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.

Figure 1

Example of a light-front perturbation theory diagram contributing to the $q\overline{q}gg$ Fock component of a photon. Energy denominators are indicated by the vertical dashed lines. The light-front time $x^{+}$ increases from the left to the right of the diagram, from $-\infty$ to 0.

Figure 2

Tree-level contribution to the wave function of a particle in light-front perturbation theory, where only the last parton splitting is specified to be a one-to-two splitting. $X$ designates here an arbitrary Fock state. The last two energy denominators are indicated by the vertical dashed lines. The light-front time $x^{+}$ increases from the left to the right of the diagram, from $-\infty$ to 0.

Figure 3

Examples of light-front perturbation theory diagrams without gluon splitting contributing to the $q\overline{q}ggg$ Fock component of a photon wave function.

Figure 4

Examples of light-front perturbation theory diagrams with gluon splittings contributing to the $q\overline{q}ggg$ Fock component of a photon wave function. These diagrams can be interpreted in the standard way, or as color-ordered contributions to the photon wave function, with the color factor $[t^{a_4}{t}^{a_2}{t}^{a_3}{]}_{{\alpha }_0{\alpha }_1}$.