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Analytic Solution for the Helicity Evolution Equations at Small $x$ and Large $N_c\&N_f$

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Pith's one-line read This paper constructs an exact analytic solution of the revised small-x helicity evolution equations in the large-$N_c\&N_f$ limit, valid for arbitrary initial conditions, and derives from it all-order resummed polarized DGLAP anomalous…

desk verdict A genuinely large analytic calculation with strong external cross-checks, but the central contour/analyticity assumption is asserted rather than proved, so the all-order results are conditional until that is checked. read the letter →

arxiv 2508.00195 v1 pith:E5XPT4GX submitted 2025-07-31 hep-ph nucl-th

Jeremy Borden , Yuri V. Kovchegov This is my paper
classification hep-phnucl-th
keywords small-xhelicityevolutionpolarizedpartondistributionfunctionsDGLAPanomalousdimensionsdoublelogarithmicapproximationprotonspinpuzzlelargeNcandNfLaplacetransformsolutionintercept
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to prove that the revised small-x helicity evolution equations in the large-$N_c\&N_f$ limit, which include the quark-to-gluon and gluon-to-quark transition operators, admit an exact analytic solution valid for arbitrary initial conditions. The authors construct the solution as double-inverse Laplace transforms and show that it yields all-order resummed expressions for the four polarized DGLAP anomalous dimensions, plus a power-law small-$x$ growth for the helicity distributions with an intercept fixed by an algebraic equation. A sympathetic reader would care because the proton spin puzzle depends on knowing the helicity parton distributions down to $x$ values no experiment will ever reach, and an exact solution gives a rigorous small-$x$ benchmark that iterative and numerical schemes can be checked against.

What carries the argument

The machinery is the double-inverse Laplace representation: each dipole amplitude is written as an integral over exponentials $e^{\omega(\eta-s_{10})+\gamma s_{10}}$, which converts the integral evolution equations into algebraic relations between Laplace images. The central objects are the four functions $\delta^{\pm\pm}_\omega$ that arise from the homogeneous part of the partial differential equations for $\Gamma$ and $\widetilde\Gamma$, and the four roots $\gamma^{\pm\pm}_\omega$ that appear as the only surviving poles in the images $G_{2\omega\gamma}$ and $\widetilde Q_{\omega\gamma}$; identifying $\gamma^{--}_\omega$ and $\gamma^{-+}_\omega$ as the eigenvalues of the polarized DGLAP anomalous-dimension matrix is what connects the solution to DGLAP.

What would settle it

Perform an independent four-loop calculation of the small-$x$ polarized DGLAP splitting functions $\Delta P_{qG}$ and $\Delta P_{Gq}$ in the $\overline{\mathrm{MS}}$ scheme and compare with the expansions in Eqs. (106); if the predicted $O(\alpha_s^4)$ coefficients do not appear, the all-order resummation extracted from the solution is incorrect. A second check is to substitute the analytic solution (84)-(85) numerically into the original integral equations (8) for a non-trivial initial condition and verify equality to machine precision; any nonzero residual would signal a dropped boundary term.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the system of seven coupled integral equations for the polarized dipole amplitudes $G_2$, $\Gamma_2$, $\widetilde G$, $\widetilde\Gamma$, $Q$, $\Gamma$, and $\widetilde Q$ is solved in closed form by the double-inverse-Laplace representations in Eqs. (84) and (85), with the Laplace images $G_{2\omega\gamma}$ and $\widetilde Q_{\omega\gamma}$ expressed through four roots $\gamma^{\pm\pm}_\omega$ and functions determined entirely by the initial conditions. From this solution the gluon and flavor-singlet quark helicity PDFs and the $g_1$ structure function follow as double-inverse Laplace transforms, the four polarized DGLAP anomalous dimensions in Eqs. (105) are resummed to all orders in $\alpha_s/\omega^2$, and the small-$x$ intercept $\alpha_h = \sqrt{\alpha_s}\,\omega_b$ is the rightmost branch point obeying the algebraic equation (109). The result is that the most general small-$x$ double-logarithmic helicity evolution currently known, including the newly added quark-to-gluon and gluon-to-quark transition operators, is exactly solvable, with all results consistent with the existing three-loop finite-order calculations.

Load-bearing premise

The whole construction rests on the contour prescription that the poles at $\gamma=\delta^{++}_\omega$ and $\gamma=\delta^{+-}_\omega$ lie to the right of the $\gamma$ contour and to the left of the $\omega$ contour (that is, $\mathrm{Re}\,\omega > \mathrm{Re}\,\gamma$), so that closing contours to discard boundary terms is legitimate; if this prescription fails for any relevant singularity or initial condition, the derived constraints and the claimed solution do not follow.

Editorial extensions

If this is right

  • The helicity PDFs $\Delta\Sigma$, $\Delta G$ and the structure function $g_1$ are obtained as explicit double-inverse Laplace transforms for arbitrary initial conditions, so any initial condition can be evolved without further approximation.
  • The four polarized DGLAP anomalous dimensions $\Delta\gamma_{qq}$, $\Delta\gamma_{qG}$, $\Delta\gamma_{Gq}$, and $\Delta\gamma_{GG}$ are resummed to all orders in $\alpha_s/\omega^2$ at large $N_c\&N_f$, giving testable predictions beyond the three known loops.
  • The small-$x$ intercept is fixed by the algebraic equation (109); the numerical values in Table I differ from the infrared-evolution-equations framework predictions by less than one percent for $N_f\le 8$ with $N_c=3$.
  • The asymptotic ratio $\Delta G/\Delta\Sigma$ for $N_f=4$, $N_c=3$ is approximately $-2.21$, lying between the earlier large-$N_c$ value $-3$ and the value from the infrared-evolution-equations framework.
  • A four-loop discrepancy with the infrared-evolution-equations framework is identified explicitly and can be resolved by a future four-loop finite-order calculation of the polarized splitting functions.

