
    G. Duplancic, A. Khodjamirian, T. Mannel, B. Melic and N. Offen, Light-cone sum rules for Bπ form factors revisited, JHEP 04 (2008) 014 [arXiv:0801.1796] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    A. Khodjamirian, T. Mannel and N. Offen, B-meson distribution amplitude from the Bπ form-factor, Phys. Lett. B 620 (2005) 52 [hep-ph/0504091] [INSPIRE].
  22. [22]
    A. Khodjamirian, T. Mannel and N. Offen, Form-factors from light-cone sum rules with B-meson distribution amplitudes, Phys. Rev. D 75 (2007) 054013 [hep-ph/0611193] [INSPIRE].
  23. [23]
    F. De Fazio, T. Feldmann and T. Hurth, Light-cone sum rules in soft-collinear effective theory, Nucl. Phys. B 733 (2006) 1 [Erratum ibid. B 800 (2008) 405] [hep-ph/0504088] [INSPIRE].
  24. [24]
    F. De Fazio, T. Feldmann and T. Hurth, SCET sum rules for BP and BV transition form factors, JHEP 02 (2008) 031 [arXiv:0711.3999] [INSPIRE].CrossRefGoogle Scholar
  25. [25]
    Y.-M. Wang and Y.-L. Shen, QCD corrections to Bπ form factors from light-cone sum rules, Nucl. Phys. B 898 (2015) 563 [arXiv:1506.00667] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Y.-M. Wang, Y.-B. Wei, Y.-L. Shen and C.-D. Lü, Perturbative corrections to BD form factors in QCD, JHEP 06 (2017) 062 [arXiv:1701.06810] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    Y.-L. Shen, Y.-B. Wei and C.-D. Lü, Renormalization group analysis of Bπ form factors with B-meson light-cone sum rules, Phys. Rev. D 97 (2018) 054004 [arXiv:1607.08727] [INSPIRE].ADSGoogle Scholar
  28. [28]
    Y.-M. Wang, Y.-L. Shen and C.-D. Lü, Λbp, Λ transition form factors from QCD light-cone sum rules, Phys. Rev. D 80 (2009) 074012 [arXiv:0907.4008] [INSPIRE].ADSGoogle Scholar
  29. [29]
    T. Feldmann and M.W.Y. Yip, Form factors for Lambda b → Λ transitions in SCET, Phys. Rev. D 85 (2012) 014035 [Erratum ibid. D 86 (2012) 079901] [arXiv:1111.1844] [INSPIRE].
  30. [30]
    Y.-M. Wang and Y.-L. Shen, Perturbative corrections to Λb → Λ form factors from QCD light-cone sum rules, JHEP 02 (2016) 179 [arXiv:1511.09036] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    J. Botts and G.F. Sterman, Hard elastic scattering in QCD: leading behavior, Nucl. Phys. B 325 (1989) 62 [INSPIRE].
  32. [32]
    H.-N. Li and G.F. Sterman, The perturbative pion form-factor with Sudakov suppression, Nucl. Phys. B 381 (1992) 129 [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    H.-N. Li, Y.-L. Shen, Y.-M. Wang and H. Zou, Next-to-leading-order correction to pion form factor in k T factorization, Phys. Rev. D 83 (2011) 054029 [arXiv:1012.4098] [INSPIRE].ADSGoogle Scholar
  34. [34]
    H.-N. Li, Y.-L. Shen and Y.-M. Wang, Next-to-leading-order corrections to Bπ form factors in k T factorization, Phys. Rev. D 85 (2012) 074004 [arXiv:1201.5066] [INSPIRE].ADSGoogle Scholar
  35. [35]
    H.-N. Li, Y.-L. Shen and Y.-M. Wang, Resummation of rapidity logarithms in B meson wave functions, JHEP 02 (2013) 008 [arXiv:1210.2978] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    H.-N. Li, Y.-L. Shen and Y.-M. Wang, Joint resummation for pion wave function and pion transition form factor, JHEP 01 (2014) 004 [arXiv:1310.3672] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    X.-G. He, T. Li, X.-Q. Li and Y.-M. Wang, PQCD calculation for Λb → Λγ in the Standard Model, Phys. Rev. D 74 (2006) 034026 [hep-ph/0606025] [INSPIRE].
