The high–energy quark–quark scattering: from Minkowskian to Euclidean theory

In this paper we consider some analytic properties of the high–energy quark–quark scattering amplitude, which, as is well known, can be described by the expectation value of two lightlike Wilson lines, running along the classical trajectories of the two colliding particles. We shall prove that the expectation value of two infinite Wilson lines, forming a certain hyperbolic angle in Minkowski space–time, and the expectation value of two infinite Euclidean Wilson lines, forming a certain angle in Euclidean four–space, are connected by an analytic continuation in the angular variables. This could open the possibility of evaluating the high–energy scattering amplitude directly on the lattice or using the stochastic vacuum model. The Abelian case (QED) is also discussed.