Evaluating single-scale and/or non-planar diagrams by differential equations

  • Johannes M. Henn
  • Alexander V. Smirnov
  • Vladimir A. Smirnov
Open Access


We apply a recently suggested new strategy to solve differential equations for Feynman integrals. We develop this method further by analyzing asymptotic expansions of the integrals. We argue that this allows the systematic application of the differential equations to single-scale Feynman integrals. Moreover, the information about singular limits significantly simplifies finding boundary constants for the differential equations. To illustrate these points we consider two families of three-loop integrals. The first are form-factor integrals with two external legs on the light cone. We introduce one more scale by taking one more leg off-shell, \( p_2^2 \) ≠ 0. We analytically solve the differential equations for the master integrals in a Laurent expansion in dimensional regularization with ϵ = (4 − D)/2. Then we show how to obtain analytic results for the corresponding one-scale integrals in an algebraic way. An essential ingredient of our method is to match solutions of the differential equations in the limit of small \( p_2^2 \) to our results at \( p_2^2 \) ≠ 0 and to identify various terms in these solutions according to expansion by regions. The second family consists of four-point non-planar integrals with all four legs on the light cone. We evaluate, by differential equations, all the master integrals for the so-called K 4 graph consisting of four external vertices which are connected with each other by six lines. We show how the boundary constants can be fixed with the help of the knowledge of the singular limits. We present results in terms of harmonic polylogarithms for the corresponding seven master integrals with six propagators in a Laurent expansion in ϵ up to weight six.


Integrable Equations in Physics Scattering Amplitudes Differential and Algebraic Geometry 


