Lawrence Berkeley National Laboratory

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(n,k) := Int_0 inf tk K_0 n(t) dt, with integers n = 1, 2, 3, 4 and k greater than or equal to 0, obtaining new results for the even moments c3,2k and c4,2k . We also derive new closed forms for the odd moments s(n,2k+1) := Int_0 inf t(2k+1) I_0(t) K_0n(t) dt,with n = 3, 4 and for t(n,2k+1) := Int_0 inf t(2k+1) I_02(t) K_0(n-2) dt, with n = 5, relating the latter to Green functions on hexagonal, diamond and cubic lattices. We conjecture the values of s(5,2k+1), make substantial progress on the evaluation of c(5,2k+1), s(6,2k+1) and t(6,2k+1) and report more limited progress regarding c(5,2k), c(6,2k+1) and c(6,2k). 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.