# Chromopolarizabilities of bottomonia from the $\Upsilon(2S,3S,4S) \to \Upsilon(1S,2S)\pi\pi$ transitions

## TL;DR

This paper systematically analyzes bottomonium dipion transitions to extract chromopolarizabilities, considering various mechanisms and final-state interactions, providing insights into bottomonium interactions with light hadrons.

## Contribution

It introduces a model-independent method to extract bottomonium chromopolarizabilities from experimental data, including effects of Z_b exchange and final-state interactions.

## Key findings

- Extracted chromopolarizability $|eta_{ ext{bottomonium}}|$ values with uncertainties.
- Found Z_b exchange has a minor impact on the results.
- Provided numerical value for $|eta_{ ext{bottomonium}}|$ considering Z_b exchange.

## Abstract

The dipion transitions $\Upsilon(2S,3S,4S) \to \Upsilon(1S,2S)\pi\pi$ are systematically studied by considering the mechanisms of the hadronization of soft gluons, exchanging the bottomoniumlike $Z_b$ states, and the bottom-meson loops. The strong pion-pion final-state interaction, especially including the channel coupling to $K\bar{K}$ in the $S$-wave, is taken into account in a model-independent way using the dispersion theory. Through fitting to the available experimental data, we extract values of the transition chromopolarizabilities $|\alpha_{\Upsilon(mS)\Upsilon(nS)}|$, which measure the chromoelectric couplings of the bottomonia with soft gluons. It is found that the $Z_b$ exchange has a slight impact on the extracted chromopolarizablity values, and the obtained $|\alpha_{\Upsilon(2S)\Upsilon(1S)}|$ considering the $Z_b$ exchange is $(0.29\pm 0.20)~\text{GeV}^{-3}$. Our results could be useful in studying the interactions of bottomonium with light hadrons.