Journal of High Energy Physics

, 2016:96 | Cite as

Two-loop planar master integrals for Higgs → 3 partons with full heavy-quark mass dependence

  • Roberto Bonciani
  • Vittorio Del Duca
  • Hjalte Frellesvig
  • Johannes M. Henn
  • Francesco MorielloEmail author
  • Vladimir A. Smirnov
Open Access
Regular Article - Theoretical Physics


We present the analytic computation of all the planar master integrals which contribute to the two-loop scattering amplitudes for Higgs→ 3 partons, with full heavy-quark mass dependence. These are relevant for the NNLO corrections to fully inclusive Higgs production and to the NLO corrections to Higgs production in association with a jet, in the full theory. The computation is performed using the differential equations method. Whenever possible, a basis of master integrals that are pure functions of uniform weight is used. The result is expressed in terms of one-fold integrals of polylogarithms and elementary functions up to transcendental weight four. Two integral sectors are expressed in terms of elliptic integrals. We show that by introducing a one-dimensional parametrization of the integrals the relevant second order differential equation can be readily solved, and the solution can be expressed to all orders of the dimensional regularization parameter in terms of iterated integrals over elliptic kernels. We express the result for the elliptic sectors in terms of two and three-fold iterated integrals, which we find suitable for numerical evaluations. This is the first time that four-point multiscale Feynman integrals have been computed in a fully analytic way in terms of elliptic integrals.


