Helicity Evolution at Small $x$: Revised Asymptotic Results at Large $N_c\&N_f$

Abstract

We present a numerical solution of the revised version of the small-$x$ helicity evolution equations at large $N_c$ and $N_f$. (Here $N_c$ and $N_f$ are the numbers of quark colors and flavors, respectively.) The evolution equations are double-logarithmic in the Bjorken $x$ variable, resumming powers of ${\alpha }_s{\ln }^2(1/x)$ with ${\alpha }_s$ the strong coupling constant. The large-$N_c\&N_f$ evolution we consider includes contributions of small-$x$ quark emissions and is thus more realistic than the large-$N_c$ one, which only involves gluon emissons. The evolution equations are written for the so-called ``polarized dipole amplitudes", which are related to the helicity distribution functions and the $g_1$ structure function. Unlike the previously reported solution of the earlier version of helicity evolution equations at large $N_c\&N_f$, our solution does not exhibit periodic oscillations in $\ln (1/x)$ for $N_f<2N_c$, while only showing occasional sign reversals. For $N_f=2N_c$, we report oscillations with $\ln (1/x)$, similar to those found earlier. We determine the intercept of our evolution for $N_f<2N_c$ as well as the parameters of the oscillatory behavior for $N_f=2N_c$. We compare our results to the existing resummation and finite-order calculations for helicity-dependent quantities in the literature.

39 pages, 16 figures

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• Proton spin from small-$x$ with constraints from the valence quark model