Transforming differential equations of multi-loop Feynman integrals into canonical form

  • Christoph MeyerEmail author
Open Access
Regular Article - Theoretical Physics


The method of differential equations has been proven to be a powerful tool for the computation of multi-loop Feynman integrals appearing in quantum field theory. It has been observed that in many instances a canonical basis can be chosen, which drastically simplifies the solution of the differential equation. In this paper, an algorithm is presented that computes the transformation to a canonical basis, starting from some basis that is, for instance, obtained by the usual integration-by-parts reduction techniques. The algorithm requires the existence of a rational transformation to a canonical basis, but is otherwise completely agnostic about the differential equation. In particular, it is applicable to problems involving multiple scales and allows for a rational dependence on the dimensional regulator. It is demonstrated that the algorithm is suitable for current multi-loop calculations by presenting its successful application to a number of non-trivial examples.


QCD Phenomenology 


Open Access

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  1. [1]
    ATLAS collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214] [INSPIRE].
  2. [2]
    CMS collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE].
  3. [3]
    G. Ossola, Automated higher-order calculations: status and prospects, PoS (DIS2015) 150 [arXiv:1508.01894] [INSPIRE].
  4. [4]
    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
  5. [5]
    A.V. Kotikov, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].ADSGoogle Scholar
  7. [7]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
  8. [8]
    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    J.M. Henn, A.V. Smirnov and V.A. Smirnov, Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation, JHEP 07 (2013) 128 [arXiv:1306.2799] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    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
  11. [11]
    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
  12. [12]
    M. Argeri et al., Magnus and Dyson series for master integrals, JHEP 03 (2014) 082 [arXiv:1401.2979] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    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
  14. [14]
    S. Caron-Huot and J.M. Henn, Iterative structure of finite loop integrals, JHEP 06 (2014) 114 [arXiv:1404.2922] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    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
  16. [16]
    F. Caola, J.M. Henn, K. Melnikov and V.A. Smirnov, Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons, JHEP 09 (2014) 043 [arXiv:1404.5590] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    Y. Li, A. von Manteuffel, R.M. Schabinger and H.X. Zhu, N 3 LO Higgs boson and Drell-Yan production at threshold: the one-loop two-emission contribution, Phys. Rev. D 90 (2014) 053006 [arXiv:1404.5839] [INSPIRE].ADSGoogle Scholar
  18. [18]
    M. Höschele, J. Hoff and T. Ueda, Adequate bases of phase space master integrals for ggh at NNLO and beyond, JHEP 09 (2014) 116 [arXiv:1407.4049] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    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
  20. [20]
    A. von Manteuffel, R.M. Schabinger and H.X. Zhu, The two-loop soft function for heavy quark pair production at future linear colliders, Phys. Rev. D 92 (2015) 045034 [arXiv:1408.5134] [INSPIRE].ADSGoogle Scholar
  21. [21]
    A. Grozin, J.M. Henn, G.P. Korchemsky and P. Marquard, Three loop cusp anomalous dimension in QCD, Phys. Rev. Lett. 114 (2015) 062006 [arXiv:1409.0023] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    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
  23. [23]
    T. Huber and S. Kränkl, Two-loop master integrals for non-leptonic heavy-to-heavy decays, JHEP 04 (2015) 140 [arXiv:1503.00735] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    T. Gehrmann, A. von Manteuffel and L. Tancredi, The two-loop helicity amplitudes for \( q\ {\overline{q}}^{\prime}\to {V}_1{V}_2\to\ 4 \) leptons, JHEP 09 (2015) 128 [arXiv:1503.04812] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    T. Gehrmann, S. Guns and D. Kara, The rare decay HZγ in perturbative QCD, JHEP 09 (2015) 038 [arXiv:1505.00561] [INSPIRE].CrossRefGoogle Scholar
  26. [26]
    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
  27. [27]
