We present a numerical solution of the revised version [J. High Energy Phys. 07 (2022) 095] of the small-$x$ helicity evolution equations from [J. High Energy Phys. 01 (2016) 072, Phys. Rev. D 99, 054032 (2019)] at large $N_c$ and $N_f$. (Here $N_c$ and $N_f$ are the numbers of quark colors and flavors, respectively.) The evolution equations are double-logarithmic in the Bjorken $x$ variable, resumming powers of ${\alpha }_s{\ln }^2(1/x)$ with ${\alpha }_s$ the strong coupling constant. The large-$N_c\&N_f$ evolution we consider includes contributions of small-$x$ quark emissions and is thus more realistic than the large-$N_c$ one, which only involves gluon emissons. The evolution equations [J. High Energy Phys. 07 (2022) 095, J. High Energy Phys. 01 (2016) 072, Phys. Rev. D 99, 054032 (2019)] are written for the so-called “polarized dipole amplitudes,” which are related to the helicity distribution functions and the $g_1$ structure function. Unlike the previously reported solution [J. High Energy Phys. 08 (2020) 014] of the earlier version of helicity evolution equations at large $N_c\&N_f$ [J. High Energy Phys. 01 (2016) 072, Phys. Rev. D 99, 054032 (2019)], our solution does not exhibit periodic oscillations in $\ln (1/x)$ for $N_f<2N_c$, while only showing occasional sign reversals. For $N_f=2N_c$, we report oscillations with $\ln (1/x)$, similar to those found earlier in [J. High Energy Phys. 08 (2020) 014]. We determine the intercept of our evolution for $N_f<2N_c$ as well as the parameters of the oscillatory behavior for $N_f=2N_c$. We compare our results to the existing resummation [Z. Phys. C 72, 627 (1996)] and finite-order calculations for helicity-dependent quantities in the literature.

• Received 12 June 2023
• Accepted 8 August 2023

DOI:https://doi.org/10.1103/PhysRevD.108.054005

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Published by the American Physical Society