The hadronic decay widths of $\eta_{c}$ and $\psi(3.1)$ are estimated using
$$
\begin{aligned}
&\eta_{c}(c \bar{c}) \rightarrow n \mathrm{~g} \rightarrow \text { hadrons } \\
&\psi(c \bar{c}) \rightarrow n^{\prime} \mathrm{g} \rightarrow \text { hadrons }
\end{aligned}
$$
where $\mathrm{g}$ is a gluon and $n$ and $n^{\prime}$ are integers. These are $\mathrm{QCD}$ analogues of the QED process of (2.80). Show that the minimum values of $n$ and $n^{\prime}$ are 2 and 3 , respectively.
Properties of the potential between the $\mathrm{c}$ and $\overline{\mathrm{c}}$ can be inferred from the charmonium spectrum. In Chapter 1, we noted that at small c and $\overline{\mathrm{c}}$ separations, QCD predicts a Coulomb-type potential $-\alpha_{s} / r$, but that at large separation $r$, we expect a confining potential which increases with $r .$ A glance at the $1 S, 2 S$, and the "center of gravity" of the $P$ levels of Fig. $2.13$ shows that the potential is in fact somewhere between Coulomb (which has $2 S$ and $P$ degenerate) and an oscillator potential $V \sim r^{2}$ (which has the $P$ level halfway between $1 S$ and $2 S$ ). A naive potential, which is phenomenologically rather satisfactory, is
$$
V(r)=-\frac{4}{3} \frac{\alpha_{s}}{r}+a r
$$
where $a$ is a constant parameter and $\frac{4}{3}$ is the color factor associated with the quark-gluon coupling $\alpha_{s}$ [see (2.98)].
Let us now repeat the steps of constructing baryons, but this time include the c quark. Combining three basic quark multiplets, we find that the analogue of $(2.59)$ is
$$
\begin{gathered}
4 \otimes 4 \otimes 4=20 \oplus 20 \oplus 20 \oplus \overline{4} . \\
s \quad M_{s} \quad M_{A} \quad A
\end{gathered}
$$
Rather than to derive this decomposition, it is better at this stage to use the elegant techniques of group theory (Young tableaux); see, for example, Close (1979). Including spin, $(2.64)$, we can as before form the required symmetric spin-flavor ground state in two ways: either the symmetric 20 with a symmetric spin $\frac{3}{2}$ or a mixed-symmetry 20 with spin $\frac{1}{2}$ constructed in exact analogy to (2.68). Extracting the flavor multiplets from a superposition of three basic (quark) tetrahedra leads to the ground-state baryons of Fig. $2.14 \mathrm{a}$.
The spin- $\frac{1}{2}$ multiplet can be viewed as three $S U(3)$ octets propping each other up and based on the edges of a fourth $S U(3)$ octet. In fact, we do not need the elegance of group theory to enumerate the states. For example, the $C=1$ spin- $\frac{1}{2}$ baryons are cqq composites with $\mathrm{q}=\mathrm{u}, \mathrm{d}$, or $\mathrm{s}$. The qq decomposition is given in (2.58), namely,
$$
3 \otimes 3=6 \oplus \overline{3},
$$
and the states are shown in Fig. $2.14 \mathrm{~b}$. The lightest charmed baryons are the $\Sigma_{c}$ isospin triplet and $\Lambda_{c}^{+}$. The observed masses are
$$
m\left(\Lambda_{c}\right)=2.28 \mathrm{GeV}, \quad m\left(\Sigma_{c}\right)=2.44 \mathrm{GeV}
$$