According to weak interaction theory, the dominant hadronic weak decay proceeds via the quark transmutations $\mathrm{c} \rightarrow \mathrm{s}$ and/or $\mathrm{u} \leftrightarrow \mathrm{d}$ (see Chapter 12). For example, an allowed charmed meson decay is $\mathrm{cu} \rightarrow \mathrm{s} \overline{\mathrm{d}}(\mathrm{u} \overline{\mathrm{u}})$.
Assuming that these, and only these, transmutations can occur, show that
$$
\mathrm{D}^0 \rightarrow \mathrm{K}^{-} \pi^{+} \quad \text { and } \quad \mathrm{K}^{-} \pi^{+} \pi^{+} \pi^{-}
$$
are allowed decay modes, but that
$$
\mathrm{D}^0 \rightarrow \pi^{+} \pi^{-}, \mathrm{K}^{+} \mathrm{K}^{-}, \mathrm{K}^{+} \pi^{-}, \quad \text { and } \mathrm{K}^{+} \pi^{-} \pi^{+} \pi^{-}
$$
are all forbidden, Further, show that $\mathrm{D}^{+} \rightarrow \mathrm{K}^{-} \pi^{+} \pi^{+}$is an allowed weak decay, but that $\mathrm{D}^{+} \rightarrow \mathrm{K}^{+} \pi^{+} \pi^{-}$is forbidden. This distinctive feature of $\mathrm{D}^{+}$ decays was in fact convincing evidence in the first ever observation of a charmed particle in 1976 , some 18 months after the revolutionary discovery of the "hidden" charm state $\psi(\mathrm{c} \tilde{\mathrm{C}})$.
Each meson multiplet contains a state. cē. of "hidden" charm. For the $J^P=0$ and $1^{-}$multiplets, it is $\eta_c(2.98)$ and the original $\psi(3.1)$, respectively. The states of the bound cẽ system can be compared with those of positronium $\mathrm{e}^{+} \mathrm{e}^{-}$. We speak of "charmonium." It is a particularly clean system to study and has revolutionized meson spectroscopy. States with $J^{P C}=1^{--}$can be directly produced $\left(\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow\right.$ virtual $\left.\gamma \rightarrow \mathrm{c} \overline{\mathrm{c}}\right)$; and, via their decays, other charmonium states can be identified. The observed states are shown in Fig. 2.13, labeled in the conventional spectroscopic manner ${ }^{2 S+1} L_J$, where $S, L$, and $J$ are, respectively, the total intrinsic spin, orbital angular momentum, and total angular momentum of the ce system. This is, of course, a nonrelativistic classification; it is the heavy mass of the $\mathrm{c}$ quark which makes it possible to use a nonrelativistic picture. We also show the $J^{P C}$ values of the states and note that the observations coincide with quark model expectations. The six $J^{P C}$ values listed in Table 2.2 are reproduced, except that the $1^{+-}\left(\right.$or $\left.{ }^1 P_1\right)$ state still awaits discovery. As in positronium, radial as well as orbital excitations are expected. In fact, the $2{ }^3 S$ and $3{ }^3 S$ excitations are seen directly as resonances in the cross section for $\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow$ hadrons (see Fig. 2.13).