DIS Dijet Production: An operator basis bridging eikonal and TMD regimes

Fei Yao; Shaswat Tiwari; Swagato Mukherjee; Tiyasa Kar; Vladimir Skokov

REVIEW

DIS Dijet Production: An operator basis bridging eikonal and TMD regimes

Tiyasa Kar, Swagato Mukherjee, Vladimir Skokov, Shaswat Tiwari, Fei Yao

High Energy Physics - Phenomenology · Submitted 11 Sep 2026 · 42 references

Pith's verdict: A genuinely new operator basis that smoothly connects the CGC and TMD descriptions of DIS dijet production; the construction is careful and the limiting checks are clean, though one load-bearing step is asserted without explicit power-counting.

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arXiv ID2609.12363 (version 1)

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CitationFei Yao, Shaswat Tiwari, Swagato Mukherjee, Tiyasa Kar, Vladimir Skokov. "DIS Dijet Production: An operator basis bridging eikonal and TMD regimes." arXiv:2609.12363 (2026). https://pith.science/paper/KMQCWUKH

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Editor's note

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This paper does something that needed doing: it builds an operator basis that interpolates between the all-twist eikonal (CGC) and the leading-twist TMD descriptions of unpolarized DIS dijet production. The endpoint transformation in Section III is the real new idea—it resums transverse kinematic corrections into endpoint factors while leaving a Hamiltonian organized by explicit field-strength insertions. That is a clean technical move. The longitudinal and transverse photon amplitudes are then derived in this basis, and the paper shows they reduce to the known CGC result in the x→0 limit and to the leading-twist gluon TMD result in the back-to-back limit. Those checks are explicit and convincing. The connection to iTMD at the end is a nice bonus, even if the non-zero-x obstruction to conventional TMD factorization ends up being a negative result.

The soft spot flagged by the stress-test is real: in going from Eq. (43) to Eq. (44), the authors commute the transverse rotation operators R^{-α} through the anticommutator {Pi, F_{-i}} and assert that the resulting commutators are “simultaneously sub-sub-eikonal and of higher twist” with no supporting power-counting argument. That claim is load-bearing for the cancellation that produces the pure TMD limit in Eqs. (66)–(68). A referee would need to see a dimensional estimate or an explicit check that those commutators are indeed beyond the stated accuracy. I do not think this is a fatal flaw—the structural consistency of the rest of the paper makes it plausible—but it is a genuine gap that should be filled in revision.

Overall, this is a serious theoretical paper by people who know the subject. It is written at a level appropriate for specialists. The derivations are mostly detailed, the accuracy criterion is stated, and the two limiting reproductions give strong confidence in the framework. I would cite this work and bring it to a reading group. It deserves a fair peer review, not a desk reject.

Referee report

Major objections (1)

[IV.A.2, Eqs. (43)–(44)] The derivation of the quark-line contribution M_q,L relies on commuting the transverse rotation operators R^{-α}(z^-,k_1^-) and R^{α}(z^-,k_1^-) through the operator {P_i(z^-),F_{-i}(z^-)}. The paper states that the commutators generated in this step are 'simultaneously sub-sub-eikonal and of higher twist and therefore lie beyond the accuracy considered here.' No explicit power-counting argument or computation is provided to support this assertion. Because the precise form of M_q,L is essential for the cancellation of the longitudinal phase derivative in the TMD limit (Eqs. (66)–(68)), a missing leading-twist contribution from these commutators would undermine the central claim that the amplitude reduces to the leading-twist gluon TMD result. The authors should either supply a detailed power-counting analysis for the twist and eikonal order of the omitted commutators, or present a direct expansion that confirms they are beyond the stated accuracy.