Reading between the lines

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

  • Beyond the paper's claims, the fully analytic solution valid for arbitrary initial conditions can serve as an exact benchmark for testing numerical solutions and alternative resummation schemes of the same equations, so the remaining small differences with the infrared-evolution-equations framework become genuine dynamical predictions of this formalism rather than numerical artifacts.
  • The technique of fixing the unknown Laplace images by requiring the absence of singularities to the right of the integration contours may transfer to other double-logarithmic evolution hierarchies, such as those for orbital angular momentum or for transverse-momentum-dependent distributions at small $x$.
  • The explicit eigenvalue structure makes sub-asymptotic corrections accessible analytically, including the diffusion-like term $\exp(-\omega_b^2 C^{(1)}(\omega_b) t^2/16y)$, which in turn allows quantitative estimates of how far down in $x$ the power-law asymptotic regime actually applies.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper claims an exact analytic solution of the small-x helicity evolution equations in the large-Nc&Nf limit (the KPS-CTT-BCL equations of [1], restated as Eqs. (8) and (10)). The solution is expressed as double-inverse Laplace transforms for the five evolving 'dipole' amplitudes (Eqs. (84)-(85)), with the double-Laplace images G2ωγ and eQωγ given in closed form in terms of the initial conditions. From this solution the authors extract, in Sec. V, all-order resummed expressions for the four polarized DGLAP anomalous dimensions Δγqq, ΔγqG, ΔγGq, and ΔγGG (Eqs. (105)), check the eigenvalue expansion against the three-loop finite-order results (Eq. (100)), and reproduce the four-loop iterative results of [1] (Eqs. (106)). In Sec. VI they extract the small-x intercept αh = √αs ωb with ωb determined by the algebraic equation (109), give an exact analytic intercept for Nf = 2Nc (Eq. (110)), and provide asymptotic forms for ΔΣ and ΔG (Eqs. (121) and (130)) and the asymptotic ratio ΔG/ΔΣ (Table II). The results reproduce the small discrepancies with the BER/IREE framework at four loops reported earlier for the large-Nc case.

Significance. If correct, this is a substantial methodological advance: it converts the iterative large-Nc&Nf small-x helicity evolution of [1] into a closed-form all-order solution, and it is the first source of all-order (in αs/ω2) predictions for all four polarized DGLAP anomalous dimensions in this limit. The paper contains several genuinely strong elements: the agreement of the eigenvalue expansion (100) with three-loop finite-order results is a real external cross-check; the exact Nf = 2Nc intercept (110) and the all-order relation ΔγGq = -(Nc/Nf)ΔγqG (107) are crisp, testable predictions; and the authors are explicit about limitations, noting that the general intercept equation is solved only numerically and that a small four-loop discrepancy with BER persists. The main caveat is auditability: the central derivation rests on a contour-analyticity prescription (text after Eq. (39)) and on two large algebraic steps announced as 'after some significant algebra' (before Eq. (79)) and 'after considerable algebra' (before Eqs. (105)), which are not displayed, so the central all-order claim cannot currently be verified from the manuscript alone.

major comments (3)
  1. [Sec. III, text after Eq. (39); Eqs. (18), (25), (41), (45), (67), (73), (82)] The prescription that the poles at γ = δ++ω and γ = δ+-ω lie to the left of the ω-contour but to the right of the γ-contour (Re ω > Re γ) is load-bearing: it is used to discard boundary terms in Eqs. (18), (25), (41), and (45), to close the γ'-contour in Eq. (67), and, via the scaling (77), to justify evaluating Eq. (73) at γ = γ++ω and γ = γ+-ω by assuming that eQωγ is regular there. The manuscript does not establish the needed analyticity of the constructed solution: for the final images (85), only the vanishing residues at γ = γ++ω and γ = γ+-ω are shown (Eq. (82)), while possible branch cuts or additional poles to the right of the γ-contour, including singularities induced by the initial conditions, are not analyzed. As written, the regularity of eQωγ at γ = γ±±ω is an assumption used before the solution is constructed, and the claim in Sec. IV that the solution holds 'for any initial conditions' requires the singularity structure of the images (85) to be controlled as a function of the initial-condition images. I recommend either a rigorous derivation of the analyticity domain or a direct numerical verification of Eqs. (84)-(85) against the original equations (10) for representative initial conditions (e.g., the Born-level conditions (94) and one analytic test function), together with an explicit check of the absence of right-of-contour singularities in (85). The agreement of the extracted anomalous dimensions with the iterative results of [1] (Eqs. (106)) is a good partial check, but it does not cover the general-initial-condition claim.
  2. [Sec. III.D.2 (before Eq. (79)); Sec. V (before Eqs. (103) and (105))] Two steps that carry the main results are presented without derivation: the reduction of the system (72) to the closed forms (79), announced as 'after some significant algebra', and the solution of the matching system (103)-(104) for the anomalous dimensions, announced as 'after considerable algebra'. Because the paper's central claim is the all-order closed form, these passages are not merely cosmetic; a referee or reader cannot check them for algebraic errors. I ask that the derivation be supplied in full, either in an appendix or in machine-checkable form (e.g., a supplementary notebook), and that the identities (103a) and (103c), which are stated to 'be verified by explicit calculation', be demonstrated explicitly for the initial conditions (94) used in the DGLAP matching.
  3. [Sec. VI.A (Eq. (109) and footnote 4); Sec. IV] The intercept is identified with the rightmost branch point of γ--ω, and footnote 4 argues that the branch points of rαβ1 and rαβ2 do not contribute because the square roots enter through products of the form γ2 - ωγ + rαβi r-α,βi. The same argument is not given for the shifted initial-condition images entering the functions f(±)(ω,γ) in Eqs. (81): terms such as Q(0)δ±±ω γ evaluate the initial-condition image at shifted arguments, and for generic initial conditions these terms will introduce ω-plane singularities at shifted locations. The identification of ωb via Eq. (109) as the leading singularity therefore requires that the initial conditions have no singularities to the right of the ω-contour, an assumption stated only in passing in Sec. VI.A ('assuming that the initial conditions contain no singularities in the complex-ω plane with a large real part of ω'). This should either be proved or be stated explicitly as a restriction on the class of allowed initial conditions, which would also qualify the 'any initial conditions' claim made in Sec. IV.
minor comments (5)
  1. [Eq. (116b)] In the denominator of the last factor, '2 - N-f/Nc' should read '2 - Nf/Nc'; the minus sign has migrated into the flavor subscript.
  2. [Eqs. (41) and (45)] The rational factor (δαβω - ω)/ω is typeset so that the denominator ω appears to sit inside the exponential argument eδαβω s10; the typesetting should clearly separate the rational factor from the exponential.
  3. [Eqs. (60)-(69), especially Eq. (64)] The notation Q(0)δ+,βω-βω√(1-n/2)γ is very hard to parse; defining a shifted-image notation (for example, writing Q(0)ab for the image evaluated at a shifted Mellin argument) and using it consistently in Eqs. (60)-(69) would greatly improve readability.
  4. [Text after Eq. (55)] The limits eδ+βω s10 eΓ(+β)ω(s10) → 0 and eδ-βω s10 ω eΓ(-β)ω(s10) → 0 are stated 'when ω → ∞'; since the contour is closed to the right, the direction Re ω → +∞ should be stated explicitly to avoid ambiguity.
  5. [Footnote 4, Sec. VI.A] The argument that the branch points of rαβ1 and rαβ2 disappear in the products rαβi r-α,βi is convincing for the individual products, but it should be spelled out for the final combinations f(eQ)(ω,γ) and f(G2)(ω,γ) in Eqs. (85c)-(85d), which contain sums of such products; a one-line verification would remove any residual doubt.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the analytic solution is an explicit functional of the stated initial conditions, and the DGLAP and intercept results are derived from it rather than fitted to external data.

