Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space

Abstract

We introduce an algebro-geometrically motived integration-by-parts (IBP) re- duction method for multi-loop and multi-scale Feynman integrals, using a framework for massively parallel computations in computer algebra. This framework combines the com- puter algebra system Singular with the workflow management system GPI-Space, which are being developed at the TU Kaiserslautern and the Fraunhofer Institute for Industrial Mathematics (ITWM), respectively. In our approach, the IBP relations are first trimmed by modern tools from computational algebraic geometry and then solved by sparse linear algebra and our new interpolation method. Modelled in terms of Petri nets, these steps are efficiently automatized and automatically parallelized by GPI-Space. We demonstrate the potential of our method at the nontrivial example of reducing two-loop five-point non- planar double-pentagon integrals. We also use GPI-Space to convert the basis of IBP reductions, and discuss the possible simplification of master-integral coefficients in a uni- formly transcendental basis.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    S. Badger, H. Frellesvig and Y. Zhang, A two-loop five-gluon helicity amplitude in QCD, JHEP 12 (2013) 045 [arXiv:1310.1051] [INSPIRE].

    ADS  Article  Google Scholar 

  2. [2]

    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 [Erratum ibid. 116 (2016) 189903] [arXiv:1511.05409] [INSPIRE].

  3. [3]

    S. Badger, C. Brønnum-Hansen, H.B. Hartanto and T. Peraro, First look at two-loop five-gluon scattering in QCD, Phys. Rev. Lett. 120 (2018) 092001 [arXiv:1712.02229] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  4. [4]

    S. Abreu et al., Planar two-loop five-gluon amplitudes from numerical unitarity, Phys. Rev. D 97 (2018) 116014 [arXiv:1712.03946] [INSPIRE].

    ADS  Article  Google Scholar 

  5. [5]

    S. Abreu et al., The two-loop five-point amplitude in 𝒩 = 4 super-Yang-Mills theory, Phys. Rev. Lett. 122 (2019) 121603 [arXiv:1812.08941] [INSPIRE].

    ADS  Article  Google Scholar 

  6. [6]

    S. Abreu et al., Planar two-loop five-parton amplitudes from numerical unitarity, JHEP 11 (2018) 116 [arXiv:1809.09067] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  7. [7]

    R.H. Boels, Q. Jin and H. Lüo, Efficient integrand reduction for particles with spin, arXiv:1802.06761 [INSPIRE].

  8. [8]

    T. Gehrmann, J.M. Henn and N.A. Lo Presti, Pentagon functions for massless planar scattering amplitudes, JHEP 10 (2018) 103 [arXiv:1807.09812] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  9. [9]

    S. Badger, C. Brønnum-Hansen, H.B. Hartanto and T. Peraro, Analytic helicity amplitudes for two-loop five-gluon scattering: the single-minus case, JHEP 01 (2019) 186 [arXiv:1811.11699] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  10. [10]

    S. Abreu et al., Analytic form of planar two-loop five-gluon scattering amplitudes in QCD, Phys. Rev. Lett. 122 (2019) 082002 [arXiv:1812.04586] [INSPIRE].

    ADS  Article  Google Scholar 

  11. [11]

    D. Chicherin et al., Analytic result for a two-loop five-particle amplitude, Phys. Rev. Lett. 122 (2019) 121602 [arXiv:1812.11057] [INSPIRE].

    ADS  Article  Google Scholar 

  12. [12]

    D. Chicherin et al., The two-loop five-particle amplitude in 𝒩 = 8 supergravity, JHEP 03 (2019) 115 [arXiv:1901.05932] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  13. [13]

    S. Abreu et al., The two-loop five-point amplitude in 𝒩 = 8 supergravity, JHEP 03 (2019) 123 [arXiv:1901.08563] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  14. [14]

    S. Abreu et al., Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD, JHEP 05 (2019) 084 [arXiv:1904.00945] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  15. [15]

    H.B. Hartanto et al., A numerical evaluation of planar two-loop helicity amplitudes for a W -boson plus four partons, JHEP 09 (2019) 119 [arXiv:1906.11862] [INSPIRE].