Minor objections (6)

[II, Eq. (3)] The definitions of Δ_⊥ and P_⊥ are given; it may be helpful to note that the sign convention is consistent with the standard TMD literature (e.g., the relative momentum is defined such that k_{1⊥} = P_⊥ + z Δ_⊥).
[III.A, Eq. (14)] The rotation operator R(x^-,k^-) is defined with the operator P_⊥^2 in the exponent. For clarity, consider adding a brief reminder that P_⊥ contains the background field, so that R is not a simple momentum-space phase but a non-trivial operator.
[IV.A.1, Eq. (38)] The approximation (z_{1⊥}|R^{-1}(y^-,\kappa^-)|z_{2⊥}) ≈ (z_{1⊥}|e^{i y^- p_⊥^2/(2\kappa^-)}|z_{2⊥}) [z_{1⊥},z_{2⊥}]_{y^-} is stated without derivation; the reader is referred to Appendix D, which provides the expansion. Consider adding a short parenthetical comment that the omitted correction is proportional to y^- F_{ij} and is sub-sub-eikonal and higher twist, consistent with the accuracy criterion.
[IV.A.4, Eq. (60)] The expansion of the staple to linear order in r_⊥ uses the Mandelstam derivative. The result [∞,y^-]_{z_{1⊥}} [z_{1⊥},z_{2⊥}]_{y^-} [y^-,∞]_{z_{2⊥}} = 1 + i r^i_⊥ ∫_{y^-}^∞ dξ^- [∞,ξ^-]_{b_⊥} F_{i-}(ξ^-,b_⊥) [ξ^-,∞]_{b_⊥} + O(r_⊥^2) is plausible but not fully derived in the main text. A brief justification or reference to Appendix B would improve readability.
[V, Eq. (97)] The final amplitude for the iTMD projection contains an integral over t and a derivative ∂/∂p_i. The evaluation of the integration by parts and the statement that p_⊥ = -t k_{1⊥} + \bar{t} k_{2⊥} should be checked for sign consistency with the definitions in Eq. (3).
[Appendix F, Eq. (F1)] The reduction of the semi-infinite dipole contribution to field-strength insertions is a useful consistency check. However, the transverse-momentum integral in Eq. (F3) would benefit from a brief comment on the ordering of integrations (the y^- integral is performed before the p_⊥ integral, as in the main text).

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The paper

The paper claims

Unified operator basis bridges eikonal and TMD regimes for DIS dijet production.

The reading

This paper constructs an operator basis for unpolarized deep-inelastic scattering (DIS) quark-antiquark dijet production that smoothly connects the all-twist eikonal (Color Glass Condensate) regime at small Bjorken x to the leading-twist transverse-momentum-dependent (TMD) regime at arbitrary x. Starting from the quark propagator in a background gluon field, the authors reorganize it using an endpoint transformation that resums transverse kinematic corrections into endpoint factors while expressing the evolution Hamiltonian in terms of explicit field-strength insertions. The resulting amplitudes reproduce the established Color Glass Condensate result in the eikonal limit and the leading-twist gluon TMD result in the back-to-back limit, with the field-strength-insertion operators being essential for canceling non-trivial longitudinal phases that appear at non-zero x. The framework also relates to improved TMD (iTMD) factorization, showing that while the eikonal limit recovers iTMD structure, non-zero x generates a continuous family of longitudinal phases that prevent factorization into a single conventional TMD operator.

What carries the argument

The central object is the background-field quark propagator reorganized via an endpoint transformation that separates the free transverse propagation (absorbed into endpoint operators R and $R^{{-1}}$) from the interacting evolution, whose Hamiltonian H_s consists of integrals over field-strength insertions ({P_i, F_{-i}} and $σ^{{ij}}$ F_{ij}). This representation yields a semi-infinite dipole operator (a staple of longitudinal Wilson lines with a transverse gauge link) combined with explicit field-strength insertions in the Wilson lines, providing a unified operator basis that organizes both the eikonal and twist expansions.

