Elliptic integral representation of Bessel moments

We record and substantially extend 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. In particular, 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 cn,k := ∫∞₀ tkKⁿ₀(t) dt with integers n = 1, 2, 3, 4 and k ≽ 0, obtaining new results for the even moments c₃,₂k and c₄,₂k. We also derive new closed forms for the odd moments sn,₂k₊₁ := ∫∞₀ t²k⁺¹I₀(t)Kⁿ₀¹₀(t) dt with n = 3, 4 and for tn,₂k₊₁ := ∫∞ ₀ t²k⁺¹I²₀(t)Kⁿ⁻²₀(t)dt with n = 5, relating the latter to Green functions on hexagonal, diamond and cubic lattices. We conjecture the values of s₅,₂k₊₁, make substantial progress on the evaluation of c₅,₂k₊₁, s₆,₂k₊₁ and t₆,₂k₊₁ and report more limited progress regarding c₅,₂k, c₆,₂k₊₁ and c₆,₂k. In the process, we obtain eight conjectural evaluations, each of which has been checked to 1200 decimal places. One of these lies deep in four-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.