A semi-infinite matrix analysis of the BFKL equation

TL;DR

The paper develops a semi-infinite matrix representation of the forward BFKL equation in discretized virtuality space, interpreting the evolution as diffusion into infrared and ultraviolet scales. A square truncation of this matrix is exponentiated to yield asymptotic eigenstates that resemble the BFKL gluon Green's function, and its eigenstructure reveals a dominant mode driving high-energy growth while subleading modes are damped. The authors establish a precise link between the square-truncated BFKL kernel and a diagonal sector of the non-compact $sl(2)$ spin chain, highlighting integrable structures in high-energy QCD. They also propose an infrared-taming modification that suppresses IR diffusion, producing energy evolution closer to the Froissart bound and offering a mechanism to control the growth inherent in BFKL dynamics. Overall, the work provides a matrix-based, integrability-inspired perspective on BFKL evolution and a practical knob to enforce unitarity-like constraints.

Abstract

The forward BFKL equation is discretised in virtuality space and it is shown that the diffusion into infrared and ultraviolet momenta can be understood in terms of a semi-infinite matrix. The square truncation of this matrix can be exponentiated leading to asymptotic eigenstates sharing many features with the BFKL gluon Green's function in the limit of large matrix size. This truncation is closely related to a representation of the XXX Heisenberg spin $= - \frac{1}{2}$ chain with SL(2) invariance where the Hamiltonian acts on a symmetric double copy of the harmonic oscillator. A simple modification of the BFKL matrix suppressing the infrared modes generates evolution with energy compatible with unitarity.

Figures (17)

• Figure 1: Initial condition vector $\phi^0_i = \frac{\delta_i^{N_0}}{\Delta}$ taking $N_0 =3$ for $\Delta = 1$ and $N=50$.
• Figure 2: Action of $\hat{\cal H}$ on $\phi^0_i = \frac{\delta_i^{N_0}}{\Delta}$ with $N=50$ and $N_0=3$.
• Figure 3: $\chi_N (\gamma)$ coincides with the BFKL eigenvalue $\chi(\gamma)$ at $N \to \infty$.
• Figure 4: (Anti-)Collinear behaviour of the gluon Green's function in the BFKL equation.
• Figure 5: The growth of the BFKL Green's function with $Y$.