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  • Letter
  • Open Access

Multiloop corrections for collider processes using auxiliary mass flow

Xiao Liu1,* and Yan-Qing Ma1,2,3,†

  • 1School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China
  • 2Center for High Energy Physics, Peking University, Beijing 100871, China
  • 3Collaborative Innovation Center of Quantum Matter, Beijing 100871, China

  • *xiao6@pku.edu.cn
  • †yqma@pku.edu.cn
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Phys. Rev. D 105, L051503 – Published 14 March, 2022

DOI: https://doi.org/10.1103/PhysRevD.105.L051503

Abstract

With a key improvement, the auxiliary mass flow method is now able to compute Feynman integrals encountered in cutting-edge collider processes. We have successfully applied it to compute some integrals involved in two-loop electroweak corrections to e+e−→HZ, two-loop QCD corrections to 3j, W/Z/H+2j, tt¯H and 4j production at hadron colliders, and three-loop QCD corrections to tt¯ production at hadron colliders, all of which are crucial for precision frontier in collider physics in the following decade. Our results are important building blocks and benchmarks for future studies of these processes.

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References (72)

  1. M. Cepeda et al., Report from working group 2: Higgs physics at the HL-LHC and HE-LHC, CERN Yellow Rep. Monogr. 7, 221 (2019).
  2. M. Dong et al. (CEPC Study Group), CEPC conceptual design report: Volume 2—physics & detector, arXiv:1811.10545.
  3. A. V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254, 158 (1991).
  4. A. V. Kotikov, Differential equation method: The calculation of N point Feynman diagrams, Phys. Lett. B 267, 123 (1991).
  5. E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cimento A 110, 1435 (1997).
  6. T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B580, 485 (2000).
  7. M. Argeri and P. Mastrolia, Feynman diagrams and differential equations, Int. J. Mod. Phys. A 22, 4375 (2007).
  8. S. Müller-Stach, S. Weinzierl, and R. Zayadeh, Picard-Fuchs equations for Feynman integrals, Commun. Math. Phys. 326, 237 (2014).
  9. J. M. Henn, Multiloop Integrals in Dimensional Regularization Made Simple, Phys. Rev. Lett. 110, 251601 (2013).
  10. J. M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48, 153001 (2015).
  11. F. Moriello, Generalised power series expansions for the elliptic planar families of Higgs+jet production at two loops, J. High Energy Phys. 01 (2020) 150.
  12. M. Hidding, DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions, Comput. Phys. Commun. 269, 108125 (2021).
  13. S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15, 5087 (2000).
  14. R. N. Lee, Space-time dimensionality D as complex variable: Calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D, Nucl. Phys. B830, 474 (2010).
  15. K. Hepp, Proof of the Bogolyubov-Parasiuk theorem on renormalization, Commun. Math. Phys. 2, 301 (1966).
  16. M. Roth and A. Denner, High-energy approximation of one loop Feynman integrals, Nucl. Phys. B479, 495 (1996).
  17. T. Binoth and G. Heinrich, An automatized algorithm to compute infrared divergent multiloop integrals, Nucl. Phys. B585, 741 (2000).
  18. G. Heinrich, Sector decomposition, Int. J. Mod. Phys. A 23, 1457 (2008).
  19. A. V. Smirnov, FIESTA4: Optimized Feynman integral calculations with GPU support, Comput. Phys. Commun. 204, 189 (2016).
  20. 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, 470 (2015).
  21. S. Borowka, G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, J. Schlenk, and T. Zirke, pySecDec: A toolbox for the numerical evaluation of multi-scale integrals, Comput. Phys. Commun. 222, 313 (2018).
  22. E. E. Boos and A. I. Davydychev, A method of evaluating massive Feynman integrals, Teor. Mat. Fiz. 89, 56 (1991) [Theor. Math. Phys. 89, 1052 (1991)].
