On the large-$N_c$ behavior of the $L_7$ coupling in $\chi$PT

It is shown that the usual large-$N_c$ counting of the coupling constant $L_7$ of the $\mathcal{O}(p^4)$ low-energy chiral $SU(3)$ Lagrangian \cite{GL85} is in conflict with general properties of QCD in the large-$N_c$ limit. The solution of this conflict within the framework of a chiral $U(3)$ Lagrangian is explained.