Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 032, 56 pages      arXiv:2009.00426      https://doi.org/10.3842/SIGMA.2021.032
Contribution to the Special Issue on Representation Theory and Integrable Systems in honor of Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th birthday

An Introduction to Motivic Feynman Integrals

Claudia Rella
Section de Mathématiques, Université de Genève, Genève, CH-1211 Switzerland

Received August 30, 2020, in final form March 03, 2021; Published online March 26, 2021

Abstract
This article gives a short step-by-step introduction to the representation of parametric Feynman integrals in scalar perturbative quantum field theory as periods of motives. The application of motivic Galois theory to the algebro-geometric and categorical structures underlying Feynman graphs is reviewed up to the current state of research. The example of primitive log-divergent Feynman graphs in scalar massless $\phi^4$ quantum field theory is analysed in detail.

Key words: scattering amplitudes; Feynman diagrams; multiple zeta values; Hodge structures; periods of motives; Galois theory; Tannakian categories.

pdf (3212 kb)   tex (2590 kb)  

References

  1. Abreu S., Britto R., Duhr C., Gardi E., Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case, J. High Energy Phys. 2017 (2017), no. 12, 090, 73 pages, arXiv:1704.07931.
  2. Abreu S., Britto R., Duhr C., Gardi E., Matthew J., Coaction for Feynman integrals and diagrams, PoS Proc. Sci. (2018), PoS(LL2018), 047, 14 pages, arXiv:1808.00069.
  3. Abreu S., Britto R., Duhr C., Gardi E., Matthew J., From positive geometries to a coaction on hypergeometric functions, J. High Energy Phys. 2020 (2020), no. 2, 122, 44 pages, arXiv:1910.08358.
  4. André Y., Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, Vol. 17, Société Mathématique de France, Paris, 2004.
  5. Bloch S., Esnault H., Kreimer D., On motives associated to graph polynomials, Comm. Math. Phys. 267 (2006), 181-225, arXiv:math.AG/0510011.
  6. Blümlein J., Broadhurst D.J., Vermaseren J.A.M., The multiple zeta value data mine, Comput. Phys. Comm. 181 (2010), 582-625, arXiv:0907.2557.
  7. Bogner C., Weinzierl S., Periods and Feynman integrals, J. Math. Phys. 50 (2009), 042302, 16 pages, arXiv:0711.4863.
  8. Bogner C., Weinzierl S., Feynman graph polynomials, Internat. J. Modern Phys. A 25 (2010), 2585-2618, arXiv:1002.3458.
  9. Bott R., Tu L.W., Differential forms in algebraic topology, Graduate Texts in Mathematics, Vol. 82, Springer-Verlag, New York - Berlin, 1982.
  10. Broadhurst D., Multiple zeta values and modular forms in quantum field theory, in Computer Algebra in Quantum Field Theory, Texts Monogr. Symbol. Comput., Springer, Vienna, 2013, 33-73.
  11. Broadhurst D.J., Kreimer D., Knots and numbers in $\phi^4$ theory to $7$ loops and beyond, Internat. J. Modern Phys. C 6 (1995), 519-524, arXiv:hep-ph/9504352.
  12. Broadhurst D.J., Kreimer D., Association of multiple zeta values with positive knots via Feynman diagrams up to $9$ loops, Phys. Lett. B 393 (1997), 403-412, arXiv:hep-th/9609128.
  13. Brown F., On the decomposition of motivic multiple zeta values, in Galois-Teichmüller theory and arithmetic geometry, Adv. Stud. Pure Math., Vol. 63, Math. Soc. Japan, Tokyo, 2012, 31-58, arXiv:1102.1310.
  14. Brown F., Feynman amplitudes, coaction principle, and cosmic Galois group, Commun. Number Theory Phys. 11 (2017), 453-556, arXiv:1512.06409.
  15. Brown F., Notes on motivic periods, Commun. Number Theory Phys. 11 (2017), 557-655, arXiv:1512.06410.
  16. Brown F., Doryn D., Framings for graph hypersurfaces, arXiv:1301.3056.
  17. Brown F., Dupont C., Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions, arXiv:1907.06603.
