Exclusive radiative decays of W and Z bosons in QCD factorization

  • Yuval Grossman
  • Matthias König
  • Matthias Neubert
Open Access
Regular Article - Theoretical Physics


We present a detailed theoretical analysis of very rare, exclusive hadronic decays of the electroweak gauge bosons V = W, Z from first principles of QCD. Our main focus is on the radiative decays V, in which M is a pseudoscalar or vector meson. At leading order in an expansion in powers of ΛQCD/m V the decay amplitudes can be factorized into convolutions of calculable hard-scattering coefficients with the leading-twist light-cone distribution amplitude of the meson M. Power corrections to the decay rates arise first at order (ΛQCD/m V ) 2 . They can be estimated in terms of higher-twist distribution amplitudes and are predicted to be tiny. We include one-loop \( \mathcal{O}\left({\alpha}_s\right) \) radiative corrections to the hard-scattering coefficients and perform the resummation of large logarithms (α s  ln(m v 2 /μ 0 2 )) n (with μ 0 ∼ 1 GeV a typical hadronic scale) to all orders in perturbation theory. Evolution effects have an important impact both numerically and conceptually, since they reduce the sensitivity to poorly determined hadronic parameters. We present detailed numerical predictions and error estimates, which can serve as benchmarks for future precision measurements. We also present an exploratory study of the weak radiative decays ZMW. Some of the decay modes studied here have branching ratios large enough to be accessible in the high-luminosity run of the LHC. Many of them can be measured with high accuracy at a future lepton collider. This will provide stringent tests of the QCD factorization formalism and enable novel searches for new physics.


Rare Decays Effective field theories Resummation Renormalization Group 


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]
    G.P. Lepage and S.J. Brodsky, Exclusive Processes in Quantum Chromodynamics: Evolution Equations for Hadronic Wave Functions and the Form-Factors of Mesons, Phys. Lett. B 87 (1979) 359 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    G.P. Lepage and S.J. Brodsky, Exclusive Processes in Perturbative Quantum Chromodynamics, Phys. Rev. D 22 (1980) 2157 [INSPIRE].ADSGoogle Scholar
  3. [3]
    A.V. Efremov and A.V. Radyushkin, Asymptotical Behavior of Pion Electromagnetic Form-Factor in QCD, Theor. Math. Phys. 42 (1980) 97 [INSPIRE].CrossRefGoogle Scholar
  4. [4]
    A.V. Efremov and A.V. Radyushkin, Factorization and Asymptotical Behavior of Pion Form-Factor in QCD, Phys. Lett. B 94 (1980) 245 [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    V.L. Chernyak and A.R. Zhitnitsky, Asymptotic Behavior of Exclusive Processes in QCD, Phys. Rept. 112 (1984) 173 [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, QCD factorization for Bππ decays: Strong phases and CP-violation in the heavy quark limit, Phys. Rev. Lett. 83 (1999) 1914 [hep-ph/9905312] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, QCD factorization for exclusive, nonleptonic B meson decays: General arguments and the case of heavy light final states, Nucl. Phys. B 591 (2000) 313 [hep-ph/0006124] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, QCD factorization in BπK, ππ decays and extraction of Wolfenstein parameters, Nucl. Phys. B 606 (2001) 245 [hep-ph/0104110] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    M. Beneke and M. Neubert, QCD factorization for BPP and BPV decays, Nucl. Phys. B 675 (2003) 333 [hep-ph/0308039] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    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].ADSGoogle Scholar
  11. [11]
    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].ADSGoogle Scholar
  12. [12]
    C.W. Bauer, S. Fleming, D. Pirjol, I.Z. Rothstein and I.W. Stewart, Hard scattering factorization from effective field theory, Phys. Rev. D 66 (2002) 014017 [hep-ph/0202088] [INSPIRE].ADSGoogle Scholar
  13. [13]
    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].ADSCrossRefMATHGoogle Scholar
  14. [14]
    S.S. Agaev, V.M. Braun, N. Offen and F.A. Porkert, Light Cone Sum Rules for the π 0 γ * γ Form Factor Revisited, Phys. Rev. D 83 (2011) 054020 [arXiv:1012.4671] [INSPIRE].ADSGoogle Scholar
  15. [15]
    M. Mangano and T. Melia, Rare exclusive hadronic W decays in a \( t\overline{t} \) environment, arXiv:1410.7475 [INSPIRE].
  16. [16]
    A. Blondel, A. Chao, W. Chou, J. Gao, D. Schulte and K. Yokoya, Report of the ICFA Beam Dynamics WorkshopAccelerators for a Higgs Factory: Linear vs. Circular(HF2012), arXiv:1302.3318 [INSPIRE].
