A nonperturbative foundation of the Euclidean-Minkowskian duality of Wilson-loop correlation functions

In this letter we discuss the analyticity properties of the Wilson-loop correlation functions relevant to the problem of soft high-energy scattering, directly at the level of the functional integral, in a genuinely nonperturbative way. The strategy is to start from the Euclidean theory and to push the dependence on the relevant variables $\theta$ (the relative angle between the loops) and $T$ (the half-length of the loops) into the action by means of a field and coordinate transformation, and then to allow them to take complex values. In particular, we determine the analyticity domain of the relevant Euclidean correlation function, and we show that the corresponding Minkowskian quantity is recovered with the usual double analytic continuation in $\theta$ and $T$ inside this domain. The formal manipulations of the functional integral are justified making use of a lattice regularisation. The new rescaled action so derived could also be used directly to get new insights (from first principles) in the problem of soft high-energy scattering.