Helicity at Small x: Oscillations, Cross-Checks and LLA Corrections

A numerical solution is constructed for the recently-derived large-$N_c \& N_f$ small-$x$ helicity evolution equations 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.) We find that adding quarks to the evolution makes quark helicity distribution oscillate as a function of $\ln(1/x)$. The typical oscillation period depends on $N_f$ and spans many units of rapidity. This result may relate to the sign variation with $x$ seen in the strange quark data. In addition, we perform a cross check on this recently-derived helicity evolution equations by analytically solving the equations with a substantially different initial condition, obtaining the same asymptotics at large $N_c$. The fact that two large-$N_c$ evolution equations resulting from two different initial conditions give the same small-$x$ asymptotics provides a validity cross-check of the calculation. Finally, we derive the single-logarithmic corrections to the double-logarithmic equations derived previously. The more complete equations, once solved, will provide a more precise estimate of the quark helicity distribution at small $x$, contributing to the resolution of the proton spin puzzle.