Exact solution of BFKL equation in jet-physics

TL;DR

This paper addresses the problem of resumming leading soft, non-Abelian logarithms in the heavy QQbar-multiplicity observable, showing that the associated evolution equation can be solved exactly. By exploiting a spectral expansion with phase shifts and hypergeometric eigenfunctions, the authors obtain a closed-form expression for the phase shift and a Mehler-Fock-type representation of the solution, revealing how a finite angular range (a boundary) alters the asymptotics compared to the BFKL equation. The work demonstrates that, while the QQbar equation and BFKL share a formal structure and a Pomeron-like intercept, boundary effects and running coupling lead to different prefactors and diffusion properties, providing new insight into jet physics versus high-energy scattering. The results include both solvable-model insights and the full analytic solution, along with numerical validation, offering a rigorous framework to compare jet observables with high-energy amplitudes.

Abstract

It has been recently found that the heavy quark-antiquark QQbar pair multiplicity, in certain phase space region (QQbar at short distance, soft and with small velocity), satisfies an evolution equation formally similar to the BFKL equation for the high energy scattering amplitude. We find the exact solution of the QQbar-equation and discuss the differences with the BFKL scattering amplitude.

Figures (3)

• Figure 1: Evolution of $\phi$ at $\rho=1$ obtained from \ref{['eq:finito']} with initial condition $I(\rho,0)=\tfrac12\rho$. The corresponding $Q\bar{Q}$ inclusive distribution $I(\rho\!=\!1,\tau)=e^{4\ln\!2\, \tau}\,\phi(0,\tau)$ is also plotted.
• Figure 2: The distribution $\phi(x,\tau)$ obtained from \ref{['eq:finito']} with initial condition $I(\rho,0)=\tfrac12\rho$
• Figure 3: Long time evolution of $\phi(x,\tau)$ obtained from \ref{['eq:finito']} with initial condition $I(\rho,0)=\tfrac12\rho$