Pion transitions from \chi_{cJ} to \eta_c

The charmonium transitions from the $\chi_{cJ}$ states to $\eta_c$ with emission of one or two pions are considered. It is shown that the only processes of such type arising in the leading order in the QCD multipole expansion are the decays $\chi_{c1} \to \eta_c \pi^+ \pi^-$ and $\chi_{c0} \to \eta_c \pi^0$. The absolute rate of the latter decay, in spite of being isotopically suppressed, is by about an order of magnitude larger than that of the former, so that the branching fractions for the two processes are approximately equal, and are expected to be significantly below the recent upper bound reported by BESIII. The rates of both decays are also related to that of another similar transition $h_c \to J/\psi \pi^0$.