    C. Anzai et al., Exact N 3 LO results for qq H + X, JHEP 07 (2015) 140 [arXiv:1506.02674] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    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
  29. [29]
    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].
  30. [30]
    O. Gituliar, Master integrals for splitting functions from differential equations in QCD, JHEP 02 (2016) 017 [arXiv:1512.02045] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    R.N. Lee and K.T. Mingulov, Total Born cross section of e + e -pair production in relativistic ion collisions from differential equations, Phys. Lett. B 757 (2016) 207 [arXiv:1602.02463] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    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
  33. [33]
    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
  34. [34]
    B. Eden and V.A. Smirnov, Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations, JHEP 10 (2016) 115 [arXiv:1607.06427] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    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
  36. [36]
    R. Bonciani, V. Del Duca, H. Frellesvig, J.M. Henn, F. Moriello and V.A. Smirnov, Two-loop planar master integrals for Higgs → 3 partons with full heavy-quark mass dependence, JHEP 12 (2016) 096 [arXiv:1609.06685] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    M. Bonetti, K. Melnikov and L. Tancredi, Two-loop electroweak corrections to Higgs-gluon couplings to higher orders in the dimensional regularization parameter, Nucl. Phys. B 916 (2017) 709 [arXiv:1610.05497] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  38. [38]
    R.N. Lee, Reducing differential equations for multiloop master integrals, JHEP 04 (2015) 108 [arXiv:1411.0911] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  40. [40]
    O. Gituliar and V. Magerya, Fuchsia and master integrals for splitting functions from differential equations in QCD, PoS (LL2016) 030 [arXiv:1607.00759] [INSPIRE].
  41. [41]
    C. Meyer, Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA, in preparation.Google Scholar
  42. [42]
    E.K. Leinartas, Factorization of rational functions of several variables into partial fractions, Izv. Vyssh. Uchebn. Zaved. Mat. 22 (1978) 47 [Soviet Math. 22 (1978) 35].
  43. [43]
    A. Raichev, Leinartass partial fraction decomposition, arXiv:1206.4740.
  44. [44]
    M. Brucherseifer, F. Caola and K. Melnikov, On the NNLO QCD corrections to single-top production at the LHC, Phys. Lett. B 736 (2014) 58 [arXiv:1404.7116] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    M. Assadsolimani, P. Kant, B. Tausk and P. Uwer, Calculation of two-loop QCD corrections for hadronic single top-quark production in the t channel, Phys. Rev. D 90 (2014) 114024 [arXiv:1409.3654] [INSPIRE].ADSGoogle Scholar
  46. [46]
    F. Cascioli et al., ZZ production at hadron colliders in NNLO QCD, Phys. Lett. B 735 (2014) 311 [arXiv:1405.2219] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    T. Gehrmann et al., W + W production at hadron colliders in next to next to leading order QCD, Phys. Rev. Lett. 113 (2014) 212001 [arXiv:1408.5243] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    M. Grazzini, S. Kallweit, D. Rathlev and M. Wiesemann, W ± Z production at hadron colliders in NNLO QCD, Phys. Lett. B 761 (2016) 179 [arXiv:1604.08576] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    G. Passarino and M.J.G. Veltman, One loop corrections for e + e annihilation into μ + μ in the Weinberg model, Nucl. Phys. B 160 (1979) 151 [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    A.I. Davydychev, A simple formula for reducing Feynman diagrams to scalar integrals, Phys. Lett. B 263 (1991) 107 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    O.V. Tarasov, Connection between Feynman integrals having different values of the space-time dimension, Phys. Rev. D 54 (1996) 6479 [hep-th/9606018] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  52. [52]
    O.V. Tarasov, Generalized recurrence relations for two loop propagator integrals with arbitrary masses, Nucl. Phys. B 502 (1997) 455 [hep-ph/9703319] [INSPIRE].
  53. [53]
    C. Anastasiou, E.W.N. Glover and C. Oleari, The two-loop scalar and tensor pentabox graph with light-like legs, Nucl. Phys. B 575 (2000) 416 [Erratum ibid. B 585 (2000) 763] [hep-ph/9912251] [INSPIRE].
  54. [54]
    E.W.N. Glover, Two loop QCD helicity amplitudes for massless quark quark scattering, JHEP 04 (2004) 021 [hep-ph/0401119] [INSPIRE].
  55. [55]
    F.V. Tkachov, A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  56. [56]
