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We construct an exact analytic solution of the revised small-$x$ helicityevolution equations derived recently. The equations we solve are obtained inthe large-$N_c$ limit (with $N_c$ the number of quark colors) and aredouble-logarithmic (summing powers of $\alpha_s \ln^2(1/x)$ with $\alpha_s$ thestrong coupling constant and $x$ the Bjorken $x$ variable). Our solutionprovides small-$x$, large-$N_c$ expressions for the flavor-singlet quark andgluon helicity parton distribution functions (PDFs) and for the $g_1$ structurefunction, with their leading small-$x$ asymptotics given by \begin{align}  \Delta \Sigma (x, Q^2) \sim \Delta G (x, Q^2)  \sim g_1 (x, Q^2) \sim \left( \frac{1}{x} \right)^{\alpha_h} , \notag\end{align} where the exact analytic expression we obtain for the intercept$\alpha_h$ can be approximated by $\alpha_h = 3.66074 \, \sqrt{\frac{\alpha_s\, N_c}{2 \pi}}$. Our solution also yields an all-order (in $\alpha_s$)resummed small-$x$ anomalous dimension $\Delta \gamma_{GG} (\omega)$ whichagrees with all the existing fixed-order calculations (to three loops).Notably, our anomalous dimension is different from that obtained in theinfrared evolution equation framework developed earlier by Bartels, Ermolaev,and Ryskin (BER), with the disagreement starting at four loops. Despite thepreviously reported agreement at two decimal points based on the numericalsolution of the same equations, the intercept of our large-$N_c$ helicityevolution and that of BER disagree beyond that precision, with the BERintercept at large $N_c$ given by a different analytic expression from ourswith the numerical value of $\alpha_h^{BER} = 3.66394 \, \sqrt{\frac{\alpha_s\, N_c}{2 \pi}}$. We speculate on the origin of this disagreement.

Analytic Solution for the Revised Helicity Evolution at Small $x$ and  Large $N_c$: New Resummed Gluon-Gluon Polarized Anomalous Dimension and  Intercept