Abstract
The associate $J/\ensuremath{\psi}+\ensuremath{\gamma}$ production at the LHC is studied completely at next-to-leading order within the framework of nonrelativistic QCD. By using three sets of color-octet long-distance matrix elements obtained in previous prompt $J/\ensuremath{\psi}$ studies, we find that only one of them can result in a positive transverse momentum (${p}_{t}$) distribution of $J/\ensuremath{\psi}$ production rate at the large ${p}_{t}$ region. Based on reasonable consideration to cut down background, our estimation is measurable up to ${p}_{t}=50\text{ }\text{ }\mathrm{GeV}$ with the present data sample collected at 8 TeV LHC. All the color-octet long-distance matrix elements in $J/\ensuremath{\psi}$ production could be fixed sensitively by including this proposed measurement and our calculation, and then a confident conclusion on the $J/\ensuremath{\psi}$ polarization puzzle could be achieved.
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