Non-local matrix elements in B(s) → {K(*), ϕ}+


We revisit the theoretical predictions and the parametrization of non-local matrix elements in rare \( {\overline{B}}_{(s)}\to \left\{{\overline{K}}^{\left(\ast \right)},\phi \right\}{\mathrm{\ell}}^{+}{\mathrm{\ell}}^{-} \) and \( {\overline{B}}_{(s)}\to \left\{{\overline{K}}^{\ast },\phi \right\}\gamma \) decays. We improve upon the current state of these matrix elements in two ways. First, we recalculate the hadronic matrix elements needed at subleading power in the light-cone OPE using B-meson light-cone sum rules. Our analytical results supersede those in the literature. We discuss the origin of our improvements and provide numerical results for the processes under consideration. Second, we derive the first dispersive bound on the non-local matrix elements. It provides a parametric handle on the truncation error in extrapolations of the matrix elements to large timelike momentum transfer using the z expansion. We illustrate the power of the dispersive bound at the hand of a simple phenomenological application. As a side result of our work, we also provide numerical results for the Bsϕ form factors from B-meson light-cone sum rules.

A preprint version of the article is available at ArXiv.


  1. [1]

    G. Buchalla, A.J. Buras and M.E. Lautenbacher, Weak decays beyond leading logarithms, Rev. Mod. Phys. 68 (1996) 1125 [hep-ph/9512380] [INSPIRE].

  2. [2]

    J. Aebischer, M. Fael, C. Greub and J. Virto, B physics Beyond the Standard Model at One Loop: Complete Renormalization Group Evolution below the Electroweak Scale, JHEP 09 (2017) 158 [arXiv:1704.06639] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  3. [3]

    S. Descotes-Genon, J. Matias and J. Virto, Understanding the BK*μ+μ Anomaly, Phys. Rev. D 88 (2013) 074002 [arXiv:1307.5683] [INSPIRE].

  4. [4]

    F. Beaujean, C. Bobeth and D. van Dyk, Comprehensive Bayesian analysis of rare (semi)leptonic and radiative B decays, Eur. Phys. J. C 74 (2014) 2897 [Erratum ibid. 74 (2014) 3179] [arXiv:1310.2478] [INSPIRE].

  5. [5]

    W. Altmannshofer and D.M. Straub, New physics in bs transitions after LHC run 1, Eur. Phys. J. C 75 (2015) 382 [arXiv:1411.3161] [INSPIRE].

    ADS  Article  Google Scholar 

  6. [6]

    S. Descotes-Genon, L. Hofer, J. Matias and J. Virto, Global analysis of bsℓℓ anomalies, JHEP 06 (2016) 092 [arXiv:1510.04239] [INSPIRE].

    ADS  Article  Google Scholar 

  7. [7]

    W. Altmannshofer, C. Niehoff, P. Stangl and D.M. Straub, Status of the BK*μ+μ anomaly after Moriond 2017, Eur. Phys. J. C 77 (2017) 377 [arXiv:1703.09189] [INSPIRE].

    ADS  Article  Google Scholar 

  8. [8]

    J. Aebischer, W. Altmannshofer, D. Guadagnoli, M. Reboud, P. Stangl and D.M. Straub, B-decay discrepancies after Moriond 2019, Eur. Phys. J. C 80 (2020) 252 [arXiv:1903.10434] [INSPIRE].

    ADS  Article  Google Scholar 

  9. [9]

    M. Algueró et al., Emerging patterns of New Physics with and without Lepton Flavour Universal contributions, Eur. Phys. J. C 79 (2019) 714 [Addendum ibid. 80 (2020) 511] [arXiv:1903.09578] [INSPIRE].

  10. [10]

    M. Ciuchini, M. Fedele, E. Franco, A. Paul, L. Silvestrini and M. Valli, Lessons from the B0,+K*0,+μ+μ angular analyses, Phys. Rev. D 103 (2021) 015030 [arXiv:2011.01212] [INSPIRE].

  11. [11]

    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].

    ADS  Article  Google Scholar 

  12. [12]

    A. Khodjamirian, T. Mannel and Y.M. Wang, BKℓ+ decay at large hadronic recoil, JHEP 02 (2013) 010 [arXiv:1211.0234] [INSPIRE].

    ADS  Article  Google Scholar 

  13. [13]

    S. Jäger and J. Martin Camalich, On BVℓℓ at small dilepton invariant mass, power corrections, and new physics, JHEP 05 (2013) 043 [arXiv:1212.2263] [INSPIRE].

