$NN$ scattering in $^3S_1-^3D_1$ channels of EFT ($\not\pi$)

The closed-form $T$ matrices in the $^3S_1$$-$$^3D_1$ channels of EFT($\not/!/!/pi$) for $NN$ scattering with the potentials truncated at order $\mathcal{O}(Q^4)$ are presented with the nonperturbative divergences parametrized in a general manner. The stringent constraints imposed by the closed form of the $T$ matrices are exploited in the underlying theory perspective and turned into virtues in the implementation of subtractions and the manifestation of power counting rules in nonperturbative regimes, leading us to the concept of EFT scenario. A number of scenarios of the EFT description of $NN$ scattering are compared with PSA data in terms of effective range expansion and $^3S_1$ phase shifts, showing that it is favorable to proceed in a scenario with conventional EFT couplings and sophisticated renormalization in order to have large $NN$ scattering lengths. The informative utilities of fine tuning are demonstrated in several examples and naturally interpreted in the underlying theory perspective. In addition, some of the approaches adopted in the recent literature are also addressed in the light of EFT scenario.