Elliptic integral evaluations of Bessel moments and applications

We record what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. More generally, we develop formulae for integrals of products of six or fewer Bessel functions. In consequence, we are able to discover and prove closed forms for c₍, ₊: =₀^ tᵏ K₀ⁿ (t) dt with integers n=1, 2, 3, 4 and k0, obtaining new results for the even moments c₃, ₂₊ and c₄, ₂₊. We also derive new closed forms for the odd moments s₍, ₂₊+₁: =₀^ t^2k+1I₀^ (t) K₀^n-1 (t) dt with n=3, 4 and for t₍, ₂₊+₁: =₀^ t^2k+1I₀² (t) K₀^n-2 (t) dt with n=5, relating the latter to Green functions on hexagonal, diamond and cubic lattices. We conjecture the values of s₅, ₂₊+₁, make substantial progress on the evaluation of c₅, ₂₊+₁, s₆, ₂₊+₁ and t₆, ₂₊+₁ and report more limited progress regarding c₅, ₂₊, c₆, ₂₊+₁ and c₆, ₂₊. In the process, we obtain 8 conjectural evaluations, each of which has been checked to 1200 decimal places. One of these lies deep in 4- dimensional quantum field theory and two are probably provable by delicate combinatorics. There remains a hard core of five conjectures whose proofs would be most instructive, to mathematicians and physicists alike.