  17. [17]
    G. Isidori, A.V. Manohar and M. Trott, Probing the nature of the Higgs-like Boson via h → Vdecays, Phys. Lett. B 728 (2014) 131 [arXiv:1305.0663] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    G.T. Bodwin, F. Petriello, S. Stoynev and M. Velasco, Higgs boson decays to quarkonia and the \( H\overline{c}c \) coupling, Phys. Rev. D 88 (2013) 053003 [arXiv:1306.5770] [INSPIRE].ADSGoogle Scholar
  19. [19]
    A.L. Kagan, G. Perez, F. Petriello, Y. Soreq, S. Stoynev and J. Zupan, An Exclusive Window onto Higgs Yukawa Couplings, Phys. Rev. Lett. 114 (2015) 101802 [arXiv:1406.1722] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    G.T. Bodwin, H.S. Chung, J.-H. Ee, J. Lee and F. Petriello, Relativistic corrections to Higgs boson decays to quarkonia, Phys. Rev. D 90 (2014) 113010 [arXiv:1407.6695] [INSPIRE].ADSGoogle Scholar
  21. [21]
    D.-N. Gao, A note on Higgs decays into Z boson and J/Ψ(Υ), Phys. Lett. B 737 (2014) 366 [arXiv:1406.7102] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    B. Bhattacharya, A. Datta and D. London, Probing New Physics in Higgs Couplings to Fermions using an Angular Analysis, Phys. Lett. B 736 (2014) 421 [arXiv:1407.0695] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  23. [23]
    ATLAS collaboration, Search for Higgs and Z Boson Decays to J/ψγ and Υ(nSwith the ATLAS Detector, arXiv:1501.03276 [INSPIRE].
  24. [24]
    L. Arnellos, W.J. Marciano and Z. Parsa, Radiative Decays W ±ρ ± γ and Z 0ρ 0 γ, Nucl. Phys. B 196 (1982) 378 [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    A.V. Manohar, The decays ZWπ and Z → γπ, Phys. Lett. B 244 (1990) 101 [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    B. Guberina, J.H. Kuhn, R.D. Peccei and R. Ruckl, Rare decays of the Z 0, Nucl. Phys. B 174 (1980) 317 [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    T.-C. Huang and F. Petriello, Rare exclusive decays of the Z-boson revisited, arXiv:1411.5924 [INSPIRE].
  28. [28]
    M. Jacob and T.T. Wu, The Decay Zπ 0 γ, Phys. Lett. B 232 (1989) 529 [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    Y.Y. Keum and X.-Y. Pham, Possible huge enhancement in the radiative decay of the weak W boson into the charmed D s meson, Mod. Phys. Lett. A 9 (1994) 1545 [hep-ph/9303300] [INSPIRE].
  30. [30]
    C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  31. [31]
    R.J. Hill and M. Neubert, Spectator interactions in soft collinear effective theory, Nucl. Phys. B 657 (2003) 229 [hep-ph/0211018] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  32. [32]
    M. Beneke and M. Neubert, Flavor singlet B decay amplitudes in QCD factorization, Nucl. Phys. B 651 (2003) 225 [hep-ph/0210085] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    M. König and M.Neubert, in preparation.Google Scholar
  34. [34]
    S.W. Bosch, R.J. Hill, B.O. Lange and M. Neubert, Factorization and Sudakov resummation in leptonic radiative B decay, Phys. Rev. D 67 (2003) 094014 [hep-ph/0301123] [INSPIRE].
