The (Q7, Q1,2) contribution to \( \overline{B}\to {X}_s\gamma \) at \( \mathcal{O}\left({\alpha}_{\mathrm{s}}^2\right) \)

  • Michał Czakon
  • Paul Fiedler
  • Tobias Huber
  • Mikołaj Misiak
  • Thomas Schutzmeier
  • Matthias Steinhauser
Open Access
Regular Article - Theoretical Physics

Abstract

Interference between the photonic dipole operator Q7 and the current-current operators Q1,2 gives one of the most important QCD corrections to the (\( \overline{B}\to {X}_s\gamma \) decay rate. So far, the \( \mathcal{O}\left({\alpha}_{\mathrm{s}}^2\right) \)) part of this correction has been known in the heavy charm quark limit only (m c m b /2). Here, we evaluate this part at m c = 0, and use both limits in an updated phenomenological study. Our prediction for the CP- and isospin-averaged branching ratio in the Standard Model reads \( {\mathrm{\mathcal{B}}}_{s\gamma}^{\mathrm{SM}}=\left(3.36\pm 0.23\right)\times 1{0}^{-4} \) for E γ > 1.6 GeV.

Keywords

Rare Decays B-Physics Standard Model Beyond Standard Model 

Notes

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.

References

  1. [1]
    CLEO collaboration, S. Chen et al., Branching fraction and photon energy spectrum for b, Phys. Rev. Lett. 87 (2001) 251807 [hep-ex/0108032] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    Belle collaboration, K. Abe et al., A measurement of the branching fraction for the inclusive BX s γ decays with BELLE, Phys. Lett. B 511(2001) 151 [hep-ex/0103042] [INSPIRE].ADSGoogle Scholar
  3. [3]
    Belle collaboration, A. Limosani et al., Measurement of inclusive radiative B-meson decays with a photon energy threshold of 1.7 GeV, Phys. Rev. Lett. 103 (2009) 241801 [arXiv:0907.1384] [INSPIRE].
  4. [4]
    BaBar collaboration, J.P. Lees et al., Precision measurement of the BX s γ photon energy spectrum, branching fraction and direct CP asymmetry A CP(BX s+d γ), Phys. Rev. Lett. 109 (2012) 191801 [arXiv:1207.2690] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    BaBar collaboration, J.P. Lees et al., Measurement of B(BX s γ), the BX s γ photon energy spectrum and the direct CP asymmetry in BX s+d γ decays, Phys. Rev. D 86 (2012) 112008 [arXiv:1207.5772] [INSPIRE].
  6. [6]
    BaBar collaboration, J.P. Lees et al., Exclusive measurements of bsγ transition rate and photon energy spectrum, Phys. Rev. D 86 (2012) 052012 [arXiv:1207.2520] [INSPIRE].
  7. [7]
    BaBar collaboration, B. Aubert et al., Measurement of the BX s γ branching fraction and photon energy spectrum using the recoil method, Phys. Rev. D 77 (2008) 051103 [arXiv:0711.4889] [INSPIRE].
  8. [8]
    Heavy Flavor Averaging Group (HFAG) collaboration, Y. Amhis et al., Averages of b-hadron, c-hadron and τ-lepton properties as of summer 2014, arXiv:1412.7515 [INSPIRE].
  9. [9]
    Belle collaboration, T. Saito et al., Measurement of the \( \overline{B}\to {X}_s\gamma \) branching fraction with a sum of exclusive decays, Phys. Rev. D 91 (2015) 052004 [arXiv:1411.7198] [INSPIRE].
  10. [10]
    T. Aushev et al., Physics at super B factory, arXiv:1002.5012 [INSPIRE].
  11. [11]
    Belle-II collaboration, T. Abe et al., Belle II technical design report, arXiv:1011.0352 [INSPIRE].
  12. [12]
    BaBar collaboration, B. Aubert et al., Measurements of the BX s γ branching fraction and photon spectrum from a sum of exclusive final states, Phys. Rev. D 72 (2005) 052004 [hep-ex/0508004] [INSPIRE].
  13. [13]
    Particle Data Group collaboration, K.A. Olive et al., Review of particle physics, Chin. Phys. C 38 (2014) 090001 [INSPIRE].
  14. [14]
    M. Benzke, S.J. Lee, M. Neubert and G. Paz, Factorization at subleading power and irreducible uncertainties in \( \overline{B}\to {X}_s\gamma \) decay, JHEP 08 (2010) 099 [arXiv:1003.5012] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  15. [15]
