The associate production at the LHC is studied completely at next-to-leading-order (NLO) within the framework of nonrelativistic QCD. By using three sets of color-octet long-distance matrix elements (LDMEs) obtained in previous prompt studies, we find that only one of them can result in a positive transverse momentum () distribution of production rate at large region. Based on reasonable consideration to cut down background, our estimation is measurable upto GeV with present data sample collected at TeV LHC. All the color-octet LDMEs in production could be fixed sensitively by including this proposed measurement and our calculation, and then confident conclusion on polarization puzzle could be achieved.