  38. [38]
    C.-D. Lü, Y.-M. Wang, H. Zou, A. Ali and G. Kramer, Anatomy of the pQCD approach to the baryonic decays Λbpπ, pK, Phys. Rev. D 80 (2009) 034011 [arXiv:0906.1479] [INSPIRE].ADSGoogle Scholar
  39. [39]
    W. Wang, B to tensor meson form factors in the perturbative QCD approach, Phys. Rev. D 83 (2011) 014008 [arXiv:1008.5326] [INSPIRE].
  40. [40]
    Y. Li, C.-D. Lü, Z.-J. Xiao and X.-Q. Yu, Branching ratio and CP asymmetry of B sπ + π decays in the perturbative QCD approach, Phys. Rev. D 70 (2004) 034009 [hep-ph/0404028] [INSPIRE].
  41. [41]
    Q. Qin, Z.-T. Zou, X. Yu, H.-N. Li and C.-D. Lü, Perturbative QCD study of B s decays to a pseudoscalar meson and a tensor meson, Phys. Lett. B 732 (2014) 36 [arXiv:1401.1028] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    H.-N. Li and Y.-M. Wang, Non-dipolar Wilson links for transverse-momentum-dependent wave functions, JHEP 06 (2015) 013 [arXiv:1410.7274] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    Y.-M. Wang, Non-dipolar gauge links for transverse-momentum-dependent pion wave functions, EPJ Web Conf. 112 (2016) 01021 [arXiv:1512.08374] [INSPIRE].CrossRefGoogle Scholar
  44. [44]
    Belle II collaboration, The Belle II physics book, arXiv:1808.10567 [INSPIRE].
  45. [45]
    V.M. Braun, Y. Ji and A.N. Manashov, Higher-twist B-meson distribution amplitudes in HQET, JHEP 05 (2017) 022 [arXiv:1703.02446] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    H. Kawamura, J. Kodaira, C.-F. Qiao and K. Tanaka, B-meson light cone distribution amplitudes in the heavy quark limit, Phys. Lett. B 523 (2001) 111 [Erratum ibid. B 536 (2002)344] [hep-ph/0109181] [INSPIRE].
  47. [47]
    A.K. Leibovich, Z. Ligeti and M.B. Wise, Comment on quark masses in SCET, Phys. Lett. B 564 (2003) 231 [hep-ph/0303099] [INSPIRE].
  48. [48]
    M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].
  49. [49]
    M. Beneke and T. Feldmann, Symmetry breaking corrections to heavy to light B meson form-factors at large recoil, Nucl. Phys. B 592 (2001) 3 [hep-ph/0008255] [INSPIRE].
  50. [50]
    A.G. Grozin and M. Neubert, Asymptotics of heavy meson form-factors, Phys. Rev. D 55 (1997) 272 [hep-ph/9607366] [INSPIRE].
  51. [51]
    M. Beneke and J. Rohrwild, B meson distribution amplitude from Bγℓν, Eur. Phys. J. C 71 (2011) 1818 [arXiv:1110.3228] [INSPIRE].
  52. [52]
    Y.-M. Wang, Factorization and dispersion relations for radiative leptonic B decay, JHEP 09 (2016) 159 [arXiv:1606.03080] [INSPIRE].
  53. [53]
    I.I. Balitsky and V.M. Braun, Evolution equations for QCD string operators, Nucl. Phys. B 311 (1989) 541 [INSPIRE].
  54. [54]
    M. Neubert, Heavy quark symmetry, Phys. Rept. 245 (1994) 259 [hep-ph/9306320] [INSPIRE].
  55. [55]
    Y.-M. Wang and Y.-L. Shen, Subleading-power corrections to the radiative leptonic Bγℓν decay in QCD, JHEP 05 (2018) 184 [arXiv:1803.06667] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    V.M. Braun, D. Yu. Ivanov and G.P. Korchemsky, The B meson distribution amplitude in QCD, Phys. Rev. D 69 (2004) 034014 [hep-ph/0309330] [INSPIRE].
  57. [57]
    V.M. Braun and I.E. Filyanov, Conformal invariance and pion wave functions of nonleading twist, Z. Phys. C 48 (1990) 239 [Sov. J. Nucl. Phys. 52 (1990) 126] [Yad. Fiz. 52 (1990) 199] [INSPIRE].
