  The closed-form Br\"ukner $G$ matrix for nuclear matter is computed in the
$^1S_0$ channel of EFT($\not\!\!\pi$) and renormalized in nonperturbative
context. The nuclear medium environment yields additional constraints that are
consistent with off-shell $T$ matrix renormalization, keeping the power
counting intact and simplifying the running behaviors of the EFT couplings.
With the $G$ obtained we computed the energy per particle for neutron and
symmetric nuclear matter to demonstrate the physical relevance of certain
'physical' parameters that arise from nonperturbative renormalization. We also
explored the pairing phenomenon in $^1S_0$ channel by examining the poles of
the closed-form $G$ matrix with a given density, where again the physical
relevance of the same set of physical parameters are clearly illustrated.
