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Analytic solution for the helicity evolution equations at small x and large Nc and Nf

Jeremy Borden* and Yuri V. Kovchegov†

  • *Contact author: borden.75@buckeyemail.osu.edu
  • †Contact author: kovchegov.1@osu.edu
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Phys. Rev. D 113, 034002 – Published 2 February, 2026

DOI: https://doi.org/10.1103/ljl6-zvrq

Abstract

We construct an exact analytic solution of the revised small-x helicity evolution equations from Borden, Kovchegov, and Li [Helicity evolution at small x: Quark to gluon and gluon to quark transition operators, J. High Energy Phys. 09 (2024) 037], 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-Nc and -Nf limit and are double-logarithmic, resumming powers of αsln2(1/x). Here Nc and Nf are the numbers of quark colors and flavors, respectively, while αs is the strong coupling constant and x is the Bjorken-x variable. Using our solution, we obtain analytic, small-x, large-Nc and -Nf expressions for the flavor singlet quark and gluon helicity parton distribution functions (PDFs) and for the g1 structure function as double-inverse Laplace transforms. We also extract analytic expressions for the eigenvalues of the matrix of polarized Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) anomalous dimensions and, subsequently, analytic expressions for each of the four individual polarized anomalous dimensions themselves (Δγqq,ΔγqG,ΔγGq, and ΔγGG): these expressions resum powers of αs/ω2 to all orders at large Nc and Nf (with ω the Mellin moment variable). We extract the leading small-x asymptotic power-law growth of the helicity distributions, given by ΔΣ(x,Q2)∼ΔG(x,Q2)∼g1(x,Q2)∼(1x)αh, where the intercept αh satisfies an algebraic equation. Although the algebraic complexity of the equation prevented us from obtaining a general analytic expression for αh, we determine αh numerically for various values of Nc and Nf (and, for the special case of Nf=2Nc, we determine αh analytically). 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 Nf/Nc). Just as for the analytic solution to small-x helicity evolution equations in the large-Nc limit constructed in Borden and 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, 014001 (2023)] and for the iterative solution to the large-Nc and -Nf helicity evolution equations constructed in Borden, Kovchegov, and Li [Helicity evolution at small x: Quark to gluon and gluon to quark transition operators, J. High Energy Phys. 09 (2024) 037], we again find that all our predictions for polarized DGLAP anomalous dimensions are fully consistent with the existing finite-order calculations. Similar to the large-Nc case [Borden and 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, 014001 (2023)], our intercept αh (evaluated at various Nc and Nf values) exhibits a very slight disagreement with the predictions made within the infrared evolution equations framework in Bartels et al. [Flavor singlet contribution to the structure function G(1) at small x, Z. Phys. C 72, 627 (1996)], Blümlein and Vogt [The singlet contribution to the structure function g1(x,Q2) at small x, Phys. Lett. B 386, 350 (1996)], Boussarie et al. [Proton spin structure at small-x, Phys. Lett. B 797, 134817 (2019)].

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Phys. Rev. D 113, 034002– Published 2 February, 2026
  • Received 12 August 2025
  • Accepted 10 December 2025
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