  35. [35]
    T. Becher, R.J. Hill and M. Neubert, Factorization in BVγ decays, Phys. Rev. D 72 (2005) 094017 [hep-ph/0503263] [INSPIRE].ADSGoogle Scholar
  36. [36]
    T. Becher, R.J. Hill and M. Neubert, Soft collinear messengers: A new mode in soft collinear effective theory, Phys. Rev. D 69 (2004) 054017 [hep-ph/0308122] [INSPIRE].ADSGoogle Scholar
  37. [37]
    T. Becher, R.J. Hill, B.O. Lange and M. Neubert, External operators and anomalous dimensions in soft collinear effective theory, Phys. Rev. D 69 (2004) 034013 [hep-ph/0309227] [INSPIRE].ADSGoogle Scholar
  38. [38]
    V.M. Braun and I.E. Filyanov, Conformal Invariance and Pion Wave Functions of Nonleading Twist, Z. Phys. C 48 (1990) 239 [INSPIRE].Google Scholar
  39. [39]
    A. Ali, V.M. Braun and H. Simma, Exclusive radiative B decays in the light cone QCD sum rule approach, Z. Phys. C 63 (1994) 437 [hep-ph/9401277] [INSPIRE].ADSGoogle Scholar
  40. [40]
    P. Ball and V.M. Braun, The ρ meson light cone distribution amplitudes of leading twist revisited, Phys. Rev. D 54 (1996) 2182 [hep-ph/9602323] [INSPIRE].ADSGoogle Scholar
  41. [41]
    P. Ball, V.M. Braun, Y. Koike and K. Tanaka, Higher twist distribution amplitudes of vector mesons in QCD: Formalism and twist 3 distributions, Nucl. Phys. B 529 (1998) 323 [hep-ph/9802299] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    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].ADSCrossRefGoogle Scholar
  43. [43]
    S. Wandzura and F. Wilczek, Sum Rules for Spin Dependent Electroproduction: Test of Relativistic Constituent Quarks, Phys. Lett. B 72 (1977) 195 [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    R. Arthur et al., Lattice Results for Low Moments of Light Meson Distribution Amplitudes, Phys. Rev. D 83 (2011) 074505 [arXiv:1011.5906] [INSPIRE].ADSGoogle Scholar
  45. [45]
    Particle Data Group collaboration, K.A. Olive et al., Review of Particle Physics, Chin. Phys. C 38 (2014) 090001.Google Scholar
  46. [46]
    M. Neubert and B. Stech, Nonleptonic weak decays of B mesons, Adv. Ser. Direct. High Energy Phys. 15 (1998) 294 [hep-ph/9705292] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    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
  48. [48]
    P. Ball and G.W. Jones, Twist-3 distribution amplitudes of K * and ϕ mesons, JHEP 03 (2007) 069 [hep-ph/0702100] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    P. Ball and R. Zwicky, SU(3) breaking of leading-twist K and K * distribution amplitudes: A reprise, Phys. Lett. B 633 (2006) 289 [hep-ph/0510338] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    P. Ball, V.M. Braun and A. Lenz, Higher-twist distribution amplitudes of the K meson in QCD, JHEP 05 (2006) 004 [hep-ph/0603063] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    P. Ball and R. Zwicky, Operator relations for SU(3) breaking contributions to K and K * distribution amplitudes, JHEP 02 (2006) 034 [hep-ph/0601086] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    A.P. Bakulev, K. Passek-Kumericki, W. Schroers and N.G. Stefanis, Pion form-factor in QCD: From nonlocal condensates to NLO analytic perturbation theory, Phys. Rev. D 70 (2004) 033014 [Erratum ibid. D 70 (2004) 079906] [hep-ph/0405062] [INSPIRE].
  53. [53]
    A.P. Bakulev, S.V. Mikhailov and N.G. Stefanis, QCD based pion distribution amplitudes confronting experimental data, Phys. Lett. B 508 (2001) 279 [Erratum ibid. B 590 (2004) 309-310] [hep-ph/0103119] [INSPIRE].
  54. [54]
    S.S. Agaev, V.M. Braun, N. Offen and F.A. Porkert, BELLE Data on the π 0 γ * γ Form Factor: A Game Changer?, Phys. Rev. D 86 (2012) 077504 [arXiv:1206.3968] [INSPIRE].ADSGoogle Scholar
  55. [55]
    BaBar collaboration, B. Aubert et al., Measurement of the γγ *π 0 transition form factor, Phys. Rev. D 80 (2009) 052002 [arXiv:0905.4778] [INSPIRE].ADSGoogle Scholar
  56. [56]
    Belle collaboration, S. Uehara et al., Measurement of γγ *π 0 transition form factor at Belle, Phys. Rev. D 86 (2012) 092007 [arXiv:1205.3249] [INSPIRE].ADSGoogle Scholar
  57. [57]
    W.E. Caswell and G.P. Lepage, Effective Lagrangians for Bound State Problems in QED, QCD and Other Field Theories, Phys. Lett. B 167 (1986) 437 [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    G.T. Bodwin, E. Braaten and G.P. Lepage, Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium, Phys. Rev. D 51 (1995) 1125 [Erratum ibid. D 55 (1997) 5853] [hep-ph/9407339] [INSPIRE].