    ADS  Article  Google Scholar 

  16. [16]

    Y. Zhang, Integrand-level reduction of loop amplitudes by computational algebraic geometry methods, JHEP 09 (2012) 042 [arXiv:1205.5707] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  17. [17]

    P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, Scattering amplitudes from multivariate polynomial division, Phys. Lett. B 718 (2012) 173 [arXiv:1205.7087] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  18. [18]

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

    ADS  Article  Google Scholar 

  19. [19]

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

    ADS  MathSciNet  MATH  Google Scholar 

  20. [20]

    H. Ita, Two-loop integrand decomposition into master integrals and surface terms, Phys. Rev. 94 (2016) 116015 [arXiv:1510.05626] [INSPIRE].

    MathSciNet  Google Scholar 

  21. [21]

    S. Abreu et al., Two-loop four-gluon amplitudes from numerical unitarity, Phys. Rev. Lett. 119 (2017) 142001 [arXiv:1703.05273] [INSPIRE].

    ADS  Article  Google Scholar 

  22. [22]

    L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  23. [23]

    L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP 12 (2013) 049 [arXiv:1308.2276] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  24. [24]

    L.J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops, JHEP 10 (2014) 065 [arXiv:1408.1505] [INSPIRE].

    ADS  Article  Google Scholar 

  25. [25]

    S. Caron-Huot, L.J. Dixon, A. McLeod and M. von Hippel, Bootstrapping a Five-Loop Amplitude Using Steinmann Relations, Phys. Rev. Lett. 117 (2016) 241601 [arXiv:1609.00669] [INSPIRE].

    ADS  Article  Google Scholar 

  26. [26]

    L.J. Dixon, M. von Hippel and A.J. McLeod, The four-loop six-gluon NMHV ratio function, JHEP 01 (2016) 053 [arXiv:1509.08127] [INSPIRE].

    Article  Google Scholar 

  27. [27]

    L.J. Dixon et al., Heptagons from the Steinmann cluster bootstrap, JHEP 02 (2017) 137 [arXiv:1612.08976] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  28. [28]

    D. Chicherin, J. Henn and V. Mitev, Bootstrapping pentagon functions, JHEP 05 (2018) 164 [arXiv:1712.09610] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  29. [29]

    S. Caron-Huot et al., Six-Gluon amplitudes in planar 𝒩 = 4 super-Yang-Mills theory at six and seven loops, JHEP 08 (2019) 016 [arXiv:1903.10890] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  30. [30]

    A. von Manteuffel and R.M. Schabinger, A novel approach to integration by parts reduction, Phys. Lett. B 744 (2015) 101 [arXiv:1406.4513] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  31. [31]

    T. Peraro, Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP 12 (2016) 030 [arXiv:1608.01902] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  32. [32]

    J. Klappert and F. Lange, Reconstructing rational functions with FireFly, Comput. Phys. Commun. 247 (2020) 106951 [arXiv:1904.00009] [INSPIRE].

    Article  Google Scholar 

  33. [33]

    T. Peraro, FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs, JHEP 07 (2019) 031 [arXiv:1905.08019] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  34. [34]

    K. Chetyrkin and F. Tkachov, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159.

    ADS  Article  Google Scholar 

  35. [35]

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

  36. [36]

    A.V. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  37. [37]

    A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  38. [38]

    A.V. Smirnov and F.S. Chuharev, FIRE6: Feynman Integral REduction with modular arithmetic, arXiv:1901.07808 [INSPIRE].

  39. [39]

    P. Maierhöfer, J. Usovitsch and P. Uwer, Kira — A Feynman integral reduction program, Comput. Phys. Commun. 230 (2018) 99 [arXiv:1705.05610] [INSPIRE].

    ADS  Article  Google Scholar 

  40. [40]

    P. Maierhöfer and J. Usovitsch, Kira 1.2 release notes, arXiv:1812.01491 [INSPIRE].

  41. [41]

    A. von Manteuffel and C. Studerus, Reduze 2 — Distributed Feynman integral reduction, arXiv:1201.4330 [INSPIRE].

  42. [42]

    J. Gluza, K. Kajda and D.A. Kosower, Towards a basis for planar two-loop integrals, Phys. Rev. D 83 (2011) 045012 [arXiv:1009.0472] [INSPIRE].