Perturbative QCD Scattering Amplitudes 


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]
    D. Graudenz, M. Spira and P.M. Zerwas, QCD corrections to Higgs boson production at proton proton colliders, Phys. Rev. Lett. 70 (1993) 1372 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    M. Spira, A. Djouadi, D. Graudenz and P.M. Zerwas, Higgs boson production at the LHC, Nucl. Phys. B 453 (1995) 17 [hep-ph/9504378] [INSPIRE].
  3. [3]
    R.K. Ellis, I. Hinchliffe, M. Soldate and J.J. van der Bij, Higgs Decay to τ + τ : A Possible Signature of Intermediate Mass Higgs Bosons at the SSC, Nucl. Phys. B 297 (1988) 221 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    R.P. Kauffman, Higgs boson p T in gluon fusion, Phys. Rev. D 44 (1991) 1415 [INSPIRE].ADSGoogle Scholar
  5. [5]
    M. Grazzini and H. Sargsyan, Heavy-quark mass effects in Higgs boson production at the LHC, JHEP 09 (2013) 129 [arXiv:1306.4581] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger, Higgs Boson Gluon-Fusion Production in QCD at Three Loops, Phys. Rev. Lett. 114 (2015) 212001 [arXiv:1503.06056] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    C. Anastasiou et al., High precision determination of the gluon fusion Higgs boson cross-section at the LHC, JHEP 05 (2016) 058 [arXiv:1602.00695] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    U. Baur and E.W.N. Glover, Higgs Boson Production at Large Transverse Momentum in Hadronic Collisions, Nucl. Phys. B 339 (1990) 38 [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    D. de Florian, M. Grazzini and Z. Kunszt, Higgs production with large transverse momentum in hadronic collisions at next-to-leading order, Phys. Rev. Lett. 82 (1999) 5209 [hep-ph/9902483] [INSPIRE].
  10. [10]
    R. Boughezal, F. Caola, K. Melnikov, F. Petriello and M. Schulze, Higgs boson production in association with a jet at next-to-next-to-leading order, Phys. Rev. Lett. 115 (2015) 082003 [arXiv:1504.07922] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    R. Boughezal, C. Focke, W. Giele, X. Liu and F. Petriello, Higgs boson production in association with a jet at NNLO using jettiness subtraction, Phys. Lett. B 748 (2015) 5 [arXiv:1505.03893] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    X. Chen, J. Cruz-Martinez, T. Gehrmann, E.W.N. Glover and M. Jaquier, NNLO QCD corrections to Higgs boson production at large transverse momentum, JHEP 10 (2016) 066 [arXiv:1607.08817] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    R.V. Harlander, T. Neumann, K.J. Ozeren and M. Wiesemann, Top-mass effects in differential Higgs production through gluon fusion at order α s4, JHEP 08 (2012) 139 [arXiv:1206.0157] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    R. Frederix, S. Frixione, E. Vryonidou and M. Wiesemann, Heavy-quark mass effects in Higgs plus jets production, JHEP 08 (2016) 006 [arXiv:1604.03017] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    F. Caola, S. Forte, S. Marzani, C. Muselli and G. Vita, The Higgs transverse momentum spectrum with finite quark masses beyond leading order, JHEP 08 (2016) 150 [arXiv:1606.04100] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    T. Neumann and C. Williams, The Higgs boson at high p T , arXiv:1609.00367 [INSPIRE].
  17. [17]
    C. Grojean, E. Salvioni, M. Schlaffer and A. Weiler, Very boosted Higgs in gluon fusion, JHEP 05 (2014) 022 [arXiv:1312.3317] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    A. Azatov and A. Paul, Probing Higgs couplings with high p T Higgs production, JHEP 01 (2014) 014 [arXiv:1309.5273] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    A. Azatov, C. Grojean, A. Paul and E. Salvioni, Resolving gluon fusion loops at current and future hadron colliders, JHEP 09 (2016) 123 [arXiv:1608.00977] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    A.V. Kotikov, Differential equation method: The Calculation of N point Feynman diagrams, Phys. Lett. B 267 (1991) 123 [Erratum ibid. B 295 (1992) 409] [INSPIRE].
  22. [22]
    Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].
  23. [23]
    E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].ADSGoogle Scholar
  24. [24]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
  25. [25]
    F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    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
  27. [27]
    S. Laporta and E. Remiddi, The analytical value of the electron (g − 2) at order α 3 in QED, Phys. Lett. B 379 (1996) 283 [hep-ph/9602417] [INSPIRE].
  28. [28]
    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
  29. [29]
    A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
  31. [31]
    A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical Polylogarithms for Amplitudes and Wilson Loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    H. Frellesvig, D. Tommasini and C. Wever, On the reduction of generalized polylogarithms to Li n and Li 2,2 and on the evaluation thereof, JHEP 03 (2016) 189 [arXiv:1601.02649] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    F. Cachazo, Sharpening The Leading Singularity, arXiv:0803.1988 [INSPIRE].