  58. [58]
    T. Feldmann, B.O. Lange and Y.-M. Wang, B-meson light-cone distribution amplitude: perturbative constraints and asymptotic behavior in dual space, Phys. Rev. D 89 (2014) 114001 [arXiv:1404.1343] [INSPIRE].ADSGoogle Scholar
  59. [59]
    T. Nishikawa and K. Tanaka, QCD sum rules for quark-gluon three-body components in the B meson, Nucl. Phys. B 879 (2014) 110 [arXiv:1109.6786] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  60. [60]
    B.O. Lange and M. Neubert, Renormalization group evolution of the B meson light cone distribution amplitude, Phys. Rev. Lett. 91 (2003) 102001 [hep-ph/0303082] [INSPIRE].
  61. [61]
    Y.-M. Wang and Y.-L. Shen, Subleading power corrections to the pion-photon transition form factor in QCD, JHEP 12 (2017) 037 [arXiv:1706.05680] [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    A. Khodjamirian, T. Mannel and M. Melcher, Flavor SU(3) symmetry in charmless B decays, Phys. Rev. D 68 (2003) 114007 [hep-ph/0308297] [INSPIRE].
  63. [63]
    S. Aoki et al., Review of lattice results concerning low-energy particle physics, Eur. Phys. J. C 77 (2017) 112 [arXiv:1607.00299] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    Particle Data Group collaboration, Review of particle physics, Phys. Rev. D 98 (2018) 030001 [INSPIRE].
  65. [65]
    M. Beneke, A. Maier, J. Piclum and T. Rauh, The bottom-quark mass from non-relativistic sum rules at NNNLO, Nucl. Phys. B 891 (2015) 42 [arXiv:1411.3132] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  66. [66]
    B. Dehnadi, A.H. Hoang and V. Mateu, Bottom and charm mass determinations with a convergence test, JHEP 08 (2015) 155 [arXiv:1504.07638] [INSPIRE].CrossRefGoogle Scholar
  67. [67]
    P. Ball and E. Kou, Bγeν transitions from QCD sum rules on the light cone, JHEP 04 (2003) 029 [hep-ph/0301135] [INSPIRE].
  68. [68]
    V.M. Braun and A. Khodjamirian, Soft contribution to Bγℓν and the B-meson distribution amplitude, Phys. Lett. B 718 (2013) 1014 [arXiv:1210.4453] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  69. [69]
    M. Beneke, V.M. Braun, Y. Ji and Y.-B. Wei, Radiative leptonic decay Bγℓν with subleading power corrections, JHEP 07 (2018) 154 [arXiv:1804.04962] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    A. Khodjamirian, T. Mannel, N. Offen and Y.-M. Wang, Bπℓν l width and |V ub| from QCD light-cone sum rules, Phys. Rev. D 83 (2011) 094031 [arXiv:1103.2655] [INSPIRE].ADSGoogle Scholar
  71. [71]
    I. Sentitemsu Imsong, A. Khodjamirian, T. Mannel and D. van Dyk, Extrapolation and unitarity bounds for the Bπ form factor, JHEP 02 (2015) 126 [arXiv:1409.7816] [INSPIRE].ADSCrossRefGoogle Scholar
  72. [72]
    A. Khodjamirian and A.V. Rusov, B sKℓν and B (s)π(K) + decays at large recoil and CKM matrix elements, JHEP 08 (2017) 112 [arXiv:1703.04765] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    G. Duplancic and B. Melic, B, B sK form factors: an update of light-cone sum rule results, Phys. Rev. D 78 (2008) 054015 [arXiv:0805.4170] [INSPIRE].ADSGoogle Scholar
  74. [74]
    P. Ball, Bπ and BK transitions from QCD sum rules on the light cone, JHEP 09 (1998) 005 [hep-ph/9802394] [INSPIRE].
  75. [75]
    C. Bourrely, I. Caprini and L. Lellouch, Model-independent description of Bπℓν decays and a determination of |V ub|, Phys. Rev. D 79 (2009) 013008 [Erratum ibid. D 82 (2010) 099902] [arXiv:0807.2722] [INSPIRE].
  76. [76]
    C.G. Boyd, B. Grinstein and R.F. Lebed, Constraints on form-factors for exclusive semileptonic heavy to light meson decays, Phys. Rev. Lett. 74 (1995) 4603 [hep-ph/9412324] [INSPIRE].
  77. [77]
    BaBar collaboration, Branching fraction and form-factor shape measurements of exclusive charmless semileptonic B decays and determination of |V ub|, Phys. Rev. D 86 (2012) 092004 [arXiv:1208.1253] [INSPIRE].
