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 $$ {\left.\Delta \Sigma \left(x,{Q}^2\right)\right|}_{\mathrm{large}\hbox{-} {N}_c\&{N}_f}\sim {\left(\frac{1}{x}\right)}^{\alpha_h^q}\cos \left[{\omega}_q\ln \left(\frac{1}{x}\right)+{\varphi}_q\right]. $$. The power $α^{q}_{h}$ exhibits a weak N f -dependence, and, for all Nf values considered, remains very close to $α^{q}_{h}$(N f =0)=$\left(4/\sqrt{3}\right)\sqrt{\alpha_s{N}_c/\left(2\pi \right)}$ obtained earlier in the large-N c limit. The novel oscillation frequency ω q and phase shift φ q depend more strongly on the number of flavors N f (with ω q = 0 in the pure-glue large-N c limit). The typical period of oscillations for ΔΣ 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.