On the Behaviour of Non--Singlet Structure Functions at Small $x$

The resummation of $O(α_s^{l+1} \ln^{2l} x)$ terms in the evolution kernels of non--singlet combinations of unpolarized and polarized structure functions is investigated. The agreement with complete calculations up to order $α_s^2$ is demonstrated, and the leading small-$x$ contributions to the three--loop non--singlet splitting functions $P^{\pm}$ are derived. The additional contributions due to the resummed terms are studied numerically for the most important non--singlet structure functions. They are found to be about $1\,\%$ or smaller in the kinematical regions accessible at present and in the foreseeable future.

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