Spin dependence of associated production of a prompt photon and a charm quark at next-to-leading order in QCD

A second order, $O\left({\alpha }_s^2\right),$ calculation in perturbative quantum chromodynamics is presented of the longitudinal spin dependence of the cross section for the two particle inclusive reaction $p+\overrightarrow{p}\gamma +c+X$ for large values of the transverse momentum of the prompt photon and charm quark. Differential distributions are provided for the spin-averaged cross section and for the two-spin longitudinal polarization asymmetry $A_{\mathrm{LL}}$ at the energy of the Brookhaven Relativistic Heavy Ion Collider. An assessment is given of the prospects for determination of the spin dependence of the charm quark density.