Extended reading notes

Core claim

The paper demonstrates that a single operator basis, built from a semi-infinite dipole operator plus explicit field-strength insertions into Wilson lines, can simultaneously describe the all-twist eikonal limit and the leading-twist TMD limit of unpolarized DIS dijet production. The key is an endpoint transformation of the background-field quark propagator that resums transverse kinematic effects into endpoint phase factors while leaving the ordered evolution kernel expressed solely in terms of field strengths. The amplitudes derived for longitudinal and transverse photon polarizations pass two independent consistency checks: in the eikonal (x→0) limit they reduce to the familiar Color Glass Condensate dipole formulas (with K0 and K1 kernels), and in the back-to-back (|Δ⊥|≪|P⊥|) limit they reduce to the leading-twist gluon TMD hard coefficients at arbitrary x. The paper further shows that the semi-infinite dipole contribution alone reproduces the full eikonal amplitude, while the field-strength-insertion terms, although sub-eikonal, are necessary to cancel derivative terms from the longitudinal phase in the TMD limit.

Load-bearing premise

The background gluon field is taken to be independent of x^+ (light-cone time), which is a standard approximation for the target but would require additional corrections if longitudinal dynamics are important.

Editorial extensions

If this is right

The operator basis provides a systematic framework for calculating sub-eikonal and higher-twist corrections to DIS dijet production, since the Hamiltonian H_s can be expanded to higher orders in field strengths.
The amplitudes derived here can be directly used for phenomenological predictions at the Electron-Ion Collider, enabling tests of saturation physics and TMD factorization across a wide range of x.
The demonstration that non-zero x introduces longitudinal phases that cannot be absorbed into a single TMD operator implies that extending iTMD-type factorization to finite x may require new multi-operator structures.
The same endpoint-transformation technique can be applied to other DIS final states (e.g., trijet, heavy-quark pair) and to polarized processes, where the leading CGC contribution is sub-eikonal.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The continuous family of longitudinal phases that obstructs conventional TMD factorization at non-zero x may be resummed into a new class of non-perturbative functions, perhaps leading to a generalized factorization scheme that reduces to both CGC and TMD in the appropriate limits.
The explicit cancellation between the semi-infinite dipole term and the field-strength-insertion terms in the back-to-back limit suggests that the operator basis is overcomplete but automatically enforces the correct kinematic constraints; a similar cancellation mechanism likely operates for other observables.
The connection to iTMD presented in Section V indicates that the iTMD framework, while successful at small x, may need to incorporate longitudinal phase information beyond the simple eikonal approximation for phenomenology at moderate x, where the EIC will have significant coverage.

