Revised Small-x Helicity Evolution
by
Zoom
A complete understanding of parton helicity inside the proton requires the knowledge about its contribution at small Bjorken-$x$, which is difficult to deduce from experiments. We attempt to fill the gap by deriving a small-$x$ helicity evolution, resumming $\alpha_s\ln^2(1/x)$, with $\alpha_s$ the strong coupling constant. Recently, a revised version of this evolution has been constructed, taking into account the observation that the sub-eikonal operator, $\overleftarrow{D}^i\, D^i$, is mixed with other helicity objects from the previous works, which are the gluon field strength, $F^{12}$, and the quark axial current, ${\bar \psi} \gamma^+ \gamma^5 \psi$. Based on the new evolution, a closed system of evolution equations can be constructed in the limits of large $N_c$ or large $N_c\& N_f$. (Here, $N_c$ and $N_f$ are the number of quark colors and flavors, respectively.) We solve the large-$N_c$ equations numerically and obtain the following small-$x$ asymptotics for the $g_1$ structure function:
\[g_1(x,Q^2) \sim \left(\frac{1}{x}\right)^{3.66\,\sqrt{\frac{\alpha_sN_c}{2\pi}}} , \]
which agrees with the earlier work by Bartels, Ermolaev and Ryskin. A preliminary numerical study of the large-$N_c\&N_f$ equations show oscillatory pattern in \ln\frac{1}{x}, with periods decreasing as $N_f$ increases. The location of the first sign flip depends on the initial condition at which the small-$x$ regime begins.