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Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops

Abstract

We obtain generalised power series expansions for a family of planar two-loop master integrals relevant for the QCD corrections to Higgs + jet production, with phys- ical heavy-quark mass. This is achieved by defining differential equations along contours connecting two fixed points, and by solving them in terms of one-dimensional generalised power series. The procedure is efficient, and can be repeated in order to reach any point of the kinematic regions. The analytic continuation of the series is straightforward, and we present new results below and above the physical thresholds. The method we use allows to compute the integrals in all kinematic regions with high precision. For example, per- forming a series expansion on a typical contour above the heavy-quark threshold takes on average O(1 second) per integral with worst relative error of O(1032), on a single CPU core. After the series is found, the numerical evaluation of the integrals in any point of the contour is virtually instant. Our approach is general, and can be applied to Feynman integrals provided that a set of differential equations is available.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    A.V. Kotikov, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett.B 254 (1991) 158 [INSPIRE].

  2. [2]

    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].

  3. [3]

    Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys.B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].

  4. [4]

    E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim.A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].

  5. [5]

    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys.B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].

  6. [6]

    F.C.S. Brown, On the periods of some Feynman integrals, arXiv:0910.0114 [INSPIRE].

  7. [7]

    E. Panzer, Feynman integrals and hyperlogarithms, Ph.D. thesis, Humboldt University, Berlin, Germany (2015), arXiv:1506.07243 [INSPIRE].

  8. [8]

    F.V. Tkachov, A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett.100B (1981) 65 [INSPIRE].

  9. [9]

    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].

  10. [10]

    S. Laporta and E. Remiddi, The analytical value of the electron (g − 2) at order α3in QED, Phys. Lett.B 379 (1996) 283 [hep-ph/9602417] [INSPIRE].

  11. [11]

    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys.A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].

  12. [12]

    K.T. Chen, Iterated path integrals, Bull. Amer. Math. Soc.83 (1977) 831.

  13. [13]

    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett.110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].

  14. [14]

    A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett.5 (1998) 497 [arXiv:1105.2076] [INSPIRE].

  15. [15]

    F. Brown and A. Levin, Multiple elliptic polylogarithms, arXiv:1110.6917.

  16. [16]

    J. Broedel, C.R. Mafra, N. Matthes and O. Schlotterer, Elliptic multiple zeta values and one-loop superstring amplitudes, JHEP07 (2015) 112 [arXiv:1412.5535] [INSPIRE].

  17. [17]

    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].

  18. [18]

    J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism, JHEP05 (2018) 093 [arXiv:1712.07089] [INSPIRE].

  19. [19]

    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].

  20. [20]

    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].

  21. [21]

    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph with arbitrary masses, J. Math. Phys.54 (2013) 052303 [arXiv:1302.7004] [INSPIRE].

  22. [22]

    S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theor.148 (2015) 328 [arXiv:1309.5865] [INSPIRE].

  23. [23]

    S. Bloch, M. Kerr and P. Vanhove, A Feynman integral via higher normal functions, Compos. Math.151 (2015) 2329 [arXiv:1406.2664] [INSPIRE].

  24. [24]

    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].

  25. [25]

    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].

  26. [26]

    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].

  27. [27]

    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].

  28. [28]

    R. Bonciani et al., Two-loop planar master integrals for Higgs → 3 partons with full heavy-quark mass dependence, JHEP12 (2016) 096 [arXiv:1609.06685] [INSPIRE].

  29. [29]

    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].

  30. [30]

    G. Passarino, Elliptic polylogarithms and basic hypergeometric functions, Eur. Phys. J.C 77 (2017) 77 [arXiv:1610.06207] [INSPIRE].

  31. [31]

    M. Harley, F. Moriello and R.M. Schabinger, Baikov-Lee representations of cut Feynman integrals, JHEP06 (2017) 049 [arXiv:1705.03478] [INSPIRE].

  32. [32]

    A. von Manteuffel and L. Tancredi, A non-planar two-loop three-point function beyond multiple polylogarithms, JHEP06 (2017) 127 [arXiv:1701.05905] [INSPIRE].

  33. [33]

    J. Ablinger et al., Iterated elliptic and hypergeometric integrals for Feynman diagrams, J. Math. Phys.59 (2018) 062305 [arXiv:1706.01299] [INSPIRE].

  34. [34]

    L.-B. Chen, Y. Liang and C.-F. Qiao, NNLO QCD corrections to γ + ηc (ηb ) exclusive production in electron-positron collision, JHEP01 (2018) 091 [arXiv:1710.07865] [INSPIRE].

