2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/154134A 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.13 pages, now includes staircase polygons and complex separatricesHigh Energy Physics - Theorytext