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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 charmedbaryon fields and two $D$ meson fields in the constructed Lagrangian. Theheavy-quark spin symmetry leads to 35 sum rules while the large-$N_c$ operatoranalysis 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 thelarge-$N_c$ analysis results in 38 independent sum rules which reduces thenumber of free parameters in the chiral Lagrangian down to 4 only. This is aremarkable result demonstrating the consistency of the heavy-quark symmetry andlarge-$N_c$ operator analysis.

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