The gluon splitting function at moderately small x

It is widely believed that at small x, the BFKL resummed gluon splitting function should grow as a power of 1/x. But in several recent calculations it has been found to decrease for moderately small-x before eventually rising. We show that this 'dip' structure is a rigorous feature of the Pgg splitting function for sufficiently small alpha_s, the miminum occurring formally at log(1/x) ~ 1/ sqrt(alpha_s). We calculate the properties of the dip, including corrections of relative order sqrt(alpha_s), and discuss how this expansion in powers of sqrt(alpha_s), which is poorly convergent, can be qualitatively matched to the fully resummed result of a recent calculation, for realistic values of alpha_s, Finally, we note that the dip position, as a function of alpha_s, provides a lower bound in x below which the NNLO fixed-order expansion of the splitting function breaks down and the resummation of small-x terms is mandatory.