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Back-to-back dijet production in DIS at arbitrary Bjorken-x: TMD gluon distributions to twist-3 accuracy

Swagato Mukherjee, Vladimir V. Skokov, Andrey Tarasov, Shaswat Tiwari, Fei Yao

Abstract

We derive the gluon transverse-momentum-dependent (TMD) operator structure of back-to-back quark-antiquark dijet production in deep inelastic scattering at arbitrary Bjorken-x to twist-3 accuracy. Working at leading order in the strong coupling and in the kinematic regime where the transverse momentum imbalance of the jets is much smaller than their individual transverse momenta, we perform a systematic gradient expansion of the quark propagator in a background gluon field. This expansion organizes multiple interactions with the target in terms of longitudinal Wilson lines and gauge-invariant field-strength insertions, yielding a TMD description valid beyond the strict high-energy eikonal (x -> 0) approximation. We obtain explicit cross sections for longitudinally and transversely polarized virtual photons, identifying all contributing gluon TMD operators up to twist-3, including structures involving F_+-, F_ij, and three-gluon correlators. The full longitudinal phase associated with Bjorken-x is retained throughout. In the small-x limit, our results reproduce the known sub-eikonal expressions obtained in the Color Glass Condensate framework, establishing a direct connection between the general-x TMD expansion and high-energy factorization. We further reduce the operator basis using equations of motion, minimizing the number of independent nonperturbative matrix elements entering the cross section. This work provides a systematic foundation for extending TMD analyses of dijet production beyond leading twist, establishing a unified operator framework valid at arbitrary Bjorken x that smoothly interpolates between moderate- and small-x descriptions of gluon TMDs.

Back-to-back dijet production in DIS at arbitrary Bjorken-x: TMD gluon distributions to twist-3 accuracy

Abstract

We derive the gluon transverse-momentum-dependent (TMD) operator structure of back-to-back quark-antiquark dijet production in deep inelastic scattering at arbitrary Bjorken-x to twist-3 accuracy. Working at leading order in the strong coupling and in the kinematic regime where the transverse momentum imbalance of the jets is much smaller than their individual transverse momenta, we perform a systematic gradient expansion of the quark propagator in a background gluon field. This expansion organizes multiple interactions with the target in terms of longitudinal Wilson lines and gauge-invariant field-strength insertions, yielding a TMD description valid beyond the strict high-energy eikonal (x -> 0) approximation. We obtain explicit cross sections for longitudinally and transversely polarized virtual photons, identifying all contributing gluon TMD operators up to twist-3, including structures involving F_+-, F_ij, and three-gluon correlators. The full longitudinal phase associated with Bjorken-x is retained throughout. In the small-x limit, our results reproduce the known sub-eikonal expressions obtained in the Color Glass Condensate framework, establishing a direct connection between the general-x TMD expansion and high-energy factorization. We further reduce the operator basis using equations of motion, minimizing the number of independent nonperturbative matrix elements entering the cross section. This work provides a systematic foundation for extending TMD analyses of dijet production beyond leading twist, establishing a unified operator framework valid at arbitrary Bjorken x that smoothly interpolates between moderate- and small-x descriptions of gluon TMDs.
Paper Structure (28 sections, 111 equations, 2 figures)

This paper contains 28 sections, 111 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General Structure of the Dijet Amplitude and Cross Section
  3. Background field propagators
  4. Scalar propagator
  5. Quark propagator
  6. On-shell limit
  7. Dijet production
  8. Amplitude
  9. Cross section
  10. Longitudinally polarized photon
  11. Transversely polarized photon
  12. Small-x limit
  13. Longitudinally polarized photon
  14. Transversely polarized photon
  15. Conclusion
  16. ...and 13 more sections

Figures (2)

  • Figure 1: Diagrammatic representation of a virtual photon $\gamma^*$ splitting into a quark--antiquark pair, which interacts with the target hadron through multiple gluon exchanges. These gluon lines represent the background-field expansion, with each line corresponding to one insertion of the background gluon field in our calculation. The vertical cut separates the amplitude from its complex conjugate; the dijet momenta are identical on the two sides of the cut.
  • Figure 2: Diagrammatic representation of Eq. \ref{['eq:scpropresumfinal']}. Horizontal dotted lines denote Wilson lines along the light cone minus direction, while the thick vertical lines represent the transverse propagation given by $(x_\perp| \exp[-i P_\perp^2(x^-)(x^- - y^-)/(2 k^-)]|y_\perp)$ and its transverse derivatives. We do not further decompose the transverse exponential into Wilson links and field-strength tensors, as such transformation becomes irrelevant after taking the on-shell limit in Sec. \ref{['sec: on-shell']}.