Small- x behaviour of initial state radiation in perturbative QCD

Abstract

We analyze in perturbative QCD the asymptotic behaviour of deep inelastic processes in the semi-hard region x→0. The study is done by extending the soft gluon insertion techniques. We confirm and extend the analysis recently performed by Ciafaloni. The main results are the following: (i) Soft gluon emission from the incoming parton takes place in a region where the angles between incoming and outgoing partons are ordered. This is due to coherent effects similar to the ones in the x→1 region. (ii) Virtual corrections involving an internal line with energy fraction x give rise, for x→0, to a new form factor of non-Sudakov type. This regularizes collinear singularities when an emitted gluon is parallel to the incoming parton. (iii) At the complete inclusive level, the new form factor plays the same role as the virtual corrections in the Lipatov equation for the Regge regime. We show that, in the semi-hard regime, the gluon anomalous dimension coincides with the Lipatov ansatz. (iv) We identify the branching structure of initial-state radiation including the semi-hard regime. The branching is formulated as a probability process given in terms of Sudakov and non-Sudakov form factors. This process, in principle, can be used to extend the existing simulations of QCD cascades to the semi-hard regime.