Original abstract

We propose an operator basis for the unpolarized deep-inelastic scattering (DIS) quark-antiquark dijet production process smoothly connecting the all-twist eikonal regime to the back-to-back leading-twist TMD regime at arbitrary Bjorken x. Starting from the background-field quark propagator we construct an operator basis that organizes the eikonal and twist expansions within a unified formulation. Utilizing this operator basis we derive the dijet production amplitudes retaining all contributions required by either the leading-eikonal or the leading-twist description. In the eikonal limit the resulting amplitudes reproduce the all-twist Color Glass Condensate result, while in the back-to-back limit they reduce to the leading-twist gluon TMD-based result at arbitrary x. The operator basis can be systematically extended to include sub-eikonal and higher-twist corrections. Further, by examining the relationship of this operator basis to the improved TMD (iTMD) factorization we recover the iTMD structure in the eikonal limit. However, we find at non-zero x the transverse resummation generates longitudinal phases that prevent factorization in terms of a conventional TMD operator.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We propose an operator basis for the unpolarized deep-inelastic scattering (DIS) quark-antiquark dijet production process smoothly connecting the all-twist eikonal regime to the back-to-back leading-twist TMD regime at arbitrary Bjorken x. Starting from the background-field quark propagator we construct an operator basis that organizes the eikonal and twist expansions within a unified formulation. Utilizing this operator basis we derive the dijet production amplitudes retaining all contributions required by either the leading-eikonal or the leading-twist description. In the eikonal limit the resulting amplitudes reproduce the all-twist Color Glass Condensate result, while in the back-to-back limit they reduce to the leading-twist gluon TMD-based result at arbitrary x. The operator basis can be systematically extended to include sub-eikonal and higher-twist corrections. Further, by examining the relationship of this operator basis to the improved TMD (iTMD) factorization we recover the iTMD structure in the eikonal limit. However, we find at non-zero x the transverse resummation generates longitudinal phases that prevent factorization in terms of a conventional TMD operator.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We propose an operator basis for the unpolarized deep-inelastic scattering (DIS) quark-antiquark dijet production process smoothly connecting the all-twist eikonal regime to the back-to-back leading-twist TMD regime at arbitrary Bjorken x.	contribution 1.00	0.93	0.70	asserted 0.92
s1	Starting from the background-field quark propagator we construct an operator basis that organizes the eikonal and twist expansions within a unified formulation.	method 0.92	0.95	0.06	asserted 1.00
s2	Utilizing this operator basis we derive the dijet production amplitudes retaining all contributions required by either the leading-eikonal or the leading-twist description.	method 0.73	0.96	0.11	asserted 1.00
s3	In the eikonal limit the resulting amplitudes reproduce the all-twist Color Glass Condensate result, while in the back-to-back limit they reduce to the leading-twist gluon TMD-based result at arbitrary x.	result 0.99	0.86	0.47	asserted 1.00
s4	The operator basis can be systematically extended to include sub-eikonal and higher-twist corrections.	future work 0.82	0.60	0.06	asserted 0.66
s5	Further, by examining the relationship of this operator basis to the improved TMD (iTMD) factorization we recover the iTMD structure in the eikonal limit.	result 0.80	0.92	0.08	asserted 1.00
s6	However, we find at non-zero x the transverse resummation generates longitudinal phases that prevent factorization in terms of a conventional TMD operator.	result 0.71	0.94	0.22	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.69 · evidence from proof or derivation 1.00.

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not examined

full text
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P abstract

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T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: Operator structures contributing to the longitudinal dijet amplitude. (a) Semi-infinite dipole contribution,

**Figure 2.** FIG. 2: Leading-twist TMD limit of Fig. 1. For nearly back-to-back jets the transverse separation of the pair is small,

Figures reproduced from arXiv:2609.12363 by the authors.

References

Works cited (42)

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Semi-infinite dipole contribution We first consider the contribution obtained by setting the twoH sσ evolution operators in Eq. (35) to unity. The semi-infinite dipole contribution to the longitudinal dijet amplitude is iM0,L =ieϵ−(q)¯u(k1)γ−v(k2) Z d4y Z d2z1⊥d2z2⊥e−iq·yei(k− 1 +k− 2 )y+ e−ik1⊥·z1⊥−ik2⊥·z2⊥ ×[∞,y −]z1⊥(z1⊥|R−1 y−,κ− |z2⊥) [y−,∞] z2⊥,(36)...

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Contributions with explicit field strength insertions The contributioniM 0,L obtained above does not by itself reproduce the complete amplitude in the kinematic limits of interest. We therefore restore the terms generated by the ordered evolution operator in Eq. (35). As we show later, these contributions are required to reproduce the fixed-twist expansio...

View on Pith
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Eikonal limit We first consider the high-energy limit of the resummed longitudinal amplitude. This provides an important con- sistency check of the construction, since the finite-energy transverse propagation retained in the preceding derivation must reduce, in the appropriate limit, to the standard eikonal propagation through the background field. In thi...

View on Pith
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Semi-infinite dipole contribution We first consider the contribution obtained by setting the ordered evolution operators to unity. The structure is closely related to its longitudinal counterpart, with the transverse polarization introducing an additional transverse momentump l together with the spin tensorT jl(z). Using Eqs. (72) and (74), we obtain iM0,...

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Contributions from the evolution operators We next restore the contributions from the ordered evolution operators. Their scalar part is independent of the photon polarization, so the corresponding field-strength insertions have the same operator structure as in the lon- gitudinal case. The transverse polarization modifies only the numerator through the fa...

