Small-$x$ Asymptotics of the Quark Helicity Distribution: Analytic Results

Abstract

In this Letter, we analytically solve the evolution equations for the small-$x$ asymptotic behavior of the (flavor singlet) quark helicity distribution in the large-$N_c$ limit. These evolution equations form a set of coupled integro-differential equations, which previously could only be solved numerically. This approximate numerical solution, however, revealed simplifying properties of the small-$x$ asymptotics, which we exploit here to obtain an analytic solution. We find that the small-$x$ power-law tail of the quark helicity distribution scales as $\Delta q^S(x,Q^2)\sim {\left(\frac{1}{x}\right)}^{{\alpha }_h}$ with ${\alpha }_h=\frac{4}{\sqrt{3}}\sqrt{\frac{{\alpha }_s{N}_c}{2\pi }}$, in excellent agreement with the numerical estimate ${\alpha }_h\approx 2.31\sqrt{\frac{{\alpha }_s{N}_c}{2\pi }}$ obtained previously. We then verify this solution by cross-checking the predicted scaling behavior of the auxiliary "neighbor dipole amplitude" against the numerics, again finding excellent agreement.

Comments: 5 pages, 2 figures