A Feynman integral via higher normal functions

By Spencer Bloch, M. Kerr, Pierre Vanhove. Compositio Mathematica, 2014

195 citations recorded by the source corpus.

We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two space-time dimensions. Two calculations are given for the Feynman integral: one based on an interpretation of the integral as an inhomogeneous solution of a classical Picard–Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a family of regulator periods associated to a family of $K3$ surfaces. We show that the integral is given by a sum of elliptic trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm value is related to the regulator of a class in the motivic cohomology of the $K3$ family. We prove a conjecture by David Broadhurst which states that at a special kinematical point the Feynman integral is given by a critical value of the Hasse–Weil $L$-function of the $K3$ surface. This result is shown to be a particular case of Deligne’s conjectures relating values of $L$-functions inside the critical strip to periods.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/31997a5c8b76fbaba64b17256b8d783c02a650b3

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Citations in the corpus

Feynman integrals and hyperlogarithms (2015, background): “Certainly this class is very rich, for it includes involved objects like elliptic polylogarithms [6, 24, 27] and special values of L-functions of modular forms [43].”
Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism (2017, background): “Our paper is not the first to consider elliptic generalisations of MPLs [14, 17, 19, 20, 24, 31, 45, 49, 50] or iterated integrals over integration kernels that involve square roots of polynomials of degree greater than two [51].”
Two-loop planar master integrals for Higgs → 3 partons with full heavy-quark mass dependence (2016, result): “See [62, 63, 67, 85] for results in terms of elliptic polylogarithms [86], and [64–66, 69] for a related class of functions.”
Reducing differential equations for multiloop master integrals (2014, background): “aster integrals the obstruction – 20 – to the reduction appears. This obstruction naturally correspondsto the fact that these master integrals can not be expressed in terms of harmonic polylogarithms [32]. The example presented in Section 8 did not require the reduction of the system to Fuchsian for”