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

## TL;DR

This paper derives an analytical solution for the small-$x$ asymptotic behavior of the quark helicity distribution, confirming previous numerical estimates and providing explicit scaling laws in the large-$N_c$ limit.

## Contribution

It provides the first analytic solution to coupled evolution equations for quark helicity at small-$x$, previously only solvable numerically.

## Key findings

- Small-$x$ quark helicity distribution scales as (1/x)^{α_h}.
- Analytic α_h matches previous numerical estimates.
- Predicted scaling verified against numerical 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 $\Delta q^S (x, Q^2) \sim \left(\tfrac{1}{x} \right)^{\alpha_h}$ with $\alpha_h = \tfrac{4}{\sqrt{3}} \sqrt{\tfrac{\alpha_s N_c}{2\pi}}$, in excellent agreement with the numerical estimate $\alpha_h \approx 2.31\sqrt{\tfrac{\alpha_s N_c}{2\pi}}$ 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.