  59. [59]
    V.V. Braguta, A.K. Likhoded and A.V. Luchinsky, The study of leading twist light cone wave function of η c meson, Phys. Lett. B 646 (2007) 80 [hep-ph/0611021] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    G.T. Bodwin, H.S. Chung, D. Kang, J. Lee and C. Yu, Improved determination of color-singlet nonrelativistic QCD matrix elements for S-wave charmonium, Phys. Rev. D 77 (2008) 094017 [arXiv:0710.0994] [INSPIRE].ADSGoogle Scholar
  61. [61]
    H.S. Chung, J. Lee and C. Yu, NRQCD matrix elements for S-wave bottomonia and Γ[η b(nS) → γγ] with relativistic corrections, Phys. Lett. B 697 (2011) 48 [arXiv:1011.1554] [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    A. Czarnecki and K. Melnikov, Two loop QCD corrections to the heavy quark pair production cross-section in e + e annihilation near the threshold, Phys. Rev. Lett. 80 (1998) 2531 [hep-ph/9712222] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    M. Beneke, A. Signer and V.A. Smirnov, Two loop correction to the leptonic decay of quarkonium, Phys. Rev. Lett. 80 (1998) 2535 [hep-ph/9712302] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    M. Beneke et al., Leptonic decay of the Υ(1S) meson at third order in QCD, Phys. Rev. Lett. 112 (2014) 151801 [arXiv:1401.3005] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    M. Neubert, Heavy quark symmetry, Phys. Rept. 245 (1994) 259 [hep-ph/9306320] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    A.G. Grozin and M. Neubert, Asymptotics of heavy meson form-factors, Phys. Rev. D 55 (1997) 272 [hep-ph/9607366] [INSPIRE].ADSGoogle Scholar
  67. [67]
    S.J. Lee and M. Neubert, Model-independent properties of the B-meson distribution amplitude, Phys. Rev. D 72 (2005) 094028 [hep-ph/0509350] [INSPIRE].ADSGoogle Scholar
  68. [68]
    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].ADSGoogle Scholar
  69. [69]
    P. Ball, G.W. Jones and R. Zwicky, BVγ beyond QCD factorisation, Phys. Rev. D 75 (2007) 054004 [hep-ph/0612081] [INSPIRE].ADSGoogle Scholar
  70. [70]
    HPQCD collaboration, R.J. Dowdall, C.T.H. Davies, R.R. Horgan, C.J. Monahan and J. Shigemitsu, B-Meson Decay Constants from Improved Lattice Nonrelativistic QCD with Physical u, d, s and c Quarks, Phys. Rev. Lett. 110 (2013) 222003 [arXiv:1302.2644] [INSPIRE].ADSCrossRefGoogle Scholar
  71. [71]
    ALPHA collaboration, F. Bernardoni et al., Decay constants of B-mesons from non-perturbative HQET with two light dynamical quarks, Phys. Lett. B 735 (2014) 349 [arXiv:1404.3590] [INSPIRE].Google Scholar
  72. [72]
    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].ADSCrossRefGoogle Scholar
  73. [73]
    G. ’t Hooft and M.J.G. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
  74. [74]
    S.A. Larin, The renormalization of the axial anomaly in dimensional regularization, Phys. Lett. B 303 (1993) 113 [hep-ph/9302240] [INSPIRE].ADSCrossRefGoogle Scholar
  75. [75]
    G. Bonneau, Preserving Canonical Ward Identities in Dimensional Regularization With a Nonanticommuting γ 5, Nucl. Phys. B 177 (1981) 523 [INSPIRE].ADSCrossRefGoogle Scholar
  76. [76]
    T.L. Trueman, Chiral Symmetry in Perturbative QCD, Phys. Lett. B 88 (1979) 331 [INSPIRE].ADSCrossRefGoogle Scholar
  77. [77]
    B. Melic, B. Nizic and K. Passek, BLM scale setting for the pion transition form-factor, Phys. Rev. D 65 (2002) 053020 [hep-ph/0107295] [INSPIRE].ADSGoogle Scholar
  78. [78]
    F.M. Dittes and A.V. Radyushkin, TWo Loop Contribution To The Evolution Of The Pion Wave Function, Phys. Lett. B 134 (1984) 359 [INSPIRE].ADSCrossRefGoogle Scholar
  79. [79]
    S.V. Mikhailov and A.V. Radyushkin, Evolution Kernels in QCD: Two Loop Calculation in Feynman Gauge, Nucl. Phys. B 254 (1985) 89 [INSPIRE].ADSCrossRefGoogle Scholar
