Skip to main content

Empirical Determinations of Feynman Integrals Using Integer Relation Algorithms

  • Chapter
  • First Online:
Anti-Differentiation and the Calculation of Feynman Amplitudes

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

Integer relation algorithms can convert numerical results for Feynman integrals to exact evaluations, when one has reason to suspect the existence of reductions to linear combinations of a basis, with rational or algebraic coefficients. Once a tentative reduction is obtained, confidence in its validity is greatly increased by computing more decimal digits of the terms and verifying the stability of the result. Here we give examples of how the PSLQ and LLL algorithms have yielded remarkable reductions of Feynman integrals to multiple polylogarithms and to the periods and quasi-periods of modular forms. Moreover, these algorithms have revealed quadratic relations between Feynman integrals. A recent application concerning black holes involves quadratic relations between combinations of Feynman integrals with algebraic coefficients.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. K. Acres, D. Broadhurst, Eta Quotients and Rademacher Sums, in Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, ed. by J. Blümlein, C. Schneider, P. Paule (Springer, 2019), pp. 1–27. https://arxiv.org/abs/1810.07478

  2. D.H. Bailey, D.J. Broadhurst, Parallel integer relation detection: techniques and applications. Math. Comp. 70, 1719–1736 (2001). https://arxiv.org/abs/math/9905048

    Article  MathSciNet  Google Scholar 

  3. D. Bailey, J. Borwein, D. Broadhurst, M. Larry Glasser, Elliptic integral evaluations of Bessel moments. J. Phys. A41, 205203 (2008). https://arxiv.org/abs/0801.0891

    MathSciNet  MATH  Google Scholar 

  4. J. Blümlein, D.J. Broadhurst, J.A.M. Vermaseren, The multiple zeta value data mine. Comput. Phys .Commun. 181, 582–625 (2010). https://arxiv.org/abs/0907.2557

    Article  MathSciNet  Google Scholar 

  5. J.M. Borwein, P.B. Borwein, Pi and the AGM: a study in analytic number theory and computational complexity, Monographies et Études de la Société Mathématique du Canada (Wiley, Toronto, 1987)

    Google Scholar 

  6. J.M. Borwein, D.J. Broadhurst, J. Kamnitzer, Central binomial sums, multiple Clausen values and zeta values. Exp. Math. 10, 25–34 (2001). https://arxiv.org/abs/hep-th/0004153

    Article  MathSciNet  Google Scholar 

  7. D.J. Broadhurst, Massless scalar Feynman diagrams: five loops and beyond. Open University report OUT-4102-18, 1985. https://arxiv.org/abs/1604.08027

  8. D.J. Broadhurst, D. Kreimer, Knots and numbers in ϕ 4 theory to 7 loops and beyond. Int. J. Mod. Phys. C6, 519–524 (1995). https://arxiv.org/abs/hep-ph/9504352

    Article  MathSciNet  Google Scholar 

  9. D.J. Broadhurst, D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops. Phys. Lett. B393, 403–412 (1997). https://arxiv.org/abs/hep-th/9609128

    Article  MathSciNet  Google Scholar 

  10. D.J. Broadhurst, Massive 3-loop Feynman diagrams reducible to SC primitives of algebras of the sixth root of unity. Eur. Phys. J. C8, 311–333 (1999). https://arxiv.org/abs/hep-th/9803091

    Article  MathSciNet  Google Scholar 

  11. D. Broadhurst, O. Schnetz, Algebraic geometry informs perturbative quantum field theory, in Loops and Legs in Quantum Field Theory, PoS (LL2014), 078 (2014). https://pos.sissa.it/211/078/pdf

  12. D. Broadhurst, A. Mellit, Perturbative quantum field theory informs algebraic geometry, in Loops and Legs in Quantum Field Theory, PoS (LL2016), 079 (2016). https://pos.sissa.it/260/079/pdf

  13. D. Broadhurst, Feynman integrals, L-series and Kloosterman moments. Commun. Number Theory Phys. 10, 527–569 (2016). https://arxiv.org/abs/1604.03057

    Article  MathSciNet  Google Scholar 

  14. D. Broadhurst, D.P. Roberts, Quadratic relations between Feynman integrals, in Loops and Legs in Quantum Field Theory, PoS (LL2018), 053 (2018). https://pos.sissa.it/303/053/pdf

