Analytic Solution for the Helicity Evolution Equations at Small $x$ and Large $N_c\&N_f$

Abstract: We construct an exact analytic solution of the revised small-$x$ helicity evolution equations, where the contributions of the quark-to-gluon and gluon-to-quark transition operators were newly included. These evolution equations are written in the large-$N_c&N_f$ limit and are double-logarithmic, resumming powers of $\alpha_s\ln^2(1/x)$. Here $N_c$ and $N_f$ are the numbers of quark colors and flavors, while $\alpha_s$ is the strong coupling constant and $x$ is the Bjorken-$x$ variable. Using our solution, we obtain analytic expressions for the flavor singlet quark and gluon helicity parton distribution functions (PDFs) and for the $g_1$ structure function as double-inverse Laplace transforms. We also extract analytic expressions for the four DGLAP polarized anomalous dimensions $\Delta \gamma_{qq}, \Delta \gamma_{qG}, \Delta \gamma_{Gq}$, and $\Delta \gamma_{GG}$: these expressions resum powers of $\alpha_s/\omega^2$ to all orders at large-$N_c&N_f$ (with $\omega$ the Mellin moment variable). We extract the leading small-$x$ growth of the helicity distributions, \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}, \end{align} where the intercept $\alpha_h$ satisfies an algebraic equation. We determine $\alpha_h$ numerically for various values of $N_c$ and $N_f$. We further obtain the explicit asymptotic expressions for the helicity distributions, which yield numerical values for the ratio of the gluon helicity PDF to the flavor singlet quark helicity PDF in the small-$x$ asymptotic limit (for different $N_f/N_c$). We find that all our predictions for polarized DGLAP anomalous dimensions are fully consistent with the existing finite-order calculations. Similar to the large-$N_c$ case, our intercept $\alpha_h$ exhibits a very slight disagreement with the predictions made within the infrared evolution equations framework.