Non-perturbative Scales in Soft Hadronic Collisions at High Energy∗)

We investigate the role of nonperturbative quark-gluon dynamics in soft high energy processes. In order to reproduce differential and total cross sections for elastic protonproton and proton-antiproton-scattering at high energy and small momentum transfer it turns out that we need two scales, the gluonic correlation length and a confinement scale. We find a small gluonic correlation length, a ≃ 0.2 fm, in accordance with recent lattice QCD results. Submitted to Phys. Lett. B ∗) Work supported in part by BMFT. †) Supported in part by Hans Boeckler and Friedrich Ebert Foundations. Introduction. Soft hadronic collisions at high energy are described successfully using Regge and Pomeron phenomenology [1, 2]. Yet an outstanding problem is the more basic understanding of pomeron exchange in terms of the underlying quark-gluon-dynamics. At large center of mass energies √ s but small momentum transfers |t| < ∼ (1− 2) GeV, one expects that non-perturbative features of QCD, such as gluonic vacuum properties, enter as important ingredients in hadron-hadron scattering amplitudes. In this context several recent papers [3, 4, 5] have emphasized specifically the role of gluonic field strength correlations in hadronic scattering processes at high energy. The aim of the present paper is to follow up on this theme and gain further insight into the basic scales which govern soft high energy collisions. In particular, the question arises whether the small gluonic correlation length recently found in lattice QCD calculations [6] is compatible with the scales deduced from the t-dependence of differential cross sections for high energy proton-proton and proton-antiproton collisions. Our starting point is the assumption that the scattering of composite hadrons can be reduced to the scattering of their quark constituents; we develop a model of quark-quark scattering in terms of nonperturbative gluon dynamics. Our basic picture is that soft interactions between quarks at high energy are described by the propagation of projectile quarks in a nonperturbative gluonic environment produced by the target quarks. The quark sources themselves are tied together by confining forces inside the hadron projectile and target. Propagation of High Energy Quarks in a Gluonic Background. We start from the QCD Lagrangian L(x) = − 2 trGμν(x)G (x) + ψ̄(x) [iγDμ(x)−m0]ψ(x) (1) with the usual field strength tensor Gμν = ∂μGν − ∂νGμ + ig[Gμ, Gν ] and the covariant derivative Dμ(x) = ∂μ+igGμ(x). Here Gμ(x) = G a μ(x)Ta are the gluon fields contracted with the SU(3)color generators Ta = λa/2. The quark field ψ(x) includes three color components, and m0 is the current quark mass. Consider the vacuum-vacuum transition amplitude Z = ∫