Advertisement

Perturbative Quantum Field Theory Meets Number Theory

  • Ivan TodorovEmail author
Conference paper
  • 39 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 314)

Abstract

Feynman amplitudes are being expressed in terms of a well structured family of special functions and a denumerable set of numbers—periods, studied by algebraic geometers and number theorists. The periods appear as residues of the poles of regularized primitively divergent multidimensional integrals. In low orders of perturbation theory (up to six loops in the massless \(\varphi ^4\) theory) the family of hyperlogarithms and multiple zeta values (MZVs) serves the job. The (formal) hyperlogarithms form a double shuffle differential graded Hopf algebra. Its subalgebra of single valued multiple polylogarithms describes a large class of euclidean Feynman amplitudes. As the grading of the double shuffle algebra of MZVs is only conjectural, mathematicians are introducing an abstract graded Hopf algebra of motivic zeta values whose weight spaces have dimensions majorizing (hopefully equal to) the dimensions of the corresponding spaces of real MZVs. The present expository notes provide an updated version of 2014’s lectures on this subject presented by the author to a mixed audience of mathematicians and theoretical physicists in Sofia and in Madrid.

Keywords

Residue Transcendental Polylogarithm Shuffle Stuffle product Formal multizeta values Single-valued hyperlogarithm 

Notes

Acknowledgements

It is a pleasure to thank Francis Brown and Herbert Gangl for enlightening discussions at different stages of this work and Kurusch Ebrahimi-Fard for his invitation to the 2014 ICMAT Research Trimester. I thank IHES for its hospitality during the completion of these notes (January, 2016). The author’s work has been supported in part by Grant DFNI T02/6 of the Bulgarian National Science Foundation.