  80. [80]
    D. Mueller, Conformal constraints and the evolution of the nonsinglet meson distribution amplitude, Phys. Rev. D 49 (1994) 2525 [INSPIRE].ADSGoogle Scholar
  81. [81]
    D. Mueller, The evolution of the pion distribution amplitude in next-to-leading-order, Phys. Rev. D 51 (1995) 3855 [hep-ph/9411338] [INSPIRE].ADSGoogle Scholar
  82. [82]
    S.J. Brodsky, P. Damgaard, Y. Frishman and G.P. Lepage, Conformal Symmetry: Exclusive Processes Beyond Leading Order, Phys. Rev. D 33 (1986) 1881 [INSPIRE].ADSGoogle Scholar
  83. [83]
    T. Feldmann, P. Kroll and B. Stech, Mixing and decay constants of pseudoscalar mesons, Phys. Rev. D 58 (1998) 114006 [hep-ph/9802409] [INSPIRE].ADSGoogle Scholar
  84. [84]
    T. Feldmann, Quark structure of pseudoscalar mesons, Int. J. Mod. Phys. A 15 (2000) 159 [hep-ph/9907491] [INSPIRE].ADSGoogle Scholar
  85. [85]
    E. Braaten, Quantum-chromodynamic corrections to meson-photon transition form factors, Phys. Rev. D 28 (1983) 524 [INSPIRE].ADSGoogle Scholar
  86. [86]
    K.G. Chetyrkin, J.H. Kuhn and M. Steinhauser, RunDec: A Mathematica package for running and decoupling of the strong coupling and quark masses, Comput. Phys. Commun. 133 (2000) 43 [hep-ph/0004189] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  87. [87]
    K.G. Chetyrkin et al., Charm and Bottom Quark Masses: An Update, Phys. Rev. D 80 (2009) 074010 [arXiv:0907.2110] [INSPIRE].ADSGoogle Scholar
  88. [88]
    K. Hagiwara, T. Kuruma and Y. Yamada, Three jet distributions from the one loop Z g g vertex at e + e colliders, Nucl. Phys. B 358 (1991) 80 [INSPIRE].ADSCrossRefGoogle Scholar
  89. [89]
    ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, SLD Electroweak Group and SLD Heavy Flavour Group collaborations, S. Schael et al., Precision electroweak measurements on the Z resonance, Phys. Rept. 427 (2006) 257 [hep-ex/0509008] [INSPIRE].ADSGoogle Scholar
  90. [90]
    UTfit collaboration, M. Bona et al., Model-independent constraints on ΔF = 2 operators and the scale of new physics, JHEP 03 (2008) 049 [arXiv:0707.0636] [INSPIRE].Google Scholar
  91. [91]
    ETM collaboration, V. Bertone et al., Kaon Mixing Beyond the SM from N f =2 tmQCD and model independent constraints from the UTA, JHEP 03 (2013) 089 [Erratum ibid. 1307 (2013) 143] [arXiv:1207.1287] [INSPIRE].
  92. [92]
    ETM collaboration, N. Carrasco et al., B-physics from N f = 2 tmQCD: the Standard Model and beyond, JHEP 03 (2014) 016 [arXiv:1308.1851] [INSPIRE].Google Scholar
  93. [93]
    R. Harnik, J. Kopp and J. Zupan, Flavor Violating Higgs Decays, JHEP 03 (2013) 026 [arXiv:1209.1397] [INSPIRE].ADSCrossRefGoogle Scholar
  94. [94]
    A. Sirlin, Current Algebra Formulation of Radiative Corrections in Gauge Theories and the Universality of the Weak Interactions, Rev. Mod. Phys. 50 (1978) 573 [Erratum ibid. 50 (1978) 905] [INSPIRE].
  95. [95]
    W.J. Marciano and A. Sirlin, Electroweak Radiative Corrections to τ Decay, Phys. Rev. Lett. 61 (1988) 1815 [INSPIRE].ADSCrossRefGoogle Scholar
  96. [96]
    E. Braaten and C.-S. Li, Electroweak radiative corrections to the semihadronic decay rate of the τ lepton, Phys. Rev. D 42 (1990) 3888 [INSPIRE].ADSGoogle Scholar
  97. [97]
    J. Erler, Calculation of the QED coupling \( \widehat{\alpha} \)(M Z) in the modified minimal subtraction scheme, Phys. Rev. D 59 (1999) 054008 [hep-ph/9803453] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Yuval Grossman
    • 1
  • Matthias König
    • 2
  • Matthias Neubert
    • 1
    • 2
  1. 1.Department of Physics, LEPPCornell UniversityIthacaUnited States
  2. 2.PRISMA Cluster of Excellence & Mainz Institute for Theoretical PhysicsJohannes Gutenberg UniversityMainzGermany

Personalised recommendations