An Introduction to Motivic Feynman Integrals

Claudia Rella

Symmetry, integrability and geometry: methods and applications, Tome 17 (2021)

Résumé

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.

Détail

Keywords: scattering amplitudes, Feynman diagrams, multiple zeta values, Hodge structures, periods of motives, Galois theory, Tannakian categories.

Bibliographie

[1] Abreu S., Britto R., Duhr C., Gardi E., “Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case”, J. High Energy Phys., 2017:12 (2017), 090, 73 pp., arXiv: 1704.07931 | DOI | MR

[2] Abreu S., Britto R., Duhr C., Gardi E., Matthew J., “Coaction for Feynman integrals and diagrams”, PoS(LL2018), PoS Proc. Sci., 2018, 047, 14 pp., 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., 2020:2 (2020), 122, 44 pp., arXiv: 1910.08358 | DOI | MR | Zbl

[4] André Y., Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, 17, Société Mathématique de France, Paris, 2004 | MR

[5] Bloch S., Esnault H., Kreimer D., “On motives associated to graph polynomials”, Comm. Math. Phys., 267 (2006), 181–225, arXiv: math.AG/0510011 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[7] Bogner C., Weinzierl S., “Periods and Feynman integrals”, J. Math. Phys., 50 (2009), 042302, 16 pp., arXiv: 0711.4863 | DOI | MR | Zbl

[8] Bogner C., Weinzierl S., “Feynman graph polynomials”, Internat. J. Modern Phys. A, 25 (2010), 2585–2618, arXiv: 1002.3458 | DOI | MR | Zbl

[9] Bott R., Tu L.W., Differential forms in algebraic topology, Graduate Texts in Mathematics, 82, Springer-Verlag, New York – Berlin, 1982 | DOI | MR | Zbl

[10] Broadhurst D., “Multiple zeta values and modular forms in quantum field theory”, Computer Algebra in Quantum Field Theory, Texts Monogr. Symbol. Comput., Springer, Vienna, 2013, 33–73 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[13] Brown F., “On the decomposition of motivic multiple zeta values”, Galois–Teichmüller theory and arithmetic geometry, Adv. Stud. Pure Math., 63, Math. Soc. Japan, Tokyo, 2012, 31–58, arXiv: 1102.1310 | DOI | MR | Zbl

[14] Brown F., “Feynman amplitudes, coaction principle, and cosmic Galois group”, Commun. Number Theory Phys., 11 (2017), 453–556, arXiv: 1512.06409 | DOI | MR | Zbl

[15] Brown F., “Notes on motivic periods”, Commun. Number Theory Phys., 11 (2017), 557–655, arXiv: 1512.06410 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[19] Brown F., Schnetz O., “A K3 in $\phi^4$”, Duke Math. J., 161 (2012), 1817–1862, arXiv: 1006.4064 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[21] Caron-Huot S., Dixon L. J., Drummond J. M., Dulat F., Foster J., Gürdoğan O., von Hippel M., McLeod A. J., Papathanasiou G., “The Steinmann cluster bootstrap for ${\mathcal N}=4$ super Yang–Mills amplitudes”, PoS(CORFU2019), PoS Proc. Sci., 2020, 003, 37 pp., 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., 2019:9 (2019), 061, 65 pp., arXiv: 1906.07116 | DOI | MR

[23] Colmez P., Serre J. P. (eds), Correspondance Grothendieck–Serre, Documents Mathématiques (Paris), 2, Société Mathématique de France, Paris, 2001 | MR | Zbl

[24] De Rham G., Sur l'analysis situs des variétés à $n$ dimensions, , 1931 http://www.numdam.org/item?id=THESE_1931__129__1_0 | MR

[25] Deligne P., “Théorie de Hodge. I”, Actes du Congrès International des Mathématiciens (Nice, 1970), v. 1, 1971, 425–430 | MR | Zbl

[26] Deligne P., “Théorie de Hodge. II”, Inst. Hautes Études Sci. Publ. Math., 40 (1971), 5–57 | DOI | MR | Zbl

[27] Deligne P., “Théorie de Hodge. III”, Inst. Hautes Études Sci. Publ. Math., 44 (1974), 5–77 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[29] Deligne P., Milne J. S., Ogus A., Shih K., Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math., 900, Springer-Verlag, Berlin – New York, 1982 | DOI | MR | Zbl

[30] Demazure M., “Motifs des variétés algébriques” (1991/92), Séminaire Bourbaki, 364–381; Lecture Notes in Math., 180, Springer-Verlag, Berlin – Heidelberg, 1971, 19–38 | DOI

[31] Dyson F. J., “The radiation theories of Tomonaga, Schwinger, and Feynman”, Phys. Rev., 75 (1949), 486–502 | DOI | MR | Zbl

[32] Dyson F. J., “Divergence of perturbation theory in quantum electrodynamics”, Phys. Rev., 85 (1952), 631–632 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[35] Gil J. I.B., Fresán J., Multiple zeta values: from numbers to motives, 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 | DOI | MR | Zbl

[38] Hartshorne R., Algebraic geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York – Heidelberg, 1977 | DOI | MR | Zbl

