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Removing infrared divergences from two-loop integrals

  • Charalampos AnastasiouEmail author
  • George Sterman
Open Access
Regular Article - Theoretical Physics

Abstract

Feynman amplitudes at higher orders in perturbation theory generically have complex singular structures. Notwithstanding the emergence of many powerful new methods, the presence of infrared divergences poses significant challenges for their evaluation. In this article, we develop a systematic method for the removal of the infrared singularities, by adding appropriate counterterms that approximate and cancel divergent limits point-by-point at the level of the integrand. We provide a proof of concept for our method by applying it to master-integrals that are found in scattering amplitudes for representative 2→2 scattering processes of massless particles. We demonstrate that, after the introduction of counterterms, the remainder is finite in four dimensions. In addition, we find in these cases that the complete singular dependence of the integrals can be obtained simply by analytically integrating the counterterms. Finally, we observe that our subtraction method can be also useful in order to extract in a simple way the asymptotic behavior of Feynman amplitudes in the limit of small mass parameters.

Keywords

NLO Computations QCD Phenomenology 

Notes

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.

References

  1. [1]
    W.T. Giele, E.W.N. Glover and D.A. Kosower, Higher order corrections to jet cross-sections in hadron colliders, Nucl. Phys. B 403 (1993) 633 [hep-ph/9302225] [INSPIRE].
  2. [2]
    S. Frixione, Z. Kunszt and A. Signer, Three jet cross-sections to next-to-leading order, Nucl. Phys. B 467 (1996) 399 [hep-ph/9512328] [INSPIRE].
  3. [3]
    S. Catani and M.H. Seymour, A General algorithm for calculating jet cross-sections in NLO QCD, Nucl. Phys. B 485 (1997) 291 [Erratum ibid. B 510 (1998) 503] [hep-ph/9605323] [INSPIRE].
  4. [4]
    C. Anastasiou, K. Melnikov and F. Petriello, A new method for real radiation at NNLO, Phys. Rev. D 69 (2004) 076010 [hep-ph/0311311] [INSPIRE].
  5. [5]
    A. Gehrmann-De Ridder, T. Gehrmann and E.W.N. Glover, Antenna subtraction at NNLO, JHEP 09 (2005) 056 [hep-ph/0505111] [INSPIRE].
  6. [6]
    A. Daleo, T. Gehrmann and D. Maître, Antenna subtraction with hadronic initial states, JHEP 04 (2007) 016 [hep-ph/0612257] [INSPIRE].
  7. [7]
    G. Somogyi, Z. Trócsányi and V. Del Duca, A Subtraction scheme for computing QCD jet cross sections at NNLO: Regularization of doubly-real emissions, JHEP 01 (2007) 070 [hep-ph/0609042] [INSPIRE].
  8. [8]
    V. Del Duca et al., Jet production in the CoLoRFulNNLO method: event shapes in electron-positron collisions, Phys. Rev. D 94 (2016) 074019 [arXiv:1606.03453] [INSPIRE].
  9. [9]
    S. Catani and M. Grazzini, An NNLO subtraction formalism in hadron collisions and its application to Higgs boson production at the LHC, Phys. Rev. Lett. 98 (2007) 222002 [hep-ph/0703012] [INSPIRE].
  10. [10]
    M. Czakon, A novel subtraction scheme for double-real radiation at NNLO, Phys. Lett. B 693 (2010) 259 [arXiv:1005.0274] [INSPIRE].
  11. [11]
    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 in perturbative QCD, JHEP 06 (2013) 072 [arXiv:1302.6216] [INSPIRE].
  12. [12]
    R. Boughezal, C. Focke, X. Liu and F. Petriello, W -boson production in association with a jet at next-to-next-to-leading order in perturbative QCD, Phys. Rev. Lett. 115 (2015) 062002 [arXiv:1504.02131] [INSPIRE].
