Improving the kinematics for low-x QCD evolution equations in coordinate space

High-energy evolution equations, such as the Balitsky-Fadin-Kuraev-Lipatov (BFKL), Balitsky-Kovchegov (BK) or Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner equations, aim at resumming the high-energy (next-to-) leading logarithms appearing in QCD perturbative series. However, the standard derivations of these equations are performed in a strict high-energy limit, whereas such equations are then applied to scattering processes at large but finite energies. For this reason, there is typically a slight mismatch between the leading logs resummed by these evolution equations without finite-energy corrections and the leading logs actually present in the perturbative expansion of any observable. That mismatch is one of the sources of large corrections at next-to-leading-order and next-to-leading-logarithmic accuracy. In the case of the BFKL equation in momentum space, this problem is solved by including a kinematical constraint in the kernel, which is the most important finite-energy correction. In this paper, such an improvement of kinematics is performed in mixed space (transverse positions and k(+)) and with a factorization scheme in the light-cone momentum k(+) (in a frame in which the projectile is right-moving and the target left-moving). This is the usual choice of variables and factorization scheme for the the BK equation. A kinematically improved version of the BK equation is provided, consistent at finite energies. The results presented here are also a necessary step towards having the high-energy limit of QCD (including gluon saturation) quantitatively under control beyond strict leading-logarithmic accuracy.