  18. Brown F., Kreimer D., Angles, scales and parametric renormalization, Lett. Math. Phys. 103 (2013), 933-1007, arXiv:1112.1180.
  19. Brown F., Schnetz O., A K3 in $\phi^4$, Duke Math. J. 161 (2012), 1817-1862, arXiv:1006.4064.
  20. Brown F., Schnetz O., Single-valued multiple polylogarithms and a proof of the zig-zag conjecture, J. Number Theory 148 (2015), 478-506, arXiv:1208.1890.
  21. Caron-Huot S., Dixon L.J., Drummond J.M., Dulat F., Foster J., Gürdougan O., von Hippel M., McLeod A.J., Papathanasiou G., The Steinmann cluster bootstrap for ${\mathcal N}=4$ super Yang-Mills amplitudes, PoS Proc. Sci. (2020), PoS(CORFU2019), 003, 37 pages, arXiv:2005.06735.
  22. Caron-Huot S., Dixon L.J., Dulat F., von Hippel M., McLeod A.J., Papathanasiou G., The cosmic Galois group and extended Steinmann relations for planar ${\mathcal N}=4$ SYM amplitudes, J. High Energy Phys. 2019 (2019), no. 9, 061, 65 pages, arXiv:1906.07116.
  23. Colmez P., Serre J.P. (Editors), Correspondance Grothendieck-Serre, Documents Mathématiques (Paris), Vol. 2, Société Mathématique de France, Paris, 2001.
  24. De Rham G., Sur l'analysis situs des variétés à $n$ dimensions, 1931, available ar http://www.numdam.org/item?id=THESE_1931__129__1_0.
  25. Deligne P., Théorie de Hodge. I, in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, 1971, 425-430.
  26. Deligne P., Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5-57.
  27. Deligne P., Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5-77.
  28. Deligne P., Goncharov A.B., Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. École Norm. Sup. (4) 38 (2005), 1-56, arXiv:math.NT/0302267.
  29. Deligne P., Milne J.S., Ogus A., Shih K., Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math., Vol. 900, Springer-Verlag, Berlin - New York, 1982.
  30. Demazure M., Motifs des variétés algébriques, in Séminaire Bourbaki, Vol. 1991/92, Exp. No. 364-381, Lecture Notes in Math., Vol. 180, Springer-Verlag, Berlin - Heidelberg, 1971, 19-38.
  31. Dyson F.J., The radiation theories of Tomonaga, Schwinger, and Feynman, Phys. Rev. 75 (1949), 486-502.
  32. Dyson F.J., Divergence of perturbation theory in quantum electrodynamics, Phys. Rev. 85 (1952), 631-632.
  33. Elvang H., Huang Y., Scattering amplitudes, arXiv:1308.1697.
  34. Feynman R.P., Space-time approach to quantum electrodynamics, Phys. Rev. 76 (1949), 769-789.
  35. Gil J.I.B., Fresán J., Multiple zeta values: from numbers to motives, available at https://javier.fresan.perso.math.cnrs.fr/mzv.pdf.
  36. Golz M., Parametric quantum electrodynamics, Ph.D. Thesis, Humboldt University, Berlin, 2018.
  37. Grothendieck A., On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 95-103.
  38. Hartshorne R., Algebraic geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York - Heidelberg, 1977.
  39. Henn J.M., Lectures on differential equations for Feynman integrals, J. Phys. A: Math. Theor. 48 (2015), 153001, 35 pages, arXiv:1412.2296.
  40. Hironaka H., Resolution of singularities of an algebraic variety over a field of characteristic zero. I, Ann. of Math. 79 (1964), 109-203.
  41. Hironaka H., Resolution of singularities of an algebraic variety over a field of characteristic zero. II, Ann. of Math. 79 (1964), 205-326.
  42. Hodge W.V.D., The theory and applications of harmonic integrals, Cambridge University Press, Cambridge, Macmillan Company, New York, 1941.
  43. Huber A., Müller-Stach S., Periods and Nori motives, textitErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Vol. 65, Springer, Cham, 2017.
  44. Kashiwara M., Schapira P., Categories and sheaves, Grundlehren der Mathematischen Wissenschaften, Vol. 332, Springer-Verlag, Berlin, 2006.