[39] Henn J. M., “Lectures on differential equations for Feynman integrals”, J. Phys. A: Math. Theor., 48 (2015), 153001, 35 pp., arXiv: 1412.2296 | DOI | MR | Zbl

[40] Hironaka H., “Resolution of singularities of an algebraic variety over a field of characteristic zero. I”, Ann. of Math., 79 (1964), 109–203 | DOI | MR | Zbl

[41] Hironaka H., “Resolution of singularities of an algebraic variety over a field of characteristic zero. II”, Ann. of Math., 79 (1964), 205–326 | DOI | MR | Zbl

[42] Hodge W. V.D., The theory and applications of harmonic integrals, Cambridge University Press, Cambridge; Macmillan Company, New York, 1941 | MR

[43] Huber A., Müller-Stach S., Periods and Nori motives, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 65, Springer, Cham, 2017 | DOI | MR | Zbl

[44] Kashiwara M., Schapira P., Categories and sheaves, Grundlehren der Mathematischen Wissenschaften, 332, Springer-Verlag, Berlin, 2006 | DOI | MR | Zbl

[45] Kaufmann R. M., Ward B. C., Feynman categories, Astérisque, 387, 2017, vii+161 pp., arXiv: 1312.1269 | MR | Zbl

[46] Kleiman S. L., “Motives”, Algebraic Geometry, Proc. Fifth Nordic Summer-School in Math. (Oslo, 1970), 1972, 53–82 | MR

[47] Kontsevich M., Zagier D., “Periods”, Mathematics Unlimited – 2001 and Beyond, Springer, Berlin, 2001, 771–808 | DOI | MR | Zbl

[48] Kreimer D., “Renormalization and knot theory”, J. Knot Theory Ramifications, 6 (1997), 479–581, arXiv: hep-th/9412045 | DOI | MR | Zbl

[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 | DOI | MR

[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 | DOI

[51] Murre J. P., Nagel J., Peters C. A.M., Lectures on the theory of pure motives, University Lecture Series, 61, Amer. Math. Soc., Providence, RI, 2013 | DOI | MR | Zbl

[52] Panzer E., Schnetz O., “The Galois coaction on $\phi^4$ periods”, Commun. Number Theory Phys., 11 (2017), 657–705, arXiv: 1603.04289 | DOI | MR | Zbl

[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., 265, Springer-Verlag, Berlin – New York, 1972 | DOI | MR

[55] Salam A., “Divergent integrals in renormalizable field theories”, Phys. Rev., 84 (1951), 426–431 | DOI | MR | Zbl

[56] Salam A., “Overlapping divergences and the $S$-matrix”, Phys. Rev., 82 (1951), 217–227 | DOI | MR | Zbl

[57] Schlotterer O., Stieberger S., “Motivic multiple zeta values and superstring amplitudes”, J. Phys. A: Math. Theor., 46 (2013), 475401, 37 pp., arXiv: 1205.1516 | DOI | MR | Zbl

[58] Schnetz O., “Quantum periods: a census of $\phi^4$-transcendentals”, Commun. Number Theory Phys., 4 (2010), 1–47, arXiv: 0801.2856 | DOI | MR | Zbl

[59] Schnetz O., “The Galois coaction on the electron anomalous magnetic moment”, Commun. Number Theory Phys., 12 (2018), 335–354, arXiv: 1711.05118 | DOI | MR | Zbl

[60] Schnetz O., “Numbers and functions in quantum field theory”, Phys. Rev. D, 97 (2018), 085018, 20 pp., arXiv: 1606.08598 | DOI | MR

[61] Serone M., Spada G., Villadoro G., “The power of perturbation theory”, J. High Energy Phys., 2017:5 (2017), 056, 41 pp., arXiv: 1702.04148 | DOI | MR

[62] Serre J. P., “Géométrie algébrique et géométrie analytique”, Ann. Inst. Fourier (Grenoble), 6 (1956), 1–42 | DOI | MR | Zbl

[63] Shanks D., “Dihedral quartic approximations and series for $\pi $”, J. Number Theory, 14 (1982), 397–423 | DOI | MR | Zbl

[64] Smirnov V. A., Feynman integral calculus, Springer-Verlag, Berlin, 2006 | DOI | MR

[65] Srednicki M., Quantum field theory, Cambridge University Press, Cambridge, 2010 | MR

[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 | DOI | MR | Zbl

[67] 't Hooft G., Veltman M., “Regularization and renormalization of gauge fields”, Nuclear Phys. B, 44 (1972), 189–213 | DOI | MR

[68] Voevodsky V., “Triangulated categories of motives over a field”, Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud., 143, Princeton University Press, Princeton, NJ, 2000, 188–238 | MR | Zbl

[69] Voisin C., Hodge theory and complex algebraic geometry, v. I, Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, Cambridge, 2002 | DOI | MR | Zbl

[70] Waldschmidt M., “Transcendence of periods: the state of the art”, Pure Appl. Math. Q, 2 (2006), 435–463 | DOI | MR | Zbl

[71] Weibel C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994 | DOI | MR | Zbl

[72] Weinberg S., “High-energy behavior in quantum field-theory”, Phys. Rev., 118 (1960), 838–849 | DOI | MR | Zbl

[73] Zee A., Quantum field theory in a nutshell, Princeton University Press, Princeton, NJ, 2003 | MR | Zbl