Effects of $Z_b$ states and bottom meson loops on $Υ(4S) \to Υ(1S,2S) π^+π^-$ transitions
We study the dipion transitions $Υ(4S) \rightarrow Υ(nS) π^+π^-$ $(n=1,2)$. In particular, we consider the effects of the two intermediate bottomoniumlike exotic states $Z_b(10610)$ and $Z_b(10650)$ as well as bottom meson loops. The strong pion-pion final-state interactions, especially including channel coupling to $K\bar{K}$ in the $S$-wave, are taken into account model-independently by using dispersion theory. Based on a nonrelativistic effective field theory we find that the contribution from the bottom meson loops is comparable to those from the chiral contact terms and the $Z_b$-exchange terms. For the $Υ(4S) \rightarrow Υ(2S) π^+π^-$ decay, the result shows that including the effects of the $Z_b$-exchange and the bottom meson loops can naturally reproduce the two-hump behavior of the $ππ$ mass spectra. Future angular distribution data are decisive for the identification of different production mechanisms. For the $Υ(4S) \rightarrow Υ(1S) π^+π^-$ decay, we show that there is a narrow dip around 1 GeV in the $ππ$ invariant mass distribution, caused by the final-state interactions. The distribution is clearly different from that in similar transitions from lower $Υ$ states, and needs to be verified by future data with high statistics. Also we predict the decay width and the dikaon mass distribution of the $Υ(4S) \rightarrow Υ(1S) K^+ K^-$ process.