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

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.