Flavor Singlet Contribution to the Structure Function $g_1$ at Small-x

TL;DR

The paper investigates the small-$x$ behavior of the flavor-singlet polarized structure function $g_1(x,Q^2)$ in the double-logarithmic approximation of QCD, revealing a power-like rise driven by DL contributions beyond standard GLAP evolution. By deriving infrared evolution equations that couple quark and gluon channels and accounting for ladder and nonladder (bremsstrahlung) diagrams, the authors show that gluon exchanges in the $t$-channel enhance the small-$x$ growth and cause significant quark–gluon mixing. The solution yields a leading small-$x$ exponent $\omega_s \approx 1.01$ (for $n_f=4$, $\alpha_s=0.18$), producing $g_1^S(x,Q^2) \sim x^{-\omega_s}$ with a sign opposite to the input distributions and a substantial gluon-driven contribution. These results have important phenomenological implications, indicating a much stronger small-$x$ rise than GLAP predictions and highlighting the sensitivity to the polarized gluon input, warranting further study and refined DL accuracy.

Abstract

The singlet contribution to the $g_1(x,Q^2)$ structure function is calculated in the double-logarithmic approximation of perturbative QCD in the region $x \ll 1$. Double logarithmic contributions of the type $(α_s \ln ^2 (1/x))^k$ which are not included in the GLAP evolution equations are shown to give a power-like rise at small-x which is much stronger than the extrapolation of the GLAP expressions. The dominant contribution is due to the gluons which, in contrast to the unpolarized case, mix with the fermions also in the region $x \ll 1$. The two main reasons why the small-x behavior of the double logarithmic approximation is so much stronger than the usual GLAP evolution are: the larger kinematical region of integration (in particular, no ordering in transverse momentum) and the contributions from non-ladder diagrams.