From Sub-eikonal DIS to Quark Distributions and their High-Energy Evolution

Relating the high-energy dipole description of deep-inelastic scattering to the standard light-ray operator formulation at finite Bjorken $x_B$ is essential for connecting the small-$x$ framework to the usual partonic description. I demonstrate that this connection already emerges at the first sub-eikonal order. At the differential level, the first sub-eikonal correction is governed by a quark TMD-like light-ray operator. In the inclusive limit, after complete phase-space integration, it reconstructs the standard nonlocal quark and helicity distributions at nonzero $x_B$. I then show independently that the same inclusive operator content follows from the high-energy limit of the leading-twist non-local operator product expansion, thereby establishing an explicit operator-level bridge between the shock-wave formalism and the non-local light-cone expansion. I further discuss the high-energy evolution of the corresponding operators at $x_B=0$. Rewriting the evolution equations in terms of dipole-type operator combinations, I identify an operator basis whose bilocal building blocks vanish in the zero-dipole-size limit, making the small-dipole behavior and the leading-logarithmic structure manifest. In the double-logarithmic approximation the evolution equations admit the usual mixed longitudinal-transverse Bessel-type solution when the transverse phase space is treated independently. When the transverse phase space is instead constrained by longitudinal ordering, the second logarithm is converted into a logarithm of energy, and in the symmetric double-logarithmic regime one recovers the fixed-coupling Kirschner-Lipatov exponent with the full finite-$N_c$ color factor $C_F$.