Possible meson decays

The decay $$D_s^{**+} rightarrow D_s^+ pi^0 $$ is a strong decay (flavors charm and strangeness being conserved). Thus, parity must be conserved.

The final state particles are both pseudo-scalars, and hence the product of their parities is positive. Thus, the final state parity equals $(-1)^{L_f}$, where $L_f$ is the final state orbital angular momentum. In turn, angular momentum conservation dictates that $L_f$ must equal the initial state total angular momentum $J_i$, since the final state particles both have spin zero (they are pseudo-scalars). But the initial state parity is $(-1)^{L_i + 1}$ = +1 for all the states. Thus, $J_i$ must be even, i.e., 0 or 2.

Thus, the decay is possible only for the $1^3P_0, 1^3P_2$ states.