full rationale

The paper's derivation chain is self-contained with respect to the target quantities. The evolution equations (8) are restated as input from [1], and the double-Laplace representation in Sec. III is a general inversion technique, not an ansatz that already encodes the DGLAP anomalous dimensions or the intercept. The final solution, Eqs. (84)-(85), expresses every dipole amplitude image (eQωγ, G2ωγ, etc.) solely in terms of the initial conditions G2(0), eG(0), Q(0), and eQ(0) plus known functions; no parameter is fitted to the finite-order DGLAP results. The anomalous dimensions in Eqs. (105) are obtained by matching the pole structure of this solution to the assumed DGLAP matrix exponential, and the comparison with the finite-order calculations in Eq. (101) is explicitly presented as a cross-check, not as an input used to fix free parameters. Similarly, the intercept ωb is defined by the branch-point condition (109) of the derived eigenvalue function γ−−ω, and the slight disagreement with the BER predictions in Table I is reported as a physical discrepancy, not as a fitted residual. The main rigor caveat—the asserted contour/analyticity prescription after Eq. (39), and the omitted algebra leading from Eq. (68) to Eq. (79)—is a correctness/checkability concern, not circularity: the solution is not defined in terms of the predicted observables, and no self-citation is used to force a conclusion that is then presented as a derivation. Self-citations to [1] and [2] provide the input equations and the solution method, but these do not reduce the paper's central results to their own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The derivation contains no fitted parameters. It solves the evolution equations of [1] under the stated double-logarithmic, large-Nc and large-Nf, fixed-coupling approximations. The numerical inputs are physical parameters such as Nc, Nf, alpha_s, and Lambda, not quantities tuned to reproduce the target results.

assumptions (4)
  • domain assumption The large-Nc and large-Nf helicity evolution equations (8), including the quark-to-gluon and gluon-to-quark transition operators, are the correct double-logarithmic small-x evolution equations.
    The paper takes Eqs. (8) from [1] as the starting point without rederiving them; if those equations are incomplete, the analytic solution applies to the wrong system.
  • domain assumption The double-logarithmic approximation at fixed coupling is sufficient; running coupling and single-logarithmic corrections are neglected.
    The whole formalism resums alpha_s ln-squared(1/x) and alpha_s ln(1/x) ln(Q-squared/Lambda-squared), so all predictions are confined to this approximation.
  • ad hoc to paper The pole and contour prescription after Eq. (39) is valid: poles at gamma equal to delta-plus-plus and delta-plus-minus lie to the left of the omega contour and to the right of the gamma contour.
    This prescription is introduced to make the double-inverse Laplace inversion well defined and is used to derive constraints (42)-(48). If wrong, the final solution and intercept do not follow.
  • domain assumption The leading singularity determining the intercept is the rightmost branch point of gamma-minus-minus, with initial conditions containing no singularities at larger real omega.
    Stated in Sec. VI A before Eq. (109). This selects the algebraic equation for the intercept and hence all numerical values in Table I.

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Cite this review

Pith. "Pith review of Analytic Solution for the Helicity Evolution Equations at Small $x$ and Large $N_c\&N_f$." pith.science (2026). https://pith.science/paper/E5XPT4GX

@misc{pith2026250800195,
  author       = {Pith},
  title        = {Pith review of: Analytic Solution for the Helicity Evolution Equations at Small $x$ and Large $N_c\&N_f$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E5XPT4GX}},
  note         = {Machine review of arXiv:2508.00195}
}
abstract

We construct an exact analytic solution of the revised small-$x$ helicity evolution equations, where the contributions of the quark-to-gluon and gluon-to-quark transition operators were newly included. These evolution equations are written in the large-$N_c\&N_f$ limit and are double-logarithmic, resumming powers of $\alpha_s\ln^2(1/x)$. Here $N_c$ and $N_f$ are the numbers of quark colors and flavors, while $\alpha_s$ is the strong coupling constant and $x$ is the Bjorken-$x$ variable. Using our solution, we obtain analytic expressions for the flavor singlet quark and gluon helicity parton distribution functions (PDFs) and for the $g_1$ structure function as double-inverse Laplace transforms. We also extract analytic expressions for the four DGLAP polarized anomalous dimensions $\Delta \gamma_{qq}, \Delta \gamma_{qG}, \Delta \gamma_{Gq}$, and $\Delta \gamma_{GG}$: these expressions resum powers of $\alpha_s/\omega^2$ to all orders at large-$N_c\&N_f$ (with $\omega$ the Mellin moment variable). We extract the leading small-$x$ growth of the helicity distributions, \begin{align} \Delta\Sigma(x,Q^2) \sim \Delta G(x,Q^2)\sim g_1(x,Q^2) \sim \left(\frac{1}{x}\right)^{\alpha_h}, \end{align} where the intercept $\alpha_h$ satisfies an algebraic equation. We determine $\alpha_h$ numerically for various values of $N_c$ and $N_f$. We further obtain the explicit asymptotic expressions for the helicity distributions, which yield numerical values for the ratio of the gluon helicity PDF to the flavor singlet quark helicity PDF in the small-$x$ asymptotic limit (for different $N_f/N_c$). We find that all our predictions for polarized DGLAP anomalous dimensions are fully consistent with the existing finite-order calculations. Similar to the large-$N_c$ case, our intercept $\alpha_h$ exhibits a very slight disagreement with the predictions made within the infrared evolution equations framework.

