  This article reports on some recent progresses in Bessel moments, which
represent a class of Feynman diagrams in 2-dimensional quantum field theory.
Many challenging mathematical problems on these Bessel moments have been
formulated as a vast set of conjectures, by David Broadhurst and collaborators,
who work at the intersection of high energy physics, number theory and
algebraic geometry. We present the main ideas behind our verifications of
several such conjectures, which revolve around linear and non-linear sum rules
of Bessel moments, as well as relations between individual Feynman diagrams and
critical values of modular $L$-functions.