    C. Bobeth, M. Misiak and J. Urban, Photonic penguins at two loops and m t dependence of BR[BX s + ], Nucl. Phys. B 574 (2000) 291 [hep-ph/9910220] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    M. Misiak and M. Steinhauser, Three loop matching of the dipole operators for bsγ and bsg, Nucl. Phys. B 683 (2004) 277 [hep-ph/0401041] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    M. Gorbahn and U. Haisch, Effective Hamiltonian for non-leptonic |F | = 1 decays at NNLO in QCD, Nucl. Phys. B 713 (2005) 291 [hep-ph/0411071] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    M. Gorbahn, U. Haisch and M. Misiak, Three-loop mixing of dipole operators, Phys. Rev. Lett. 95 (2005) 102004 [hep-ph/0504194] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    M. Czakon, U. Haisch and M. Misiak, Four-loop anomalous dimensions for radiative flavour-changing decays, JHEP 03 (2007) 008 [hep-ph/0612329] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    T. Hermann, M. Misiak and M. Steinhauser, \( \overline{B}\to {X}_s\gamma \) in the two Higgs doublet model up to next-to-next-to-leading order in QCD, JHEP 11 (2012) 036 [arXiv:1208.2788] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    A.J. Buras, M. Misiak, M. Münz and S. Pokorski, Theoretical uncertainties and phenomenological aspects of BX s γ decay, Nucl. Phys. B 424 (1994) 374 [hep-ph/9311345] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    K.G. Chetyrkin, M. Misiak and M. Münz, Weak radiative B meson decay beyond leading logarithms, Phys. Lett. B 400 (1997) 206 [Erratum ibid. B 425 (1998) 414] [hep-ph/9612313] [INSPIRE].
  23. [23]
    M. Kamiński, M. Misiak and M. Poradzinski, Tree-level contributions to BX s γ, Phys. Rev. D 86 (2012) 094004 [arXiv:1209.0965] [INSPIRE].ADSGoogle Scholar
  24. [24]
    T. Huber, M. Poradzinski and J. Virto, Four-body contributions to \( \overline{B}\to {X}_s\gamma \) at NLO, JHEP 01 (2015) 115 [arXiv:1411.7677] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    I.R. Blokland, A. Czarnecki, M. Misiak, M. Slusarczyk and F. Tkachov, The electromagnetic dipole operator effect on \( \overline{B}\to {X}_s\gamma \) at O(α s2), Phys. Rev. D 72 (2005) 033014 [hep-ph/0506055] [INSPIRE].ADSGoogle Scholar
  26. [26]
    K. Melnikov and A. Mitov, The photon energy spectrum in BX s + γ in perturbative QCD through O(α s2), Phys. Lett. B 620 (2005) 69 [hep-ph/0505097] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    H.M. Asatrian et al., NNLL QCD contribution of the electromagnetic dipole operator to \( \Gamma \left(\overline{B}\to {X}_s\gamma \right) \), Nucl. Phys. B 749 (2006) 325 [hep-ph/0605009] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    H.M. Asatrian, T. Ewerth, A. Ferroglia, P. Gambino and C. Greub, Magnetic dipole operator contributions to the photon energy spectrum in \( \overline{B}\to {X}_s\gamma \) at O(α s2), Nucl. Phys. B 762 (2007) 212 [hep-ph/0607316] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  29. [29]
    H.M. Asatrian, T. Ewerth, H. Gabrielyan and C. Greub, Charm quark mass dependence of the electromagnetic dipole operator contribution to \( \overline{B}\to {X}_s\gamma \) at O(α s2), Phys. Lett. B 647 (2007) 173 [hep-ph/0611123] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    K. Bieri, C. Greub and M. Steinhauser, Fermionic NNLL corrections to b, Phys. Rev. D 67 (2003) 114019 [hep-ph/0302051] [INSPIRE].ADSGoogle Scholar
  31. [31]
    Z. Ligeti, M.E. Luke, A.V. Manohar and M.B. Wise, The \( \overline{B}\to {X}_s\gamma \) photon spectrum, Phys. Rev. D 60 (1999) 034019 [hep-ph/9903305] [INSPIRE].ADSGoogle Scholar
  32. [32]
    R. Boughezal, M. Czakon and T. Schutzmeier, NNLO fermionic corrections to the charm quark mass dependent matrix elements in \( \overline{B}\to {X}_s\gamma \), JHEP 09 (2007) 072 [arXiv:0707.3090] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    S.J. Brodsky, G.P. Lepage and P.B. Mackenzie, On the elimination of scale ambiguities in perturbative quantum chromodynamics, Phys. Rev. D 28 (1983) 228 [INSPIRE].ADSGoogle Scholar
  34. [34]
    M. Misiak and M. Steinhauser, NNLO QCD corrections to the \( \overline{B}\to {X}_s\gamma \) matrix elements using interpolation in m c, Nucl. Phys. B 764 (2007) 62 [hep-ph/0609241] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    M. Misiak and M. Steinhauser, Large-m c asymptotic behaviour of O(α s2) corrections to BX s γ, Nucl. Phys. B 840 (2010) 271[arXiv:1005.1173][INSPIRE].ADSCrossRefMATHGoogle Scholar
