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

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