The solution of the LO BFKL evolution describes a scattering amplitude
that grows proportionally to a positive power of the center-of-mass energy of the hadronic
scattering processes: at this order the kernel of the evolution equation respects the conformal symmetry of
the SL(2,C) Mobius group and the eigenfunctions are power-like functions of transverse distance
in coordinate space (or, in momentum space, powers of transverse momenta), while the eigenvalue of
the kernel is related to the Pomeron intercept. At the NLO there appears also a contribution to
the evolution kernel due to the running of the QCD coupling constant and the conformal property
of the LO BFKL is lost. Consequently, the LO BFKL kernels conformal eigenfunctions are not
eigenfunctions of the NLO BFKL kernel: at this order the power-law growth of the amplitudes with
energy also seems to be lost because of the non-Regge terms appearing due to the running
coupling effects. Despite a number of efforts, an exact analytical solution of the NLO BFKL equation
was still lacking. This is in stark contrast to the DGLAP evolution equation, which is a
renormalization group equation in the virtuality Q2: the eigenfunctions of that evolution equation are
simple powers of Bjorken-x variable for the kernel calculated to any order in the coupling
constant. The general form of the solution for DGLAP equation is well-known with the higher-order
corrections in the powers of the coupling constant entering into the anomalous dimension of the operator
at hand.
We derive the solution of the NLO BFKL equation by constructing its
eigenfunctions perturbatively, using an expansion around the LO BFKL (conformal) eigenfunctions. As a
result we, not only have a perturbative expansion of the intercept (eigenvalues of the kernel),
but also a perturbative expansion of the eigenfunctions thus, restoring the power-law growth of
the amplitudes with energy.