Helicity at Small $x$: Bringing Back the Quarks

We find the small-$x$ asymptotics of the quark helicity distribution in the large-$N_{c}${\&}$N_{f}$ limit by numerically solving small-$x$ evolution equations derived in earlier works, where $N_{c}$ is the number of quark colors and $N_{f}$ is the number of quark flavors. Previously, those evolution equations were solved only in the large-$N_{c}$ limit. We find that $\Delta q$ oscillates as a function of $\ln {(1/x)}$ at small $x$, with the oscillation frequency being dependent on the number of quark flavors, $N_{f}$. Our result may account for the apparent oscillation in the strange quark helicity distribution $\Delta s$ as a function of Bjorken $x$. For $N_{f}=0$, these oscillations disappear; this is why they were not seen in the earlier large-$N_{c}$ studies. Our work presents the most precise theoretical determination of the small-$x$ asymptotics of the quark helicity distribution based on the Wilson line approach to small-$x$ evolution.~When combined with the future EIC data, our approach should allow for a precise determination of the amount of the proton spin coming from small-$x$ partons, thus contributing to the resolution of the proton spin puzzle.