Helicity at Small x: Oscillations and LLA Corrections

Proton spin puzzle is a longstanding problem in high-energy and nuclear physics: how is the proton spin distributed between quarks and gluons in the proton? A missing piece of the puzzle is the amount of spin coming from the quarks inside the proton at small Bjorken-$x$. Integral equations which predict quark helicity distributions at small $x$ were derived only recently. In this work, we construct a numerical solution of these equations at large-$N_c \& N_f$, with the aim to establish the small-$x$ asymptotics of the quark helicity distribution. (Here $N_c$ and $N_f$ are the numbers of quark colors and flavors.) Our main result is the following: we find that the quark helicity distribution should oscillate as a function of $\ln(1/x)$. The oscillation period depends on $N_f$ and spans many units of rapidity. This result may relate to the sign variation of the strange quark helicity distribution with $x$ seen in phenomenology. Our solution provides a better constraint on the quark's helicity distribution at small $x$, contributing to the resolution of the proton spin puzzle.