Proton spin from small-$x$ with constraints from the valence quark model

TL;DR

This work tackles the proton spin problem by focusing on the small-$x$ region through helicity evolution and non-perturbative initial conditions. The authors constrain the initial helicity dipole amplitudes using a valence-quark model, fixing 16 of 24 parameters and leaving 8 free, while exploring two coordinate-space running-coupling prescriptions. Global fits to polarized DIS and SIDIS data yield a good description with $ ext{χ}^2/ ext{N}_{ ext{pts}} obreak\approx obreak 1.26$–$1.28$, and robust predictions, notably a negative $g_1^p$ at $x o 10^{-3}$ and a positive small-$x$ spin contribution dominated by $rac{1}{2}\Delta ext{Σ}+ ext{ΔG}$, whose value depends on the running-coupling scale. These results significantly tighten the small-$x$ spin budgets and suggest substantial spin carried by small-$x$ partons, with the exact amount sensitive to single-logarithmic corrections and the scale choice, highlighting the pivotal role of the valence-quark constraint for future analyses and EIC-era data.

Abstract

We apply the valence quark model recently calculated for polarized proton to constrain the non-perturbative initial condition for the small-$x$ helicity evolution. The remaining free parameters are constrained by performing a global analysis to the available polarized small-$x$ deep inelastic scattering data. A good description of the world data is obtained with only 8 free parameters. The model parameters are tightly constrained by the data, allowing us to predict the proton polarized structure function $g_1^p$ to be negative at small $x$. Furthermore, we obtain the small-$x$ quark and gluon spins to give a contribution $\int_{10^{-5}}^{0.1} dx \left( \frac{1}{2}ΔΣ+ ΔG \right) = 0.76 \pm 0.13$ or $1.70\pm 0.20$ to the proton spin, depending on the applied running coupling prescription.

Figures (6)

• Figure 1: The resulting $g_1$ structure function for the proton, together with the $1\sigma$ uncertainty bands, calculated from the fit of small-$x$ helicity evolution to the polarized DIS and SIDIS measurements. The dark green and pink bands correspond to the initial condition with constraints from the valence quark model using the coordinate-space running coupling with $C^2=1$ and $C^2=e^{-2\gamma_E}$, respectively. The light green band corresponds to the Born-inspired initial condition employed in Adamiak:2023yhz.
• Figure 2: The resulting truncated moment of the quark-singlet (left panel) and gluon (right panel) hPDFs, together with the $1\sigma$ uncertainty bands, calculated from the fit of small-$x$ helicity evolution to the polarized DIS and SIDIS measurements. The dark blue and pink bands correspond to the initial condition with constraints from the valence quark model using the coordinate-space running coupling with $C^2=1$ and $C^2=e^{-2\gamma_E}$, respectively. The light blue band corresponds to the Born-inspired initial condition employed in Adamiak:2023yhz.
• Figure S1: Comparison plots for asymmetry obervables in polarized DIS processes. Here, the coordinate-space running coupling is employed with $C^2=1$ for both initial conditions.
• Figure S2: Comparison plots for asymmetry obervables in polarized SIDIS processes. Here, the coordinate-space running coupling is employed with $C^2=1$ for both initial conditions.
• Figure S3: Comparison plots for asymmetry obervables in polarized DIS processes. Here, the coordinate-space running coupling is employed with $C^2=e^{-2\gamma_E}$ for both initial conditions.