Figures

Figures reproduced from arXiv: 2508.00195 by the authors.

Figure 1
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Forward citations

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Reference graph

Works this paper leans on

108 extracted references · 12 canonical work pages · cited by 3 Pith papers

  1. [1]

    Four Straightforward Constraints The four equations (42a), (44), (46a), (48) constitute a system of equations we can straightforwardly solve for all four of the eΓ(αβ) ω (s10) functions as integrals over γ. First we rewrite the common structure shared by each as eΓ(αβ) ω (s10) = e−δαβ ω s10 Z dγ 2πi eγs10 eΓ(αβ) ωγ (49) (note the distinction of eΓ(αβ) ωγ ...

  2. [2]

    G2ωγ γ − r++ 1 γ − r−+ 1 − G(0) 2ωγ γ − δ++ ω γ − δ−+ ω + 2 − 1 2 √ 4 − 2n # − n 4 ω

    Two Remaining Constraints Two constraints that remain to be satisfied are given by Eqs. (42b) and (46b). Solving these will allow us to obtain expressions for our final two remaining unknowns, the double-Laplace images G2ωγ and eQωγ , and will complete our solution of the evolution equations (10). We rewrite the two constraints below: 0 = e−ωs10 Z dγ 2πi ...

  3. [3]

    Bartels, B

    J. Bartels, B. Ermolaev and M. Ryskin, Flavor singlet contribution to the structure function G(1) at small x , Z.Phys. C72 (1996) 627–635, [ hep-ph/9603204]

    arXiv 1996

  4. [4]

    Bl¨ umlein and A

    J. Bl¨ umlein and A. Vogt,The Singlet contribution to the structure function g1 (x, Q**2) at small x , Phys. Lett. B 386 (1996) 350–358, [ hep-ph/9606254]

    arXiv 1996

  5. [5]

    (84f) + Z dω 2πi Z dγ 2πi e ω ln(z′sx2 21)+γ ln 1 x2 21Λ2 Aωγ − 2e ω ln(z′sx2 21)+γ ln 1 x2 10Λ2 G2ωγ − G(0) 2ωγ − 2e ω ln(z′sx2 10)+γ ln 1 x2 10Λ2 G(0) 2ωγ , Γ(x2 10, x2 21, z′s) = Z dω 2πi eω ln(z′sx2

  6. [6]

    X α,β=+,− e δαβ ω ln 1 x2 21Λ2 1 + β q 1 − Nf 2 Nc Nf /(4Nc) eΓ(αβ) ω (x2

  7. [7]

    (84g) + Z dω 2πi Z dγ 2πi e ω ln(z′sx2 21)+γ ln 1 x2 21Λ2 Aωγ − 2e ω ln(z′sx2 21)+γ ln 1 x2 10Λ2 G2ωγ − G(0) 2ωγ − 2e ω ln(z′sx2 10)+γ ln 1 x2 10Λ2 G(0) 2ωγ , with eQωγ = f ( eQ)(ω, γ) − 2(γ−γ+− ω )(γ−γ−− ω ) αs √ s2(ω) f ( eQ)(ω, γ= γ++ ω ) + 2(γ−γ++ ω )(γ−γ−+ ω ) αs √ s2(ω) f ( eQ)(ω, γ= γ+− ω ) 2 q 4 − 2Nf Nc γ − γ++ω γ − γ+−ω γ − γ−+ω γ − γ−−ω , (85a)...

  8. [8]

    8 + 11 Nf Nc − 7 Nf Nc 2# + 16 ω δ−− ω − δ−+ ω r 4 − 2 Nf Nc

    = e −δαβ ω ln 1 x2 10Λ2 Z dγ 2πi e γ ln 1 x2 10Λ2 β 8(δαβ ω − δ−α,β ω ) q 4 − 2Nf Nc (85m) × n Aωγ 2Nf Nc − 4Aωγ 2 − β q 4 − 2Nf Nc ) (δ−α,β ω − γ) + Nf Nc ω eQωγ − eQ(0) ωγ 3 − β q 4 − 2Nf Nc − 2 αs γ δα,β ω + 2 ω G2ωγ − G(0) 2ωγ 2 − β q 4 − 2Nf Nc 4 − 1 αs γ δα,β ω − 2Nf Nc o , Aωγ = 1 γ − δ++ω γ − δ+−ω γ − δ−+ω γ − δ−−ω ( ωγ 2 h αs 3 − Nf 2Nc − 3 γ (γ ...

    1984

Show all 108 references
  1. [9]

    Borden, Y

    J. Borden, Y. V. Kovchegov and M. Li, Helicity evolution at small x: quark to gluon and gluon to quark transition operators, JHEP 09 (2024) 037, [ 2406.11647]

    2024 arXiv

  2. [10]

    → √αs eΓ(αβ) ω (x2 10), δ±± ω → δ±± ω√αs , γ±± ω → γ±± ω√αs , with the arguments ω, γof the double inverse Laplace transform images of the dipole amplitudes not reflecting the rescaling of ω, γ. Recalling that n = Nf /Nc we write eQ(x2 10, zs) = Z dω 2πi Z dγ 2πi e ω ln(zsx2 1...