    ADS  Article  Google Scholar 

  14. [14]

    S. Descotes-Genon, L. Hofer, J. Matias and J. Virto, On the impact of power corrections in the prediction of BK*μ+μ observables, JHEP 12 (2014) 125 [arXiv:1407.8526] [INSPIRE].

    ADS  Article  Google Scholar 

  15. [15]

    M. Ciuchini et al., BK* + decays at large recoil in the Standard Model: a theoretical reappraisal, JHEP 06 (2016) 116 [arXiv:1512.07157] [INSPIRE].

  16. [16]

    C. Bobeth, M. Chrzaszcz, D. van Dyk and J. Virto, Long-distance effects in BK* ℓℓ from analyticity, Eur. Phys. J. C 78 (2018) 451 [arXiv:1707.07305] [INSPIRE].

  17. [17]

    V.G. Chobanova, T. Hurth, F. Mahmoudi, D. Martinez Santos and S. Neshatpour, Large hadronic power corrections or new physics in the rare decay BK*μ+μ?, JHEP 07 (2017) 025 [arXiv:1702.02234] [INSPIRE].

    ADS  Article  Google Scholar 

  18. [18]

    M. Dimou, J. Lyon and R. Zwicky, Exclusive Chromomagnetism in heavy-to-light FCNCs, Phys. Rev. D 87 (2013) 074008 [arXiv:1212.2242] [INSPIRE].

  19. [19]

    J. Lyon and R. Zwicky, Isospin asymmetries in B → (K*, ρ)γ/l+l and BKl+l in and beyond the standard model, Phys. Rev. D 88 (2013) 094004 [arXiv:1305.4797] [INSPIRE].

  20. [20]

    M. Beneke, T. Feldmann and D. Seidel, Systematic approach to exclusive BVl+l, Vγ decays, Nucl. Phys. B 612 (2001) 25 [hep-ph/0106067] [INSPIRE].

  21. [21]

    M. Beneke, T. Feldmann and D. Seidel, Exclusive radiative and electroweak bd and bs penguin decays at NLO, Eur. Phys. J. C 41 (2005) 173 [hep-ph/0412400] [INSPIRE].

  22. [22]

    S. Descotes-Genon, A. Khodjamirian and J. Virto, Light-cone sum rules for BKπ form factors and applications to rare decays, JHEP 12 (2019) 083 [arXiv:1908.02267] [INSPIRE].

    ADS  Article  Google Scholar 

  23. [23]

    B. Grinstein and D. Pirjol, Exclusive rare BK* + decays at low recoil: Controlling the long-distance effects, Phys. Rev. D 70 (2004) 114005 [hep-ph/0404250] [INSPIRE].

  24. [24]

    M. Beylich, G. Buchalla and T. Feldmann, Theory of BK(*) + decays at high q2: OPE and quark-hadron duality, Eur. Phys. J. C 71 (2011) 1635 [arXiv:1101.5118] [INSPIRE].

  25. [25]

    H.H. Asatryan, H.M. Asatrian, C. Greub and M. Walker, Calculation of two loop virtual corrections to bsl+l in the standard model, Phys. Rev. D 65 (2002) 074004 [hep-ph/0109140] [INSPIRE].

  26. [26]

    C. Greub, V. Pilipp and C. Schupbach, Analytic calculation of two-loop QCD corrections to bsl+l in the high q2 region, JHEP 12 (2008) 040 [arXiv:0810.4077] [INSPIRE].

    ADS  Article  Google Scholar 

  27. [27]

    A. Ghinculov, T. Hurth, G. Isidori and Y.P. Yao, The rare decay BXsl+l to NNLL precision for arbitrary dilepton invariant mass, Nucl. Phys. B 685 (2004) 351 [hep-ph/0312128] [INSPIRE].

  28. [28]

    G. Bell and T. Huber, Master integrals for the two-loop penguin contribution in non-leptonic B-decays, JHEP 12 (2014) 129 [arXiv:1410.2804] [INSPIRE].

    ADS  Article  Google Scholar 

  29. [29]

    S. de Boer, Two loop virtual corrections to b → (d, s)+ and cuℓ+ for arbitrary momentum transfer, Eur. Phys. J. C 77 (2017) 801 [arXiv:1707.00988] [INSPIRE].

