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Back-to-back dijet production in DIS at arbitrary Bjorken-x: TMD gluon distributions to twist-3 accuracy
High Energy Physics - Phenomenology · Submitted 16 Feb 2026
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.99 | 0.97 | 0.39 | asserted 1.00 |
| s1 | 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. | method 0.99 | 0.95 | 0.03 | asserted 0.99 |
| s2 | 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. | method 0.70 | 0.89 | 0.14 | asserted 1.00 |
| s3 | 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. | result 0.91 | 0.96 | 0.17 | asserted 1.00 |
| s4 | The full longitudinal phase associated with Bjorken-x is retained throughout. | method 0.50 | 0.88 | 0.06 | asserted 1.00 |
| s5 | 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. | result 0.62 | 0.88 | 0.12 | asserted 1.00 |
| s6 | We further reduce the operator basis using equations of motion, minimizing the number of independent nonperturbative matrix elements entering the cross section. | method 0.59 | 0.96 | 0.06 | asserted 1.00 |
| s7 | 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. | interpretation 0.88 | 0.84 | 0.85 | asserted 0.99 |
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The paper
Original 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.
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References
How the literature uses it
7 arXiv papers cite this in 23 sentences
Papers citing it · 16 sentences without a confident label
Uses its method or tool 1 paper
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The derivation follows the method of Ref. [ref] and is summarized in Appendix [ref]; here we only present the terms needed for the twist-three cross section.
Back-to-back dijet production in DIS at arbitrary Bjorken-x: TMD quark distributions to twist-3 accuracy · Background field propagators in quark background · 1.00
Cites it as background 3 papers
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The total cross-section of the inclusive dijet production in DIS or photoproduction has long been considered in the small-$x$ literature [ref] , including both next-to-leading logarithmic [ref] and next-to-eikonal corrections [ref] .
Quantum entanglement in electron-nucleus collisions: Role of the linearly polarized gluon distribution · The spin-density matrix for the inclusive dijet production at small-$x$ · 0.99
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A variety of approaches have been developed to address this problem, and the associated literature is by now quite broad [ref] .
From Sub-eikonal DIS to Quark Distributions and their High-Energy Evolution · Introduction · 0.99
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In deep-inelastic scattering (DIS), these developments have been extensively applied to observables in the current fragmentation region (CFR) [ref] , including single-hadron (or jet) production [ref] , dihadron (or dijet) correlations [ref] , as well as lepton-jet correlations [ref] .
Gluonic nucleon energy correlators and fracture functions for Color Glass Condensate · Introduction · 0.85
What it cites · 41 sentences without a confident label
Builds on 3 papers
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The methods developed here provide a foundation for extending MSTT framework [ref] through systematic inclusion of higher-twist contributions, such as $ψ̅Fψ$, $FFF$ etc.
A unified description of DGLAP, CSS, and BFKL: TMD factorization bridging large and small x · Conclusion · 0.86
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The methods developed here provide a foundation for extending MSTT framework [ref] through systematic inclusion of higher-twist contributions, such as $ψ̅Fψ$, $FFF$ etc.
Perturbative Corrections to Quark TMDPDFs in the Background-Field Method: Gauge Invariance, Equations of Motion, and Multiple Interactions · Conclusion · 0.86
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Inclusion of quark background fields will allow treatment of quark–gluon dijet production (this was considered at small $x$ in Ref. [ref] ), involving operators such as $ψ̅γ^μψ$ and $ψ̅Fψ$ at twist-3 accuracy.
Probing quark transverse momentum distributions in the Color Glass Condensate: quark-gluon dijets in Deep Inelastic Scattering at next-to-eikonal accuracy · Conclusion · 0.86
Uses the method or tool of 9 papers
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In this work, we address this issue using the TMD factorization framework recently proposed in Refs. [ref] , hereafter referred to as the MSTT framework.
Perturbative Corrections to Quark TMDPDFs in the Background-Field Method: Gauge Invariance, Equations of Motion, and Multiple Interactions · Introduction · 0.95
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In this work, we address this issue using the TMD factorization framework recently proposed in Refs. [ref] , hereafter referred to as the MSTT framework.
A unified description of DGLAP, CSS, and BFKL: TMD factorization bridging large and small x · Introduction · 0.95
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In this appendix, following Appendix C of Ref. [ref] , we show that a specific higher-order kinematic contribution arising from the expansion of leading-twist operators can be rewritten in terms of genuine twist-3 operator structures.
Back-to-back dijet production in DIS at next-to-eikonal accuracy and twist-3 gluon TMDs · Relation between kinematic and dynamic twists from the Bianchi identity · 0.80
Compares with 6 papers
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In the small-$x$ limit, we find exact agreement with the results of Ref. [ref] .
Back-to-back dijet production in DIS at next-to-eikonal accuracy and twist-3 gluon TMDs · Longitudinally polarized photon · 1.00
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In the appropriate limit, we find full agreement with existing sub-eikonal CGC results [ref] , while providing a unified description valid for an arbitrary Bjorken-$x$.
DIS dijet production at next-to-eikonal accuracy in the CGC · Introduction · 0.99
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Our results agree with the back-to-back limit of CGC calculations [ref] , while providing a systematic determination of the Fourier phase for twist-3 TMDs—information that was not accessible in the sub-eikonal approach of Ref. [ref] .
Universality of Unintegrated Gluon Distributions at small x · Conclusion · 1.00
Cites as background 14 papers
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Back-to-back dijet production in DIS, including leading-order (LO) as well as next-to-leading-order (NLO) quantum corrections, was calculated in the strict eikonal limit corresponding to $x=0$ in Refs. [ref] .
Back-to-back inclusive dijets in DIS at small $x$: Gluon Weizs\"acker-Williams distribution at NLO · Introduction · 0.81
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Back-to-back dijet production in DIS, including leading-order (LO) as well as next-to-leading-order (NLO) quantum corrections, was calculated in the strict eikonal limit corresponding to $x=0$ in Refs. [ref] .
The importance of kinematic twists and genuine saturation effects in dijet production at the Electron-Ion Collider · Introduction · 0.81
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Back-to-back dijet production in DIS, including leading-order (LO) as well as next-to-leading-order (NLO) quantum corrections, was calculated in the strict eikonal limit corresponding to $x=0$ in Refs. [ref] .
Dijet impact factor in DIS at next-to-leading order in the Color Glass Condensate · Introduction · 0.81
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