For an excited state $D_s^{**+}$ of the $D_s^+$ meson, a possible decay is $$D_s^{**+} \rightarrow D_s^+ \pi^0 $$

For which of the $1P$ mesons, i.e. $1^1P_1, 1^3P_0, 1^3P_1,1^3P_2$, is this decay possible?


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.


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