  In a previous paper we proposed a factorization principle for the correlation
functions of Berenstein-Maldacena-Nastase (BMN) operators in free
$\mathcal{N}=4$ super-Yang-Mills theory. These correlators are conjectured to
described physical string amplitudes in an infinitely curved Ramond-Ramond
pp-wave background. There was a puzzle that the factorization seems to break
down for $S$-channel in the $2\rightarrow 2$ scattering process. Here we
resolve this puzzle by including some diagrams missed in the previous paper. We
also observe some interesting relations which further support the
interpretation of higher genus correlators as physical string loop amplitudes.
