One-loop gauge invariant amplitudes with a space-like gluon

Nowadays the particle physics has entered an era where high precision calculations are required in order to compare the theoretical predictions with the experimental data. In this paper, we explicitly compute the virtual contributions for the space-like one-jet processes, $\text{procgg},\text{procqq},\text{procH}$ and $\text{procee}$ within the auxiliary parton method. Our results, which are expected to play an important role in high precision description of small $x$ physics, explicitly confirm the conjecture developed in Ref. \cite{vanHameren:2022mtk}, thus helping to bridge the gap between lowest order calculations and NLO corrections within hybrid $k_T$-factorization scheme.