  30. [30]

    H.M. Asatrian, C. Greub and J. Virto, Exact NLO matching and analyticity in bsℓℓ, JHEP 04 (2020) 012 [arXiv:1912.09099] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  31. [31]

    G. Buchalla, G. Isidori and S.J. Rey, Corrections of order \( {\Lambda}_{QCD}^2/{m}_c^2 \) to inclusive rare B decays, Nucl. Phys. B 511 (1998) 594 [hep-ph/9705253] [INSPIRE].

  32. [32]

    HPQCD collaboration, Rare decay BKℓ+ form factors from lattice QCD, Phys. Rev. D 88 (2013) 054509 [Erratum ibid. 88 (2013) 079901] [arXiv:1306.2384] [INSPIRE].

  33. [33]

    R.R. Horgan, Z. Liu, S. Meinel and M. Wingate, Lattice QCD calculation of form factors describing the rare decays BK* ℓℓ and Bsϕℓ+, Phys. Rev. D 89 (2014) 094501 [arXiv:1310.3722] [INSPIRE].

  34. [34]

    R.R. Horgan, Z. Liu, S. Meinel and M. Wingate, Rare B decays using lattice QCD form factors, PoS LATTICE2014 (2015) 372 [arXiv:1501.00367] [INSPIRE].

  35. [35]

    N. Gubernari, A. Kokulu and D. van Dyk, BP and BV Form Factors from B-Meson Light-Cone Sum Rules beyond Leading Twist, JHEP 01 (2019) 150 [arXiv:1811.00983] [INSPIRE].

  36. [36]

    A. Bharucha, D.M. Straub and R. Zwicky, BVℓ+ in the Standard Model from light-cone sum rules, JHEP 08 (2016) 098 [arXiv:1503.05534] [INSPIRE].

    ADS  Article  Google Scholar 

  37. [37]

    J. Lyon and R. Zwicky, Resonances gone topsy turvy — the charm of QCD or new physics in bsℓ+?, arXiv:1406.0566 [INSPIRE].

  38. [38]

    A. Arbey, T. Hurth, F. Mahmoudi and S. Neshatpour, Hadronic and New Physics Contributions to bs Transitions, Phys. Rev. D 98 (2018) 095027 [arXiv:1806.02791] [INSPIRE].

  39. [39]

    LHCb collaboration, Measurement of the phase difference between short- and long-distance amplitudes in the B+K+μ+μ decay, Eur. Phys. J. C 77 (2017) 161 [arXiv:1612.06764] [INSPIRE].

  40. [40]

    S. Braß, G. Hiller and I. Nisandzic, Zooming in on BK**ℓℓ decays at low recoil, Eur. Phys. J. C 77 (2017) 16 [arXiv:1606.00775] [INSPIRE].

    ADS  Article  Google Scholar 

  41. [41]

    T. Blake, U. Egede, P. Owen, K.A. Petridis and G. Pomery, An empirical model to determine the hadronic resonance contributions to \( {\overline{B}}^0\to {\overline{K}}^{\ast 0}{\mu}^{+}{\mu}^{-} \) transitions, Eur. Phys. J. C 78 (2018) 453 [arXiv:1709.03921] [INSPIRE].

    ADS  Article  Google Scholar 

  42. [42]

    M. Chrzaszcz, A. Mauri, N. Serra, R. Silva Coutinho and D. van Dyk, Prospects for disentangling long- and short-distance effects in the decays BK*μ+μ, JHEP 10 (2019) 236 [arXiv:1805.06378] [INSPIRE].

  43. [43]

    A. Mauri, N. Serra and R. Silva Coutinho, Towards establishing lepton flavor universality violation in \( \overline{B}\to {\overline{K}}^{\ast }{\mathrm{\ell}}^{+}{\mathrm{\ell}}^{-} \) decays, Phys. Rev. D 99 (2019) 013007 [arXiv:1805.06401] [INSPIRE].

  44. [44]

    I.I. Balitsky and V.M. Braun, Evolution Equations for QCD String Operators, Nucl. Phys. B 311 (1989) 541 [INSPIRE].

    ADS  Article  Google Scholar 

  45. [45]

    A. Kozachuk and D. Melikhov, Revisiting nonfactorizable charm-loop effects in exclusive FCNC B-decays, Phys. Lett. B 786 (2018) 378 [arXiv:1805.05720] [INSPIRE].