  13. [13]
    J. Currie, E.W.N. Glover and S. Wells, Infrared Structure at NNLO Using Antenna Subtraction, JHEP 04 (2013) 066 [arXiv:1301.4693] [INSPIRE].
  14. [14]
    J. Gaunt, M. Stahlhofen, F.J. Tackmann and J.R. Walsh, N-jettiness Subtractions for NNLO QCD Calculations, JHEP 09 (2015) 058 [arXiv:1505.04794] [INSPIRE].
  15. [15]
    M. Cacciari, F.A. Dreyer, A. Karlberg, G.P. Salam and G. Zanderighi, Fully Differential Vector-Boson-Fusion Higgs Production at Next-to-Next-to-Leading Order, Phys. Rev. Lett. 115 (2015) 082002 [Erratum ibid. 120 (2018) 139901] [arXiv:1506.02660] [INSPIRE].
  16. [16]
    F. Caola, K. Melnikov and R. Röntsch, Nested soft-collinear subtractions in NNLO QCD computations, Eur. Phys. J. C 77 (2017) 248 [arXiv:1702.01352] [INSPIRE].
  17. [17]
    J. Currie, A. Gehrmann-De Ridder, T. Gehrmann, N. Glover, A. Huss and J. Pires, Jet cross sections at the LHC with NNLOJET, PoS(LL2018)001 (2018) [arXiv:1807.06057] [INSPIRE].
  18. [18]
    F. Herzog, Geometric IR subtraction for final state real radiation, JHEP 08 (2018) 006 [arXiv:1804.07949] [INSPIRE].
  19. [19]
    M. Grazzini, S. Kallweit and M. Wiesemann, Fully differential NNLO computations with MATRIX, Eur. Phys. J. C 78 (2018) 537 [arXiv:1711.06631] [INSPIRE].
  20. [20]
    R. Boughezal, A. Isgrò and F. Petriello, Next-to-leading-logarithmic power corrections for N -jettiness subtraction in color-singlet production, Phys. Rev. D 97 (2018) 076006 [arXiv:1802.00456] [INSPIRE].
  21. [21]
    A. Behring, M. Czakon and R. Poncelet, Sector-improved residue subtraction: Improvements and Applications, PoS(LL2018)024 (2018) [arXiv:1808.07656] [INSPIRE].
  22. [22]
    M.L. Czakon and A. Mitov, A simplified expression for the one-loop soft-gluon current with massive fermions, arXiv:1804.02069 [INSPIRE].
  23. [23]
    L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati, Factorisation and Subtraction beyond NLO, JHEP 12 (2018) 062 [arXiv:1809.05444] [INSPIRE].
  24. [24]
    L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati, Local analytic sector subtraction at NNLO, JHEP 12 (2018) 107 [Erratum ibid. 06 (2019) 013] [arXiv:1806.09570] [INSPIRE].
  25. [25]
    B. Mistlberger, Higgs boson production at hadron colliders at N 3 LO in QCD, JHEP 05 (2018) 028 [arXiv:1802.00833] [INSPIRE].
  26. [26]
    F.A. Dreyer and A. Karlberg, Vector-Boson Fusion Higgs Production at Three Loops in QCD, Phys. Rev. Lett. 117 (2016) 072001 [arXiv:1606.00840] [INSPIRE].
  27. [27]
    F. Dulat, B. Mistlberger and A. Pelloni, Precision predictions at N 3 LO for the Higgs boson rapidity distribution at the LHC, Phys. Rev. D 99 (2019) 034004 [arXiv:1810.09462] [INSPIRE].
  28. [28]
    L. Cieri, X. Chen, T. Gehrmann, E.W.N. Glover and A. Huss, Higgs boson production at the LHC using the q T subtraction formalism at N 3 LO QCD, JHEP 02 (2019) 096 [arXiv:1807.11501] [INSPIRE].
  29. [29]
    J. Currie, T. Gehrmann, E.W.N. Glover, A. Huss, J. Niehues and A. Vogt, N 3 LO corrections to jet production in deep inelastic scattering using the Projection-to-Born method, JHEP 05 (2018) 209 [arXiv:1803.09973] [INSPIRE].
  30. [30]
    F.A. Dreyer and A. Karlberg, Vector-Boson Fusion Higgs Pair Production at N 3 LO, Phys. Rev. D 98 (2018) 114016 [arXiv:1811.07906] [INSPIRE].
  31. [31]
    B. Ruijl, F. Herzog, T. Ueda, J.A.M. Vermaseren and A. Vogt, The R -operation and five-loop calculations, PoS(RADCOR2017)011 (2018) [arXiv:1801.06084] [INSPIRE].
