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Within the Color Glass Condensate (CGC) effective field theory, we derive thenext-to-leading order (NLO) cross-section for the single-jet semi-inclusivecross-section in deep inelastic scattering (DIS) at small $x$, for bothlongitudinally and transversely polarized virtual photons. We provide analyticexpressions, valid at finite $N_c$ and suitable for numerical evaluation, forboth the cross-section differential in rapidity and transverse momentum and thecross-section differential in rapidity only. Our NLO formulae demonstrate thatthe very forward rapidity regime is plagued by large double logarithmiccorrections coming from phase space constraints on soft gluons close to thekinematic threshold for jet production. A joint resummation of small-$x$ andthreshold logarithms at single logarithmic accuracy is proposed to remedy theinstability of the cross-section in this regime. By integrating over thesingle-jet phase space, we recover known results for the NLO DIS structurefunctions at small $x$, previously obtained using the optical theorem.

Semi-inclusive single-jet production in DIS at next-to-leading order in the Color Glass Condensate