 Four-flavour meson multiplet

Four quark flavours in the defining representation of flavour SU(4) and their antiquarks give sixteen meson flavour states. The trace of $M^i{}_j=q^i{\bar{q}}_j$ is the trivial representation; its traceless part is the Adjoint representation. Under light-flavour SU(3), the full content is $8\oplus 3\oplus \overline{3}\oplus 21$. Relative to the light-flavour nonet, the new states are $u\bar{c},d\bar{c},s\bar{c},c\bar{u},c\bar{d},c\bar{s},c\bar{c}$. Inside the traceless fifteen-dimensional multiplet the additional singlet is instead $(u\bar{u}+d\bar{d}+s\bar{s}-3c\bar{c})/\sqrt{12}$. The pure $c\bar{c}$ state is a linear combination of this state and the full SU(4) singlet. The classification concerns flavour; it does not imply degenerate physical masses or fix spin.