Understand
In order to explain the observed unusually large dipion transition rates of $\Upsilon(10870)$, the scalar resonance contributions in the re-scattering model to the dipion transitions of $\Upsilon(4S)$ and $\Upsilon(5S)$ are studied.
- Since the imaginary part of the re-scattering amplitude is expected to be dominant, the large ratios of the transition rates of $\Upsilon(10870)$, which is identified with $\Upsilon(5S)$, to that of $\Upsilon(4S)$ can be understood as mainly coming from the difference between the $p$-values in their decays into open bottom channels, and the ratios are estimated numerically to be about 200-600 with reasonable choices of parameters.
- The absolute and relative rates of $\Upsilon(5S)\to\Upsilon(1S,2S,3S)\pi^+\pi^-$ and $\Upsilon(5S)\to\Upsilon(1S)K^+K^-$ are roughly consistent with data.
- We emphasize that the dipion transitions observed for some of the newly discovered $Y$ states associated with charmonia may have similar features to the dipion transitions of $\Upsilon(5S)$.