Helicity at Small $x$: Oscillations Generated by Bringing Back the Quarks

We construct a numerical solution of 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 beyond the large-$N_c$ limit explored previously in the same framework. (Here $N_c$ and $N_f$ are the numbers of quark colors and flavors.) While the large-$N_c$ helicity evolution involves gluons only, the large-$N_c \& N_f$ evolution includes contributions from quarks as well. We find that adding quarks to the evolution makes quark helicity distribution oscillate as a function of $x$. Our numerical results in the large-$N_c \& N_f$ limit lead to the $x$-dependence of the flavor-singlet quark helicity distribution which is well-approximated by \begin{align} \Delta \Sigma (x, Q^2)\bigg|_{\mbox{large-}N_c \& N_f} \sim \left( \frac{1}{x} \right)^{\alpha_h^q} \, \cos \left[ \omega_q \, \ln \left( \frac{1}{x} \right) + \varphi_q \right]. \end{align} The power $\alpha_h^q$ exhibits a weak $N_f$-dependence, and, for all $N_f$ values considered, remains very close to $\alpha_h^q (N_f=0) = (4/\sqrt{3}) \sqrt{\alpha_s N_c/(2 \pi)}$ obtained earlier in the large-$N_c$ limit. The novel oscillation frequency $\omega_q$ and phase shift $\varphi_q$ depend more strongly on the number of flavors $N_f$ (with $\omega_q =0$ in the pure-glue large-$N_c$ limit). The typical period of oscillations for $\Delta \Sigma$ is rather long, spanning many units of rapidity. We speculate whether the oscillations we find are related to the sign variation with $x$ seen in the strange quark helicity distribution extracted from the data.