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Systematic approximation of multi-scale Feynman integrals

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  • Published: 20 August 2018
  • Volume 2018, article number 111, (2018)
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Systematic approximation of multi-scale Feynman integrals
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  • Sophia Borowka  ORCID: orcid.org/0000-0003-4482-19621,
  • Thomas Gehrmann  ORCID: orcid.org/0000-0001-7009-432X2 &
  • Daniel Hulme  ORCID: orcid.org/0000-0001-7649-23612 

  • 359 Accesses

  • 8 Citations

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A preprint version of the article is available at arXiv.

Abstract

An algorithm for the systematic analytical approximation of multi-scale Feynman integrals is presented. The algorithm produces algebraic expressions as functions of the kinematical parameters and mass scales appearing in the Feynman integrals, allowing for fast numerical evaluation. The results are valid in all kinematical regions, both above and below thresholds, up to in principle arbitrary orders in the dimensional regulator. The scope of the algorithm is demonstrated by presenting results for selected two-loop threepoint and four-point integrals with an internal mass scale that appear in the two-loop amplitudes for Higgs+jet production.

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References

  1. S. Weinberg, The quantum theory of fields, volume 1, Cambridge University Press, Cambridge U.K. (2008), pg. 497.

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

    Article  ADS  MathSciNet  Google Scholar 

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

    ADS  Google Scholar 

  4. M. Caffo, H. Czyz, S. Laporta and E. Remiddi, Master equations for master amplitudes, Acta Phys. Polon. B 29 (1998) 2627 [hep-th/9807119] [INSPIRE].

    Google Scholar 

  5. 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].

    ADS  Google Scholar 

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

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

  8. R.N. Lee, A.V. Smirnov and V.A. Smirnov, Solving differential equations for Feynman integrals by expansions near singular points, JHEP 03 (2018) 008 [arXiv:1709.07525] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. X. Liu, Y.-Q. Ma and C.-Y. Wang, A Systematic and Efficient Method to Compute Multi-loop Master Integrals, Phys. Lett. B 779 (2018) 353 [arXiv:1711.09572] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  10. H. Poincaré, Sur les groupes des équations lineéaires, Acta Math. 4 (1883) 215.

    Google Scholar 

  11. E. Kummer, Über die Transzendenten, welche aus wiederholten Integrationen rationaler Formeln entstehen, J. Reine Angew. Math. 21 (1840) 74.

    Article  MathSciNet  Google Scholar 

  12. N. Nielsen, Der eulersche dilogarithmus und seine verallgemeinerungen, Nova Acta Leopoldina (Halle) 90 123 (1909).

  13. A.B. Goncharov, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math. 114 (1995) 197.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].

  16. J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].

  17. 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].

    Article  ADS  MathSciNet  Google Scholar 

  18. J. Ablinger, J. Blumlein and C. Schneider, Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials, J. Math. Phys. 52 (2011) 102301 [arXiv:1105.6063] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  20. 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].

  21. 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].

  22. R. Bonciani, P. Mastrolia and E. Remiddi, Master integrals for the two loop QCD virtual corrections to the forward backward asymmetry, Nucl. Phys. B 690 (2004) 138 [hep-ph/0311145] [INSPIRE].

  23. C. Anastasiou, S. Beerli, S. Bucherer, A. Daleo and Z. Kunszt, Two-loop amplitudes and master integrals for the production of a Higgs boson via a massive quark and a scalar-quark loop, JHEP 01 (2007) 082 [hep-ph/0611236] [INSPIRE].

  24. T. Gehrmann, L. Tancredi and E. Weihs, Two-loop master integrals for \( q\overline{q}\to VV \) : the planar topologies, JHEP 08 (2013) 070 [arXiv:1306.6344] [INSPIRE].

    Article  ADS  Google Scholar 

  25. 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].

    Article  ADS  Google Scholar 

  26. 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].

    Article  ADS  Google Scholar 

  27. T. Gehrmann, A. von Manteuffel, L. Tancredi and E. Weihs, The two-loop master integrals for \( q\overline{q}\to VV \), JHEP 06 (2014) 032 [arXiv:1404.4853] [INSPIRE].

