Infrared finitness and analyticity properties of the loop–loop scattering amplitudes in gauge theories

Differently from the parton–parton scattering amplitudes, which are known to be affected by infrared (IR) divergences, the elastic scattering amplitude of two colourless states in gauge theories, e.g., two qq̄ meson states, is expected to be an IR–finite physical quantity. It was shown in Refs. [1,2,3] that the high–energy meson–meson elastic scattering amplitude can be approximately reconstructed by first evaluating, in the eikonal approximation, the elastic scattering amplitude of two qq̄ pairs (usually called “dipoles”), of given transverse sizes ~ R1⊥ and ~ R2⊥ respectively, and then averaging this amplitude over all possible values of ~ R1⊥ and ~ R2⊥ with two proper squared wave functions |ψ1(~ R1⊥)| 2 and |ψ2(~ R2⊥)| , describing the two interacting mesons. The high– energy elastic scattering amplitude of two dipoles is governed by the correlation function of twoWilson loops W1 and W2, which follow the classical straight lines for quark (antiquark) trajectories: