First study of polarized proton-proton scattering with small-$x$ helicity evolution

We perform a phenomenological study of helicity-dependent parton distribution functions (PDFs) using small-$x$ helicity evolution equations, incorporating for the first time single-inclusive jet production data in polarized proton-proton ($pp$) scattering at parton momentum fractions $x<0.1$. We also simultaneously include double-longitudinal spin asymmetries in inclusive and semi-inclusive deep-inelastic scattering probing $x<0.1$. Employing the polarized small-$x$ pure-glue calculation of $pp\to gX$ for the jet production cross section, we modify the large-$N_c\&N_f$ KPS-CTT evolution equations by setting $N_f = 0$ to replicate the large-$N_c$ (pure-glue) limit, while retaining external quark flavors for the spinor field operators. We find that the $pp$ data have a considerable impact on the helicity PDFs at small $x$, reducing their uncertainties and leading to a total quark and gluon helicity in the proton for $x<0.1$ of $-0.04 \pm 0.23$. Combining our analysis with the a recent JAM helicity PDF analysis of the world polarized data, which includes $x>0.1$, we find a total quark and gluon helicity contribution for $x>10^{-7}$ of between 0.02 and 0.51.