  A proof is found for the elliptic integral evaluation of the Bessel moment
$$M:=\int_0^\infty t I_0^2(t)K_0^2(t)K_0(2t) {\rm d}t ={1/12} {\bf
K}(\sin(\pi/12)){\bf K}(\cos(\pi/12))
=\frac{\Gamma^6(\frac13)}{64\pi^22^{2/3}}$$ 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 $$\int_0^\infty(I_0(a
t)K_0(a t)-\frac{2}{\pi} K_0(4a t) K_0(t))K_0(t) {\rm d}t=0$$ with $a>0$,
proven by analytic continuation of an identity from Bailey's work, and
$$\int_0^\infty t I_0(a t)(I_0^3(a t)K_0(8t)- \frac{1}{4\pi^2} I_0(t)K_0^3(t))
{\rm d}t=0$$ 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.
