An introduction to leading and next-to-leading BFKL

Some twenty-five years ago Balitsky, Fadin, Kuraev and Lipatov (BFKL) set out to determine the high-energy behaviour of the scattering of hadronic objects within perturbative QCD. They found terms going as (αs ln s) n, where s is the squared centre-of-mass energy. Since ln s is large it can compensate the smallness of ᾱs and thus it was necessary to sum this whole series of leading logarithmic (LL) terms. The result was that the cross section should grow as a power of the squared centre-of-mass energy s [1]. For the values of αs ≃ 0.2 that are typically relevant, this power comes out as being of the order of 0.5. Over the past few years much experimental effort has been devoted towards observing this phenomenon, and the conclusion has consistently been that while the cross sections do rise, that rise is much slower than s0.5 (see for example [2–6]). The solution to this problem was to have been in the next-to-leading corrections to the BFKL equation, terms αs(αs ln s) n, which have been calculated over the past ten years [7]. The various contributions were put together last year [8, 9], and to the consternation of the community turned