Peskin & Schröder say on page 216:
The poles in $p^2$ come only from one-particle intermediate states, while multiparticle intermediate states give weaker branch cut singularities.
In order to figure out what "branch cut singularities" are in this context, I have come across many terms like "branch poins", "branch cut" or "branch point which is a singularity", but I am still confused about what a branch cut singularity is.
I know what a branch cut is - a curve that you can not perform a integral around - but I don't understand how a branch cut could be a singularity. Furthermore, why should we examine its properties in a physical context? What is the physical meaning of such a branch cut singularity?