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 $lambda_{theta}$, $lambda_{thetaphi}$, $lambda_{phi}$ 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 $lambda_{thetaphi}$ 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.