Extracting $\pi \pi$ $S$-wave scattering lengths from cusp effect in heavy quarkonium dipion transitions

Abstract

Charge-exchange rescattering ${\pi }^{+}{\pi }^{-}\to {\pi }^0{\pi }^0$ leads to a cusp effect in the ${\pi }^0{\pi }^0$ invariant mass spectrum of processes with ${\pi }^0{\pi }^0$ in the final state which can be used to measure $\pi \pi$ $S$-wave scattering lengths. Employing a non-relativistic effective field theory, we discuss the possibility of extracting the scattering lengths in heavy quarkonium ${\pi }^0{\pi }^0$ transitions. The transition ${\rm Y}(3S)\to {\rm Y}(2S){\pi }^0{\pi }^0$ is studied in details. We discuss the precision that can be reached in such an extraction for a certain number of events.

Comments: 15 pages, 7 figures, 1 table