    Article  ADS  Google Scholar 

  28. C.G. Papadopoulos, D. Tommasini and C. Wever, The Pentabox Master Integrals with the Simplified Differential Equations approach, JHEP 04 (2016) 078 [arXiv:1511.09404] [INSPIRE].

    ADS  Google Scholar 

  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 [arXiv:1511.05409] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  30. J.M. Henn, A.V. Smirnov and V.A. Smirnov, Analytic results for planar three-loop integrals for massive form factors, JHEP 12 (2016) 144 [arXiv:1611.06523] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. 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].

  32. M. Becchetti and R. Bonciani, Two-Loop Master Integrals for the Planar QCD Massive Corrections to Di-photon and Di-jet Hadro-production, JHEP 01 (2018) 048 [arXiv:1712.02537] [INSPIRE].

    Article  ADS  Google Scholar 

  33. F.C.S. Brown and A. Levin, Multiple Elliptic Polylogarithms, arXiv:1110.6917.

  34. 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].

  35. 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].

    ADS  Google Scholar 

  36. E. Remiddi and L. Tancredi, An Elliptic Generalization of Multiple Polylogarithms, Nucl. Phys. B 925 (2017) 212 [arXiv:1709.03622] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. 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].

    Article  ADS  MathSciNet  Google Scholar 

  38. 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].

    Article  ADS  Google Scholar 

  39. B. Mistlberger, Higgs boson production at hadron colliders at N 3 LO in QCD, JHEP 05 (2018) 028 [arXiv:1802.00833] [INSPIRE].

  40. K. Hepp, Proof of the bogoliubov-parasiuk theorem on renormalization, Commun. Math. Phys. 2 (1966) 301.

    Article  ADS  MATH  Google Scholar 

  41. M. Roth and A. Denner, High-energy approximation of one loop Feynman integrals, Nucl. Phys. B 479 (1996) 495 [hep-ph/9605420] [INSPIRE].

  42. T. Binoth and G. Heinrich, An automatized algorithm to compute infrared divergent multiloop integrals, Nucl. Phys. B 585 (2000) 741 [hep-ph/0004013] [INSPIRE].

  43. G. Heinrich, Sector Decomposition, Int. J. Mod. Phys. A 23 (2008) 1457 [arXiv:0803.4177] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. C. Bogner and S. Weinzierl, Resolution of singularities for multi-loop integrals, Comput. Phys. Commun. 178 (2008) 596 [arXiv:0709.4092] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. 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].

    Article  ADS  MATH  Google Scholar 

  46. 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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. A.V. Smirnov, FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions, Comput. Phys. Commun. 185 (2014) 2090 [arXiv:1312.3186] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  48. J. Gluza, K. Kajda, T. Riemann and V. Yundin, Numerical Evaluation of Tensor Feynman Integrals in Euclidean Kinematics, Eur. Phys. J. C 71 (2011) 1516 [arXiv:1010.1667] [INSPIRE].

    ADS  Google Scholar 

  49. J. Carter and G. Heinrich, SecDec: A general program for sector decomposition, Comput. Phys. Commun. 182 (2011) 1566 [arXiv:1011.5493] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  50. S. Borowka, J. Carter and G. Heinrich, Numerical Evaluation of Multi-Loop Integrals for Arbitrary Kinematics with SecDec 2.0, Comput. Phys. Commun. 184 (2013) 396 [arXiv:1204.4152] [INSPIRE].

  51. S. Borowka, G. Heinrich, S.P. Jones, M. Kerner, J. Schlenk and T. Zirke, SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop, Comput. Phys. Commun. 196 (2015) 470 [arXiv:1502.06595] [INSPIRE].

  52. 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].

  53. S. Borowka, T. Hahn, S. Heinemeyer, G. Heinrich and W. Hollik, Momentum-dependent two-loop QCD corrections to the neutral Higgs-boson masses in the MSSM, Eur. Phys. J. C 74 (2014) 2994 [arXiv:1404.7074] [INSPIRE].

  54. S. Borowka et al., Higgs Boson Pair Production in Gluon Fusion at Next-to-Leading Order with Full Top-Quark Mass Dependence, Phys. Rev. Lett. 117 (2016) 012001 [arXiv:1604.06447] [INSPIRE].

