  The leading order single inclusive soft gluon production in high energy
proton-nucleus (pA) collisions has been studied by various approaches for more
than two decades. The first correction due to the gluon saturation in proton
was analytically attempted recently through a diagrammatic approach in which
only partial results were obtained. An important feature of the first
saturation correction is that it includes both initial state and final state
interactions. In this paper, we present the complete result derived from the
Color Glass Condensate framework. Our approach is to analytically solve the
classical Yang-Mills equations in the dilute-dense regime and then use the
Lehmann-Symanzik-Zimmermann (LSZ) reduction formula to obtain gluon production
from classical gluon fields.
