Rational Function Simplification for Integration-by-Parts Reduction and Beyond


  • Kirill S. Mokrov
  • Alexander V. Smirnov
  • Mao Zeng


Feynman integrals
integration by parts
computer algebra


We present FUEL (Fractional Universal Evaluation Library), a C++ library for performingrational function arithmetic with a flexible choice of third-party computer algebra systems as simplifiers. FUEL is an outgrowth of a C++ interface to Fermat which was originally part of the FIRE code for integration-by-parts (IBP) reduction for Feynman integrals, now promoted to be a standalone library with access to simplifiers other than Fermat. We compare the performance of various simplifiers for standalone benchmark problems as well as IBP reduction runs with FIRE. A speedup of more than 10 times is achieved for an example IBP problem related to calculation of the off-shell three-particle form factors in N = 4 supersymmetric Yang-Mills theory.






Methods and algorithms of computational mathematics and their applications

Author Biographies

Kirill S. Mokrov

Alexander V. Smirnov

Mao Zeng

University of Edinburgh,
Higgs Centre for Theoretical Physics
James Clark Maxwell Building, Peter Guthrie Tait Road, EH9 3FD, Edinburgh, United Kingdom
• Royal Society University Research Fellow


  1. K. G. Chetyrkin and F. V. Tkachov, “Integration by Parts: The Algorithm to Calculate β-Functions in 4 Loops,” Nucl. Phys. B 192 (1), 159-204 (1981).
    doi 10.1016/0550-3213(81)90199-1
  2. S. Laporta, “High-Precision Calculation of Multiloop Feynman Integrals by Difference Equations,” Int. J. Mod. Phys. A 15 (32), 5087-5159 (2000).
    doi 10.1142/S0217751X00002159
  3. J. Abbott, A. M. Bigatti, and L. Robbiano, CoCoA System: Computations in Commutative Algebra. . Cited October 4, 2023.
  4. J. Abbott and A. M. Bigatti, CoCoALib: A C++ Library for Doing Computations in Commutative Algebra. Cited October 4, 2023.
  5. R. H. Lewis, Fermat: A Computer Algebra System for Polynomial and Matrix Computation. Cited September 30, 2023.
  6. J. A. M. Vermaseren, The Main Directory for FORM. form/maindir/maindir.html . Cited September 30, 2023.
  7. GiNaC is Not a CAS. Cited September 30, 2023.
  8. D. R. Grayson and M. E. Stillman. Macaulay2Doc: Macaulay2 Documentation. Cited October 4, 2023.
  9. Maplesoft: The Essential Tool for Mathematics. Cited October 4, 2023.
  10. Maxima: A Computer Algebra System. Version 5.43.2. Cited September 30, 2023.
  11. C. Fieker, W. Hart, T. Hofmann, and F. Johansson, “Nemo/Hecke: Computer Algebra and Number Theory Packages for the Julia Programming Language,” in Proc. 2017 ACM Int. Symposium on Symbolic and Algebraic Computation, Kaiserslautern, Germany, July 23-28, 2017 (ACM Press, New York, 2017), pp. 157-164.
    doi 10.1145/3087604.3087611
  12. PARI/GP Version 2.11.2, Univ. Bordeaux, 2022. Cited October 1, 2023.
  13. Symbolica. Cited October 1, 2023.
  14. Wolfram Mathematica, Version 13.1. . Cited October 1, 2023.
  15. A. V. Smirnov and V. A. Smirnov, “FIRE4, LiteRed and Accompanying Tools to Solve Integration by Parts Relations,” Comput. Phys. Commun. 184 (12), 2820-2827 (2013).
    doi 10.1016/j.cpc.2013.06.016
