Large-$N_c$ constraints on parity-violating low-energy constants in three-derivative pionless effective field theory

A combination of pionless effective field theory and large-$N_c$ expansion has demonstrated great potential for analyzing few-body hadronic parity violation (PV) at low energies. Here we focus on three-derivative nucleon-nucleon interactions that contribute to $P$-$D$ transitions as well as corrections to $S$-$P$ transitions. We show that the large-$N_c$ expansion can systematically separate low-energy constants (LECs) into those that occur at leading order in $N_c$ and those that occur at next-to-leading order in $N_c$. Relationships among the PV LECs emerge at leading order in $N_c$. Details of their subtraction-point dependent behaviors based on the current understanding of parity-conserving LECs are also presented. Finally, we discuss how our analysis may impact the prioritization of future PV experiments and lattice QCD calculations in this sector.