    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
  57. [57]
    J. Gluza, K. Kajda and D.A. Kosower, Towards a basis for planar two-loop integrals, Phys. Rev. D 83 (2011) 045012 [arXiv:1009.0472] [INSPIRE].ADSGoogle Scholar
  58. [58]
    R.M. Schabinger, A new algorithm for the generation of unitarity-compatible integration by parts relations, JHEP 01 (2012) 077 [arXiv:1111.4220] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    P. Kant, Finding linear dependencies in integration-by-parts equations: a Monte Carlo approach, Comput. Phys. Commun. 185 (2014) 1473 [arXiv:1309.7287] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  60. [60]
    A. von Manteuffel and R.M. Schabinger, A novel approach to integration by parts reduction, Phys. Lett. B 744 (2015) 101 [arXiv:1406.4513] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    K.J. Larsen and Y. Zhang, Integration-by-parts reductions from unitarity cuts and algebraic geometry, Phys. Rev. D 93 (2016) 041701 [arXiv:1511.01071] [INSPIRE].ADSMathSciNetGoogle Scholar
  62. [62]
    A.V. Smirnov and A.V. Petukhov, The number of master integrals is finite, Lett. Math. Phys. 97 (2011) 37 [arXiv:1004.4199] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    R.N. Lee and A.A. Pomeransky, Critical points and number of master integrals, JHEP 11 (2013) 165 [arXiv:1308.6676] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
  65. [65]
    C. Anastasiou and A. Lazopoulos, Automatic integral reduction for higher order perturbative calculations, JHEP 07 (2004) 046 [hep-ph/0404258] [INSPIRE].
  66. [66]
    C. Studerus, Reduze-Feynman integral reduction in C++, Comput. Phys. Commun. 181 (2010) 1293 [arXiv:0912.2546] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  67. [67]
    A. von Manteuffel and C. Studerus, Reduze 2 — distributed Feynman integral reduction, arXiv:1201.4330 [INSPIRE].
  68. [68]
    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
  69. [69]
    A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  70. [70]
    R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
  71. [71]
    K.-T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977) 831 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  73. [73]
    D.A. Cox, J. Little and D. O’Shea, Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra, Undergrad. Texts Math., Springer-Verlag, Secaucus NJ U.S.A., (2007).Google Scholar
  74. [74]
    B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal (in German), Ph.D. thesis, University of Innsbruck, Innsbruck Austria, (1965).Google Scholar
  75. [75]
    B. Buchberger, Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems (in German), Aequat. Math. 4 (1970) 374.CrossRefzbMATHGoogle Scholar
  76. [76]
    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
  77. [77]
    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].
  78. [78]
    S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theor. 148 (2015) 328 [arXiv:1309.5865] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  79. [79]
    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
  80. [80]
    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
  81. [81]
    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
  82. [82]
    S. Bloch, M. Kerr and P. Vanhove, Local mirror symmetry and the sunset Feynman integral, arXiv:1601.08181 [INSPIRE].
  83. [83]
    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
  84. [84]
    L. Adams, C. Bogner, A. Schweitzer and S. Weinzierl, The kite integral to all orders in terms of elliptic polylogarithms, arXiv:1607.01571 [INSPIRE].
  85. [85]
    T. Gehrmann, L. Tancredi and E. Weihs, Two-loop master integrals for \( q\overline{q}\to V\ V \) : the planar topologies, JHEP 08 (2013) 070 [arXiv:1306.6344] [INSPIRE].ADSCrossRefGoogle Scholar
  86. [86]
    C.G. Papadopoulos, D. Tommasini and C. Wever, Two-loop master integrals with the simplified differential equations approach, JHEP 01 (2015) 072 [arXiv:1409.6114] [INSPIRE].ADSCrossRefGoogle Scholar
  87. [87]
    D. Binosi, J. Collins, C. Kaufhold and L. Theussl, JaxoDraw: a graphical user interface for drawing Feynman diagrams. Version 2.0 release notes, Comput. Phys. Commun. 180 (2009) 1709 [arXiv:0811.4113] [INSPIRE].ADSCrossRefGoogle Scholar
  88. [88]
    J.A.M. Vermaseren, Axodraw, Comput. Phys. Commun. 83 (1994) 45 [INSPIRE].

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© The Author(s) 2017

Authors and Affiliations

  1. 1.Institut für PhysikHumboldt-Universität zu BerlinBerlinGermany

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