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]
    A.V. Kotikov, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  2. [2]
    A.V. Kotikov, Differential equation method: the Calculation of N point Feynman diagrams, Phys. Lett. B 267 (1991) 123 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  3. [3]
    E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].ADSGoogle Scholar
  4. [4]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  5. [5]
    T. Gehrmann and E. Remiddi, Two loop master integrals for γ → 3 jets: the Planar topologies, Nucl. Phys. B 601 (2001) 248 [hep-ph/0008287] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    T. Gehrmann and E. Remiddi, Two loop master integrals for γ → 3 jets: the Nonplanar topologies, Nucl. Phys. B 601 (2001) 287 [hep-ph/0101124] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    K.-T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1997) 831.CrossRefGoogle Scholar
  9. [9]
    A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    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].zbMATHGoogle Scholar
  11. [11]
    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].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    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
  13. [13]
    T. Gehrmann, G. Heinrich, T. Huber and C. Studerus, Master integrals for massless three-loop form-factors: one-loop and two-loop insertions, Phys. Lett. B 640 (2006) 252 [hep-ph/0607185] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    G. Heinrich, T. Huber and D. Maître, Master integrals for fermionic contributions to massless three-loop form-factors, Phys. Lett. B 662 (2008) 344 [arXiv:0711.3590] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    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
  16. [16]
    G. Heinrich, T. Huber, D.A. Kosower and V.A. Smirnov, Nine-Propagator Master Integrals for Massless Three-Loop Form Factors, Phys. Lett. B 678 (2009) 359 [arXiv:0902.3512] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    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].ADSCrossRefGoogle Scholar
  18. [18]
    R.N. Lee and V.A. Smirnov, Analytic ϵ-expansions of Master Integrals Corresponding to Massless Three-Loop Form Factors and Three-Loop g-2 up to Four-Loop Transcendentality Weight, JHEP 02 (2011) 102 [arXiv:1010.1334] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    V.G. Knizhnik and A.B. Zamolodchikov, Current Algebra and Wess-Zumino Model in Two-Dimensions, Nucl. Phys. B 247 (1984) 83 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    V.A. Smirnov and E.R. Rakhmetov, The Strategy of regions for asymptotic expansion of two loop vertex Feynman diagrams, Theor. Math. Phys. 120 (1999) 870 [hep-ph/9812529] [INSPIRE].CrossRefzbMATHGoogle Scholar
  22. [22]
    V.A. Smirnov, Problems of the strategy of regions, Phys. Lett. B 465 (1999) 226 [hep-ph/9907471] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    V.A. Smirnov, Applied asymptotic expansions in momenta and masses, Springer Tracts Mod. Phys. 177 (2002) 1.ADSCrossRefGoogle Scholar
  24. [24]
    T. Schutzmeier, Matrix elements for the \( \overline{B} \)X sγ decay at NNLO, Ph.D. Thesis, Universität Würzburg (2009).Google Scholar
  25. [25]
    Z. Bern, J.J. Carrasco, L.J. Dixon, H. Johansson, D.A. Kosower and R. Roiban, Three-Loop Superfiniteness of N = 8 Supergravity, Phys. Rev. Lett. 98 (2007) 161303 [hep-th/0702112] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    C. Bogner and M. Luders, Multiple polylogarithms and linearly reducible Feynman graphs, arXiv:1302.6215 [INSPIRE].
  27. [27]
    F. Brown, The Massless higher-loop two-point function, Commun. Math. Phys. 287 (2009) 925 [arXiv:0804.1660] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  28. [28]
    E. Panzer, On the analytic computation of massless propagators in dimensional regularization, arXiv:1305.2161 [INSPIRE].
  29. [29]
    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].ADSCrossRefMathSciNetGoogle Scholar
  30. [30]
    R.N. Lee, A.V. Smirnov and V.A. Smirnov, Master Integrals for Four-Loop Massless Propagators up to Transcendentality Weight Twelve, Nucl. Phys. B 856 (2012) 95 [arXiv:1108.0732] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  31. [31]
    V.A. Smirnov, Analytic tools for Feynman integrals, Springer Tracts Mod. Phys. 250 (2012) 1.CrossRefGoogle Scholar
  32. [32]
    A.V. Smirnov, Algorithm FIREFeynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    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].ADSCrossRefGoogle Scholar
  34. [34]
    E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  35. [35]
    A. Pak and A. Smirnov, Geometric approach to asymptotic expansion of Feynman integrals, Eur. Phys. J. C 71 (2011) 1626 [arXiv:1011.4863] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    B. Jantzen, A.V. Smirnov and V.A. Smirnov, Expansion by regions: revealing potential and Glauber regions automatically, Eur. Phys. J. C 72 (2012) 2139 [arXiv:1206.0546] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    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
  38. [38]
    R.N. Lee and A.A. Pomeransky, Critical points and number of master integrals, JHEP 11 (2013) 165 [arXiv:1308.6676] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  39. [39]
    A.V. Smirnov and A.V. Petukhov, The Number of Master Integrals is Finite, Lett. Math. Phys. 97 (2011) 37 [arXiv:1004.4199] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  40. [40]
    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
  41. [41]
    J.B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  42. [42]
    A.V. Smirnov, FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions, arXiv:1312.3186 [INSPIRE].
  43. [43]
    J.A.M. Vermaseren, Harmonic sums, Mellin transforms and integrals, Int. J. Mod. Phys. A 14 (1999) 2037 [hep-ph/9806280] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  44. [44]
    S. Moch and P. Uwer, XSummer: transcendental functions and symbolic summation in form, Comput. Phys. Commun. 174 (2006) 759 [math-ph/0508008] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  45. [45]
    M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  46. [46]
    A.V. Smirnov and V.A. Smirnov, On the Resolution of Singularities of Multiple Mellin-Barnes Integrals, Eur. Phys. J. C 62 (2009) 445 [arXiv:0901.0386] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    M. Czakon, MBasymptotics.m,
  48. [48]
    D. Kosower, barnesroutines.m,
  49. [49]
    D. Maître, HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    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
  51. [51]
    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].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  52. [52]
    F. Brown and O. Schnetz, Proof of the zig-zag conjecture, arXiv:1208.1890 [INSPIRE].
  53. [53]
    J.M. Drummond, J. Henn, V.A. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  54. [54]
    J.M. Drummond, J.M. Henn and J. Trnka, New differential equations for on-shell loop integrals, JHEP 04 (2011) 083 [arXiv:1010.3679] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  55. [55]
    D.J. Broadhurst and D. Kreimer, Knots and numbers in ϕ 4 theory to 7 loops and beyond, Int. J. Mod. Phys. C 6 (1995) 519 [hep-ph/9504352] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Johannes M. Henn
    • 1
  • Alexander V. Smirnov
    • 2
  • Vladimir A. Smirnov
    • 3
    • 4
  1. 1.Institute for Advanced StudyPrincetonU.S.A.
  2. 2.Scientific Research Computing CenterMoscow State UniversityMoscowRussia
  3. 3.Skobeltsyn Institute of Nuclear Physics of Moscow State UniversityMoscowRussia
  4. 4.Institut für Theoretische Teilchenphysik, KITKarlsruheGermany

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