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

Abstract

A proof is found for the elliptic integral evaluation of the Bessel moment $$M:={\int }_0^{\infty }t{I}_0^2(t)K_0^2(t)K_0(2t)\mathrm{d}t=1/12\mathbf{K}(\sin (\pi /12))\mathbf{K}(\cos (\pi /12))=\frac{{\Gamma }^6(\frac{1}{3})}{64{\pi }^2{2}^{2/3}}$$ 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 $${\int }_0^{\infty }(I_0(at)K_0(at)-\frac{2}{\pi }{K}_0(4at)K_0(t))K_0(t)\mathrm{d}t=0$$ with $a>0$, proven by analytic continuation of an identity from Bailey's work, and $${\int }_0^{\infty }t{I}_0(at)(I_0^3(at)K_0(8t)-\frac{1}{4{\pi }^2}{I}_0(t)K_0^3(t))\mathrm{d}t=0$$ with $2\ge a\ge 0$, proven by showing that a Feynman diagram in two spacetime dimensions generates the enumeration of staircase polygons in four dimensions.

13 pages, now includes staircase polygons and complex separatrices

References in corpus (2)

• Elliptic integral evaluations of Bessel moments
• Lectures on configuration space methods for sunrise-type diagrams

Cited by in corpus (2)

• Elliptic integral evaluations of Bessel moments
• Solutions by radicals at singular values k_N from new class invariants for N \equiv 3 mod 8