On the structure of the ππ invariant mass spectra of the ϒ(4S)→ϒ(1S,2S)π+π− decays

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