  36. [36]
    T. Ewerth, Fermionic corrections to the interference of the electro- and chromomagnetic dipole operators in \( \overline{B}\to {X}_s\gamma \) at O(α s2), Phys. Lett. B 669 (2008) 167 [arXiv:0805.3911] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    H.M. Asatrian, T. Ewerth, A. Ferroglia, C. Greub and G. Ossola, Complete (O 7 , O 8) contribution to BX s γ at order α s2, Phys. Rev. D 82 (2010) 074006 [arXiv:1005.5587] [INSPIRE].ADSGoogle Scholar
  38. [38]
    A. Ferroglia and U. Haisch, Chromomagnetic dipole-operator corrections in \( \overline{B}\to {X}_s\gamma \) at O(β 0 α s2), Phys. Rev. D 82 (2010) 094012 [arXiv:1009.2144] [INSPIRE].ADSGoogle Scholar
  39. [39]
    M. Misiak and M. Poradzinski, Completing the calculation of BLM corrections to \( \overline{B}\to {X}_s\gamma \), Phys. Rev. D 83 (2011) 014024 [arXiv:1009.5685] [INSPIRE].ADSGoogle Scholar
  40. [40]
    T. Ewerth, P. Gambino and S. Nandi, Power suppressed effects in \( \overline{B}\to {X}_s\gamma \) at O(α s), Nucl. Phys. B 830 (2010) 278 [arXiv:0911.2175] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  41. [41]
    A. Alberti, P. Gambino and S. Nandi, Perturbative corrections to power suppressed effects in semileptonic B decays, JHEP 01 (2014) 147 [arXiv:1311.7381] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    M. Misiak et al., Estimate of \( B\left(\overline{B}\to {X}_s\gamma \right) \) at O(α s2), Phys. Rev. Lett. 98 (2007) 022002 [hep-ph/0609232] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    A.J. Buras, A. Czarnecki, M. Misiak and J. Urban, Completing the NLO QCD calculation of \( \overline{B}\to {X}_s\gamma \), Nucl. Phys. B 631 (2002) 219 [hep-ph/0203135] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B 646 (2002) 220 [hep-ph/0207004] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    K.G. Chetyrkin and F.V. Tkachov, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    M. Czakon, DiaGen/IdSolver, unpublished.Google Scholar
  47. [47]
    J. Kuipers, T. Ueda, J.A.M. Vermaseren and J. Vollinga, FORM version 4.0, Comput. Phys. Commun. 184 (2013) 1453 [arXiv:1203.6543] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  48. [48]
    M. Benzke, S.J. Lee, M. Neubert and G. Paz, Long-distance dominance of the CP asymmetry in BX s,d + γ decays, Phys. Rev. Lett. 106 (2011) 141801 [arXiv:1012.3167] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    A. Alberti, P. Gambino, K.J. Healey and S. Nandi, Precision determination of the Cabibbo-Kobayashi-Maskawa element V cb, Phys. Rev. Lett. 114 (2015) 061802 [arXiv:1411.6560] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    A. Pak and A. Czarnecki, Mass effects in muon and semileptonic bc decays, Phys. Rev. Lett. 100 (2008) 241807 [arXiv:0803.0960] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    M. Misiak et al., Updated NNLO QCD predictions for the weak radiative B-meson decays, arXiv:1503.01789 [INSPIRE].
  52. [52]
    M. Neubert, Renormalization-group improved calculation of the BX s γ branching ratio, Eur. Phys. J. C 40 (2005) 165 [hep-ph/0408179] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    P. Gambino, private communication.Google Scholar
  54. [54]
    G. Buchalla, G. Isidori and S.J. Rey, Corrections of order ΛQCD2/m c2 to inclusive rare B decays, Nucl. Phys. B 511 (1998) 594 [hep-ph/9705253] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    P. Gambino and U. Haisch, Complete electroweak matching for radiative B decays, JHEP 10 (2001) 020 [hep-ph/0109058] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    T. Huber, Master integrals for massless three-loop form factors, PoS(RADCOR2009)038 [arXiv:1001.3132] [INSPIRE].
  57. [57]
    A. Gehrmann-De Ridder, T. Gehrmann and G. Heinrich, Four particle phase space integrals in massless QCD, Nucl. Phys. B 682 (2004) 265 [hep-ph/0311276] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  58. [58]
