Journal of High Energy Physics

, 2019:169 | Cite as

All orders structure and efficient computation of linearly reducible elliptic Feynman integrals

  • Martijn HiddingEmail author
  • Francesco Moriello
Open Access
Regular Article - Theoretical Physics


We define linearly reducible elliptic Feynman integrals, and we show that they can be algorithmically solved up to arbitrary order of the dimensional regulator in terms of a 1-dimensional integral over a polylogarithmic integrand, which we call the inner polylogarithmic part (IPP). The solution is obtained by direct integration of the Feynman parametric representation. When the IPP depends on one elliptic curve (and no other algebraic functions), this class of Feynman integrals can be algorithmically solved in terms of elliptic multiple polylogarithms (eMPLs) by using integration by parts identities. We then elaborate on the differential equations method. Specifically, we show that the IPP can be mapped to a generalized integral topology satisfying a set of differential equations in ϵ-form. In the examples we consider the canonical differential equations can be directly solved in terms of eMPLs up to arbitrary order of the dimensional regulator. The remaining 1-dimensional integral may be performed to express such integrals completely in terms of eMPLs. We apply these methods to solve two- and three-points integrals in terms of eMPLs. We analytically continue these integrals to the physical region by using their 1-dimensional integral representation.


Perturbative QCD Scattering Amplitudes 


Open Access

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  1. [1]
    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].
  2. [2]
    F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].
  3. [3]
    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
  4. [4]
    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].
  5. [5]
    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
  6. [6]
    A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
  8. [8]
    A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].
  9. [9]
    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].
  10. [10]
    Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].
  11. [11]
    E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].
  12. [12]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
  13. [13]
    F.C.S. Brown, On the periods of some Feynman integrals, arXiv:0910.0114 [INSPIRE].
  14. [14]
    E. Panzer, Feynman integrals and hyperlogarithms, Ph.D. thesis, Humboldt U., Berlin, Inst. Math., 2015. arXiv:1506.07243. 10.18452/17157 [INSPIRE].
  15. [15]
    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    M. Prausa, epsilon: A tool to find a canonical basis of master integrals, Comput. Phys. Commun. 219 (2017) 361 [arXiv:1701.00725] [INSPIRE].
  17. [17]
    O. Gituliar and V. Magerya, Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form, Comput. Phys. Commun. 219 (2017) 329 [arXiv:1701.04269] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    C. Meyer, Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA, Comput. Phys. Commun. 222 (2018) 295 [arXiv:1705.06252] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
  20. [20]
    A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
  21. [21]
    P. Maierhöfer, J. Usovitsch and P. Uwer, Kira — A Feynman integral reduction program, Comput. Phys. Commun. 230 (2018) 99 [arXiv:1705.05610] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188 (2015) 148 [arXiv:1403.3385] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  23. [23]
    J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].
  24. [24]
    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].
  25. [25]
    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].
  26. [26]
    B.A. Kniehl, A.V. Kotikov, A. Onishchenko and O. Veretin, Two-loop sunset diagrams with three massive lines, Nucl. Phys. B 738 (2006) 306 [hep-ph/0510235] [INSPIRE].
  27. [27]
    M. Yu. Kalmykov and B.A. Kniehl, Towards all-order Laurent expansion of generalized hypergeometric functions around rational values of parameters, Nucl. Phys. B 809 (2009) 365 [arXiv:0807.0567] [INSPIRE].
  28. [28]
    S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theor. 148 (2015) 328 [arXiv:1309.5865] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    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
  30. [30]
    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
  31. [31]
    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].
  32. [32]
    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].
  33. [33]
    S. Bloch, M. Kerr and P. Vanhove, Local mirror symmetry and the sunset Feynman integral, Adv. Theor. Math. Phys. 21 (2017) 1373 [arXiv:1601.08181] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    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].