    Article  ADS  Google Scholar 

  55. S. Borowka et al., Full top quark mass dependence in Higgs boson pair production at NLO, JHEP 10 (2016) 107 [arXiv:1608.04798] [INSPIRE].

    Article  ADS  Google Scholar 

  56. S.P. Jones, M. Kerner and G. Luisoni, Next-to-Leading-Order QCD Corrections to Higgs Boson Plus Jet Production with Full Top-Quark Mass Dependence, Phys. Rev. Lett. 120 (2018)162001 [arXiv:1802.00349] [INSPIRE].

  57. S. Borowka, S. Paßehr and G. Weiglein, Complete two-loop QCD contributions to the lightest Higgs-boson mass in the MSSM with complex parameters, Eur. Phys. J. C 78 (2018) 576 [arXiv:1802.09886] [INSPIRE].

    Article  ADS  Google Scholar 

  58. E. Panzer, On hyperlogarithms and Feynman integrals with divergences and many scales, JHEP 03 (2014) 071 [arXiv:1401.4361] [INSPIRE].

    Article  ADS  Google Scholar 

  59. 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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. A. von Manteuffel and C. Studerus, Reduze 2 — Distributed Feynman Integral Reduction, arXiv:1201.4330 [INSPIRE].

  61. C.W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language, J. Symb. Comput. 33 (2000) 1 [cs/0004015] [INSPIRE].

  62. Mathematica, Copyright by Wolfram Research.

  63. J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].

  64. J. Kuipers, T. Ueda and J.A.M. Vermaseren, Code Optimization in FORM, Comput. Phys. Commun. 189 (2015) 1 [arXiv:1310.7007] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  65. F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65.

    Article  ADS  MathSciNet  Google Scholar 

  66. 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].

    Article  ADS  Google Scholar 

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

  68. T. Kaneko and T. Ueda, A Geometric method of sector decomposition, Comput. Phys. Commun. 181 (2010) 1352 [arXiv:0908.2897] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  69. T. Kaneko and T. Ueda, Sector Decomposition Via Computational Geometry, PoS(ACAT2010)082 [arXiv:1004.5490] [INSPIRE].

  70. H. Cheng and T. Wu, Expanding Protons: Scattering at High Energies, MIT Press, Cambridge U.S.A. (1987).

    Google Scholar 

  71. V.A. Smirnov, Feynman integral calculus, Springer, Berlin Germany (2006).

    Google Scholar 

  72. R. Bonciani, P. Mastrolia and E. Remiddi, Vertex diagrams for the QED form-factors at the two loop level, Nucl. Phys. B 661 (2003) 289 [Erratum ibid. B 702 (2004) 359] [hep-ph/0301170] [INSPIRE].

  73. T. Gehrmann, S. Guns and D. Kara, The rare decay H → Zγ in perturbative QCD, JHEP 09 (2015) 038 [arXiv:1505.00561] [INSPIRE].

    Article  Google Scholar 

  74. 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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  75. K. Melnikov, L. Tancredi and C. Wever, Two-loop amplitudes for qg → Hq and \( q\overline{q}\to Hg \) mediated by a nearly massless quark, Phys. Rev. D 95 (2017) 054012 [arXiv:1702.00426] [INSPIRE].

    ADS  Google Scholar 

  76. G.P. Lepage, A New Algorithm for Adaptive Multidimensional Integration, J. Comput. Phys. 27 (1978) 192 [INSPIRE].

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  1. Theoretical Physics Department, CERN, CH-1211, Geneva 23, Switzerland

    Sophia Borowka

  2. Physik-Institut, Universität Zürich, Winterthurerstr. 190, CH-8057, Zürich, Switzerland

    Thomas Gehrmann & Daniel Hulme

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Correspondence to Daniel Hulme.

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ArXiv ePrint: 1804.06824

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Borowka, S., Gehrmann, T. & Hulme, D. Systematic approximation of multi-scale Feynman integrals. J. High Energ. Phys. 2018, 111 (2018). https://doi.org/10.1007/JHEP08(2018)111

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  • Received: 24 April 2018

  • Revised: 09 July 2018

  • Accepted: 26 July 2018

  • Published: 20 August 2018

  • DOI: https://doi.org/10.1007/JHEP08(2018)111

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Keywords

  • Perturbative QCD
  • Scattering Amplitudes
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