  32. [32]
    K.G. Chetyrkin, G. Falcioni, F. Herzog and J.A.M. Vermaseren, The method of global R and its applications, PoS(RADCOR2017)004 (2018) [arXiv:1801.03024] [INSPIRE].
  33. [33]
    Z. Nagy and D.E. Soper, General subtraction method for numerical calculation of one loop QCD matrix elements, JHEP 09 (2003) 055 [hep-ph/0308127] [INSPIRE].
  34. [34]
    Z. Nagy and D.E. Soper, Numerical integration of one-loop Feynman diagrams for N-photon amplitudes, Phys. Rev. D 74 (2006) 093006 [hep-ph/0610028] [INSPIRE].
  35. [35]
    W. Gong, Z. Nagy and D.E. Soper, Direct numerical integration of one-loop Feynman diagrams for N-photon amplitudes, Phys. Rev. D 79 (2009) 033005 [arXiv:0812.3686] [INSPIRE].
  36. [36]
    S. Becker, C. Reuschle and S. Weinzierl, Numerical NLO QCD calculations, JHEP 12 (2010) 013 [arXiv:1010.4187] [INSPIRE].
  37. [37]
    S. Becker, D. Goetz, C. Reuschle, C. Schwan and S. Weinzierl, Numerical evaluation of NLO multiparton processes, arXiv:1209.2846 [INSPIRE].
  38. [38]
    S. Becker, C. Reuschle and S. Weinzierl, Efficiency Improvements for the Numerical Computation of NLO Corrections, JHEP 07 (2012) 090 [arXiv:1205.2096] [INSPIRE].
  39. [39]
    T. Binoth and G. Heinrich, An automatized algorithm to compute infrared divergent multiloop integrals, Nucl. Phys. B 585 (2000) 741 [hep-ph/0004013] [INSPIRE].
  40. [40]
    C. Anastasiou, S. Beerli and A. Daleo, Evaluating multi-loop Feynman diagrams with infrared and threshold singularities numerically, JHEP 05 (2007) 071 [hep-ph/0703282] [INSPIRE].
  41. [41]
    A. Lazopoulos, T. McElmurry, K. Melnikov and F. Petriello, Next-to-leading order QCD corrections to ttZ production at the LHC, Phys. Lett. B 666 (2008) 62 [arXiv:0804.2220] [INSPIRE].
  42. [42]
    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].
  43. [43]
    V.A. Smirnov, Analytical result for dimensionally regularized massless on shell double box, Phys. Lett. B 460 (1999) 397 [hep-ph/9905323] [INSPIRE].
  44. [44]
    J.B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [INSPIRE].
  45. [45]
    C. Anastasiou and A. Daleo, Numerical evaluation of loop integrals, JHEP 10 (2006) 031 [hep-ph/0511176] [INSPIRE].
  46. [46]
    M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [INSPIRE].
  47. [47]
    E. Panzer, On hyperlogarithms and Feynman integrals with divergences and many scales, JHEP 03 (2014) 071 [arXiv:1401.4361] [INSPIRE].
  48. [48]
    A. von Manteuffel, E. Panzer and R.M. Schabinger, A quasi-finite basis for multi-loop Feynman integrals, JHEP 02 (2015) 120 [arXiv:1411.7392] [INSPIRE].
  49. [49]
    J. Collins, Foundations of perturbative QCD, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 32 (2011) 1 [INSPIRE].
  50. [50]
    S.B. Libby and G.F. Sterman, Jet and Lepton Pair Production in High-Energy Lepton-Hadron and Hadron-Hadron Scattering, Phys. Rev. D 18 (1978) 3252 [INSPIRE].
  51. [51]
    C.W. Bauer, D. Pirjol and I.W. Stewart, Power counting in the soft collinear effective theory, Phys. Rev. D 66 (2002) 054005 [hep-ph/0205289] [INSPIRE].
  52. [52]
    O. Erdoğan and G. Sterman, Ultraviolet divergences and factorization for coordinate-space amplitudes, Phys. Rev. D 91 (2015) 065033 [arXiv:1411.4588] [INSPIRE].
  53. [53]
    S. Coleman and R.E. Norton, Singularities in the physical region, Nuovo Cim. 38 (1965) 438 [INSPIRE].
  54. [54]