  35. [35]
    L. Adams, C. Bogner, A. Schweitzer and S. Weinzierl, The kite integral to all orders in terms of elliptic polylogarithms, J. Math. Phys. 57 (2016) 122302 [arXiv:1607.01571] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle 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].
  37. [37]
    A. Primo and L. Tancredi, On the maximal cut of Feynman integrals and the solution of their differential equations, Nucl. Phys. B 916 (2017) 94 [arXiv:1610.08397] [INSPIRE].
  38. [38]
    A. von Manteuffel and L. Tancredi, A non-planar two-loop three-point function beyond multiple polylogarithms, JHEP 06 (2017) 127 [arXiv:1701.05905] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    J. Ablinger, J. Blümlein, A. De Freitas, C. Schneider and K. Schönwald, The two-mass contribution to the three-loop pure singlet operator matrix element, Nucl. Phys. B 927 (2018) 339 [arXiv:1711.06717] [INSPIRE].
  40. [40]
    C. Bogner, A. Schweitzer and S. Weinzierl, Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral, Nucl. Phys. B 922 (2017) 528 [arXiv:1705.08952] [INSPIRE].
  41. [41]
    E. Remiddi and L. Tancredi, An Elliptic Generalization of Multiple Polylogarithms, Nucl. Phys. B 925 (2017) 212 [arXiv:1709.03622] [INSPIRE].
  42. [42]
    J.L. Bourjaily, A.J. McLeod, M. Spradlin, M. von Hippel and M. Wilhelm, Elliptic Double-Box Integrals: Massless Scattering Amplitudes beyond Polylogarithms, Phys. Rev. Lett. 120 (2018) 121603 [arXiv:1712.02785] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    L.-B. Chen, J. Jiang and C.-F. Qiao, Two-Loop integrals for CP-even heavy quarkonium production and decays: Elliptic Sectors, JHEP 04 (2018) 080 [arXiv:1712.03516] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism, JHEP 05 (2018) 093 [arXiv:1712.07089] [INSPIRE].
  45. [45]
    J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral, Phys. Rev. D 97 (2018) 116009 [arXiv:1712.07095] [INSPIRE].
  46. [46]
    L. Adams and S. Weinzierl, The ε-form of the differential equations for Feynman integrals in the elliptic case, Phys. Lett. B 781 (2018) 270 [arXiv:1802.05020] [INSPIRE].
  47. [47]
    L. Adams, E. Chaubey and S. Weinzierl, Analytic results for the planar double box integral relevant to top-pair production with a closed top loop, JHEP 10 (2018) 206 [arXiv:1806.04981] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    H. Frellesvig and C.G. Papadopoulos, Cuts of Feynman Integrals in Baikov representation, JHEP 04 (2017) 083 [arXiv:1701.07356] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    J. Bosma, M. Sogaard and Y. Zhang, Maximal Cuts in Arbitrary Dimension, JHEP 08 (2017) 051 [arXiv:1704.04255] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    M. Harley, F. Moriello and R.M. Schabinger, Baikov-Lee Representations Of Cut Feynman Integrals, JHEP 06 (2017) 049 [arXiv:1705.03478] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    L. Adams, E. Chaubey and S. Weinzierl, Planar Double Box Integral for Top Pair Production with a Closed Top Loop to all orders in the Dimensional Regularization Parameter, Phys. Rev. Lett. 121 (2018) 142001 [arXiv:1804.11144] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    F.C.S. Brown and A. Levin, Multiple Elliptic Polylogarithms, arXiv:1110.6917.
  53. [53]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series, JHEP 08 (2018) 014 [arXiv:1803.10256] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    C. Bogner and S. Weinzierl, Feynman graph polynomials, Int. J. Mod. Phys. A 25 (2010) 2585 [arXiv:1002.3458] [INSPIRE].
  56. [56]
    N. Nakanishi, Graph theory and Feynman integrals, Gordon and Breach, Philadelphia U.S.A. (1971).Google Scholar
  57. [57]
    H. Cheng and T.T. Wu, Expanding Protons: Scattering at High Energies, MIT Press, Cambridge U.S.A. (1987).Google Scholar
  58. [58]
    G. ’t Hooft and M.J.G. Veltman, Scalar One Loop Integrals, Nucl. Phys. B 153 (1979) 365 [INSPIRE].
  59. [59]
    C. Duhr, Mathematical aspects of scattering amplitudes, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: Journeys Through the Precision Frontier: Amplitudes for Colliders (TASI 2014), Boulder U.S.A. (2014), pg. 419 [arXiv:1411.7538] [INSPIRE].
  60. [60]
    E. Panzer, On hyperlogarithms and Feynman integrals with divergences and many scales, JHEP 03 (2014) 071 [arXiv:1401.4361] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    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
  62. [62]
    C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    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].

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© The Author(s) 2019

Authors and Affiliations

  1. 1.Hamilton Mathematics InstituteTrinity CollegeDublin 2Ireland
  2. 2.School of MathematicsTrinity CollegeDublin 2Ireland
  3. 3.ETH Zürich, Institut fur theoretische PhysikZürichSwitzerland

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