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Power corrections to heavy-quark spin correlations in small-$x$ deep-inelastic scattering
hep-ph 2026-11 conditional novelty 7.0 of 10

The azimuthally averaged heavy-quark spin density matrix in small-x DIS obtains target-dependent corrections at quartic order in dipole size, with mass-dependent asymptotics and a 3.5-3.7% increase in log-likelihood r...
Quark TMDPDFs from Large to Small $x$: A Unified Description of CSS, DGLAP, and Small-$x$ Double Logarithms
hep-ph 2026-10 accept novelty 6.0 of 10

A unified NLO framework for quark TMDPDFs that reproduces both DGLAP and small-x double logarithms is presented.
Transverse momentum dependent parton distributions at small $x$
hep-ph 2026-10 unverdicted

A comprehensive review of small-x TMDs in the CGC framework, covering their theoretical foundation, evolution, and phenomenology.

Discussion

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Semi-infinite dipole contribution We first consider the contribution obtained by setting the twoH sσ evolution operators in Eq. (35) to unity. The semi-infinite dipole contribution to the longitudinal dijet amplitude is iM0,L =ieϵ−(q)¯u(k1)γ−v(k2) Z d4y Z d2z1⊥d2z2⊥e−iq·yei(k− 1 +k− 2 )y+ e−ik1⊥·z1⊥−ik2⊥·z2⊥ ×[∞,y −]z1⊥(z1⊥|R−1 y−,κ− |z2⊥) [y−,∞] z2⊥,(36)...

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Contributions with explicit field strength insertions The contributioniM 0,L obtained above does not by itself reproduce the complete amplitude in the kinematic limits of interest. We therefore restore the terms generated by the ordered evolution operator in Eq. (35). As we show later, these contributions are required to reproduce the fixed-twist expansio...

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Eikonal limit We first consider the high-energy limit of the resummed longitudinal amplitude. This provides an important con- sistency check of the construction, since the finite-energy transverse propagation retained in the preceding derivation must reduce, in the appropriate limit, to the standard eikonal propagation through the background field. In thi...

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TMD limit We next consider the opposite kinematic expansion relevant for nearly back-to-back dijets. In this regime the relative transverse momentumP⊥ is hard, whereas the total transverse momentum∆ ⊥ =k 1⊥ +k 2⊥ remains parametrically smaller. In coordinate space, this hierarchy corresponds to a small transverse separation between the quark and antiquark...

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[5] [5]

Semi-infinite dipole contribution We first consider the contribution obtained by setting the ordered evolution operators to unity. The structure is closely related to its longitudinal counterpart, with the transverse polarization introducing an additional transverse momentump l together with the spin tensorT jl(z). Using Eqs. (72) and (74), we obtain iM0,...

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Contributions from the evolution operators We next restore the contributions from the ordered evolution operators. Their scalar part is independent of the photon polarization, so the corresponding field-strength insertions have the same operator structure as in the lon- gitudinal case. The transverse polarization modifies only the numerator through the fa...

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Eikonal limit We first verify that the transverse amplitude reproduces the expected high-energy result. Since the dependence ony− in Eq. (75) is carried by the same staple integral as for longitudinal polarization, its eikonal reduction can be taken over directly from the preceding subsection. At leading order, Z dy−eik+ p⊥y− S(y−) = i k+p⊥ (1−S(−∞)) +O(s...

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TMD limit We finally consider the back-to-back limit. Using the transverse coordinates introduced in the longitudinal analysis, we expand the staple to first order in the relative separationr ⊥: S(y−;z 1⊥,z 2⊥) = 1 +iri ⊥ Z ∞ y− dξ− [∞,ξ−]b⊥Fi−(ξ−,b⊥)[ξ−,∞] b⊥ +O(r 2 ⊥).(82) The unit term is proportional toδ(k + p⊥) and does not contribute fork + p⊥ >0. K...

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