  78. [78]
    Belle collaboration, Study of exclusive BX u ℓν decays and extraction ofV ubusing full reconstruction tagging at the Belle experiment, Phys. Rev. D 88 (2013) 032005 [arXiv:1306.2781] [INSPIRE].
  79. [79]
    BaBar collaboration, Study of Bπℓν and Bρℓν decays and determination of |V ub|, Phys. Rev. D 83 (2011) 032007 [arXiv:1005.3288] [INSPIRE].
  80. [80]
    BaBar collaboration, Measurement of the B 0π + ν and B +η () + ν branching fractions, the B 0π + ν and B +ηℓ + ν form-factor shapes and determination of |V ub|, Phys. Rev. D 83 (2011) 052011 [arXiv:1010.0987] [INSPIRE].
  81. [81]
    Belle collaboration, Measurement of the decay B 0π + ν and determination of |V ub|, Phys. Rev. D 83 (2011) 071101 [arXiv:1012.0090] [INSPIRE].
  82. [82]
    M. Beneke, T. Feldmann and D. Seidel, Systematic approach to exclusive BVℓ + , Vγ decays, Nucl. Phys. B 612 (2001) 25 [hep-ph/0106067] [INSPIRE].
  83. [83]
    A. Khodjamirian, T. Mannel, A.A. Pivovarov and Y.-M. Wang, Charm-loop effect in BK (*) + and BK * γ, JHEP 09 (2010) 089 [arXiv:1006.4945] [INSPIRE].
  84. [84]
    A. Khodjamirian, T. Mannel and Y.M. Wang, BKℓ + decay at large hadronic recoil, JHEP 02 (2013) 010 [arXiv:1211.0234] [INSPIRE].ADSCrossRefGoogle Scholar
  85. [85]
    M. Dimou, J. Lyon and R. Zwicky, Exclusive chromomagnetism in heavy-to-light FCNCs, Phys. Rev. D 87 (2013) 074008 [arXiv:1212.2242] [INSPIRE].ADSGoogle Scholar
  86. [86]
    J. Lyon and R. Zwicky, Isospin asymmetries in B → (K * , ρ)γ/ℓ + and BKℓ + in and beyond the Standard Model, Phys. Rev. D 88 (2013) 094004 [arXiv:1305.4797] [INSPIRE].ADSGoogle Scholar
  87. [87]
    G. Buchalla and A.J. Buras, The rare decays \( K\to \pi \nu \overline{\nu},\kern0.5em B\to X\nu \overline{\nu} \) and B + : an update, Nucl. Phys. B 548 (1999) 309 [hep-ph/9901288] [INSPIRE].
  88. [88]
    G. Buchalla and A.J. Buras, QCD corrections to the sdZ vertex for arbitrary top quark mass, Nucl. Phys. B 398 (1993) 285 [INSPIRE].ADSCrossRefGoogle Scholar
  89. [89]
    M. Misiak and J. Urban, QCD corrections to FCNC decays mediated by Z penguins and W boxes, Phys. Lett. B 451 (1999) 161 [hep-ph/9901278] [INSPIRE].
  90. [90]
    J. Brod, M. Gorbahn and E. Stamou, Two-loop electroweak corrections for the \( K\to \pi \nu \overline{\nu} \) decays, Phys. Rev. D 83 (2011) 034030 [arXiv:1009.0947] [INSPIRE].ADSGoogle Scholar
  91. [91]
    M. Bartsch, M. Beylich, G. Buchalla and D.-N. Gao, Precision flavour physics with \( B\to K\nu \overline{\nu} \) and BKℓ + , JHEP 11 (2009) 011 [arXiv:0909.1512] [INSPIRE].ADSCrossRefGoogle Scholar
  92. [92]
    BaBar collaboration, Search for \( B\to {K}^{\left(*\right)}\nu \overline{\nu} \) and invisible quarkonium decays, Phys. Rev. D 87 (2013) 112005 [arXiv:1303.7465] [INSPIRE].
  93. [93]
    D. Bigi and P. Gambino, Revisiting BDℓν, Phys. Rev. D 94 (2016) 094008 [arXiv:1606.08030] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Institute of High Energy PhysicsCASBeijingChina
  2. 2.School of PhysicsUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.College of Information Science and EngineeringOcean University of ChinaQingdaoP.R. China
  4. 4.School of PhysicsNankai UniversityTianjinChina

Personalised recommendations