Combined large-$N_c$ and heavy-quark operator analysis for the chiral Lagrangian with charmed baryons

The chiral $SU(3)$ Lagrangian with charmed baryons of spin $J^P=1/2^{+}$ and $J^P=3/2^{+}$ is analyzed. We consider all counterterms that are relevant at next-to-next-to-next-to-leading order (${\mathrm{N}}^3\mathrm{LO}$) in a chiral extrapolation of the charmed baryon masses. At ${\mathrm{N}}^2\mathrm{LO}$ we find 16 low-energy parameters. There are three mass parameters for the antitriplet and the two sextet baryons, six parameters describing the meson-baryon vertices and seven symmetry breaking parameters. The heavy-quark spin symmetry predicts four sum rules for the meson-baryon vertices and degenerate masses for the two baryon sextet fields. Here a large-$N_c$ operator analysis at next-to-leading order (NLO) suggests the relevance of one further spin-symmetry breaking parameter. Going from ${\mathrm{N}}^2\mathrm{LO}$ to ${\mathrm{N}}^3\mathrm{LO}$ adds $53=17+36$ parameters. For the leading symmetry conserving two-body counterterms involving two baryon fields and two Goldstone boson fields we find 36 terms. While the heavy-quark spin symmetry leads to $36-16=20$ sum rules, an expansion in $1/N_c$ at NLO generates $36-7=29$ parameter relations. A combined expansion leaves six unknown parameters only. For the symmetry breaking counterterms we find 17 terms, for which there are $17-9=8$ sum rules from the heavy-quark spin symmetry and $17-5=12$ sum rules from a $1/N_c$ expansion at NLO.