- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

From integrability to the Galois coaction on Feynman periods

Ömer Gürdoğan*

  • Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom and School of Physics & Astronomy, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom

  • *o.c.gurdogan@soton.ac.uk
PDF

Phys. Rev. D 103, L081703 – Published 22 April, 2021

DOI: https://doi.org/10.1103/PhysRevD.103.L081703

Abstract

We argue that the description of Feynman loop integrals as integrable systems is intimately connected with their motivic properties and the action of the Cosmic Galois Group. We show how in the case of a family of fishnet graphs, coaction relations between them follow directly from iterative constructions of Q-functions in the quantum spectral curve formalism. Using this observation we conjecture a “differential equation for numbers” that enter these periods.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (31)

  1. A. B. Goncharov, M. Spradlin, C. Vergu, and A. Volovich, Phys. Rev. Lett. 105, 151605 (2010); L. J. Dixon, J. Drummond, T. Harrington, A. J. McLeod, G. Papathanasiou, and M. Spradlin, J. High Energy Phys. 02 (2017) 137; L. J. Dixon, J. M. Drummond, and J. M. Henn, 11 (2011) 023; J. Drummond, J. Foster, Ö. Gürdoğan, and G. Papathanasiou, 03 (2019) 087; J. M. Drummond, G. Papathanasiou, and M. Spradlin, 03 (2015) 072; S. Caron-Huot, L. J. Dixon, F. Dulat, M. Von Hippel, A. J. McLeod, and G. Papathanasiou, 09 (2019) 061;
  2. F. Brown, arXiv:1102.1312.
  3. A. Goncharov, Duke Math. J. 128, 209 (2002).
  4. F. Brown, arXiv:1102.1310.
  5. F. Brown, Commun. Numer. Theor. Phys. 11, 453 (2017).
  6. F. Brown and O. Schnetz, Duke Math. J. 161, 1817 (2012).
  7. E. Panzer and O. Schnetz, Commun. Numer. Theor. Phys. 11, 657 (2017).
  8. S. Abreu, R. Britto, C. Duhr, E. Gardi, and J. Matthew, Proc. Sci. RADCOR2019 (2019) 067 [arXiv:1912.03205]; LL2018 (2018) 047 [arXiv:1808.00069]; S. Abreu, R. Britto, C. Duhr, and E. GardiJ. High Energy Phys. 12 (2017) 090; Phys. Rev. Lett. 119, 051601 (2017).
  9. M. Tapušković, arXiv:1911.01540.
  10. Ö. Gürdoğan and V. Kazakov, Phys. Rev. Lett. 117, 201602 (2016); 117, 259903(A) (2016).
  11. A. Zamolodchikov, Phys. Lett. 97B, 63 (1980).
  12. A. Isaev, Nucl. Phys. B662, 461 (2003).
  13. F. Loebbert, J. Miczajka, D. Müller, and H. Münkler, Phys. Rev. Lett. 125, 091602 (2020).
  14. N. Gromov, F. Levkovich-Maslyuk, and G. Sizov, Phys. Rev. Lett. 115, 251601 (2015).
  15. N. Gromov, V. Kazakov, S. Leurent, and D. Volin, Phys. Rev. Lett. 112, 011602 (2014).
  16. N. Gromov, V. Kazakov, G. Korchemsky, S. Negro, and G. Sizov, J. High Energy Phys. 01 (2018) 095.

  17. Cfr equation (2.3.9) of [18] where it is apparent that the definition of periods is identical to that of anomalous dimensions.

  18. E. Panzer, Ph.D. thesis, Humboldt University, 2015.
  19. D. J. Broadhurst, Phys. Lett. 164B, 356 (1985).

  20. In contrast with [16], we use μ to denote the parameter m therein.

  21. S. Leurent and D. Volin, Nucl. Phys. B875, 757 (2013).

  22. With coefficients at most linear in u.

  23. It is straightforward to write ηw(u) in terms of other MHZFs with argument u+i, and this does not change the alphabet of their indices.

  24. O. Schnetz, Hyperlogprocedures (2020), https://www.math.fau.de/person/oliver-schnetz/.

  25. We should note that “the f-alphabet” satisfying (11) is not unique.

  26. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevD.103.L081703, for Computer-readable lists of wheel periods and wheel MZVs.

  27. We defer their study to a future work.

  28. O. Schlotterer and S. Stieberger, J. Phys. A 46, 475401 (2013); J. Drummond and E. Ragoucy, J. High Energy Phys. 08 (2013) 135.
  29. F. Brown and C. Dupont, arXiv:1907.06603.
  30. J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisonek, arXiv:math/9812020.
  31. S. Charlton, J. Number Theory 148, 463 (2015).

Outline

Information

Phys. Rev. D 103, L081703– Published 22 April, 2021
  • Received 11 January 2021
  • Accepted 9 March 2021
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation