  Various features of the even order poles appearing in the S-matrices of
simply-laced affine Toda field theories are analysed in some detail. In
particular, the coefficients of first- and second-order singularities appearing
in the Laurent expansion of the S-matrix around a general $2N^{\rm th}$ order
pole are derived in a universal way using perturbation theory at one loop. We
show how to cut loop diagrams contributing to the pole into particular products
of tree-level graphs that depend on the on-shell geometry of the loop; in this
way, we recover the coefficients of the Laurent expansion around the pole
exploiting tree-level integrability properties of the theory. The analysis is
independent of the particular simply-laced theory considered, and all the
results agree with those obtained in the conjectured bootstrapped S-matrices of
the ADE series of theories.