References

  1. 1.
    Abreu, S., Britto, R., Duhr, C., Gardi, E.: From multiple unitarity cuts to the coproduct of Feynman integrals. arXiv:1401.3546v2 [hep-th]
  2. 2.
    Adams, L., Bogner, C., Weinzierl, S.: The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms. J. Math. Phys. 55, 102301 (2014). arXiv:1405.5640 [hep-ph]; see also arXiv:1504.03255, arXiv:1512.05630 [hep-ph]
  3. 3.
    Askey, R.: Polylogarithms and associated functions, by Leonard Lewin. Bull. Amer. Math. Soc. 6(2), 248–251 (1982)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ayoub, R.: Euler and the zeta function. Amer. Math. Monthly 81, 1067–1086 (1974)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bloch, S.: Applications of the dilogarithm function in algebraic K-theory and algebraic geometry, In: Proceedings of the Internat. Symposium on Algebraic Geometry. Kinokuniya, Tokyo (1978)Google Scholar
  6. 6.
    Bloch, S.: Feynman amplitudes in mathematics and physics, August 2014 lectures at ICMAT, Madrid. arXiv:1509.00361 [math.AG]
  7. 7.
    Bloch, S., Esnault, H., Kreimer, D.: On motives and graph polynomials. Commun. Math. Phys. 267, 181–225 (2006). [math/0510011]Google Scholar
  8. 8.
    Bloch, S., Kerr, M., Vanhove, P.: A Feynman integral via higher normal functions. arXiv:1406.2664v3 [hep-th]
  9. 9.
    Bloch, S., Kreimer, D.: Mixed Hodge structures and renormalization in physics. Commun. Number Theory Phys. 2, 637–718 (2008). arXiv:0804.4399 [hep-th]; Feynman amplitudes and Landau singularities for 1-loop graphs, arXiv:1007.0338 [hep-th]
  10. 10.
    Bloch, S., Vanhove, P.: The elliptic dilogarithm for the sunset graph. J. Number Theory 148, 328–364 (2015). arXiv:1309.5865 [hep-th]
  11. 11.
    Bogner, C., Weinzierl, S.: Periods and Feynman integrals. J. Math. Phys. 50, 042302 (2009). arXiv:0711.4863v2 [hep-th]
  12. 12.
    Bogoliubov, N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields, 3d edn. Wiley (1980) (first Russian edition, 1957)Google Scholar
  13. 13.
    Broadhurst, D.J.: Feynman’s sunshine numbers. arXiv:1004.4238 [physics.pop-ph]
  14. 14.
    Broadhurst, D.J.: Multiple Deligne values: a data mine with empirically tamed denominators. arXiv:1409.7204 [hep-th]
  15. 15.
    Broadhurst, D.J., Kreimer, D.: Knots and numbers in \(\phi ^4\) to 7 loops and beyond. Int. J. Mod. Phys. 6C 519–524 (1995); Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops. Phys. Lett. B393, 403–412 (1997). [hep-th/9609128]Google Scholar
  16. 16.
    Broadhurst, D.J., Schnetz, O.: Algebraic geometry informs perturbative quantum field theory. Proc. Sci. 211, 078 (2014). arXiv:1409.5570
  17. 17.
    Brown, F.: Single-valued hyperlogarithms and unipotent differential equations. IHES notes (2004)Google Scholar
  18. 18.
    Brown, F.: Single valued multiple polylogarithms in one variable. C.R. Acad. Sci. Paris Ser. I 338, 522–532 (2004)Google Scholar
  19. 19.
    Brown, F.: Iterated integrals in quantum field theory. In: Cardona, A. et al. (eds.) Geometric and Topological Methods for Quantum Field Theory, Proceedings of the 2009 Villa de Leyva Summer School, pp.188–240. Cambridge Univ. Press (2013)Google Scholar
  20. 20.
    Brown, F.: On the decomposition of motivic multiple zeta values. Adv. Stud. Pure Math. 63, 31–58 (2012). arXiv:1102.1310v2 [math.NT]
  21. 21.
    Brown, F.: Mixed Tate motives over \({\mathbb{Z}}\). Annals of math. 175, 949–976 (2012). arXiv:1102.1312 [math.AG]
  22. 22.
    Brown, F.: Single-valued periods and multiple zeta values. arXiv:1309.5309 [math.NT]
  23. 23.
    Brown, F.: Motivic periods and \({\mathbb{P}}^1\setminus \{0, 1, \infty \}\). In: Jang, S.Y. et al. (eds.) Proc. ICM, Invited Lectures II, 295–318. Seoul (2014). arXiv:1407.5165 [math.NT]; see also arXiv:1512.09265 [math-ph]
  24. 24.
    Brown, F.: Multiple modular values for \(SL(2,{\mathbb{Z}})\). arXiv:1407.5167
  25. 25.
    Brown, F.: Zeta elements of depth 3 and the fundamental Lie algebra of a punctured elliptic curve. arXiv:1504.04737 [math.NT]
  26. 26.
    Brown, F.: Periods and Feynman amplitudes, Talk at the ICMP, Santiago de Chile. arXiv:1512.09265 [math-ph]; – Notes on motivic periods, arXiv:1512.06409v2 [math-ph], arXiv:1512.06410 [math.NT]
  27. 27.
    Brown, F., Levin, A.: Multiple elliptic polylogarithms. arXiv:1110.6917v2 [math.NT]
  28. 28.
    Brown, F., Schnetz, O.: A K3 in \(\phi ^4\). Duke Math. Jour. 161(10), 1817–1862 (2012). arXiv:1006.4064v5 [math.AG]
  29. 29.
    Brown, F., Schnetz, O.: Proof of the zig-zag conjecture. arXiv:1208.1890v2 [math.NT]
  30. 30.