    ADS  Google Scholar 

  43. [43]

    R.M. Schabinger, A new algorithm for the generation of unitarity-compatible integration by parts relations, JHEP 01 (2012) 077 [arXiv:1111.4220] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  44. [44]

    K.J. Larsen and Y. Zhang, Integration-by-parts reductions from unitarity cuts and algebraic geometry, Phys. Rev. D 93 (2016) 041701 [arXiv:1511.01071] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  45. [45]

    Z. Bern, M. Enciso, H. Ita and M. Zeng, Dual conformal symmetry, integration-by-parts reduction, differential equations and the nonplanar sector, Phys. Rev. D 96 (2017) 096017 [arXiv:1709.06055] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  46. [46]

    R.N. Lee and A.A. Pomeransky, Critical points and number of master integrals, JHEP 11 (2013) 165 [arXiv:1308.6676] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  47. [47]

    A. Georgoudis, K.J. Larsen and Y. Zhang, Azurite: an algebraic geometry based package for finding bases of loop integrals, Comput. Phys. Commun. 221 (2017) 203 [arXiv:1612.04252] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  48. [48]

    T. Bitoun, C. Bogner, R.P. Klausen and E. Panzer, Feynman integral relations from parametric annihilators, Lett. Math. Phys. 109 (2019) 497 [arXiv:1712.09215] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  49. [49]

    H.A. Chawdhry, M.A. Lim and A. Mitov, Two-loop five-point massless QCD amplitudes within the integration-by-parts approach, Phys. Rev. D 99 (2019) 076011 [arXiv:1805.09182] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  50. [50]

    S. Badger et al., Analytic form of the full two-loop five-gluon all-plus helicity amplitude, Phys. Rev. Lett. 123 (2019) 071601 [arXiv:1905.03733] [INSPIRE].

    ADS  Article  Google Scholar 

  51. [51]

    D.A. Kosower, Direct solution of integration-by-parts systems, Phys. Rev. D 98 (2018) 025008 [arXiv:1804.00131] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  52. [52]

    P. Mastrolia and S. Mizera, Feynman integrals and intersection theory, JHEP 02 (2019) 139 [arXiv:1810.03818] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  53. [53]

    H. Frellesvig et al., Decomposition of Feynman integrals on the maximal cut by intersection numbers, JHEP 05 (2019) 153 [arXiv:1901.11510] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  54. [54]

    H. Frellesvig et al., Vector space of Feynman integrals and multivariate intersection numbers, Phys. Rev. Lett. 123 (2019) 201602 [arXiv:1907.02000] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  55. [55]

    X. Liu and Y.-Q. Ma, Determining arbitrary Feynman integrals by vacuum integrals, Phys. Rev. D 99 (2019) 071501 [arXiv:1801.10523] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  56. [56]

    X. Liu, Y.-Q. Ma and C.-Y. Wang, A systematic and efficient method to compute multi-loop master integrals, Phys. Lett. 779 (2018) 353 [arXiv:1711.09572] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  57. [57]

    Y. Wang, Z. Li and N. Ul Basat, Direct reduction of amplitude, arXiv:1901.09390 [INSPIRE].

  58. [58]

    A. Kardos, A new reduction strategy for special negative sectors of planar two-loop integrals without Laporta algorithm, arXiv:1812.05622 [INSPIRE].

  59. [59]

    Y. Zhang, Lecture notes on multi-loop integral reduction and applied algebraic geometry, arXiv:1612.02249 [INSPIRE].

  60. [60]

    J. Böhm et al., Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections, JHEP 09 (2018) 024 [arXiv:1805.01873] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  61. [61]

    J. Böhm et al., Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals, Phys. Rev. D 98 (2018) 025023 [arXiv:1712.09737] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  62. [62]

    W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-1-1 — A computer algebra system for polynomial computations, http://www.singular.uni-kl.de (2018).

  63. [63]

    F.J. Pfreundt and M. Rahn, GPI-Space, Fraunhofer ITWM Kaiserslautern, http://www.gpi-space.de/ (2018).

