Remarks on the amplitude of the decay Upsilon(3S) -> Upsilon(1S) pi pi

TL;DR

This paper analyzes the amplitude of the decay Upsilon(3S) to Upsilon(1S) pi pi, highlighting inconsistencies in recent experimental form factor results and proposing a simple correlation measurement to resolve ambiguities.

Contribution

It critically examines the phase assumptions in recent form factor analyses and suggests a straightforward experimental approach to determine all relevant decay amplitude terms.

Findings

The reported complex phase is inconsistent with final state interaction models.

Polarization information is essential for complete amplitude analysis.

A correlation between pion momentum and beam axis can resolve form factor ambiguities.

Abstract

We point out some properties of the amplitude of the dipion transition $\Upsilon(3S) \to \Upsilon(1S) \pi \pi$ in relation to the recently reported results of a CLEO analysis of form factors in this amplitude. We find that the reported significant complex phase between two of the form factors under the assumption that the third form factor is zero, is not consistent with the picture where the phase shifts arise due to the final state interaction in the $\pi \pi$ channel. It is also shown that in an analysis that uses no information on the polarization of both the initial and the final $\Upsilon$ resonances it is impossible in principle to determine all the relevant terms. We suggest that a study of a simple correlation between the direction of the total momentum of the two pions and the axis of the initial beams is sufficient to resolve the ambiguity in the fit for the form factors.