  35. [35]

    M. Hidding and F. Moriello, All orders structure and efficient computation of linearly reducible elliptic Feynman integrals, JHEP01 (2019) 169 [arXiv:1712.04441] [INSPIRE].

  36. [36]

    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].

  37. [37]

    J.L. Bourjaily et al., Elliptic double-box integrals: massless scattering amplitudes beyond polylogarithms, Phys. Rev. Lett.120 (2018) 121603 [arXiv:1712.02785] [INSPIRE].

  38. [38]

    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].

  39. [39]

    S. Laporta, High-precision calculation of the 4-loop contribution to the electron g − 2 in QED, Phys. Lett.B 772 (2017) 232 [arXiv:1704.06996] [INSPIRE].

  40. [40]

    J. Broedel et al., Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series, JHEP08 (2018) 014 [arXiv:1803.10256] [INSPIRE].

  41. [41]

    B. Mistlberger, Higgs boson production at hadron colliders at N3LO in QCD, JHEP05 (2018) 028 [arXiv:1802.00833] [INSPIRE].

  42. [42]

    R.N. Lee, Symmetric - and (+ 1/2)-forms and quadratic constraints in “elliptic” sectors, JHEP10 (2018) 176 [arXiv:1806.04846] [INSPIRE].

  43. [43]

    J. Broedel et al., Elliptic Feynman integrals and pure functions, JHEP01 (2019) 023 [arXiv:1809.10698] [INSPIRE].

  44. [44]

    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].

  45. [45]

    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, JHEP10 (2018) 206 [arXiv:1806.04981] [INSPIRE].

  46. [46]

    J. Broedel et al., Elliptic polylogarithms and Feynman parameter integrals, JHEP05 (2019) 120 [arXiv:1902.09971] [INSPIRE].

  47. [47]

    C. Bogner, S. Müller-Stach and S. Weinzierl, The unequal mass sunrise integral expressed through iterated integrals on \( {\overline{\mathcal{M}}}_{1,3}, \) arXiv:1907.01251 [INSPIRE].

  48. [48]

    B.A. Kniehl, A.V. Kotikov, A.I. Onishchenko and O.L. Veretin, Two-loop diagrams in non-relativistic QCD with elliptics, Nucl. Phys.B 948 (2019) 114780 [arXiv:1907.04638] [INSPIRE].

  49. [49]

    J. Broedel et al., An analytic solution for the equal-mass banana graph, JHEP09 (2019) 112 [arXiv:1907.03787] [INSPIRE].

  50. [50]

    S. Borowka et al., SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop, Comput. Phys. Commun.196 (2015) 470 [arXiv:1502.06595] [INSPIRE].

  51. [51]

    A.V. Smirnov, FIESTA4: optimized Feynman integral calculations with GPU support, Comput. Phys. Commun.204 (2016) 189 [arXiv:1511.03614] [INSPIRE].

  52. [52]

    R. Boughezal, M. Czakon and T. Schutzmeier, NNLO fermionic corrections to the charm quark mass dependent matrix elements in \( \overline{B}\to {X}_s\upgamma, \)JHEP09 (2007) 072 [arXiv:0707.3090] [INSPIRE].

  53. [53]

    M. Czakon and T. Schutzmeier, Double fermionic contributions to the heavy-quark vacuum polarization, JHEP07 (2008) 001 [arXiv:0712.2762] [INSPIRE].

  54. [54]

    M. Czakon, Tops from light quarks: full mass dependence at two-loops in QCD, Phys. Lett.B 664 (2008) 307 [arXiv:0803.1400] [INSPIRE].

  55. [55]

    M.K. Mandal and X. Zhao, Evaluating multi-loop Feynman integrals numerically through differential equations, JHEP03 (2019) 190 [arXiv:1812.03060] [INSPIRE].

  56. [56]

    S. Pozzorini and E. Remiddi, Precise numerical evaluation of the two loop sunrise graph master integrals in the equal mass case, Comput. Phys. Commun.175 (2006) 381 [hep-ph/0505041] [INSPIRE].

  57. [57]

    U. Aglietti, R. Bonciani, L. Grassi and E. Remiddi, The Two loop crossed ladder vertex diagram with two massive exchanges, Nucl. Phys.B 789 (2008) 45 [arXiv:0705.2616] [INSPIRE].

  58. [58]

    R. Mueller and D.G. Öztürk, On the computation of finite bottom-quark mass effects in Higgs boson production, JHEP08 (2016) 055 [arXiv:1512.08570] [INSPIRE].

  59. [59]

    R.N. Lee, A.V. Smirnov and V.A. Smirnov, Solving differential equations for Feynman integrals by expansions near singular points, JHEP03 (2018) 008 [arXiv:1709.07525] [INSPIRE].

