BFkl

BFKL stands for the Balitsky–Fadin–Kuraev–Lipatov equation, a perturbative QCD framework developed to describe high-energy scattering in the Regge limit where the center-of-mass energy is large and the momentum transfer is fixed. It resums large logarithms of 1/x that arise at small Bjorken x, providing a description of how gluon densities evolve with energy.

The BFKL equation is an integral-differential linear evolution equation for the gluon density or the gluon Green's function as a function of rapidity Y ≈ ln(1/x) and transverse momentum k. In practical form it relates the derivative with respect to ln(1/x) to an integral over transverse momenta with a process-dependent kernel K that encodes real emissions and virtual corrections. The equation admits eigenfunctions proportional to powers of k^2, (k^2)^(γ-1), with eigenvalues χ(γ) known as the BFKL kernel. At leading order, the maximum of χ(γ) occurs at γ = 1/2 and gives the well-known LL intercept χ0(1/2) = 4 ln 2, leading to a growth of cross sections with energy: αP − 1 ≈ (αs Nc/π) χ0(1/2).

Beyond leading order, next-to-leading-log (NLL) corrections are large and sensitive to scale choices. This spurred approaches to resumm collinear contributions and implement running coupling effects to stabilize the evolution. BFKL is connected to but distinct from DGLAP evolution; in the small-x regime it complements DGLAP by describing rapidity evolution rather than solely virtuality ordering. Phenomenologically, it informs small-x behavior of structure functions, forward jet production, and Mueller–Navelet jet correlations, though practical predictions depend on resummation schemes and higher-order effects.

Extensions include nonlinear generalizations for parton saturation (e.g., BK/JIMWLK), while various numerical implementations and Monte Carlo formalisms model BFKL dynamics in event generation.