  36. [36]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local Integrals for Planar Scattering Amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    J.M. Henn, Modern methods for scattering amplitudes, lectures given at Nordita School on Integrability, Nordita, Stockholm, 4-12 August 2014,
  38. [38]
    J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  39. [39]
    R.N. Lee, Reducing differential equations for multiloop master integrals, JHEP 04 (2015) 108 [arXiv:1411.0911] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    J.M. Henn and V.A. Smirnov, Analytic results for two-loop master integrals for Bhabha scattering I, JHEP 11 (2013) 041 [arXiv:1307.4083] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    J.M. Henn, A.V. Smirnov and V.A. Smirnov, Evaluating single-scale and/or non-planar diagrams by differential equations, JHEP 03 (2014) 088 [arXiv:1312.2588] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    M. Argeri et al., Magnus and Dyson Series for Master Integrals, JHEP 03 (2014) 082 [arXiv:1401.2979] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    J.M. Henn, K. Melnikov and V.A. Smirnov, Two-loop planar master integrals for the production of off-shell vector bosons in hadron collisions, JHEP 05 (2014) 090 [arXiv:1402.7078] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    T. Gehrmann, A. von Manteuffel, L. Tancredi and E. Weihs, The two-loop master integrals for \( q\overline{q}\to V\ V \), JHEP 06 (2014) 032 [arXiv:1404.4853] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    F. Caola, J.M. Henn, K. Melnikov and V.A. Smirnov, Non-planar master integrals for th production of two off-shell vector bosons in collisions of massless partons, JHEP 09 (2014) 043 [arXiv:1404.5590] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    S. Di Vita, P. Mastrolia, U. Schubert and V. Yundin, Three-loop master integrals for ladder-box diagrams with one massive leg, JHEP 09 (2014) 148 [arXiv:1408.3107] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    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].ADSCrossRefGoogle Scholar
  48. [48]
    R. Bonciani, V. Del Duca, H. Frellesvig, J.M. Henn, F. Moriello and V.A. Smirnov, Next-to-leading order QCD corrections to the decay width HZγ , JHEP 08 (2015) 108 [arXiv:1505.00567] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    T. Gehrmann, S. Guns and D. Kara, The rare decay HZγ in perturbative QCD, JHEP 09 (2015) 038 [arXiv:1505.00561] [INSPIRE].CrossRefGoogle Scholar
  50. [50]
    A. Grozin, J.M. Henn, G.P. Korchemsky and P. Marquard, The three-loop cusp anomalous dimension in QCD and its supersymmetric extensions, JHEP 01 (2016) 140 [arXiv:1510.07803] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    T. Gehrmann, J.M. Henn and N.A. Lo Presti, Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD, Phys. Rev. Lett. 116 (2016) 062001 [Erratum ibid. 116 (2016)189903] [arXiv:1511.05409] [INSPIRE].
  52. [52]
    J.M. Henn, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, A planar four-loop form factor and cusp anomalous dimension in QCD, JHEP 05 (2016) 066 [arXiv:1604.03126] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    R. Bonciani, S. Di Vita, P. Mastrolia and U. Schubert, Two-Loop Master Integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering, JHEP 09 (2016) 091 [arXiv:1604.08581] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    J.M. Henn and B. Mistlberger, Four-Gluon Scattering at Three Loops, Infrared Structure and the Regge Limit, Phys. Rev. Lett. 117 (2016) 171601 [arXiv:1608.00850] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    R.N. Lee and V.A. Smirnov, Evaluating the last missing ingredient for the three-loop quark static potential by differential equations, JHEP 10 (2016) 089 [arXiv:1608.02605] [INSPIRE].ADSGoogle Scholar
  56. [56]
    K.-T. Chen, Iterated path integrals, Bull. Am. Math. Soc. 83 (1977) 831 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    S. Caron-Huot and J.M. Henn, Iterative structure of finite loop integrals, JHEP 06 (2014) 114 [arXiv:1404.2922] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    L. Tancredi, Integration by parts identities in integer numbers of dimensions. A criterion for decoupling systems of differential equations, Nucl. Phys. B 901 (2015) 282 [arXiv:1509.03330] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    M. Caffo, H. Czyz, S. Laporta and E. Remiddi, The Master differential equations for the two loop sunrise selfmass amplitudes, Nuovo Cim. A 111 (1998) 365 [hep-th/9805118] [INSPIRE].ADSGoogle Scholar
  61. [61]
    S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph, Nucl. Phys. B 704 (2005) 349 [hep-ph/0406160] [INSPIRE].
  62. [62]
    S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theor. 148 (2015) 328 [arXiv:1309.5865] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    S. Bloch, M. Kerr and P. Vanhove, A Feynman integral via higher normal functions, Compos. Math. 151 (2015) 2329 [arXiv:1406.2664] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms, J. Math. Phys. 55 (2014) 102301 [arXiv:1405.5640] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case, J. Math. Phys. 56 (2015) 072303 [arXiv:1504.03255] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    L. Adams, C. Bogner and S. Weinzierl, The iterated structure of the all-order result for the two-loop sunrise integral, J. Math. Phys. 57 (2016) 032304 [arXiv:1512.05630] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  67. [67]
