BalitskyFadinKuraevLipatov

Balitsky–Fadin–Kuraev–Lipatov equation, commonly known as the BFKL equation, is a key result in quantum chromodynamics (QCD) that describes the evolution of parton densities in the high-energy or small-x limit. It sums leading logarithms of 1/x in perturbation theory, providing a framework to study scattering processes at fixed momentum transfer but large center-of-mass energy. The equation was derived independently by Balitsky and Fadin, and by Kuraev and Lipatov, in the mid to late 1970s.

Formally, the BFKL equation is an integral (or differential) equation for the unintegrated gluon distribution as a function of transverse momentum k_T and rapidity Y, or equivalently for the scattering amplitude in the Regge limit. Its solutions reveal a Reggeized gluon exchange and lead to a BFKL pomeron, predicting a power-like rise of gluon densities with energy characterized by an intercept α_P(0) = 1 + (4 α_s N_c/π) ln 2 at leading order.

Although LO BFKL captures the main energy growth, next-to-leading order corrections are sizable and higher-order effects must be resummed to obtain stable predictions. The equation has wide use in describing small-x phenomena in deep inelastic scattering and in processes such as Mueller-Navelet jets and forward particle production at high-energy colliders, where transverse momentum diffusion plays a role.

Over the years, refinements include collinear-improved BFKL and various resummation schemes to tame the perturbative series. The BFKL framework complements the DGLAP evolution, providing a complementary description of QCD dynamics at high energy.