  16. A. V. Smirnov, “FIRE5: A C++ Implementation of Feynman Integral REduction,” Comput. Phys. Commun. 189, 182-191 (2015).
    doi 10.1016/j.cpc.2014.11.024
  17. A. V. Smirnov and F. S. Chukharev, “FIRE6: Feynman Integral REduction with Modular Arithmetic,” Comput. Phys. Commun. 247, Article Number 106877 (2020).
    doi 10.1016/j.cpc.2019.106877
  18. C. Anastasiou and A. Lazopoulos, “Automatic Integral Reduction for Higher Order Perturbative Calculations,” J. High Energy Phys. No. 7, Article Identifier 046 (2004).
    doi 10.1088/1126-6708/2004/07/046
  19. A. von Manteuffel and C. Studerus, “Reduze 2 -- Distributed Feynman Integral Reduction,” . Cited October 1, 2023.
  20. R. N. Lee, “LiteRed 1.4: A Powerful Tool for Reduction of Multiloop Integrals,” J. Phys. Conf. Ser. 523 (1), Article Identifier 012059 (2014).
    doi 10.1088/1742-6596/523/1/012059
  21. P. Maierhöfer, J. Usovitsch, and P. Uwer, “Kira -- A Feynman Integral Reduction Program,” Comput. Phys. Commun. 230, 99-112 (2018).
    doi 10.1016/j.cpc.2018.04.012
  22. P. Maierhöfer and J. Usovitsch, “Kira 1.2 Release Notes,” . Cited October 1, 2023.
  23. J. Klappert, F. Lange, P. Maierhöfer, and J. Usovitsch, “Integral Reduction with Kira 2.0 and Finite Field Methods,” Comput. Phys. Commun. 266, Article Number 108024 (2021).
    doi 10.1016/j.cpc.2021.108024
  24. J. M. Henn, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, “A Planar Four-Loop Form Factor and Cusp Anomalous Dimension in QCD,” J. High Energy Phys. No. 5, Article Identifier 066 (2016).
    doi 10.1007/JHEP05(2016)066
  25. J. Henn, R. N. Lee, A. V. Smirnov, et al., “Four-Loop Photon Quark Form Factor and Cusp Anomalous Dimension in the Large-N_c Limit of QCD,” J. High Energy Phys. No. 3, Article Identifier 139 (2017).
    doi 10.1007/JHEP03(2017)139
  26. R. N. Lee, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, “n_f² Contributions to Fermionic Four-Loop Form Factors,” Phys. Rev. D 96 (1), Article Identifier 014008 (2017).
    doi 10.1103/PhysRevD.96.014008
  27. D. Bendle, J. Böhm, W. Decker, et al., “Integration-by-Parts Reductions of Feynman Integrals Using Singular and GPI-Space,” J. High Energy Phys. No. 2, Article Identifier 079 (2020).
    doi 10.1007/JHEP02(2020)079
  28. R. N. Lee, “Group Structure of the Integration-by-Part Identities and Its Application to the Reduction of Multiloop Integrals,” J. High Energy Phys. No. 7, Article Identifier 031 (2008).
    doi 10.1088/1126-6708/2008/07/031
  29. B. Ruijl, T. Ueda, and J. A. M. Vermaseren, “Forcer, a FORM Program for the Parametric Reduction of Four-Loop Massless Propagator Diagrams,” Comput. Phys. Commun. 253, Article Number 107198 (2020).
    doi 10.1016/j.cpc.2020.107198
  30. A. V. Smirnov and V. A. Smirnov, “How to Choose Master Integrals,” Nucl. Phys. B 960, Article Number 115213 (2020).
    doi 10.1016/j.nuclphysb.2020.115213
  31. J. Usovitsch, “Factorization of Denominators in Integration-by-Parts Reductions,” . Cited October 1, 2023.
  32. X. Liu and Y.-Q. Ma, “Determining Arbitrary Feynman Integrals by Vacuum Integrals,” Phys. Rev. D 99 (7), Article Identifier 071501 (2019).
    doi 10.1103/PhysRevD.99.071501
