An Updated Study for $\Upsilon$ Production and Polarization at the Tevatron and LHC

Following the nonrelativistic QCD factorization scheme, by taking latest available measurement on $\chi_b(3P)$ into consideration, we present an updated study on the yield and polarization of $\Upsilon(1S,2S,3S)$ hadroproduction, and the fractions of $\chi_b(mP)$ feed-down in $\Upsilon(nS)$ production at QCD next-to-leading order. In the fitting, three schemes are applied with different choice of $\chi_b(mP)$ feed-down ratios and NRQCD factorization scale. The results can explain the measurements on yield very well as in our previous work. The polarization puzzle to $\Upsilon(3S)$ is now solved by considering the $\chi_b(3P)$ feed-down contributions. The ratio of $\sigma[\chi_{b2}(1P)]/\sigma[\chi_{b1}(1P)]$ measurements from CMS can also be reproduced in our prediction. Among the different schemes, the results show little difference, but there are sizeable difference for the fitted long-distance color-octet matrix elements. It may bring large uncertainty when the values are applied in other experiments such as in $ee,~ep$ colliders.