  60. [60]

    R.N. Lee, A.V. Smirnov and V.A. Smirnov, Evaluating ‘elliptic’ master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points, JHEP07 (2018) 102 [arXiv:1805.00227] [INSPIRE].

  61. [61]

    R. Bonciani, G. Degrassi, P.P. Giardino and R. Gröber, A numerical routine for the crossed vertex diagram with a massive-particle loop, Comput. Phys. Commun.241 (2019) 122 [arXiv:1812.02698] [INSPIRE].

  62. [62]

    M. Caffo, H. Czyz and E. Remiddi, Numerical evaluation of the general massive 2 loop sunrise selfmass master integrals from differential equations, Nucl. Phys.B 634 (2002) 309 [hep-ph/0203256] [INSPIRE].

  63. [63]

    M. Caffo, H. Czyz, M. Gunia and E. Remiddi, BOKASUN: a fast and precise numerical program to calculate the master integrals of the two-loop sunrise diagrams, Comput. Phys. Commun.180 (2009) 427 [arXiv:0807.1959] [INSPIRE].

  64. [64]

    K. Melnikov, L. Tancredi and C. Wever, Two-loop gg → H g amplitude mediated by a nearly massless quark, JHEP11 (2016) 104 [arXiv:1610.03747] [INSPIRE].

  65. [65]

    K. Melnikov, L. Tancredi and C. Wever, Two-loop amplitudes for qg → H q and qq̄ → H g mediated by a nearly massless quark, Phys. Rev.D 95 (2017) 054012 [arXiv:1702.00426] [INSPIRE].

  66. [66]

    R. Bonciani, G. Degrassi, P.P. Giardino and R. Gröber, Analytical method for next-to-leading-order QCD corrections to double-Higgs production, Phys. Rev. Lett.121 (2018) 162003 [arXiv:1806.11564] [INSPIRE].

  67. [67]

    R. Brüser, S. Caron-Huot and J.M. Henn, Subleading Regge limit from a soft anomalous dimension, JHEP04 (2018) 047 [arXiv:1802.02524] [INSPIRE].

  68. [68]

    J. Davies, G. Mishima, M. Steinhauser and D. Wellmann, Double-Higgs boson production in the high-energy limit: planar master integrals, JHEP03 (2018) 048 [arXiv:1801.09696] [INSPIRE].

  69. [69]

    J. Davies, G. Mishima, M. Steinhauser and D. Wellmann, Double Higgs boson production at NLO in the high-energy limit: complete analytic results, JHEP01 (2019) 176 [arXiv:1811.05489] [INSPIRE].

  70. [70]

    M. Heller, A. von Manteuffel and R.M. Schabinger, Multiple polylogarithms with algebraic arguments and the two-loop EW-QCD Drell-Yan master integrals, arXiv:1907.00491 [INSPIRE].

  71. [71]

    R. Bonciani et al., Evaluating two-loop non-planar master integrals for Higgs + jet production with full heavy-quark mass dependence, arXiv:1907.13156.

  72. [72]

    K.-T. Chen, Iterated path integrals, Bull. Am. Math. Soc.83 (1977) 831 [INSPIRE].

  73. [73]

    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].

  74. [74]

    C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP10 (2012) 075 [arXiv:1110.0458] [INSPIRE].

  75. [75]

    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].

  76. [76]

    J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys.A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].

  77. [77]

    V.A. Smirnov, Asymptotic expansions in limits of large momenta and masses, Commun. Math. Phys.134 (1990) 109 [INSPIRE].

  78. [78]

    W. Wasow, Asymptotic expansions for ordinary differential equations, Dover Publications, U.S.A. (2002).

  79. [79]

    E. Coddington, Theory of ordinary differential equations, McGraw-Hill Book Company Inc., U.S.A. (1955).

  80. [80]

    T. Hahn, CUBA: A Library for multidimensional numerical integration, Comput. Phys. Commun.168 (2005) 78 [hep-ph/0404043] [INSPIRE].

  81. [81]

    S. Borowka et al., pySecDec: a toolbox for the numerical evaluation of multi-scale integrals, Comput. Phys. Commun.222 (2018) 313 [arXiv:1703.09692] [INSPIRE].

  82. [82]

    S. Borowka et al., A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec, Comput. Phys. Commun.240 (2019) 120 [arXiv:1811.11720] [INSPIRE].

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Moriello, F. Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops. J. High Energ. Phys. 2020, 150 (2020). https://doi.org/10.1007/JHEP01(2020)150

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Keywords

  • Higgs Physics
  • Perturbative QCD