The elliptic dilogarithm for the sunset graph

By Spencer Bloch, Pierre Vanhove. 2013

272 citations recorded by the source corpus.

Abstract We study the sunset graph defined as the scalar two-point self-energy at two-loop order. We evaluated the sunset integral for all identical internal masses in two dimensions. We give two calculations for the sunset amplitude; one based on an interpretation of the amplitude 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 amplitude in this case is a family of periods associated to the universal family of elliptic curves over the modular curve X 1 ( 6 ) . We show that the integral is given by an elliptic dilogarithm evaluated at a sixth root of unity modulo periods. We explain as well how this elliptic dilogarithm value is related to the regulator of a class in the motivic cohomology of the universal elliptic family.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/462d50f270dd5d82f63e9b4863b6f5b14393e7cc

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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].”
Lectures on differential equations for Feynman integrals (2014, unknown): “in [38, 39].”
Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph (2017, unknown): “05 46”
Local mirror symmetry and the sunset Feynman integral (2016, background): “^ 2(x) isnotinvariantunderx!xq(seeequation(3.72)), reﬂectingthemultivaluednatureoftheFeynmanintegral. Thiswasalreadythecase forellipticpolylogarithmexpansionsoftheFeynmanintegralsforthetwo-loopsunset [BV]andthree-loopbanana[BKV]withequalmasses. Theresultin(1.2)generalizes the expression for the all ”
Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral (2016, methodology): “…an impressive increase in our knowledge of the mathematical structures that appear in multiloop Feynman integrals, thanks to the combined use of various computational techniques, such as to the method of differential equations [1–3], the introduction of a class of special functions (dubbed…”
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.”