Single Inclusive Jet Production in $pA$ Collisions at NLO in the small-$x$ regime

We present the first complete NLO prediction with full jet algorithm implementation for the single inclusive jet production in $pA$ collisions within the CGC effective theory. Our prediction is fully differential over the final state physical kinematics, which allows the implementation of any IR safe observable including the jet clustering procedure. The NLO calculation is organized with the aid of the power counting proposed in [1] which gives rise to the novel soft contributions in the CGC factorization. We achieve the fully-differential calculation by constructing suitable subtraction terms to handle the singularities in the real corrections. The subtraction contributions can be exactly integrated analytically. We present the NLO cross section with the jets constructed using the anti-$k_T$ algorithm. The NLO calculation demonstrates explicitly the validity of the CGC factorization in jet production. Furthermore, as a byproduct of the subtraction method, we also derive the fully analytic cross section for the forward jet production in the small-$R$ limit. We show that in the small-$R$ approximation, the forward jet cross section can be factorized into a semi-hard cross section that produces a parton and the semi-inclusive jet functions. We argue that this feature holds for generic jet production and jet substructure observables in the CGC framework. Last, we show numerical analyses of the derived formula to validate our calculations. We justify when the small-$R$ approximation is appropriate. Like forward hadron production, the obtained NLO result also exhibits the negativity of the cross section in the large jet transverse regime, which signals the need for the threshold resummation. A sketch of the threshold resummation in the CGC framework is presented based on the multiple emission picture.