    L.D. Landau, On analytic properties of vertex parts in quantum field theory, Nucl. Phys. 13 (1959) 181 [INSPIRE].
  55. [55]
    G.F. Sterman, Mass Divergences in Annihilation Processes. 1. Origin and Nature of Divergences in Cut Vacuum Polarization Diagrams, Phys. Rev. D 17 (1978) 2773 [INSPIRE].
  56. [56]
    A. Sen, Asymptotic Behavior of the Wide Angle On-Shell Quark Scattering Amplitudes in Nonabelian Gauge Theories, Phys. Rev. D 28 (1983) 860 [INSPIRE].
  57. [57]
    G.F. Sterman and M.E. Tejeda-Yeomans, Multiloop amplitudes and resummation, Phys. Lett. B 552 (2003) 48 [hep-ph/0210130] [INSPIRE].
  58. [58]
    G.F. Sterman, An Introduction to quantum field theory, Cambridge University Press (1993).Google Scholar
  59. [59]
    J.C. Collins and D.E. Soper, Back-To-Back Jets in QCD, Nucl. Phys. B 193 (1981) 381 [Erratum ibid. B 213 (1983) 545] [INSPIRE].
  60. [60]
    N. Kidonakis, G. Oderda and G.F. Sterman, Evolution of color exchange in QCD hard scattering, Nucl. Phys. B 531 (1998) 365 [hep-ph/9803241] [INSPIRE].
  61. [61]
    S. Becker and S. Weinzierl, Direct numerical integration for multi-loop integrals, Eur. Phys. J. C 73 (2013) 2321 [arXiv:1211.0509] [INSPIRE].
  62. [62]
    I. Bierenbaum, S. Catani, P. Draggiotis and G. Rodrigo, A Tree-Loop Duality Relation at Two Loops and Beyond, JHEP 10 (2010) 073 [arXiv:1007.0194] [INSPIRE].
  63. [63]
    S. Buchta, G. Chachamis, P. Draggiotis and G. Rodrigo, Numerical implementation of the loop-tree duality method, Eur. Phys. J. C 77 (2017) 274 [arXiv:1510.00187] [INSPIRE].
  64. [64]
    Z. Capatti, V. Hirschi, D. Kermanschah and B. Ruijl, Loop Tree Duality for multi-loop numerical integration, arXiv:1906.06138 [INSPIRE].
  65. [65]
    C. Anastasiou and A. Banfi, Loop lessons from Wilson loops in N = 4 supersymmetric Yang-Mills theory, JHEP 02 (2011) 064 [arXiv:1101.4118] [INSPIRE].
  66. [66]
    C. Anastasiou, C. Duhr, F. Dulat and B. Mistlberger, Soft triple-real radiation for Higgs production at N3LO, JHEP 07 (2013) 003 [arXiv:1302.4379] [INSPIRE].
  67. [67]
    V.A. Smirnov and O.L. Veretin, Analytical results for dimensionally regularized massless on-shell double boxes with arbitrary indices and numerators, Nucl. Phys. B 566 (2000) 469 [hep-ph/9907385] [INSPIRE].
  68. [68]
    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].
  69. [69]
    U. Baur and E.W.N. Glover, Higgs Boson Production at Large Transverse Momentum in Hadronic Collisions, Nucl. Phys. B 339 (1990) 38 [INSPIRE].
  70. [70]
    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].
  71. [71]
    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].
  72. [72]
    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].
  73. [73]
    M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].
  74. [74]
    V.A. Smirnov, Asymptotic expansions of two loop Feynman diagrams in the Sudakov limit, Phys. Lett. B 404 (1997) 101 [hep-ph/9703357] [INSPIRE].
  75. [75]
    V.A. Smirnov, Applied asymptotic expansions in momenta and masses, Springer Tracts Mod. Phys. 177 (2002) 1 [INSPIRE].
  76. [76]
    B. Jantzen, Foundation and generalization of the expansion by regions, JHEP 12 (2011) 076 [arXiv:1111.2589] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Institute for Theoretical Physics, ETH ZurichZürichSwitzerland
  2. 2.C.N. Yang Institute for Theoretical Physics and Department of Physics and AstronomyStony Brook UniversityStony BrookU.S.A.

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