Comparison of improved TMD and CGC frameworks in forward quark dijet production

For studying small-$x$ gluon saturation in forward dijet production in high-energy dilute-dense collisions, the improved TMD (ITMD) factorization formula was recently proposed. In the Color Glass Condensate (CGC) framework, it represents the leading term of an expansion in inverse powers of the hard scale. It contains the leading-twist TMD factorization formula relevant for small gluon's transverse momentum $k_t$, but also incorporates an all-order resummation of kinematical twists, resulting in a proper matching to high-energy factorization at large $k_t$. In this paper, we evaluate the accuracy of the ITMD formula quantitatively, for the case of quark dijet production in high-energy proton-proton($p+p$) and proton-nucleus ($p+A$) collisions at LHC energies. We do so by comparing the quark-antiquark azimuthal angle $\Delta\phi$ distribution to that obtained with the CGC formula. For a dijet with each quark momentum $p_t$ much larger than the target saturation scale, $Q_s$, the ITMD formula is a good approximation to the CGC formula in a wide range of azimuthal angle. It becomes less accurate as the jet $p_t$'s are lowered, as expected, due to the presence of genuine higher-twists contributions in the CGC framework, which represent multi-body scattering effects absent in the ITMD formula. We find that, as the hard jet momenta are lowered, the accuracy of ITMD start by deteriorating at small angles, in the high-energy-factorization regime, while in the TMD regime near $\Delta\phi=\pi$, very low values of $p_t$ are needed to see differences between the CGC and the ITMD formula. In addition, the genuine twists corrections to ITMD become visible for higher values of $p_t$ in $p+A$ collisions, compared to $p+p$ collisions, signaling that they are enhanced by the target saturation scale.