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We compute the differential cross-section for inclusive prompt photon$+$quarkproduction in deeply inelastic scattering of electrons off nuclei at small $x$($e+A$ DIS) in the framework of the Color Glass Condensate effective fieldtheory. The result is expressed as a convolution of the leading order (in thestrong coupling $\alpha_{\mathrm{s}}$) impact factor for the process anduniversal dipole matrix elements, in the limit of hard photon transversemomentum relative to the nuclear saturation scale $Q_{s,A}(x)$. We perform anumerical study of this process for the kinematics of the Electron-Ion Collider(EIC), exploring in particular the azimuthal angle correlations between thefinal state photon and quark. We observe a systematic suppression andbroadening pattern of the back-to-back peak in the relative azimuthal angledistribution, as the saturation scale is increased by replacing proton targetswith gold nuclei. Our results suggest that photon+jet final states in inclusive$e+A$ DIS at high energies are in general a promising channel for exploringgluon saturation that is complementary to inclusive and diffractive dijetproduction. They also provide a sensitive empirical test of the universality ofdipole matrix elements when compared to identical measurements inproton-nucleus collisions. However because photon+jet correlations at small $x$in EIC kinematics require jet reconstruction at small $k_\perp$, it will beimportant to study their feasibility relative to photon-hadron correlations.

Inclusive prompt photon-jet correlations as a probe of gluon saturation in electron-nucleus scattering at small $x$