    P.A. Baikov, K.G. Chetyrkin, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Quark and gluon form factors to three loops, Phys. Rev. Lett. 102 (2009) 212002 [arXiv:0902.3519] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    R.N. Lee, A.V. Smirnov and V.A. Smirnov, Analytic results for massless three-loop form factors, JHEP 04 (2010) 020 [arXiv:1001.2887] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  60. [60]
    T. Gehrmann, E.W.N. Glover, T. Huber, N. Ikizlerli and C. Studerus, Calculation of the quark and gluon form factors to three loops in QCD, JHEP 06 (2010) 094 [arXiv:1004.3653] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  61. [61]
    V.A. Smirnov, Analytical result for dimensionally regularized massless on shell double box, Phys. Lett. B 460 (1999) 397 [hep-ph/9905323] [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    J.B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  63. [63]
    C. Anastasiou and A. Daleo, Numerical evaluation of loop integrals, JHEP 10 (2006) 031 [hep-ph/0511176] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  64. [64]
    V.A. Smirnov, Evaluating Feynman integrals, Springer Tracts Mod. Phys. 211 (2004) 1 [INSPIRE].MathSciNetMATHGoogle Scholar
  65. [65]
    P.A. Baikov and K.G. Chetyrkin, Four loop massless propagators: an algebraic evaluation of all master integrals, Nucl. Phys. B 837 (2010) 186 [arXiv:1004.1153] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  66. [66]
    A.V. Smirnov and M. Tentyukov, Four loop massless propagators: a numerical evaluation of all master integrals, Nucl. Phys. B 837 (2010) 40 [arXiv:1004.1149] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  67. [67]
    M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  68. [68]
    T. Huber and D. Maître, HypExp: a Mathematica package for expanding hypergeometric functions around integer-valued parameters, Comput. Phys. Commun. 175 (2006) 122 [hep-ph/0507094] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  69. [69]
    T. Huber and D. Maître, HypExp 2, expanding hypergeometric functions about half-integer parameters, Comput. Phys. Commun. 178 (2008) 755 [arXiv:0708.2443] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  70. [70]
    P. Gambino and C. Schwanda, Inclusive semileptonic fits, heavy quark masses and V cb, Phys. Rev. D 89 (2014) 014022 [arXiv:1307.4551] [INSPIRE].ADSGoogle Scholar
  71. [71]
    K.G. Chetyrkin et al., Charm and bottom quark masses: an update, Phys. Rev. D 80 (2009) 074010 [arXiv:0907.2110] [INSPIRE].ADSGoogle Scholar
  72. [72]
    P. Gambino and M. Misiak, Quark mass effects in \( \overline{B}\to {X}_s\gamma \), Nucl. Phys. B 611 (2001) 338 [hep-ph/0104034] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    J. Charles et al., Current status of the standard model CKM fit and constraints onF = 2 new physics, Phys. Rev. D 91 (2015) 073007 [arXiv:1501.05013] [INSPIRE].ADSMathSciNetGoogle Scholar
  74. [74]
    A. Kapustin, Z. Ligeti and H.D. Politzer, Leading logarithms of the b quark mass in inclusive BX s γ decay, Phys. Lett. B 357 (1995) 653 [hep-ph/9507248] [INSPIRE].ADSCrossRefGoogle Scholar
  75. [75]
    H.M. Asatrian and C. Greub, Tree-level contribution to \( \overline{B}\to {X}_s\gamma \) using fragmentation functions, Phys. Rev. D 88 (2013) 074014 [arXiv:1305.6464] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Michał Czakon
    • 1
  • Paul Fiedler
    • 1
  • Tobias Huber
    • 2
  • Mikołaj Misiak
    • 3
  • Thomas Schutzmeier
    • 4
  • Matthias Steinhauser
    • 5
  1. 1.Institut für Theoretische Teilchenphysik und KosmologieRWTH Aachen UniversityAachenGermany
  2. 2.Theoretische Physik 1, Naturwissenschaftlich-Technische FakultätUniversität SiegenSiegenGermany
  3. 3.Institute of Theoretical PhysicsUniversity of WarsawWarsawPoland
  4. 4.Physics DepartmentFlorida State UniversityTallahasseeUnited States
  5. 5.Institut für Theoretische TeilchenphysikKarlsruhe Institute of Technology (KIT)KarlsruheGermany

Personalised recommendations