  3. [11]

    Borden and Y

    J. Borden and Y. V. Kovchegov, Analytic solution for the revised helicity evolution at small x and large Nc: New resummed gluon-gluon polarized anomalous dimension and intercept , Phys. Rev. D 108 (2023) 014001, [ 2304.06161]

    2023 arXiv

  4. [12]

    Leader and C

    E. Leader and C. Lorc´ e,The angular momentum controversy: What’s it all about and does it matter? , Phys. Rept. 541 (2014) 163–248, [ 1309.4235]

    2014 arXiv

  5. [13]

    Boussarie, Y

    R. Boussarie, Y. Hatta and F. Yuan, Proton Spin Structure at Small- x, Phys. Lett. B797 (2019) 134817, [ 1904.02693]

    2019 arXiv

  6. [14]

    Ashman, B

    J. Ashman, B. Badelek, G. Baum, J. Beaufays, C. Bee, C. Benchouk et al., A measurement of the spin asymmetry and determination of the structure function g1 in deep inelastic muon-proton scattering , Physics Letters B 206 (1988) 364–370

    1988

  7. [15]

    R. L. Jaffe and A. Manohar, The G(1) Problem: Fact and Fantasy on the Spin of the Proton , Nucl. Phys. B337 (1990) 509–546

    1990

  8. [16]

    Ji, Gauge-Invariant Decomposition of Nucleon Spin , Phys

    X.-D. Ji, Gauge-Invariant Decomposition of Nucleon Spin , Phys. Rev. Lett. 78 (1997) 610–613, [ hep-ph/9603249]

    1997 arXiv

  9. [17]

    Boer et al., Gluons and the quark sea at high energies: Distributions, polarization, tomography , 1108.1713

    D. Boer et al., Gluons and the quark sea at high energies: Distributions, polarization, tomography , 1108.1713

    arXiv

  10. [18]

    C. A. Aidala, S. D. Bass, D. Hasch and G. K. Mallot, The Spin Structure of the Nucleon , Rev. Mod. Phys. 85 (2013) 655–691, [1209.2803]

    2013 arXiv

  11. [19]

    Accardi et al., Electron Ion Collider: The Next QCD Frontier , Eur

    A. Accardi et al., Electron Ion Collider: The Next QCD Frontier , Eur. Phys. J. A52 (2016) 268, [ 1212.1701]

    2016 arXiv

  12. [20]

    Kirschner and L

    R. Kirschner and L. Lipatov, Double Logarithmic Asymptotics and Regge Singularities of Quark Amplitudes with Flavor Exchange, Nucl.Phys. B213 (1983) 122–148

    1983

  13. [21]

    X α,β=+,− e δαβ ω ln 1 x2 10Λ2 eΓ(αβ) ω (x2

  14. [22]

    E. C. Aschenauer et al., The RHIC Spin Program: Achievements and Future Opportunities , 1304.0079

    arXiv

  15. [23]

    Aschenauer et al., The RHIC SPIN Program: Achievements and Future Opportunities , 1501.01220

    E.-C. Aschenauer et al., The RHIC SPIN Program: Achievements and Future Opportunities , 1501.01220

    arXiv

  16. [24]

    Prokudin, Y

    A. Prokudin, Y. Hatta, Y. Kovchegov and C. Marquet, eds., Proceedings, Probing Nucleons and Nuclei in High Energy Collisions: Dedicated to the Physics of the Electron Ion Collider: Seattle (WA), United States, October 1 - November 16, 2018 , WSP, 2020. 10.1142/11684

    2018 doi

  17. [25]

    X. Ji, F. Yuan and Y. Zhao, What we know and what we don ’t know about the proton spin after 30 years , Nature Rev. Phys. 3 (2021) 27–38, [ 2009.01291]

    2021 arXiv

  18. [26]

    Abdul Khalek et al., Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report , 2103.05419

    R. Abdul Khalek et al., Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report , 2103.05419

    arXiv

  19. [27]

    Bartels, B

    J. Bartels, B. Ermolaev and M. Ryskin, Nonsinglet contributions to the structure function g1 at small x , Z.Phys. C70 (1996) 273–280, [ hep-ph/9507271]

    1996 arXiv

  20. [28]

    V. G. Gorshkov, V. N. Gribov, L. N. Lipatov and G. V. Frolov, Doubly logarithmic asymptotic behavior in quantum electrodynamics, Sov. J. Nucl. Phys. 6 (1968) 95

    1968

  21. [29]

    Kirschner, Reggeon interactions in perturbative QCD , Z.Phys

    R. Kirschner, Reggeon interactions in perturbative QCD , Z.Phys. C65 (1995) 505–510, [ hep-th/9407085]

    1995 arXiv

  22. [30]

    Kirschner, Regge asymptotics of scattering with flavor exchange in QCD , Z.Phys

    R. Kirschner, Regge asymptotics of scattering with flavor exchange in QCD , Z.Phys. C67 (1995) 459–466, [hep-th/9404158]

    1995 arXiv

  23. [31]

    Griffiths and D

    S. Griffiths and D. Ross, Studying the perturbative Reggeon, Eur.Phys.J. C12 (2000) 277–286, [ hep-ph/9906550]

    2000 arXiv

  24. [32]

    Blumlein and A

    J. Blumlein and A. Vogt, On the behavior of nonsinglet structure functions at small x , Phys. Lett. B 370 (1996) 149–155, [hep-ph/9510410]

    1996 arXiv

  25. [33]

    B. I. Ermolaev, M. Greco and S. I. Troian, QCD running coupling effects for the nonsinglet structure function at small x, Nucl. Phys. B571 (2000) 137–150, [ hep-ph/9906276]

    2000 arXiv

  26. [34]

    B. I. Ermolaev, M. Greco and S. I. Troyan, Intercepts of the nonsinglet structure functions , Nucl. Phys. B594 (2001) 71–88, [hep-ph/0009037]

    2001 arXiv

  27. [35]

    B. I. Ermolaev, M. Greco and S. I. Troyan, Running coupling effects for the singlet structure function g1 at small x, Phys. Lett. B579 (2004) 321–330, [ hep-ph/0307128]

    2004 arXiv

  28. [36]

