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

Understand

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].