Large-$N_c$ and renormalization group constraints on parity-violating low-energy coefficients for three-derivative operators in pionless effective field theory

We extend from operators with one derivative to operators with three derivatives the analysis of two-body hadronic parity violation in a combined pionless effective field theory (EFT$_{pi!/}$) and large-$N_c$ expansion, where $N_c$ is the number of colors in quantum chromodynamics (QCD). In elastic scattering, these operators contribute to $S$-$P$ and $P$-$D$ wave transitions, with five operators and their accompanying low energy coefficients (LECs) characterizing the $S$-$P$ transitions and six operators and LECs those in $P$-$D$ transitions. We show that the large-$N_c$ analysis separates them into leading order in $N_c$, next-to-leading order in $N_c$, etc. Relationships among EFT$_{pi!/}$ LECs emerge in the large-$N_c$ expansion. We also discuss the renormalization scale dependence of these LECs. Our analysis can complement lattice QCD calculations and help prioritize future parity-violating experiments.