Abstract
We prove Laporta’s conjecture
which relates the 4-loop sunrise diagram in 2-dimensional quantum field theory to Watson’s integral for 4-dimensional hypercubic lattice. We also establish several related integral identities proposed by Laporta, including a reduction of the 4-loop sunrise diagram to special values of Euler’s gamma function and generalized hypergeometric series:
Similar content being viewed by others
Notes
It is arguable whether \( L(f_{4,6},2)\) should be counted as a closed-form evaluation in its own right. As one may recall, Bloch–Kerr–Vanhove [7] and Samart [30] have expressed the 3-loop sunrise diagram \( 2^3\int _0^\infty I_0(t)[K_0(t)]^4t{{{\,\mathrm{d}\,}}}t\) as \( \frac{12\pi }{\sqrt{15}}L(f_{3,15},2)\), for a modular form \(f_{3,15}(z)=[\eta (3z)\eta (5z)]^3+[\eta (z)\eta (15z)]^3\) of weight 3 and level 15. Meanwhile, according to the work of Rogers–Wan–Zucker [28], such a special L-value can be reduced to a product of gamma values at rational arguments, thus leaving us a formula \( 2^3\int _0^\infty I_0(t)[K_0(t)]^4t{{{\,\mathrm{d}\,}}}t=\frac{1}{30\sqrt{5}}\Gamma \left( \frac{1}{15} \right) \Gamma \left( \frac{2}{15} \right) \Gamma \left( \frac{4}{15} \right) \Gamma \left( \frac{8}{15} \right) \) (see [39, Theorem 2.2.2] for a simplified proof of this integral identity). At the time of writing, it is not clear to us if the special L-value \( L(f_{4,6},2)\) admits a similar reduction.
References
Bailey, D.H., Borwein, J.M., Broadhurst, D., Glasser, M.L.: Elliptic integral evaluations of Bessel moments and applications. J. Phys. A, 41(20):205203 (2008). arXiv:0801.0891v2 [hep-th]
Bailey, W.N.: Some transformations of generalized hypergeometric series, and contour integrals of Barnes’s type. Q. J. Math. 3, 168–182 (1932)
Bailey, W.N.: Some infinite integrals involving Bessel functions (II). J. Lond. Math. Soc. S1–11(1), 16–20 (1936)
Bateman, H.: Higher Transcendental Functions, vol I. McGraw-Hill, New York, NY (1953). (compiled by staff of the Bateman Manuscript Project: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, Francesco G. Tricomi, David Bertin, W. B. Fulks, A. R. Harvey, D. L. Thomsen, Jr., Maria A. Weber and E. L. Whitney)
Bateman, H.: Table of Integral Transforms, vol II. McGraw-Hill, New York, NY (1954). (compiled by staff of the Bateman Manuscript Project: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, Francesco G. Tricomi, David Bertin, W. B. Fulks, A. R. Harvey, D. L. Thomsen, Jr., Maria A. Weber and E. L. Whitney)
Berndt, B.C.: Ramanujan’s Notebooks (Part V). Springer, New York, NY (1998)
Bloch, Spencer, Kerr, Matt, Vanhove, Pierre: A Feynman integral via higher normal functions. Compos. Math. 151(12), 2329–2375 (2015). arXiv:1406.2664v3 [hep-th]
Borwein, J.M., Straub, A., Wan, J.: Three-step and four-step random walk integrals. Exp. Math. 22(1), 1–14 (2013)
Borwein, J.M., Straub, A., Wan, J., Zudilin, W.: Densities of short uniform random walks. Canad. J. Math. 64(5), 961–990 (2012). (With an appendix by Don Zagier) arXiv:1103.2995v2 [math.CA]
Broadhurst, D.: Elliptic integral evaluation of a Bessel moment by contour integration of a lattice Green function (2008). arXiv:0801.4813v3 [hep-th]
Broadhurst, D.: Multiple zeta values and modular forms in quantum field theory. In: Schneider, C., Blümlein, J. (eds.) Computer Algebra in Quantum Field Theory, Texts & Monographs in Symbolic Computation, pp. 33–73. Springer, Vienna (2013)
Broadhurst, D.: Feynman integrals, \({L}\)-series and Kloosterman moments. Commun. Number Theory Phys. 10(3), 527–569 (2016). arXiv:1604.03057v1 [physics.gen-ph]
Broadhurst, D: \({L}\)-series from Feynman diagrams with up to 22 loops. In: Workshop on Multi-loop Calculations: Methods and Applications, Paris, France, June 7, 2017. Séminaires Internationaux de Recherche de Sorbonne Universités. https://multi-loop-2017.sciencesconf.org/data/program/Broadhurst.pdf
Broadhurst, D.: Combinatorics of Feynman integrals. In: Combinatoire Algébrique, Résurgence, Moules et Applications, Marseille-Luminy, France, June 28, 2017. Centre International de Rencontres Mathématiques. http://library.cirm-math.fr/Record.htm?idlist=29&record=19282814124910000969
Broadhurst, D.: Feynman integrals, beyond polylogs, up to 22 loops. In: Amplitudes 2017, Edinburgh, Scotland, UK, July 12, 2017. Higgs Centre for Theoretical Physics. https://indico.ph.ed.ac.uk/event/26/contribution/21/material/slides/0.pdf
Broadhurst, D.: Combinatorics of feynman integrals. In: Programme on “Algorithmic and Enumerative Combinatorics”, Vienna, Austria, Oct 17, 2017. Erwin Schrödinger International Institute for Mathematics and Physics. http://www.mat.univie.ac.at/~kratt/esi4/broadhurst.pdf
