I was reading QFT & Standard Model by Schwartz, Chapter 20 which is about IR divergences. He says that
IR divergences only cancel cross sections for processes involving different initial or final states are combined …… Although the cross section for the $2rightarrow2$ process is IR divergent at order $e{_R}^{4}$, as is the cross section for the related $2rightarrow3$ process (like $e^{+}e^{-}rightarrowmu^{+}mu^{-} gamma$), their sum is IR finite.
I’d like to know how we know the next-to-leading diagrams’ cross sections cancel IR divergences? Also, I’d be appreciate if someone could mention why we need cross sections for cancellation, indeed? Why S matrix not enough?