  64. [64]

    P. Wasser, Analytic properties of Feynman integrals for scattering amplitudes, M.Sc. thesis,Johannes Gutenberg-Universität Mainz, Mainz, Germany (2016).

    Google Scholar 

  65. [65]

    P.A. Baikov, Explicit solutions of the three loop vacuum integral recurrence relations, Phys. Lett. B 385 (1996) 404 [hep-ph/9603267] [INSPIRE].

  66. [66]

    R.N. Lee, Modern techniques of multiloop calculations, in the proceedings of the 49th Rencontres de Moriond on QCD and High Energy Interactions, March 22–29, La Thuile, Italy (2014), arXiv:1405.5616 [INSPIRE].

  67. [67]

    The_SpaSM_group, SpaSM: a Sparse direct Solver Modulo p, v1.2 ed., http://github.com/cbouilla/spasm (2017).

  68. [68]

    E.W. Mayr and A.R. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. Math. 46 (1982) 305.

    MathSciNet  MATH  Article  Google Scholar 

  69. [69]

    J. Böhm et al., Towards massively parallel computations in algebraic geometry, arXiv:1808.09727.

  70. [70]

    C. Jordan, M. Joswig and L. Kastner, Parallel enumeration of triangulations, Electron. J. Combin. 25 (2018) 27.

    MathSciNet  MATH  Article  Google Scholar 

  71. [71]

    M. Rahn, GPI-Space whitepaper, Fraunhofer ITWM Kaiserslautern, http://gpi-space.com/wp-content/uploads/2014/06/GPISpaceWhitepaper.pdf (2014).

  72. [72]

    L. Ristau, Using Petri nets to parallelize algebraic algorithms, Ph.D. Thesis, TU Kaiserslautern, Kaiserslautern, Germany (2019).

  73. [73]

    C. Reinbold, Computation of the GIT-fan using a massively parallel implementation, Master’s Thesis (2018).

  74. [74]

    D. Bendle, Massively parallel computation of tropical varieties, Bachelor’s Thesis (2018).

  75. [75]

    Z. Bern et al., Logarithmic singularities and maximally supersymmetric amplitudes, JHEP 06 (2015) 202 [arXiv:1412.8584] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  76. [76]

    D. Chicherin et al., All master integrals for three-jet production at next-to-next-to-leading order, Phys. Rev. Lett. 123 (2019) 041603 [arXiv:1812.11160] [INSPIRE].

    ADS  Article  Google Scholar 

  77. [77]

    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local integrals for planar scattering amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  78. [78]

    D. Chicherin et al., Analytic result for the nonplanar hexa-box integrals, JHEP 03 (2019) 042 [arXiv:1809.06240] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  79. [79]

    J. Lykke Jacobsen, Y. Jiang and Y. Zhang, Torus partition function of the six-vertex model from algebraic geometry, JHEP 03 (2019) 152 [arXiv:1812.00447] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  80. [80]

    Y. Jiang and Y. Zhang, Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE, JHEP 03 (2018) 087 [arXiv:1710.04693] [INSPIRE].

  81. [81]

    R. Zippel, Probabilistic algorithms for sparse polynomials, in International Symposium on Symbolic and Algebraic Manipulation, E.W. Ng ed., Lecture Notes in Computer Science volume 72, Springer, Germany (1979).

  82. [82]

    M. Ben Or and P. Tiwari, A deterministic algorithm for sparse multivariate polynomial interpolation, in the proceedings of the 20th annual ACM symposium on Theory of computing (STOC’88), Chicago, U.S.A. (1988).

    Google Scholar 

  83. [83]

    E. Kaltofen, W.S. Lee and A. Lobo, Early Termination in Ben-Or/Tiwari Sparse Interpolation and a Hybrid of Zippel’s Algorithm*, in the proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC01), London, Canada (2001).

    Google Scholar 

Download references

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

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yang Zhang.

Additional information

ArXiv ePrint: 1908.04301

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bendle, D., Böhm, J., Decker, W. et al. Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space. J. High Energ. Phys. 2020, 79 (2020). https://doi.org/10.1007/JHEP02(2020)079

Download citation

Keywords

  • Scattering Amplitudes
  • Differential and Algebraic Geometry