Closed-form Br\"uckner $G$-matrix and nuclear matter in EFT($\not\!\pi$)

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.