The role of intermediate $\mathrm{\Delta }\mathrm{\Delta }$ states in nucleon-nucleon scattering in the large-$N_c$ and unitary limits, and $\mathrm{\Delta }\mathrm{\Delta }$ and $\mathrm{\Omega }\mathrm{\Omega }$ scattering

We explore potential explanations for why using large-$N_c$ ($N_c$ is the number of colors) scaling to determine the relative size of few-nucleon low-energy operators agrees with experiment even when dynamical $\mathrm{\Delta }$’s are not explicitly included. Given that the large-$N_c$ analysis is predicated on the nucleons and $\mathrm{\Delta }$’s being degenerate, this is a curious result. We show that for purely $S$-wave interactions the relationships dictated by large-$N_c$ scaling are unaffected whether the $\mathrm{\Delta }$ is included or not. In the case of higher partial waves that do not mix with $S$-waves, the impact of the $\mathrm{\Delta }$ is perturbative, which makes the agreement with naive ($\mathrm{\Delta }$-less) large-$N_c$ ordering unsurprising. For higher partial waves that mix with $S$-waves, the nucleon and $\mathrm{\Delta }$ would need to decouple to get agreement with naive large-$N_c$ ordering. We find all $NN$, $\mathrm{\Delta }N$, and $\mathrm{\Delta }\mathrm{\Delta }$ low energy coefficients for leading-order baryon-baryon scattering in $\mathrm{\Delta }$-full pionless effective field theory in terms of the two independent parameters dictated by the SU($2F$) spin-flavor symmetry that arises in the $N_c\to \mathrm{\infty }$ limit. Because of recent lattice QCD results and experimental interest, we extend our analysis to the three-flavor case to study $\mathrm{\Omega }\mathrm{\Omega }$ scattering. We show that in the unitary limit (where scattering lengths become infinite) one of the two SU($2F$) parameters is driven to zero, resulting in enhanced symmetries, which agree with those found in spin-1/2 entanglement studies.