    Brown, F., Schnetz, O.: Modular forms in quantum field theory. arXiv:1304.5342v2 [math.AG]
  31. 31.
    Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623–661 (2000). [math-ph/990328]Google Scholar
  32. 32.
    Carr, S., Gangl, H., Schneps, L.: On the Broadhurst-Kreimer generating series for multiple zeta values. In: Proceedings of the Madrid-ICMAT conference on Multizetas (2015)Google Scholar
  33. 33.
    Chen, K.T.: Iterated path integrals. Bull. Amer. Math. Soc. 83, 831–879 (1977)Google Scholar
  34. 34.
    Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann-Hilbert problem I, II. Commun. Math. Phys. 210, 249–273 (2000), 216, 215–241 (2001). [hep-th/9912092, hep-th/0003188]; Insertion and elimination: the doubly infinite algebra of Feynman graphs. Ann. Inst. Henri Poincaré 3, 411–433 (2002). [hep-th/0201157]Google Scholar
  35. 35.
    Deligne, P.: Multizetas d’aprés Francis Brown. Séminaire Bourbaki 64ème année, 1048Google Scholar
  36. 36.
    de Medeiros, P., Hollands, S.: Superconformal quantum field theory in curved spacetime. arXiv:1305.5191 [gr-qc]
  37. 37.
    Drummond, J., Duhr, C., Eden, B., Heslop, P., Pennington, J., Smirnov, V.A.: Leading singularities and off shell conformal amplitudes. JHEP 1308, 133 (2013). arXiv:1303.6909v2 [hep-th]
  38. 38.
    Drummond, J.M., Henn, J., Korchemsky, G.P., Sokatchev, E.: Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory. arXiv:0807.1095v2 [hep-th]
  39. 39.
    Drummond, J.M., Henn, J., Smirnov, V.A., Sokatchev, E.: Magic identities for conformal four-point integrals. JHEP 0701, 064 (2007). arXiv:hep-th/0607160
  40. 40.
    Duhr, C.: Mathematical aspects of scattering amplitudes. arXiv:1411.7538 [hep-ph]
  41. 41.
    Dütsch, M., Fredenhagen, K.: Causal perturbation theory in terms of retarded products, and a proof of the action Ward identity. Rev. Math. Phys. 16(10), 1291–1348 (2004). arXiv:hep-th/0403213v3MathSciNetCrossRefGoogle Scholar
  42. 42.
    Dyson, F.J.: Missed opportunities. Bull. Amer. Math. Soc. 78(5), 635–652 (1972)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Epstein, H., Glaser, V.: The role of locality in perturbation theory. Ann. Inst. H. Poincaré A19(3), 211–295 (1973)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Flory, M., Helling, R.C., Sluka, C.: How I learned to stop worrying and love QFT. arXiv:1201.2714
  45. 45.
    Gautschi, W.: Leonhard Euler: his life, the man, and his works. SIAM Rev. 50(1), 3–33 (2008)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Golden, J.K., Goncharov, A.B., Spradlin, M., Vergu, C., Volovich, A.: Motivic amplitudes and cluster coordinates. arXiv:1305.1617 [hep-th]; Golden, J.K., Spradlin, M.: The differential of all two-loop MHV amplitudes in N=4 Yang Mills theory. arXiv:1306.1833 [hep-th]
  47. 47.
    Goncharov, A.: Galois symmetry of fundamental groupoids and noncommutative geometry. Duke Math. J. 128(2), 209–284 (2005). [math/0208144v4]MathSciNetCrossRefGoogle Scholar
  48. 48.
    Gracia-Bondia, J.M., Gutierrez-Garro, H., Varilly, J.C.: Improved Epstein-Glaser renormalization in x-space. III Versus differential renormalization. Nucl. Phys. B886, 824–826 (2014). arXiv:1403.1785v3
  49. 49.
    Henn, J.M.: Lectures on differential equations for Feynman integrals. J. Phys. A. arXiv:1412.2296v3 [hep-ph]
  50. 50.
    Hollands, S., Wald, R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309–345 (2002). [gr-qc/0111108]Google Scholar
  51. 51.
    Kontsevich, M., Zagier, D.: Periods. In: Engquist, B., Schmid, W. (eds.) Mathematics - 20101 and beyond, pp. 771–808. Springer, Berlin (2001)Google Scholar
  52. 52.
    Källen, G., Sabry, A.: Forth order vacuum polarization. Dan. Mat. Fys. Med. 29(17), 1–20 (1955)zbMATHGoogle Scholar
  53. 53.
    Kinoshita, T.: Tenth-order QED contribution to the electron \(g-2\) and high precision test of quantum electrodynamics. In: Proceedings of the Conference in Honor of the 90th Birthday of Freeman Dyson, pp.148–172. World Scientific (2014)Google Scholar
  54. 54.
    Lagarias, J.C.: Euler’s constant: Euler’s work and modern developments. Bull. Amer. Math. Soc. 50(4), 527–628 (2013)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Laporta, S.: High precision calculation of the 4-loop contribution to the electron \(g-2\) in QED. arXiv:1704.06996 [hep-ph]
  56. 56.
    Laporta, S., Remiddi, E.: The analytical value of the electron \(g-2\) at order \(\alpha ^3\) in QED. Phys. Lett. B379, 283–291 (1996). arXiv:hep-ph/9602417
  57. 57.
