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Considering the defects of the previous work for estimating the anomalousproduction rates of $e^+e^-\to \Upsilon(1S)\pi^+\pi^-,\,\Upsilon(2S)\pi^+\pi^-$near the peak of the $\Upsilon(5S)$ resonance at $\sqrt s=10.87$ GeV [K.F. Chen{\it et al}. (Belle Collaboration), Phys. Rev. Lett. {\bf 100}, 112001 (2008)],we suggest a new scenario where the contributions from the direct dipiontransition and the final state interactions interfere to result in not only theanomalously large production rates, but also the lineshapes of the differentialwidths consistent with the experimental measurement when assuming the reactionsare due to the dipion emission of $\Upsilon(5S)$. At the end, we raise a newpuzzle that the predicted differential width$d\Gamma(\Upsilon(5S)\to\Upsilon(2S)\pi^+\pi^-)/d\cos\theta$ has a discrepanttrend from the data while other predictions are well in accord with the data.It should be further clarified by more accurate measurements carried by futureexperiments.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Dipion invariant mass distribution of the anomalous $\Upsilon(1S)  \pi^{+} \pi^{-}$ and $\Upsilon(2S) \pi^{+} \pi^{-}$ production near the peak  of $\Upsilon(10860)$