  45. Kaufmann R.M., Ward B.C., Feynman categories, Astérisque 387 (2017), vii+161 pages, arXiv:1312.1269.
  46. Kleiman S.L., Motives, in Algebraic Geometry, Oslo 1970 (Proc. Fifth Nordic Summer-School in Math., Oslo, 1970), 1972, 53-82.
  47. Kontsevich M., Zagier D., Periods, in Mathematics Unlimited - 2001 and Beyond, Springer, Berlin, 2001, 771-808.
  48. Kreimer D., Renormalization and knot theory, J. Knot Theory Ramifications 6 (1997), 479-581, arXiv:hep-th/9412045.
  49. Laporta S., High-precision calculation of the 4-loop contribution to the electron $g-2$ in QED, Phys. Lett. B 772 (2017), 232-238, arXiv:1704.06996.
  50. Laporta S., Remiddi E., The Analytical value of the electron $g-2$ at order $\alpha^3$ in QED, Phys. Lett. B 379 (1996), 283-291, arXiv:hep-ph/9602417.
  51. Murre J.P., Nagel J., Peters C.A.M., Lectures on the theory of pure motives, University Lecture Series, Vol. 61, Amer. Math. Soc., Providence, RI, 2013.
  52. Panzer E., Schnetz O., The Galois coaction on $\phi^4$ periods, Commun. Number Theory Phys. 11 (2017), 657-705, arXiv:1603.04289.
  53. Petermann A., Fourth order magnetic moment of the electron, Helv. Phys. Acta 30 (1957), 407-408.
  54. Saavedra Rivano N., Catégories Tannakiennes, Lecture Notes in Math., Vol. 265, Springer-Verlag, Berlin - New York, 1972.
  55. Salam A., Divergent integrals in renormalizable field theories, Phys. Rev. 84 (1951), 426-431.
  56. Salam A., Overlapping divergences and the $S$-matrix, Phys. Rev. 82 (1951), 217-227.
  57. Schlotterer O., Stieberger S., Motivic multiple zeta values and superstring amplitudes, J. Phys. A: Math. Theor. 46 (2013), 475401, 37 pages, arXiv:1205.1516.
  58. Schnetz O., Quantum periods: a census of $\phi^4$-transcendentals, Commun. Number Theory Phys. 4 (2010), 1-47, arXiv:0801.2856.
  59. Schnetz O., The Galois coaction on the electron anomalous magnetic moment, Commun. Number Theory Phys. 12 (2018), 335-354, arXiv:1711.05118.
  60. Schnetz O., Numbers and functions in quantum field theory, Phys. Rev. D 97 (2018), 085018, 20 pages, arXiv:1606.08598.
  61. Serone M., Spada G., Villadoro G., The power of perturbation theory, J. High Energy Phys. 2017 (2017), no. 5, 056, 41 pages, arXiv:1702.04148.
  62. Serre J.P., Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1956), 1-42.
  63. Shanks D., Dihedral quartic approximations and series for $\pi $, J. Number Theory 14 (1982), 397-423.
  64. Smirnov V.A., Feynman integral calculus, Springer-Verlag, Berlin, 2006.
  65. Srednicki M., Quantum field theory, Cambridge University Press, Cambridge, 2010.
  66. Stieberger S., Taylor T.R., Closed string amplitudes as single-valued open string amplitudes, Nuclear Phys. B 881 (2014), 269-287, arXiv:1401.1218.
  67. 't Hooft G., Veltman M., Regularization and renormalization of gauge fields, Nuclear Phys. B 44 (1972), 189-213.
  68. Voevodsky V., Triangulated categories of motives over a field, in Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud., Vol. 143, Princeton University Press, Princeton, NJ, 2000, 188-238.
  69. Voisin C., Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, Vol. 76, Cambridge University Press, Cambridge, 2002.
  70. Waldschmidt M., Transcendence of periods: the state of the art, Pure Appl. Math. Q. 2 (2006), 435-463.
  71. Weibel C.A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994.
  72. Weinberg S., High-energy behavior in quantum field-theory, Phys. Rev. 118 (1960), 838-849.
  73. Zee A., Quantum field theory in a nutshell, Princeton University Press, Princeton, NJ, 2003.

Previous article  Next article  Contents of Volume 17 (2021)