Does $\psi (4660)$ exist?

Abstract

The nature of the vector charmoniumlike states $\psi (4360)$ and $\psi (4660)$ is controversial, and the existence of the $\psi (4660)$ has been questioned: the BESIII measurement of $e^{+}{e}^{-}\to {\Lambda }_c^{+}{\bar{\Lambda }}_c^{-}$ reveals a nearly flat line shape around 4.66 GeV with a nonzero value at threshold, in contrast to the near-threshold enhancement reported earlier by Belle. We address this puzzle in an S-wave coupled-channel framework for $e^{+}{e}^{-}$ annihilation in $\sqrt{s}\in [4.0,5.0]\mathrm{GeV}$. The formalism includes the open-charm channels ${\Lambda }_c^{+}{\bar{\Lambda }}_c^{-}$, ${\Xi }_c^{+}{\bar{\Xi }}_c^{-}$, ${\Xi }_c^0{\bar{\Xi }}_c^0$ (via SU(3) symmetry), the inelastic $\psi (2S){\pi }^{+}{\pi }^{-}$ channel mediated by $\sigma (500)$ and $f_0(980)$, and the bare $\psi (4360)$ and $\psi (4660)$ as the 4S and 5S excitations. Short-range contact potentials in the heavy-quark limit are used, and the amplitudes follow from solving the Lippmann-Schwinger equation. A combined fit to Belle and BESIII data on $e^{+}{e}^{-}\to {\Lambda }_c^{+}{\bar{\Lambda }}_c^{-}$ and $e^{+}{e}^{-}\to \psi (2S){\pi }^{+}{\pi }^{-}$ yields two poles, at $(4401.93\pm 16.96i)\mathrm{MeV}$ and $(4631.39\pm 62.96i)\mathrm{MeV}$, for the $\psi (4360)$ and $\psi (4660)$. Tracing the pole trajectories as the coupling to the bare states is switched off, we show that both poles originate from the bare $4S$ and $5S$ states strongly renormalized by coupled-channel dynamics. In particular, the $\psi (4660)$ emerges as a broad resonance about 58~MeV above the ${\Lambda }_c^{+}{\bar{\Lambda }}_c^{-}$ threshold; its flat line shape follows from this threshold interplay, providing evidence for its existence. Results are cutoff-robust.

arXiv comment17 pages, 8 figures