The Small $x$ Behavior of $g_1$ in the Resummed Approach

The double logarithmic terms ${\alpha }_s{\ln }^{2}x$ are important to predict precisely the small $x$ behavior of the spin structure function $g_1$. We numerically analyze the evolution of the flavor non-singlet $g_1$ including the all-order resummed effect of these terms. It is pointed out that the next-to-leading logarithmic corrections produce an unexpectedly large suppression factor over the experimentally accessible range of $x$ and $Q^2$. This implies that the next-to-leading logarithmic contributions are very important in order to obtain a definite prediction.