Elliptic integral evaluation of a Bessel moment by contour integration of a lattice Green function

TL;DR

The paper proves the elliptic-integral evaluation of the Bessel moment M by relating it to a hexagonal-lattice Green function and a pair of Bessel moments through contour integration and a network of modular transformations. Central to the approach are cubic and sesquiplicate modular transformations, Bailey’s reduction of Appell-series expressions, and representations in terms of lattice Green functions (diamond and honeycomb). The main result is M = (1/12)K_3K_3', accompanied by numerous related sum rules, identities, and connections to combinatorial lattice walk counts. This work uncovers deep links among Feynman diagrams, arithmetic-geometric means, and elliptic-integral evaluations, broadening the toolkit for evaluating challenging multi-loop integrals and their special values.

Abstract

A proof is found for the elliptic integral evaluation of the Bessel moment $$M:=\int_0^\infty t I_0^2(t)K_0^2(t)K_0(2t) {\rm d}t ={1/12} {\bf K}(\sin(π/12)){\bf K}(\cos(π/12)) =\frac{Γ^6(\frac13)}{64π^22^{2/3}}$$ resulting from an angular average of a 2-loop 4-point massive Feynman diagram, with one internal mass doubled. This evaluation follows from contour integration of the Green function for a hexagonal lattice, thereby relating $M$ to a linear combination of two more tractable moments, one given by the Green function for a diamond lattice and both evaluated by using W.N. Bailey's reduction of an Appell double series to a product of elliptic integrals. Cubic and sesquiplicate modular transformations of an elliptic integral from the equal-mass Dalitz plot are proven and used extensively. Derivations are given of the sum rules $$\int_0^\infty(I_0(a t)K_0(a t)-\frac{2}π K_0(4a t) K_0(t))K_0(t) {\rm d}t=0$$ with $a>0$, proven by analytic continuation of an identity from Bailey's work, and $$\int_0^\infty t I_0(a t)(I_0^3(a t)K_0(8t)- \frac{1}{4π^2} I_0(t)K_0^3(t)) {\rm d}t=0$$ with $2\ge a\ge0$, proven by showing that a Feynman diagram in two spacetime dimensions generates the enumeration of staircase polygons in four dimensions.