Amplitude analysis of e⁺e^-→Υ(nS)π⁺π^- at $\sqrt{s}$ = 10.886 GeV

We report results on studies of the $e^{+}{e}^{-}$ annihilation into three-body ${\rm Y}\left(nS\right){\pi }^{+}{\pi }^{-}$ ($n=1,2,3$) final states including measurements of cross sections and the full amplitude analysis. The cross sections measured at $\sqrt{s}=10.866\mathrm{GeV}$ and corrected for the initial state radiation are $\sigma (e^{+}{e}^{-}\to {\rm Y}(1S\left){\pi }^{+}{\pi }^{-}\right)=(2.27\pm 0.12\pm 0.14)\mathrm{pb}$, $\sigma (e^{+}{e}^{-}\to {\rm Y}(2S\left){\pi }^{+}{\pi }^{-}\right)=(4.07\pm 0.16\pm 0.45)\mathrm{pb}$, and $\sigma (e^{+}{e}^{-}\to {\rm Y}(3S\left){\pi }^{+}{\pi }^{-}\right)=(1.46\pm 0.09\pm 0.16)\mathrm{pb}$. Amplitude analysis of the three-body ${\rm Y}\left(nS\right){\pi }^{+}{\pi }^{-}$ final states strongly favors $I^G\left(J^P\right)=1^{+}\left(1^{+}\right)$ quantum-number assignments for the two bottomonium-like $Z_b^{\pm }$ states, recently observed in the ${\rm Y}\left(nS\right){\pi }^{\pm }$ and $h_b\left(mP\right){\pi }^{\pm }$ ($m=1,2$) decay channels. The results are obtained with a $121.4{\mathrm{fb}}^{-1}$ data sample collected with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider.