We analyze the world polarized deep-inelastic scattering (DIS) and semi-inclusive DIS (SIDIS) data at low values of $x<0.1$, using small-$x$ evolution equations for the flavor singlet and nonsinglet helicity parton distribution functions (hPDFs), which resum all powers of both ${\alpha }_s{\ln }^2(1/x)$ and ${\alpha }_s\ln (1/x)\ln (Q^2/Q_0^2)$, with ${\alpha }_s$ being the strong coupling constant. The hPDFs for quarks, antiquarks, and gluons are extracted and evolved to lower values of $x$ to make predictions for the future Electron-Ion Collider (EIC). We improve on our earlier work by employing the more realistic large-$N_c\&N_f$ limit of the revised small-$x$ helicity evolution, and by incorporating running coupling corrections along with SIDIS data into the fit. We find an anticorrelation between the signs of the gluon and $C$-even quark hPDFs, as well as the $g_1$ structure function. While the existing low-$x$ polarized DIS and SIDIS data are insufficient to constrain the initial conditions for the polarized dipole amplitudes in the helicity evolution equations, future EIC data will allow more precise predictions for hPDFs and the $g_1$ structure function for $x$ values beyond those probed at the EIC. Using the obtained hPDFs, we discuss the contributions to the proton spin from quark and gluon spins at small $x$.

• Received 24 August 2023
• Accepted 19 October 2023

DOI:https://doi.org/10.1103/PhysRevD.108.114007

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Published by the American Physical Society