NLO BFKL equation, running coupling, and renormalization scales

I examine the solution of the BFKL equation with NLO corrections relevant for deep inelastic scattering. Particular emphasis is placed on the part played by the running of the coupling. It is shown that the solution factorizes into a part describing the evolution in $Q^2,$ and a constant part describing the input distribution. The latter is infrared dominated, being described by a coupling which grows as $x$ decreases, and thus being contaminated by infrared renormalons. Hence, for this part we agree with previous assertions that predictive power breaks down for small enough $x$ at any $Q^2.$ However, the former is ultraviolet dominated, being described by a coupling which falls like $1/(\ln {(Q}^2/{\Lambda }^2)+A\left[{\overline{\alpha }}_s{(Q}^2\right)\ln \mathrm{}(1/x)\mathrm{}{]}^{1/2})$ with decreasing $x,$ and thus is perturbatively calculable at all $x.\mathrm{}$ Therefore, although the BFKL equation is unable to predict the input for a structure function for small $x,$ it is able to predict its evolution in $Q^2,$ as we would expect from the factorization theory. The evolution at small $x$ has no true powerlike behavior due to the fall of the coupling, but does have significant differences from that predicted from a standard NLO in ${\alpha }_s$ treatment. Application of the resummed splitting functions with the appropriate coupling constant to an analysis of data, i.e., a global fit, is very successful.