  15. F. Brown, A class of non-holomorphic modular forms III: real analytic cusp forms for SL 2(Z). Res. Math. Sci. 5, 34 (2018). https://doi.org/10.1007/s40687-018-0151-3

    Article  Google Scholar 

  16. P. Candelas, X. de la Ossa, M. Elmi, D. van Straten, A one parameter family of Calabi-Yau manifolds with attractor points of rank two. J. High Energy Phys. 2020, 202 (2020). https://arxiv.org/abs/1912.06146

    Article  MathSciNet  Google Scholar 

  17. H.R.P. Ferguson, R.W. Forcade, Generalization of the euclidean algorithm for real numbers to all dimensions higher than two. Bull. Am. Math. Soc. 1, 912–914 (1979). https://doi.org/10.1090/S0273-0979-1979-14691-3

    Article  MathSciNet  Google Scholar 

  18. H.R.P. Ferguson, D.H. Bailey, A polynomial time, numerically stable integer relation algorithm. RNR technical report RNR-91-032, 1992. https://www.davidhbailey.com/dhbpapers/pslq.pdf

  19. J. Fresán, C. Sabbah, J.-D. Yu, Quadratic relations between Bessel moments. https://arxiv.org/abs/2006.02702

  20. M.L. Glasser, I.J. Zucker, Extended Watson integrals for the cubic lattices. PNAS 74, 1800–1801 (1977). https://doi.org/10.1073/pnas.74.5.1800

    Article  MathSciNet  Google Scholar 

  21. S. Laporta, High-precision calculation of the 4-loop contribution to the electron g − 2 in QED. Phys. Lett. B772, 232–238 (2017). https://arxiv.org/abs/1704.06996

    Article  Google Scholar 

  22. A.K. Lenstra, H.W. Lenstra Jr., L. Lovász, Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515–534 (1982). https://doi.org/10.1007/BF01457454

    Article  MathSciNet  Google Scholar 

  23. E. Panzer, Feynman integrals and hyperlogarithms. Thesis, Humboldt-Universität zu Berlin, 2015. https://arxiv.org/abs/1506.07243

    MATH  Google Scholar 

  24. Pari-GP version 2.13.1, Univ. Bordeaux, 2021. http://pari.math.u-bordeaux.fr

  25. O. Schnetz, Quantum periods: a census of ϕ 4 transcendentals. Commun. Number Theory Phys. 4, 1–47 (2010). https://arxiv.org/abs/0801.2856

    Article  MathSciNet  Google Scholar 

  26. O. Schnetz, Graphical functions and single-valued multiple polylogarithms. https://arxiv.org/abs/1302.6445

  27. G.N. Watson, Three triple integrals. Q. J. Math. 10, 266–276 (1939). https://doi.org/10.1093/qmath/os-10.1.266

    Article  MathSciNet  Google Scholar 

  28. Y. Zhou, Wronskian factorizations and Broadhurst-Mellit determinant formulae. Commun. Number Theory Phys. 12, 355–407 (2018). https://arxiv.org/abs/1711.01829

    Article  MathSciNet  Google Scholar 

  29. Y. Zhou, Some algebraic and arithmetic properties of Feynman diagrams, in Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, ed. by J. Blümlein, C. Schneider, P. Paule (Springer, 2019), pp. 485–509. https://arxiv.org/abs/1801.05555

  30. Y. Zhou, On Laporta’s 4-loop sunrise formulae. Ramanujan J. 50, 465–503 (2019). https://arxiv.org/abs/1801.02182

    Article  MathSciNet  Google Scholar 

  31. Y. Zhou, Wrońskian algebra and Broadhurst-Roberts quadratic relations. https://arxiv.org/abs/2012.03523

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to David Broadhurst .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Acres, K., Broadhurst, D. (2021). Empirical Determinations of Feynman Integrals Using Integer Relation Algorithms. In: Blümlein, J., Schneider, C. (eds) Anti-Differentiation and the Calculation of Feynman Amplitudes. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-80219-6_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-80219-6_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-80218-9

  • Online ISBN: 978-3-030-80219-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics