On the structure of the pi pi invariant mass spectra of the Upsilon(4S)-->Upsilon(1S,2S) pi^+ pi^- decays

We perform a model-independent analysis for recently reported data of the $\pi^+\pi^-$ invariant mass spectra in the $\Upsilon(4S)\to\Upsilon(1S,2S)\pi^+\pi^-$ decays and point out that there does exist a broad peak below 0.6 GeV in the data of the $\Upsilon(4S)\to\Upsilon(1S)\pi^+\pi^-$ decay, which is analogous to that in the $\Upsilon(3S)\to\Upsilon(1S)\pi^+\pi^-$ decay. With the data of $\Upsilon(4S)$ decays, we further test our model developed for studying the puzzle in the $\Upsilon(3S)\to\Upsilon(1S)\pi^+\pi^-$ decay. The result shows that with such a model, all the $\pi^+\pi^-$ invariant mass spectra of $\Upsilon(4S)$ decays can be described. We also predict the $\cos\theta_{\pi}^*$ distributions of $\Upsilon(4S)\to\Upsilon(1S,2S)\pi^+\pi^-$ decays, which can be used to justify our model prediction.