  23. V. A. Smirnov, Analytical result for dimensionally regularized massless on shell double box, Phys. Lett. B 460, 397 (1999).
  24. J. B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469, 225 (1999).
  25. M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175, 559 (2006).
  26. A. V. Smirnov and V. A. Smirnov, On the resolution of singularities of multiple Mellin-Barnes integrals, Eur. Phys. J. C 62, 445 (2009).
  27. J. Gluza, K. Kajda, and T. Riemann, AMBRE: A mathematica package for the construction of Mellin-Barnes representations for Feynman integrals, Comput. Phys. Commun. 177, 879 (2007).
  28. X. Liu, Y.-Q. Ma, and C.-Y. Wang, A systematic and efficient method to compute multi-loop master integrals, Phys. Lett. B 779, 353 (2018).
  29. K. G. Chetyrkin and F. V. Tkachov, Integration by parts: The algorithm to calculate beta functions in 4 loops, Nucl. Phys. B192, 159 (1981).
  30. J. Gluza, K. Kajda, and D. A. Kosower, Towards a basis for planar two-loop integrals, Phys. Rev. D 83, 045012 (2011).
  31. R. M. Schabinger, A new algorithm for the generation of unitarity-compatible integration by parts relations, J. High Energy Phys. 01 (2012) 077.
  32. A. von Manteuffel and C. Studerus, Reduze 2—distributed Feynman integral reduction, arXiv:1201.4330.
  33. R. N. Lee, LiteRed 1.4: A powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523, 012059 (2014).
  34. A. von Manteuffel and R. M. Schabinger, A novel approach to integration by parts reduction, Phys. Lett. B 744, 101 (2015).
  35. K. J. Larsen and Y. Zhang, Integration-by-parts reductions from unitarity cuts and algebraic geometry, Phys. Rev. D 93, 041701 (2016).
  36. T. Peraro, Scattering amplitudes over finite fields and multivariate functional reconstruction, J. High Energy Phys. 12 (2016) 030.
  37. P. Mastrolia and S. Mizera, Feynman integrals and intersection theory, J. High Energy Phys. 02 (2019) 139.
  38. X. Liu and Y.-Q. Ma, Determining arbitrary Feynman integrals by vacuum integrals, Phys. Rev. D 99, 071501 (2019).
  39. X. Guan, X. Liu, and Y.-Q. Ma, Complete reduction of two-loop five-light-parton scattering amplitudes, Chin. Phys. C 44, 093106 (2020).
  40. J. Klappert and F. Lange, Reconstructing rational functions with FireFly, Comput. Phys. Commun. 247, 106951 (2020).
  41. T. Peraro, FiniteFlow: Multivariate functional reconstruction using finite fields and dataflow graphs, J. High Energy Phys. 07 (2019) 031.
  42. H. Frellesvig, F. Gasparotto, S. Laporta, M. K. Mandal, P. Mastrolia, L. Mattiazzi, and S. Mizera, Decomposition of Feynman integrals on the maximal cut by intersection numbers, J. High Energy Phys. 05 (2019) 153.
  43. Y. Wang, Z. Li, and N. Ul Basat, Direct reduction of multiloop multiscale scattering amplitudes, Phys. Rev. D 101, 076023 (2020).
  44. A. V. Smirnov and F. S. Chuharev, FIRE6: Feynman integral reduction with modular arithmetic, Comput. Phys. Commun. 247, 106877 (2020).
  45. J. Klappert, F. Lange, P. Maierhöfer, and J. Usovitsch, Integral reduction with Kira 2.0 and finite field methods, Comput. Phys. Commun. 266, 108024 (2021).
  46. J. Boehm, M. Wittmann, Z. Wu, Y. Xu, and Y. Zhang, IBP reduction coefficients made simple, J. High Energy Phys. 12 (2020) 054.
  47. M. Heller and A. von Manteuffel, MultivariateApart: Generalized partial fractions, Comput. Phys. Commun. 271, 108174 (2022).
  48. D. Bendle, J. Boehm, M. Heymann, R. Ma, M. Rahn, L. Ristau, M. Wittmann, Z. Wu, and Y. Zhang, Two-loop five-point integration-by-parts relations in a usable form, arXiv:2104.06866.
  49. Q. Song and A. Freitas, On the evaluation of two-loop electroweak box diagrams for e+e−→HZ production, J. High Energy Phys. 04 (2021) 179.