    S. Bloch, M. Kerr and P. Vanhove, Local mirror symmetry and the sunset Feynman integral, arXiv:1601.08181 [INSPIRE].
  68. [68]
    E. Remiddi and L. Tancredi, Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral, Nucl. Phys. B 907 (2016) 400 [arXiv:1602.01481] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    L. Adams, C. Bogner, A. Schweitzer and S. Weinzierl, The kite integral to all orders in terms of elliptic polylogarithms, arXiv:1607.01571 [INSPIRE].
  70. [70]
    C.G. Papadopoulos, Simplified differential equations approach for Master Integrals, JHEP 07 (2014) 088 [arXiv:1401.6057] [INSPIRE].ADSCrossRefGoogle Scholar
  71. [71]
    A.V. Smirnov, Algorithm FIREFeynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  72. [72]
    A.V. Smirnov and V.A. Smirnov, FIRE4, LiteRed and accompanying tools to solve integration by parts relations, Comput. Phys. Commun. 184 (2013) 2820 [arXiv:1302.5885] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  73. [73]
    A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  74. [74]
    R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
  75. [75]
    F.C.S. Brown, Multiple zeta values and periods of moduli spaces \( {\mathfrak{M}}_{0,n} \), Annales Sci. Ecole Norm. Sup. 42 (2009) 371 [math/0606419] [INSPIRE].
  76. [76]
    V.A. Smirnov, Asymptotic expansions in limits of large momenta and masses, Commun. Math. Phys. 134 (1990) 109 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    V.A. Smirnov, Asymptotic expansions in momenta and masses and calculation of Feynman diagrams, Mod. Phys. Lett. A 10 (1995) 1485 [hep-th/9412063] [INSPIRE].ADSCrossRefGoogle Scholar
  78. [78]
    V.A. Smirnov, Applied asymptotic expansions in momenta and masses, Springer Tracts Mod. Phys. 177 (2002) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  79. [79]
    R.K. Ellis and G. Zanderighi, Scalar one-loop integrals for QCD, JHEP 02 (2008) 002 [arXiv:0712.1851] [INSPIRE].ADSCrossRefGoogle Scholar
  80. [80]
    A.V. Smirnov and M.N. Tentyukov, Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA), Comput. Phys. Commun. 180 (2009) 735 [arXiv:0807.4129] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  81. [81]
    A.V. Smirnov, V.A. Smirnov and M. Tentyukov, FIESTA 2: Parallelizeable multiloop numerical calculations, Comput. Phys. Commun. 182 (2011) 790 [arXiv:0912.0158] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  82. [82]
    A.V. Smirnov, FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions, Comput. Phys. Commun. 185 (2014) 2090 [arXiv:1312.3186] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  83. [83]
    B.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th edition, Cambridge University Press, London (1958), pg. 206-208.Google Scholar
  84. [84]
    L. Tancredi, Differential equations and dispersion relations for Feynman amplitudes, talk given at LoopFest 2016,
  85. [85]
    J. Broedel, C.R. Mafra, N. Matthes and O. Schlotterer, Elliptic multiple zeta values and one-loop superstring amplitudes, JHEP 07 (2015) 112 [arXiv:1412.5535] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  86. [86]
    F.C.S. Brown and A. Levin, Multiple Elliptic Polylogarithms, arXiv:1110.6917.
  87. [87]
    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].
  88. [88]
    J.A.M. Vermaseren, Axodraw, Comput. Phys. Commun. 83 (1994) 45 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  89. [89]
    D.A. Kosower and K.J. Larsen, Maximal Unitarity at Two Loops, Phys. Rev. D 85 (2012) 045017 [arXiv:1108.1180] [INSPIRE].ADSGoogle Scholar
  90. [90]
    L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [INSPIRE].
  91. [91]
    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  92. [92]
    A. Hodges, The Box Integrals in Momentum-Twistor Geometry, JHEP 08 (2013) 051 [arXiv:1004.3323] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  93. [93]
    M. Søgaard and Y. Zhang, Elliptic Functions and Maximal Unitarity, Phys. Rev. D 91 (2015) 081701 [arXiv:1412.5577] [INSPIRE].ADSMathSciNetGoogle Scholar

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© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Roberto Bonciani
    • 1
    • 2
  • Vittorio Del Duca
    • 3
    • 4
  • Hjalte Frellesvig
    • 5
  • Johannes M. Henn
    • 6
  • Francesco Moriello
    • 1
    • 2
    • 3
    Email author
  • Vladimir A. Smirnov
    • 7
  1. 1.Dipartimento di Fisica, Sapienza — Università di RomaRomeItaly
  2. 2.INFN Sezione di RomaRomeItaly
  3. 3.ETH Zurich, Institut fur theoretische PhysikZurichSwitzerland
  4. 4.INFN Laboratori Nazionali di FrascatiFrascati, RomaItaly
  5. 5.Institute of Nuclear and Particle Physics, NCSR DemokritosAgia ParaskeviGreece
  6. 6.PRISMA Cluster of Excellence, Johannes Gutenberg UniversityMainzGermany
  7. 7.Skobeltsyn Inst. of Nuclear Physics of Moscow State UniversityMoscowRussia

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