ASYMPTOTICALLY FREE PARTONS AT HIGH ENERGY Richard

We describe the application of renormalization group improved perturbative QCD to inelastic lepton-hadron scattering at high center-of-mass energy but comparatively low photon virtuality. We construct a high energy factorization theorem which complements the mass factorization theorem used for processes with high virtualities. From it we derive a renormalization group equation which resums all large logarithms at high energy, thereby extending to this regime asymptotic freedom and thus the full range of perturbative computational techniques. We discuss the solution of this equation in various limits, and in particular show that the high energy behaviour of physical cross-sections is consistent with phenomenological expectations and unitarity bounds. March 1997 * Royal Society University Research Fellow Asymptotic freedom [1] in deep inelastic processes, derived using mass factorization and the renormalization group, lies at the heart of our understanding of perturbative QCD. Attempts to use perturbative QCD to understand processes with high centre-ofmass energies but relatively low virtualities are as old [2,3], but have never had as firm a theoretical foundation. In these processes distances from the light-cone are no longer short, but longitudinal distances are compressed by the large relative longitudinal momenta of the interacting particles [4]. At leading order in perturbation theory cross-sections are asymptotically flat, while at higher orders logarithms of energy (or rapidity) generate a rise. However these large logarithms eventually spoil the perturbative expansion. Attempts to sum them [5,6] lead to cross-sections which seem to rise too quickly to agree with current data [7], and eventually violate unitarity bounds [8]. Here we describe an approach to these processes based on a high energy factorization theorem for inelastic lepton-hadron scattering which, when combined with the renormalization group, implies that QCD becomes asymptotically free at high energies. The large energy logarithms can then be resummed to give a self-consistent perturbative description of inelastic scattering at high energy. A partonic picture is developed similar in many ways to the conventional one [9] useful at high virtualities. We find that cross-sections then rise only logarithmically, consistent both with phenomenological expectations and the behaviour of unitarity bounds. In this paper we will outline our main results; detailed proofs and a wider range of applications will be presented in a forthcoming publication. Perturbative logarithmic corrections to cross-sections arise (in physical gauges) from two particle reducible diagrams: two particle irreducible diagrams can be at most constant at high scales in a nonabelian gauge theory because they contain no logarithmic singularities [10]. The two particle reducible contributions to an inelastic virtual photon-hadron cross-section satisfy the fundamental factorization [11]: