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Combined heavy-quark symmetry and large-$N_c$ operator analysis for 2-body counterterms in the chiral Lagrangian with $D$ mesons and charmed baryons

We construct, in the work, chiral $SU(3)$ Lagrangian with $D$ mesons of spin $J^P=0^-$ and $J^P=1^-$ and charmed baryons of spin $J^P=1/2^+$ and $J^P=3/2^+$. There are 42 leading two-body counter-terms involving two charmed baryon fields and two $D$ meson fields in the constructed Lagrangian. The heavy-quark spin symmetry leads to 35 sum rules while the large-$N_c$ operator analysis predicts 29 ones at the next-to leading order of $1/N_c$ expansion. The combination of the sum rules from both the heavy-quark symmetry and the large-$N_c$ analysis results in 38 independent sum rules which reduces the number of free parameters in the chiral Lagrangian down to 4 only. This is a remarkable result demonstrating the consistency of the heavy-quark symmetry and large-$N_c$ operator analysis.