  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 $c_{n,k}:=\int_0^\infty t^k
K_0^n(t) {\rm d}t$ with integers $n=1,2,3,4$ and $k\ge0$, obtaining new results
for the even moments $c_{3,2k}$ and $c_{4,2k}$. We also derive new closed forms
for the odd moments $s_{n,2k+1}:=\int_0^\infty t^{2k+1}I_0^{}(t) K_0^{n-1}(t)
{\rm d}t$ with $n=3,4$ and for $t_{n,2k+1}:=\int_0^\infty t^{2k+1}I_0^2(t)
K_0^{n-2}(t) {\rm d}t$ 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.
