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

Giovanni Antonio Chirilli

Abstract

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 $C_F$.

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

Abstract

Relating the high-energy dipole description of deep-inelastic scattering to the standard light-ray operator formulation at finite Bjorken is essential for connecting the small- 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 . 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 . 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- color factor .

Paper Structure

This paper contains 19 sections, 140 equations, 4 figures.

Table of Contents

  1. Introduction
  2. DIS at sub-eikonal level
  3. DIS at sub-eikonal level: Diagrams with 1-point in the shock-wave
  4. Longitudinal photon polarization
  5. Transverse photon polarization
  6. Quark-TMD cross-section
  7. Naive collinear quark pdf
  8. Quark pdf operator
  9. Helicity distributions from asymmetry
  10. From non-local OPE to High-energy OPE
  11. Evolution equation
  12. Double-logarithmic solution of the $Q_1$ and $Q_5$ evolution
  13. Independent transverse phase space: BFKL/DGLAP double log
  14. Longitudinally constrained transverse phase space: pure double logarithm of energy.
  15. Evolution equation with kinematic constraint and its solution
  16. ...and 4 more sections

Figures (4)

  • Figure 1: Diagrams contributing to the transition amplitude $\gamma^*(q)\to q(k)$ in the right panel, and $\gamma^*(q)\to {\bar{q}}(k)$ in the left panel. The blue fermionic lines are the quantum fields, while the red ones are the classical fields.
  • Figure 2: The quark starts its propagation in the shock-wave and ends it outside. In this case, the pure gauge is only on one side of the shock-wave.
  • Figure 3: Diagrams contributing to the calculation of $T_{\mu\nu}$
  • Figure 4: Diagrams with $\hat{Q}_{1x}$ and $\hat{Q}_{5x}$ quantum.