The Singlet Contribution to the Structure Function $g_1(x,Q^2)$ at Small $x$

Abstract

The resummation of $O({\alpha }_s^{l+1}{\ln }^{2l}x)$ terms in the evolution equation of the singlet part of $g_1(x,Q^2)$ is carried out. The corresponding singlet evolution kernels are calculated explicitely. The leading small-$x$ contribution to the three-loop splitting function matrix is determined in the $\overline{\mathrm{MS}}$ scheme. Relations are derived for the case of $\mathcal{N}=1$ supersymmetric Yang--Mills field theory. Numerical results are presented for the polarized singlet and gluon densities, and the structure functions $g_1^p(x,Q^2)$ and $g_1^n(x,Q^2)$. They are compared for different assumptions on the non--perturbative input distributions, and the stability of the results against presently unknown subleading contributions is investigated.

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