The complete study on polarization of $\Upsilon(nS)$ hadroproduction at QCD next-to-leading order

TL;DR

This paper provides a comprehensive next-to-leading order QCD analysis of the polarization of $ S)$ bottomonium states in hadroproduction, comparing theoretical predictions with experimental data across different polarization frames.

Contribution

It offers the first complete NLO QCD analysis of $9S)$ polarization parameters using NRQCD, including extraction of long-distance matrix elements from data.

Findings

Good agreement with measured polarization parameters $ $ and $ $ in both frames.

Frame invariant parameter $ ilde{ }$ is consistent across frames.

Discrepancies remain between theoretical predictions and experimental data for $ ilde{ }$.

Abstract

Applying the nonrelativistic quantum chromodynamics factorization formalism to the $\Upsilon(1S,2S,3S)$ hadroproduction, a complete analysis on the polarization parameters $\lambda_{\theta}$, $\lambda_{\theta\phi}$, $\lambda_{\phi}$ for the production are presented at QCD next-to-leading order. With the long-distance matrix elements extracted from experimental data for the production rate and polarization parameter $\lambda_{\theta}$ of $\Upsilon$ hadroproduction, our results provide a good description for the measured parameters $\lambda_{\theta\phi}$ and $\lambda_{\phi}$ in both the helicity and the Collins-Soper frames. In our calculations the frame invariant parameter $\tilde{\lambda}$ is consistent in the two frames. Finally, it is pointed out that there are discrepancies for $\tilde{\lambda}$ between available experimental data and corresponding theoretical predictions.