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Past exam of the particle physics course of the University of Oxford / 2015 / 3 / d / Solution by

Codex  0 2026-10-25

For a neutral

c \overset{c}{ˉ}

bound state, meson parity and charge conjugation give

P = (- 1)^{L + 1}

and

C = (- 1)^{L + S}

. In spectroscopic notation

n^{2 S + 1} L_{J}

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:

$η_{c}$ : $1^{1} S_{0}$ , $(L, S, J) = (0, 0, 0)$ , $0^{-+}$ .
$J / ψ$ : $1^{3} S_{1}$ , $(0, 1, 1)$ , $1^{--}$ .
$h_{c}$ : $1^{1} P_{1}$ , $(1, 0, 1)$ , $1^{+-}$ .
$χ_{c 0}$ : $1^{3} P_{0}$ , $(1, 1, 0)$ , $0^{++}$ .
$χ_{c 1}$ : $1^{3} P_{1}$ , $(1, 1, 1)$ , $1^{++}$ .
$χ_{c 2}$ : $1^{3} P_{2}$ , $(1, 1, 2)$ , $2^{++}$ .
$η_{c} (2 S)$ : $2^{1} S_{0}$ , $(0, 0, 0)$ , $0^{-+}$ .
$ψ (2 S)$ : $2^{3} S_{1}$ , $(0, 1, 1)$ , $1^{--}$ .

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.

Figure 1.
Charmonium assignments for the eight surviving source levels
. 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. Details.

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