    Lautrup, B.E., Peterman, A., de Rafael, E.: Recent developments in the comparison between theory and experiment in quantum electrodynamics. Phys. Rep. 3(4), 193–260 (1972)CrossRefGoogle Scholar
  58. 58.
    Lewin, L.: Polylogarithms and Associated Functions, North Holland, Amsterdam (1981); Structural Properties of Polylogarithms, Mathematical Surveys and Monographs, vol. 37. AMS, Providence, R.I. (1991)Google Scholar
  59. 59.
    Maximon, L.C.: The dilogarithm function for complex argument. Proc. Roy. Soc. Lond. A 459, 2807–2819 (2003)Google Scholar
  60. 60.
    Milnor, J.W.: Hyperbolic geometry: the first 150 years. Bull. Amer. Math. Soc. 5(1) (1982)Google Scholar
  61. 61.
    Müller-Stach, S.: What is a period?. Not. AMS (2014). arXiv:1407.2388 [math.NT]
  62. 62.
    Nikolov, N.M., Stora, R., Todorov, I.: Euclidean configuration space renormalization, residues and dilation anomaly. In: Dobrev, V.K. (eds.) Lie Theory and Its Applications in Physics (LT9), pp. 127–147. Springer, Japan, Tokyo (2013). CERN-TH-PH/2012-076, LAPTH-Conf-016/12Google Scholar
  63. 63.
    Nikolov, N.M., Stora, R., Todorov, I.: Renormalization of massless Feynman amplitudes as an extension problem for associate homogeneous distributions. Rev. Math. Phys. 26(4), 1430002 (65 pages) (2014). CERN-TH-PH/2013-107; arXiv:1307.6854 [hep-th]
  64. 64.
    Panzer, E.: Feynman integrals via hyperlogarithms. Proc. Sci. 211, 049 (2014). arXiv:1407.0074 [hep-ph]; Feynman integrals and hyperlogarithms, 220 pp. Ph.D. thesis. arXiv:1506.07243 [math-ph]
  65. 65.
    Remiddi, E., Vermaseren, J.A.M.: Harmonic polylogarithms. Int. J. Mod. Phys. A15, 725–754 (2000). arXiv:hep-ph/9905237
  66. 66.
    Rosner, J.: Sixth order contribution to \(Z_3\) in finite quantum electrodynamics. Phys. Rev. Letters 17(23), 1190–1192 (1966)CrossRefGoogle Scholar
  67. 67.
    Schneps, L.: Survey of the theory of multiple zeta values (2011)Google Scholar
  68. 68.
    Schnetz, O.: Natural renormalization. J. Math. Phys. 38, 738-758 (1997). arXiv:9610025
  69. 69.
    Schnetz, O.: Quantum periods: a census of \(\phi ^4\) transcendentals. Commun. Number Theory Phys. 4(1), 1–48 (2010). arXiv:0801.2856v2
  70. 70.
    Schnetz, O.: Graphical functions and single-valued multiple polylogarithms. Commun. Number Theory Phys. 8(4), 589-685 (2014). arXiv:1302.6445v2 [math.NT]
  71. 71.
    Steuding, J.: An Introduction to the theory of L-functions. Course Giv. Madr 06 (2005)Google Scholar
  72. 72.
    Styer, D.: Calculation of the anomalous magnetic moment of the electron (2012) (available electronically)Google Scholar
  73. 73.
    Todorov, I.: Polylogarithms and multizeta values in massless Feynman amplitudes, In: Dobrev, V. (eds.) Lie Theory and Its Applications in Physics (LT10). Springer Proceedings in Mathematics and Statistics, vol. 111, pp. 155–176. Springer, Tokyo (2014). Bures-sur-Yvette, IHES/P/14/10Google Scholar
  74. 74.
    Todorov, I.: Hyperlogarithms and periods in Feynman amplitudes, Chapter 10. In: Dobrev, V.K. (eds.) Springer Proceedings in Mathematics and Statistics, International Workshop on Lie Theory and Its Applications in Physics (LT-11), vol. 191, pp. 151–167 June 2015, Varna, Bulgaria. Springer, Tokyo-Heidelberg (2016). arXiv:1611.09323 [math-ph]
  75. 75.
    Todorov, I.: Renormalization of position space amplitudes in a massless QFT, PEPAN 48(2), 227–236 (2017) (Special Issue); (see also CERN-PH-TH-2015-016)Google Scholar
  76. 76.
    Ussyukina, N.I., Davyddychev, A.I.: An approach to the evaluation of 3- and 4-point ladder diagrams. Phys. Letters B 298, 363–370 (1993)Google Scholar
  77. 77.
    Waldschmidt, M.: Lectures on multiple zeta values. IMSc, Chennai (2011)Google Scholar
  78. 78.
    Weil, A.: Number Theory-An Approach through history from Hammurapi to Legendre. Birkhäuser, Basel 1983 (2007)Google Scholar
  79. 79.
    Weil, A.: Prehistory of the zeta-function. Number Theory, Trace Formula and Discrete Groups, pp. 1–9. Academic Press, N.Y (1989)Google Scholar
  80. 80.
    Zagier, D.: Introduction to modular forms. In: From Number Theory to Physics, pp. 238–291. Springer, Berlin (1992)(Les Houches,1989)Google Scholar
  81. 81.
    Zagier, D.: The dilogarithm function. In: Frontiers in Number Theory, Physics and Geometry II, pp. 3–65.Springer, Berlin (2006)Google Scholar
  82. 82.
    Zhao, J.: Multiple Polylogarithms. In: Notes for the Workshop Polylogarithms as a Bridge between Number Theory and Particle Physics, Durham, July 3–13 2013Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance
  2. 2.Institute for Nuclear Research and Nuclear EnergySofiaBulgaria

Personalised recommendations