Small- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>x</mml:mi> </mml:math> Factorization in the Target Fragmentation Region

We consider the differential cross-section for single-inclusive jet production with transverse momentum P_{⊥} in deep inelastic scattering at small Bjorken x_{Bj}, mediated by a virtual photon with virtuality Q^{2}. Unlike most studies at small x, which focus on particle production in the current fragmentation region, we investigate the kinematic regime where the jet is produced in the target fragmentation hemisphere of the Breit frame, and with P_{⊥}≪Q. For a longitudinally polarized photon, we demonstrate that this cross section is not power suppressed in P_{⊥}/Q and we derive a factorized expression in terms of extended quark and gluon jet fracture functions. Our formula, valid at next-to-leading order in α_{s} at small x, is akin to the Altarelli-Martinelli identity for the longitudinal DIS structure function. Numerical estimates show that the extended quark jet fracture function is the most sensitive to saturation effects in large nuclei.