  We study the processes $e^+ e^- \to Y(4260) \to J/\psi \pi\pi(K\bar{K})$. The
strong final-state interactions, especially the coupled-channel ($\pi\pi$ and
$K\bar{K}$) final-state interaction in the $S$-wave are taken into account in a
model-independent way using dispersion theory. It is found that the light-quark
SU(3) octet state plays a significant role in these transitions, implying that
the $Y(4260)$ contains a large light-quark component. Our findings suggest that
the $Y(4260)$ is neither a hybrid nor a conventional charmonium state.
Furthermore, through an analysis of the ratio of the light-quark SU(3) octet
and singlet components, we show that the $Y(4260)$ does not behave like a pure
$\bar D D_1$ hadronic molecule as well.
