  We describe a general formalism based on the partial-wave decomposition to
compute the iterative $s$-channel discontinuity of four-point amplitudes at any
loop order. As an application, we focus on the low-energy expansions of type I
and II superstring amplitudes. Besides providing new results for their leading
and sub-leading logarithmic contributions beyond genus one, our approach
elucidates the general structure of non-analytic threshold terms. In the case
of open strings, the use of orthogonal colour projectors allows us to
efficiently compute all contributions from different worldsheet topologies at a
given loop order.