    B. I. Ermolaev, M. Greco and S. I. Troyan, Overview of the spin structure function g1 at arbitrary x and Q2, Riv. Nuovo Cim. 33 (2010) 57–122, [ 0905.2841]

    2010 arXiv

  29. [37]

    A. H. Mueller, Soft gluons in the infinite momentum wave function and the BFKL pomeron , Nucl. Phys. B415 (1994) 373–385

    1994

  30. [38]

    A. H. Mueller and B. Patel, Single and double BFKL pomeron exchange and a dipole picture of high-energy hard processes, Nucl. Phys. B425 (1994) 471–488, [ hep-ph/9403256]

    1994 arXiv

  31. [39]

    A. H. Mueller, Unitarity and the BFKL pomeron , Nucl. Phys. B437 (1995) 107–126, [ hep-ph/9408245]

    1995 arXiv

  32. [40]

    Balitsky, Operator expansion for high-energy scattering , Nucl

    I. Balitsky, Operator expansion for high-energy scattering , Nucl. Phys. B463 (1996) 99–160, [ hep-ph/9509348]

    1996 arXiv

  33. [41]

    Balitsky, Factorization and high-energy effective action , Phys

    I. Balitsky, Factorization and high-energy effective action , Phys. Rev. D60 (1999) 014020, [ hep-ph/9812311]

    1999 arXiv

  34. [42]

    Y. V. Kovchegov, Small-x F2 structure function of a nucleus including multiple pomeron exchanges , Phys. Rev. D60 (1999) 034008, [ hep-ph/9901281]

    1999 arXiv

  35. [43]

    Y. V. Kovchegov, Unitarization of the BFKL pomeron on a nucleus , Phys. Rev. D61 (2000) 074018, [ hep-ph/9905214]

    2000 arXiv

  36. [44]

    Jalilian-Marian, A

    J. Jalilian-Marian, A. Kovner and H. Weigert, The Wilson renormalization group for low x physics: Gluon evolution at 37 finite parton density , Phys. Rev. D59 (1998) 014015, [ hep-ph/9709432]

    1998 arXiv

  37. [45]

    Jalilian-Marian, A

    J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, The Wilson renormalization group for low x physics: Towards the high density regime , Phys. Rev. D59 (1998) 014014, [ hep-ph/9706377]

    1998 arXiv

  38. [46]

    Weigert, Unitarity at small Bjorken x , Nucl

    H. Weigert, Unitarity at small Bjorken x , Nucl. Phys. A703 (2002) 823–860, [ hep-ph/0004044]

    2002 arXiv

  39. [47]

    Iancu, A

    E. Iancu, A. Leonidov and L. D. McLerran, The renormalization group equation for the color glass condensate , Phys. Lett. B510 (2001) 133–144

    2001

  40. [48]

    Iancu, A

    E. Iancu, A. Leonidov and L. D. McLerran, Nonlinear gluon evolution in the color glass condensate. I , Nucl. Phys. A692 (2001) 583–645, [ hep-ph/0011241]

    2001 arXiv

  41. [49]

    Ferreiro, E

    E. Ferreiro, E. Iancu, A. Leonidov and L. McLerran, Nonlinear gluon evolution in the color glass condensate. II , Nucl. Phys. A703 (2002) 489–538, [ hep-ph/0109115]

    2002 arXiv

  42. [50]

    Y. V. Kovchegov, D. Pitonyak and M. D. Sievert, Helicity Evolution at Small-x , JHEP 01 (2016) 072, [ 1511.06737]

    2016 arXiv

  43. [51]

    Hatta, Y

    Y. Hatta, Y. Nakagawa, F. Yuan, Y. Zhao and B. Xiao, Gluon orbital angular momentum at small- x, Phys. Rev. D95 (2017) 114032, [ 1612.02445]

    2017 arXiv

  44. [52]

    Y. V. Kovchegov, D. Pitonyak and M. D. Sievert, Helicity Evolution at Small x: Flavor Singlet and Non-Singlet Observables, Phys. Rev. D95 (2017) 014033, [ 1610.06197]

    2017 arXiv

  45. [53]

    Y. V. Kovchegov, D. Pitonyak and M. D. Sievert, Small-x asymptotics of the quark helicity distribution , Phys. Rev. Lett. 118 (2017) 052001, [ 1610.06188]

    2017 arXiv

  46. [54]

    Y. V. Kovchegov, D. Pitonyak and M. D. Sievert, Small-x Asymptotics of the Quark Helicity Distribution: Analytic Results, Phys. Lett. B772 (2017) 136–140, [ 1703.05809]

    2017 arXiv

  47. [55]

    Y. V. Kovchegov, D. Pitonyak and M. D. Sievert, Small-x Asymptotics of the Gluon Helicity Distribution , JHEP 10 (2017) 198, [ 1706.04236]

    2017 arXiv

  48. [56]

    Y. V. Kovchegov and M. D. Sievert, Small-x Helicity Evolution: an Operator Treatment , Phys. Rev. D99 (2019) 054032, [1808.09010]

    2019 arXiv

  49. [57]

    Y. V. Kovchegov, Orbital Angular Momentum at Small x, JHEP 03 (2019) 174, [ 1901.07453]

    2019 arXiv

  50. [58]

    Cougoulic and Y

    F. Cougoulic and Y. V. Kovchegov, Helicity-dependent generalization of the JIMWLK evolution , Phys. Rev. D100 (2019) 114020, [ 1910.04268]

    2019 arXiv

  51. [59]

    Y. V. Kovchegov and Y. Tawabutr, Helicity at Small x: Oscillations Generated by Bringing Back the Quarks , JHEP 08 (2020) 014, [ 2005.07285]

    2020 arXiv

  52. [60]

    Cougoulic and Y

    F. Cougoulic and Y. V. Kovchegov, Helicity-dependent extension of the McLerran-Venugopalan model , Nucl. Phys. A 1004 (2020) 122051, [ 2005.14688]

    2020 arXiv

  53. [61]

    G. A. Chirilli, High-energy operator product expansion at sub-eikonal level , JHEP 06 (2021) 096, [ 2101.12744]

    2021 arXiv

  54. [62]