  33. X. Guan, X. Liu, and Y.-Q. Ma, “Complete Reduction of Integrals in Two-Loop Five-Light-Parton Scattering Amplitudes,” Chinese Phys. C 44 (9), Article Identifier 093106 (2020).
    doi 10.1088/1674-1137/44/9/093106
  34. J. Gluza, K. Kajda, and D. A. Kosower, “Towards a Basis for Planar Two-Loop Integrals,” Phys. Rev. D 83 (4), Article Identifier 045012 (2011).
    doi 10.1103/PhysRevD.83.045012
  35. R. M. Schabinger, “A New Algorithm for the Generation of Unitarity-Compatible Integration by Parts Relations,” J. High Energy Phys. No. 1, Article Identifier 077 (2012).
    doi 10.1007/JHEP01(2012)077
  36. K. J. Larsen and Y. Zhang, “Integration-by-Parts Reductions from Unitarity Cuts and Algebraic Geometry,” Phys. Rev. D 93 (4), Article Identifier 041701 (2016).
    doi 10.1103/PhysRevD.93.041701
  37. J. Böhm, A. Georgoudis, K. J. Larsen, et al., “Complete Integration-by-Parts Reductions of the Non-Planar Hexagon-Box via Module Intersections,” J. High Energy Phys. No. 9, Article Identifier 024 (2018).
    doi 10.1007/JHEP09(2018)024
  38. H. Ita, “Two-Loop Integrand Decomposition into Master Integrals and Surface Terms,” Phys. Rev. D 94 (11), Article Identifier 116015 (2016).
    doi 10.1103/PhysRevD.94.116015
  39. S. Abreu, F. Febres Cordero, H. Ita, et al., “Two-Loop Four-Gluon Amplitudes from Numerical Unitarity,” Phys. Rev. Lett. 119 (14), Article Identifier 142001 (2017).
    doi 10.1103/PhysRevLett.119.142001
  40. S. Abreu, F. Febres Cordero, H. Ita, et al., “Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity,” Phys. Rev. D 97 (11), Article Identifier 116014 (2018).
    doi 10.1103/PhysRevD.97.116014
  41. A utility to accurately report the core memory usage for a program. . Cited October 1, 2023.
  42. A. V. Belitsky, L. V. Bork, A. F. Pikelner, and V. A. Smirnov, “Exact Off Shell Sudakov Form Factor in mathcalN=4 Supersymmetric Yang-Mills Theory,” Phys. Rev. Lett. 130 (9), Article Identifier 091605 (2023).
    doi 10.1103/PhysRevLett.130.091605
  43. A. V. Belitsky, L. V. Bork, and V. A. Smirnov, “Off-Shell Form Factor in mathcalN=4 sYM at Three Loops,” . Cited October 1, 2023.
  44. M. Monagan and R. Pearce, “Fermat Benchmarks for Rational Expressionals in Maple,” ACM Commun. Comput. Algebra 50 (4), 188-190 (2016).
    doi 10.1145/3055282.3055299
  45. Pynac -- symbolic computation with Python objects, Sage Math support library. . Cited October 1, 2023.
  46. P. Zimmermann, A. Casamayou, N. Cohen, et al., Computational Mathematics with SageMath (SIAM Press, Philadelphia, 2018).
  47. Y. Zhang, Lecture Notes on Multi-Loop Integral Reduction and Applied Algebraic Geometry. . Cited October 1, 2023.
  48. M. Monagan and R. Pearce, “Poly: A New Polynomial Data Structure for Maple 17,” ACM Commun. Comput. Algebra 46 (3/4), 164-167 (2013).
    doi 10.1145/2429135.2429173
  49. M. Monagan, “Speeding up Polynomial GCD, a Crucial Operation in Maple,” Maple Trans. 2 (1), Article Number 14457 (2022).
    doi 10.5206/mt.v2i1.14452
  50. J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, “Julia: A Fresh Approach to Numerical Computing,” SIAM Rev. 59 (1), 65-98 (2017).
    doi 10.1137/141000671
  51. W. B. Hart, “Flint Fast Library for Number Theory,” Computeralgebra-Rundbrief, No. 49, 15-17 (2011). . Cited October 3, 2023.