  50. D. Chicherin, T. Gehrmann, J. M. Henn, P. Wasser, Y. Zhang, and S. Zoia, All Master Integrals for Three-Jet Production at Next-to-Next-to-Leading Order, Phys. Rev. Lett. 123, 041603 (2019).
  51. C. G. Papadopoulos, D. Tommasini, and C. Wever, The pentabox master integrals with the simplified differential equations approach, J. High Energy Phys. 04 (2016) 078.
  52. T. Gehrmann, J. M. Henn, and N. A. Lo Presti, Pentagon functions for massless planar scattering amplitudes, J. High Energy Phys. 10 (2018) 103.
  53. D. Chicherin, T. Gehrmann, J. M. Henn, N. A. Lo Presti, V. Mitev, and P. Wasser, Analytic result for the nonplanar hexa-box integrals, J. High Energy Phys. 03 (2019) 042.
  54. D. Chicherin and V. Sotnikov, Pentagon functions for scattering of five massless particles, J. High Energy Phys. 12 (2020) 167.
  55. S. Abreu, H. Ita, F. Moriello, B. Page, W. Tschernow, and M. Zeng, Two-loop integrals for planar five-point one-mass processes, J. High Energy Phys. 11 (2020) 117.
  56. D. D. Canko, C. G. Papadopoulos, and N. Syrrakos, Analytic representation of all planar two-loop five-point Master Integrals with one off-shell leg, J. High Energy Phys. 01 (2021) 199.
  57. A. I. Davydychev and J. B. Tausk, Two loop selfenergy diagrams with different masses and the momentum expansion, Nucl. Phys. B397, 123 (1993).
  58. D. J. Broadhurst, Massive three—loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, Eur. Phys. J. C 8, 311 (1999).
  59. Y. Schröder and A. Vuorinen, High-precision epsilon expansions of single-mass-scale four-loop vacuum bubbles, J. High Energy Phys. 06 (2005) 051.
  60. T. Luthe, Fully massive vacuum integrals at 5 loops, Ph.D. thesis, Bielefeld University, 2015, https://pub.uni-bielefeld.de/publication/2776013.
  61. B. A. Kniehl, A. F. Pikelner, and O. L. Veretin, Three-loop massive tadpoles and polylogarithms through weight six, J. High Energy Phys. 08 (2017) 024.
  62. T. Luthe, A. Maier, P. Marquard, and Y. Schroder, Complete renormalization of QCD at five loops, J. High Energy Phys. 03 (2017) 020.
  63. C. Brønnum-Hansen and C.-Y. Wang, Contribution of third generation quarks to two-loop helicity amplitudes for W boson pair production in gluon fusion, J. High Energy Phys. 01 (2021) 170.
  64. C. Brønnum-Hansen and C.-Y. Wang, Top quark contribution to two-loop helicity amplitudes for Z boson pair production in gluon fusion, J. High Energy Phys. 05 (2021) 244.
  65. M. Beneke and V. A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B522, 321 (1998).
  66. V. A. Smirnov, Problems of the strategy of regions, Phys. Lett. B 465, 226 (1999).
  67. A. Georgoudis, V. Goncalves, E. Panzer, R. Pereira, A. V. Smirnov, and V. A. Smirnov, Glue-and-cut at five loops, J. High Energy Phys. 09 (2021) 098.
  68. X. Liu and Y.-Q. Ma, AMFlow: A Mathematica package for Feynman integrals computation via Auxiliary Mass Flow, arXiv:2201.11669.
  69. R. Bonciani, V. Del Duca, H. Frellesvig, J. M. Henn, M. Hidding, L. Maestri, F. Moriello, G. Salvatori, and V. A. Smirnov, Evaluating a family of two-loop non-planar master integrals for Higgs+jet production with full heavy-quark mass dependence, J. High Energy Phys. 01 (2020) 132.
  70. H. Frellesvig, M. Hidding, L. Maestri, F. Moriello, and G. Salvatori, The complete set of two-loop master integrals for Higgs+jet production in QCD, J. High Energy Phys. 06 (2020) 093.
  71. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevD.105.L051503 for numerical results of Feynman integrals calculated in this work.
  72. D. H. Bailey and D. J. Broadhurst, Parallel integer relation detection: Techniques and applications, Math. Comput. 70, 1719 (2001).

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Phys. Rev. D 105, L051503– Published 14 March, 2022
  • Received 8 July 2021
  • Accepted 18 February 2022
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