= Solution
For a neutral $c\bar c$ <bound state>, <meson parity and charge conjugation> give $P=(-1)^{L+1}$ and $C=(-1)^{L+S}$. In spectroscopic notation $n\,{}^{2S+1}L_J$, the integer $n$ labels radial excitation, $L$ is <orbital angular momentum>, $S$ the total constituent <spin>, and $J$ their total <angular momentum>. The surviving level diagram contains eight states:
* $\eta_c$: $1\,{}^1S_0$, $(L,S,J)=(0,0,0)$, $0^{-+}$.
* $J/\psi$: $1\,{}^3S_1$, $(0,1,1)$, $1^{--}$.
* $h_c$: $1\,{}^1P_1$, $(1,0,1)$, $1^{+-}$.
* $\chi_{c0}$: $1\,{}^3P_0$, $(1,1,0)$, $0^{++}$.
* $\chi_{c1}$: $1\,{}^3P_1$, $(1,1,1)$, $1^{++}$.
* $\chi_{c2}$: $1\,{}^3P_2$, $(1,1,2)$, $2^{++}$.
* $\eta_c(2S)$: $2\,{}^1S_0$, $(0,0,0)$, $0^{-+}$.
* $\psi(2S)$: $2\,{}^3S_1$, $(0,1,1)$, $1^{--}$.
The <spin>-dependent splittings distinguish the singlet and triplet levels and the three triplet P levels. The radial excitation changes neither <parity> nor <charge conjugation>.
\Image[/past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2015-charmonium-levels.png]
{title=<Charmonium> assignments for the eight surviving source levels}
{description=Original redraw with approximate vertical positions from the intact right-hand source diagram. Positions are schematic; labels and quantum-number assignments, rather than precision masses, are the content.}
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