    Y. V. Kovchegov, A. Tarasov and Y. Tawabutr, Helicity evolution at small x: the single-logarithmic contribution , JHEP 03 (2022) 184, [ 2104.11765]

    2022 arXiv

  55. [63]

    Cougoulic, Y

    F. Cougoulic, Y. V. Kovchegov, A. Tarasov and Y. Tawabutr, Quark and gluon helicity evolution at small x: revised and updated, JHEP 07 (2022) 095, [ 2204.11898]

    2022 arXiv

  56. [64]

    Adamiak, Y

    D. Adamiak, Y. V. Kovchegov and Y. Tawabutr, Helicity evolution at small x: Revised asymptotic results at large Nc and Nf , Phys. Rev. D 108 (2023) 054005, [ 2306.01651]

    2023 arXiv

  57. [65]

    Altinoluk, N

    T. Altinoluk, N. Armesto, G. Beuf, M. Martinez and C. A. Salgado, Next-to-eikonal corrections in the CGC: gluon production and spin asymmetries in pA collisions , JHEP 07 (2014) 068, [ 1404.2219]

    2014 arXiv

  58. [66]

    Balitsky and A

    I. Balitsky and A. Tarasov, Rapidity evolution of gluon TMD from low to moderate x , JHEP 10 (2015) 017, [1505.02151]

    2015 arXiv

  59. [67]

    Balitsky and A

    I. Balitsky and A. Tarasov, Gluon TMD in particle production from low to moderate x , JHEP 06 (2016) 164, [1603.06548]

    2016 arXiv

  60. [68]

    G. A. Chirilli, Sub-eikonal corrections to scattering amplitudes at high energy , JHEP 01 (2019) 118, [ 1807.11435]

    2019 arXiv

  61. [69]

    Jalilian-Marian, Quark jets scattering from a gluon field: from saturation to high pt, Phys

    J. Jalilian-Marian, Quark jets scattering from a gluon field: from saturation to high pt, Phys. Rev. D99 (2019) 014043, [1809.04625]

    2019 arXiv

  62. [70]

    Jalilian-Marian, Rapidity loss, spin, and angular asymmetries in the scattering of a quark from the color field of a proton or nucleus , Phys

    J. Jalilian-Marian, Rapidity loss, spin, and angular asymmetries in the scattering of a quark from the color field of a proton or nucleus , Phys. Rev. D 102 (2020) 014008, [ 1912.08878]

    2020 arXiv

  63. [71]

    Altinoluk, G

    T. Altinoluk, G. Beuf, A. Czajka and A. Tymowska, Quarks at next-to-eikonal accuracy in the CGC: Forward quark-nucleus scattering, Phys. Rev. D 104 (2021) 014019, [ 2012.03886]

    2021 arXiv

  64. [72]

    Boussarie and Y

    R. Boussarie and Y. Mehtar-Tani, Gauge invariance of transverse momentum dependent distributions at small x, Phys. Rev. D 103 (2021) 094012, [ 2001.06449]

    2021 arXiv

  65. [73]

    Boussarie and Y

    R. Boussarie and Y. Mehtar-Tani, A novel formulation of the unintegrated gluon distribution for DIS , Phys. Lett. B 831 (2022) 137125, [ 2006.14569]

    2022 arXiv

  66. [74]

    Y. V. Kovchegov and M. G. Santiago, Quark sivers function at small x: spin-dependent odderon and the sub-eikonal evolution, JHEP 11 (2021) 200, [ 2108.03667]

    2021 arXiv

  67. [75]

    Altinoluk and G

    T. Altinoluk and G. Beuf, Quark and scalar propagators at next-to-eikonal accuracy in the CGC through a dynamical background gluon field , Phys. Rev. D 105 (2022) 074026, [ 2109.01620]

    2022 arXiv

  68. [76]

    Y. V. Kovchegov and M. G. Santiago, T-odd leading-twist quark TMDs at small x , JHEP 11 (2022) 098, [ 2209.03538]

    2022 arXiv

  69. [77]

    Altinoluk, G

    T. Altinoluk, G. Beuf, A. Czajka and A. Tymowska, DIS dijet production at next-to-eikonal accuracy in the CGC , Phys. Rev. D 107 (2023) 074016, [ 2212.10484]

    2023 arXiv

  70. [78]

    Altinoluk, N

    T. Altinoluk, N. Armesto and G. Beuf, Probing quark transverse momentum distributions in the color glass condensate: Quark-gluon dijets in deep inelastic scattering at next-to-eikonal accuracy , Phys. Rev. D 108 (2023) 074023, [2303.12691]. 38

    2023 arXiv

  71. [79]

    Altinoluk, G

    T. Altinoluk, G. Beuf and J. Jalilian-Marian, Renormalization of the gluon distribution function in the background field formalism, 2305.11079

    arXiv

  72. [80]

    Li, Small x physics beyond eikonal approximation: an effective Hamiltonian approach , JHEP 07 (2023) 158, [2304.12842]

    M. Li, Small x physics beyond eikonal approximation: an effective Hamiltonian approach , JHEP 07 (2023) 158, [2304.12842]

    2023 arXiv

  73. [81]

    Altinoluk, G

    T. Altinoluk, G. Beuf and S. Mulani, Forward parton-nucleus scattering at next-to-eikonal accuracy in the color glass condensate, Phys. Rev. D 111 (2025) 034028, [ 2411.15047]

    2025 arXiv

  74. [82]

    Altinoluk, G

    T. Altinoluk, G. Beuf, E. Blanco and S. Mulani, Quark TMDs from back-to-back dijet production at forward rapidities in pA collisions beyond eikonal accuracy in the CGC , JHEP 06 (2025) 097, [ 2412.08485]

    2025

  75. [83]

    Altinoluk, F

    T. Altinoluk, F. Bergabo, J. Jalilian-Marian, C. Marquet and Y. Shi, SIDIS at small x at next-to-leading order: transverse photon, 2505.04557

    arXiv

  76. [84]

    Altinoluk, G

    T. Altinoluk, G. Beuf and J. Jalilian-Marian, One-loop renormalization of quark TMD in the light-cone gauge: CSS evolution, 2505.20467

    arXiv

  77. [85]