Broadhurst, D.: Feynman integrals, \({L}\)-series and Kloosterman moments. In: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Zeuthen, Germany, Oct 23, 2017. KMPB Conference at DESY. https://indico.desy.de/getFile.py/access?contribId=3&resId=0&materialId=slides&confId=18291
Broadhurst, D., Mellit, A.: Perturbative quantum field theory informs algebraic geometry. In: Loops and Legs in Quantum Field Theory. PoS (LL2016) 079 (2016). https://pos.sissa.it/archive/conferences/260/079/LL2016_079.pdf
Glasser, M.L., Guttmann, A.J.: Lattice Green function (at \(0\)) for the \(4\)D hypercubic lattice. J. Phys. A 27(21), 7011–7014 (1994)
Glasser, M.L., Montaldi, E.: Staircase polygons and recurrent lattice walks. Phys. Rev. E 48, R2339–R2342 (1993)
Guttmann, A.J.: Lattice Green functions and Calabi–Yau differential equations. J. Phys. A: Math. Theor. 42, 232001 (2009)
Joyce, G.S.: Singular behaviour of the lattice Green function for the \(d\)-dimensional hypercubic lattice. J. Phys. A 36(4), 911–921 (2003)
Joyce, G.S., Zucker, I.J.: On the evaluation of generalized Watson integrals. Proc. Am. Math. Soc. 133(1), 71–81 (2005)
Laporta, S.: Analytical expressions of three- and four-loop sunrise Feynman integrals and four-dimensional lattice integrals. Int. J. Mod. Phys. A 23(31), 5007–5020 (2008). arXiv:0803.1007v4 [hep-ph]
Laporta, S.: High-precision calculation of the 4-loop contribution to the electron \(g-2\) in QED. Phys. Lett. B 772(Supplement C), 232–238 (2017). arXiv:1704.06996 [hep-th]
Meijer, C.S.: On the \({G}\)-function. II. Proc. Kon. Ned. Akad. v. Wetensch. 49, 344–356 (1946)
Nielsen, N.: Handbuch der Theorie der Gammafunktion. Teubner, Leipzig (1906)
Rogers, M., Wan, J.G., Zucker, I.J.: Moments of elliptic integrals and critical \({L}\)-values. Ramanujan J. 37(1), 113–130 (2015). arXiv:1303.2259v2 [math.NT]
Rogers, M.D.: New \(_5F_4\) hypergeometric transformations, three-variable Mahler measures, and formulas for \(1/\pi \). Ramanujan J. 18(3), 327–340 (2009). arXiv:0704.2438v4 [math.NT]
Samart, D.: Feynman integrals and critical modular \(L\)-values. Commun. Number Theory Phys. 10(1), 133–156 (2016). arXiv:1511.07947v2 [math.NT]
Slater, Lucy Joan: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966)
Stein, E.M., Shakarchi, R.: Complex Analysis. Princeton Lectures in Analysis, vol. II. Princeton University Press, Princeton, NJ (2003)
Vanhove, P.: The physics and the mixed Hodge structure of Feynman integrals. In: String-Math 2013, Proceedings of Symposia in Pure Mathematics, vol 88, pp. 161–194. American Mathematical Society, Providence, RI, (2014). arXiv:1401.6438 [hep-th]
Wan, J.G.: Moments of products of elliptic integrals. Adv. Appl. Math. 48, 121–141 (2012)
Watson, G.N.: Three triple integrals. Q. J. Math. 10, 266–276 (1939)
Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944)
Zhou, Y.: Kontsevich-Zagier integrals for automorphic Green’s functions. I. Ramanujan J. 38(2), 227–329 (2015). arXiv:1312.6352v4 [math.CA]
Zhou, Y.: Hilbert transforms and sum rules of Bessel moments. Ramanujan J. (2017). https://doi.org/10.1007/s11139-017-9945-y
Zhou, Y.: Wick rotations, Eichler integrals, and multi-loop Feynman diagrams. Comm. Number Theory Phys. 12(1), 127–192 (2018). arXiv:1706.08308 [math.NT]
Zhou, Y.: Wrońskian factorizations and Broadhurst–Mellit determinant formulae. Comm. Number Theory Phys. 12(2), 355–407 (2018). arXiv:1711.01829 [math.CA]
Zucker, I.J.: 70\(+\) Years of the Watson integrals. J. Stat. Phys. 145, 591–612 (2011)
Zudilin, W.: Very well-poised hypergeometric series and multiple integrals. Russ. Math. Surv., 57, 824–826 (2002). ==В. В. Зудилин. Совершенно уравновешенные гипергеометрические ряды и кратные интегралы. Успехи матем. наук, 57(4), 177–178 (2002)
Zudilin, W.: Arithmetic of linear forms involving odd zeta values. J. Théor. Nombres Bordeaux 16(1), 251–291 (2004)
Acknowledgements
A large proportion of this work has been assembled from my research notes on hypergeometric series, which were prepared at Princeton in 2012. I thank Prof. Weinan E (Princeton University and Peking University) for running a seminar on mathematical problems in quantum fields at Princeton, covering both 2-dimensional and \((4-\varepsilon )\)-dimensional theories. I am grateful to Dr. David Broadhurst for many fruitful communications on recent progress in the arithmetic properties of Feynman diagrams. In particular, I thank him for suggesting the challenging integral identity in (1.11).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research was supported in part by the Applied Mathematics Program within the Department of Energy (DOE) Office of Advanced Scientific Computing Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4).
Rights and permissions
About this article
Cite this article
Zhou, Y. On Laporta’s 4-loop sunrise formulae. Ramanujan J 50, 465–503 (2019). https://doi.org/10.1007/s11139-018-0090-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-018-0090-z