Summation of Leading Logarithms at Small x

Abstract

We show how perturbation theory may be reorganized to give splitting functions which include order by order convergent sums of all leading logarithms of $x$. This gives a leading twist evolution equation for parton distributions which sums all leading logarithms of $x$ and $Q^2$, allowing stable perturbative evolution down to arbitrarily small values of $x$. Perturbative evolution then generates the double scaling rise of $F_2$ observed at HERA, while in the formal limit $x\to 0$ at fixed $Q^2$ the Lipatov $x^{-λ}$ behaviour is eventually reproduced. We are thus able to explain why leading order perturbation theory works so well in the HERA region.