    ’t Hooft, Can We Make Sense Out of Quantum Chromodynamics? , Subnucl

    G. ’t Hooft, Can We Make Sense Out of Quantum Chromodynamics? , Subnucl. Ser. 15 (1979) 943

    1979

  78. [86]

    Veneziano, Some Aspects of a Unified Approach to Gauge, Dual and Gribov Theories , Nucl

    G. Veneziano, Some Aspects of a Unified Approach to Gauge, Dual and Gribov Theories , Nucl. Phys. B 117 (1976) 519–545

    1976

  79. [87]

    V. N. Gribov and L. N. Lipatov, Deep inelastic e p scattering in perturbation theory , Sov. J. Nucl. Phys. 15 (1972) 438–450

    1972

  80. [88]

    Altarelli and G

    G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language , Nucl. Phys. B126 (1977) 298

    1977

  81. [89]

    Y. L. Dokshitzer, Calculation of the Structure Functions for Deep Inelastic Scattering and e+e− Annihilation by Perturbation Theory in Quantum Chromodynamics , Sov. Phys. JETP 46 (1977) 641–653

    1977

  82. [90]

    E. B. Zijlstra and W. L. van Neerven, Order-α2 s corrections to the polarized structure function g1(x, Q2), Nucl. Phys. B 417 (1994) 61–100

    1994

  83. [91]

    Mertig and W

    R. Mertig and W. L. van Neerven, The Calculation of the two loop spin splitting functions P(ij)(1)(x) , Z. Phys. C 70 (1996) 637–654, [ hep-ph/9506451]

    1996 arXiv

  84. [92]

    Moch and J

    S. Moch and J. A. M. Vermaseren, Deep inelastic structure functions at two loops , Nucl. Phys. B 573 (2000) 853–907, [hep-ph/9912355]

    2000 arXiv

  85. [93]

    W. L. van Neerven and A. Vogt, NNLO evolution of deep inelastic structure functions: The Singlet case , Nucl. Phys. B 588 (2000) 345–373, [ hep-ph/0006154]

    2000 arXiv

  86. [94]

    J. A. M. Vermaseren, A. Vogt and S. Moch, The Third-order QCD corrections to deep-inelastic scattering by photon exchange, Nucl. Phys. B 724 (2005) 3–182, [ hep-ph/0504242]

    2005 arXiv

  87. [95]

    S. Moch, J. A. M. Vermaseren and A. Vogt, The Three-Loop Splitting Functions in QCD: The Helicity-Dependent Case , Nucl. Phys. B 889 (2014) 351–400, [ 1409.5131]

    2014 arXiv

  88. [96]

    Bl¨ umlein, P

    J. Bl¨ umlein, P. Marquard, C. Schneider and K. Sch¨ onwald,The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements , JHEP 01 (2022) 193, [ 2111.12401]

    2022 arXiv

  89. [97]

    Bl¨ umlein and M

    J. Bl¨ umlein and M. Saragnese,The N3LO scheme-invariant QCD evolution of the non-singlet structure functions F2NS(x,Q2) and g1NS(x,Q2) , Phys. Lett. B 820 (2021) 136589, [ 2107.01293]

    2021 arXiv

  90. [98]

    Davies, C

    J. Davies, C. H. Kom, S. Moch and A. Vogt, Resummation of small-x double logarithms in QCD: inclusive deep-inelastic scattering, JHEP 08 (2022) 135, [ 2202.10362]

    2022 arXiv

  91. [99]

    Bl¨ umlein, P

    J. Bl¨ umlein, P. Marquard, C. Schneider and K. Sch¨ onwald,The massless three-loop Wilson coefficients for the deep-inelastic structure functions F 2, F L, xF 3 and g 1, JHEP 11 (2022) 156, [ 2208.14325]

    2022 arXiv

  92. [100]

    G. P. Lepage and S. J. Brodsky, Exclusive processes in perturbative quantum chromodynamics , Phys. Rev. D22 (1980) 2157

    1980

  93. [101]

    S. J. Brodsky, H.-C. Pauli and S. S. Pinsky, Quantum chromodynamics and other field theories on the light cone , Phys.Rept. 301 (1998) 299–486, [ hep-ph/9705477]

    1998 arXiv

  94. [102]

    Adamiak, Y

    Jefferson Lab Angular Momentumcollaboration, D. Adamiak, Y. V. Kovchegov, W. Melnitchouk, D. Pitonyak, N. Sato and M. D. Sievert, First analysis of world polarized DIS data with small-x helicity evolution , Phys. Rev. D 104 (2021) L031501, [ 2102.06159]

    2021 arXiv

  95. [103]

    Adamiak, N

    Jefferson Lab Angular Momentum (JAM)collaboration, D. Adamiak, N. Baldonado, Y. V. Kovchegov, W. Melnitchouk, D. Pitonyak, N. Sato et al., Global analysis of polarized DIS and SIDIS data with improved small-x helicity evolution , Phys. Rev. D 108 (2023) 114007, [ 2308.07461]

    2023 arXiv

  96. [104]

    Y. V. Kovchegov and B. Manley, Orbital angular momentum at small x revisited , JHEP 02 (2024) 060, [ 2310.18404]

    2024 arXiv

  97. [105]

    Hatta and D.-J

    Y. Hatta and D.-J. Yang, On the small- x behavior of the orbital angular momentum distributions in QCD , Phys. Lett. B781 (2018) 213–219, [ 1802.02716]

    2018 arXiv

  98. [106]

    B. I. Ermolaev, Parton helicities at arbitrary x and Q2 in double-logarithmic approximation , 2505.07359

  99. [107]

    E. A. Kuraev, L. N. Lipatov and V. S. Fadin, The Pomeranchuk singlularity in non-Abelian gauge theories , Sov. Phys. JETP 45 (1977) 199–204

    1977

  100. [108]

    Balitsky and L

    I. Balitsky and L. Lipatov, The Pomeranchuk Singularity in Quantum Chromodynamics , Sov.J.Nucl.Phys. 28 (1978) 822–829

    1978

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