High–energy scattering and Euclidean–Minkowskian duality

We shall discuss how some relevant analyticity and crossing-symmetry properties of the “eikonal scattering amplitudes” of two Wilson loops in QCD, when going from Euclidean to Minkowskian theory, can be related to the still unsolved problem of the asymptotic s–dependence of the hadron–hadron total cross–sections. In particular, we critically discuss the question if (and how) a pomeron–like behaviour can be derived from this Euclidean–Minkowskian duality. 1 Loop–loop and meson–meson scattering amplitudes It was shown in Refs. [1, 2] (for a review see Refs. [3–5] and references therein) that the high– energy meson–meson elastic scattering amplitude can be approximately reconstructed in two steps: i) one first evaluates, in the functional–integral approach, the high–energy elastic scattering amplitude of two qq̄ pairs (usually called dipoles), of given transverse sizes ~ R1⊥ and ~ R2⊥ and given longitudinal–momentum fractions f1 and f2 of the two quarks in the two dipoles respectively; ii) one then averages this amplitude over all possible values of ~ R1⊥, f1 and ~ R2⊥, f2 with two proper squared wave functions |ψ1(~ R1⊥, f1)| and |ψ2(~ R2⊥, f2)|, describing the two interacting mesons. The high–energy elastic scattering amplitude of two dipoles (defined in Eq. (8) below) is governed by the following (properly normalized) connected correlation function of two Wilson loops forming an hyperbolic angle χ in the longitudinal plane (see Eq. (4) below) and separated by a distance ~z⊥ = (z2, z3) in the transverse plane (impact parameter): CM(χ, ~z⊥; 1, 2) ≡ lim T→∞ [ 〈W ) 1 W (T ) 2 〉 〈W ) 1 〉〈W (T ) 2 〉 − 1 ]