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

We apply the valence quark model [1] to constrain the non-perturbative initial condition for the small-$x$ helicity evolution. The remaining free parameters are constrained by performing a global analysis akin to Adamiak et al. [2] 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 ${\int }_{{10}^{-5}}^{0.1}dx(\frac{1}{2}\Delta \Sigma +\Delta G)=0.63\pm 0.10$ or $1.35\pm 0.16$ to the proton spin, depending on the applied running coupling prescription.