    ADS  Article  Google Scholar 

  46. [46]

    D. Melikhov, Charming loops in exclusive rare FCNC B-decays, EPJ Web Conf. 222 (2019) 01007 [arXiv:1911.03899] [INSPIRE].

  47. [47]

    B. Geyer and O. Witzel, B-meson distribution amplitudes of geometric twist vs. dynamical twist, Phys. Rev. D 72 (2005) 034023 [hep-ph/0502239] [INSPIRE].

  48. [48]

    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].

    ADS  Article  Google Scholar 

  49. [49]

    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].

  50. [50]

    A. Bazavov et al., B- and D-meson leptonic decay constants from four-flavor lattice QCD, Phys. Rev. D 98 (2018) 074512 [arXiv:1712.09262] [INSPIRE].

  51. [51]

    N. Carrasco et al., Leptonic decay constants fK, fD, and \( {f}_{D_s} \) with Nf = 2 + 1 + 1 twisted-mass lattice QCD, Phys. Rev. D 91 (2015) 054507 [arXiv:1411.7908] [INSPIRE].

  52. [52]

    Particle Data Group collaboration, Review of Particle Physics, Phys. Rev. D 98 (2018) 030001 [INSPIRE].

  53. [53]

    V.M. Braun, D.Y. Ivanov and G.P. Korchemsky, The B meson distribution amplitude in QCD, Phys. Rev. D 69 (2004) 034014 [hep-ph/0309330] [INSPIRE].

  54. [54]

    M. Bordone, N. Gubernari, D. van Dyk and M. Jung, Heavy-Quark expansion for \( {\overline{B}}_s\to {D}_s^{\left(\ast \right)} \) form factors and unitarity bounds beyond the SU(3)F limit, Eur. Phys. J. C 80 (2020) 347 [arXiv:1912.09335] [INSPIRE].

  55. [55]

    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].

    ADS  Article  Google Scholar 

  56. [56]

    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].

  57. [57]

    A. Khodjamirian, R. Mandal and T. Mannel, Inverse moment of the Bs-meson distribution amplitude from QCD sum rule, JHEP 10 (2020) 043 [arXiv:2008.03935] [INSPIRE].

    ADS  Article  Google Scholar 

  58. [58]

    D. van Dyk et al., EOS — A HEP program for Flavor Observables, (2020),

  59. [59]

    A.G. Grozin and M. Neubert, Asymptotics of heavy meson form-factors, Phys. Rev. D 55 (1997) 272 [hep-ph/9607366] [INSPIRE].

  60. [60]

    A. Djouadi and P. Gambino, Electroweak gauge bosons selfenergies: Complete QCD corrections, Phys. Rev. D 49 (1994) 3499 [Erratum ibid. 53 (1996) 4111] [hep-ph/9309298] [INSPIRE].

  61. [61]

    Particle Data Group collaboration, Review of Particle Physics, PTEP 2020 (2020) 083C01 [INSPIRE].

  62. [62]

    A. Bharucha, T. Feldmann and M. Wick, Theoretical and Phenomenological Constraints on Form Factors for Radiative and Semi-Leptonic B-Meson Decays, JHEP 09 (2010) 090 [arXiv:1004.3249] [INSPIRE].

    ADS  Article  Google Scholar 

  63. [63]

    C.G. Boyd, B. Grinstein and R.F. Lebed, Model independent determinations of \( \overline{B}\to {D}^{\left(\ast \right)}{\mathrm{\ell}}^{-}\overline{\nu} \) form-factors, Nucl. Phys. B 461 (1996) 493 [hep-ph/9508211] [INSPIRE].

  64. [64]

    W. Rudin, Real and complex analysis, 3 ed., McGraw-Hill, (1987).

  65. [65]

    B. Simon, Orthogonal Polynomials on the Unit Circle, no. v. 54, no. 1 in American Mathematical Society colloquium publications, American Mathematical Society (2005).

  66. [66]

    E.S. Eberhard, Extending dispersive bounds to include sub-threshold branch cuts, MSc Thesis, Technical University of Munich, Germany (2020).

  67. [67]

    I. Caprini, Functional Analysis and Optimization Methods in Hadron Physics, SpringerBriefs in Physics, Springer (2019), [DOI] [INSPIRE].

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Gubernari, N., van Dyk, D. & Virto, J. Non-local matrix elements in B(s) → {K(*), ϕ}+. J. High Energ. Phys. 2021, 88 (2021).

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  • Beyond Standard Model
  • Heavy Quark Physics
  • Nonperturbative Effects