Z. Phys. C - Particles and Fields 17, 69-84 (1983) Particles ~r Physik C andFieids 9 Springer-Verlag 1983 Improved Description of the Small-x Limit of Deep Inelastic Scattering Marcelo Loewe 1 II. Institut flit Theoretische Physik der Universit~it Hamburg, D-2000 Hamburg, Federal Republic of Germany Received 28 September 1982 Abstract. In this paper we attempt to find, within the framework of perturbative QCD, an improved de- scription of the small-x region of deep-inelastic scat- tering. Motivated by results on the Pomeron in QCD, we investigate diagrams with a system of an arbitrary number of gluon lines in the t-channel. The gluon lines are allowed to interact pairwise in all possible combinations. We find that the leading behaviour of these diagrams comes from configu- rations where the gluon lines arrange themselves into non interacting, non-forward ladders. We then analyse couplings between gluon ladders, in particu- lar non-planar ones. We compare our results with a previous study of Gribov, Levin, and Ryskin. In agreement with them we conclude that the sum of these diagrams leads to a tempered increase of the structure function in the small-x region. I. Introduction It is well-known that the standard approximation of perturbative QCD which nicely describes deep in- elastic scattering in the Bjorken limit becomes in- appropriate when the scaling variable x tends to zero. One might hope that by including more and more Feynman diagrams into the analysis, the per- turbative description in the small-x region becomes better, although the limit x~0, is expected to re- quire non perturbative effects. It is the aim of this paper to find an improved description within per- turbative QCD. A leading logarithmic analysis in the axial gauge 1 On leave from Universidad Cat61ica de Chile, Santiago, Chile Permanent address after 1-10-82. Facultad de Fisica, Pontificia Universidad Cat61ica de Chile, Casilla l14-D Santiago Chile [1, 2] shows us that the relevant Feynman diagram for the structure function in the small-x case, but x not too small, corresponds to a forward gluon lad- der, more precisely to the discontinuity of a gluon ladder which begins at a valence quark and ends with a quark loop, that interacts with the external photon. The virtualities of the partons along the ladder are strongly ordered when going from the valence quarks to the photon. In the limit x--,0 this diagram, however, leads to an answer for the struc- ture function which violates the Froissart bound. This means that the naive single ladder approxima- tion is not sufficient, and, therefore, it is mandatory to include another type of diagrams which could be relevant in the small-x region. The motivation for the choice of these diagrams in the hard regime comes from the Regge limit [3]. We know that at the level of a single ladder, the hard regime in the small-x region and the Regge limit are intimately connected. By studying the Po- meron kernel of the Regge limit in the regime of high transverse momenta, Dokshitzer [1] has shown that this kernel turns into the small-x kernel in Bjorken's regime. In the Regge limit it is known [3] that diagrams with only two gluon lines in the t-channel violate the unitarity bound. In order to improve this behaviour we have to take into account diagrams with more gluon lines in the t-channel. This fact motivates us to study, now in the hard regime, diagrams with n gluon lines. Following the Regge limit analysis, we allow these gluon lines to interact in pairs, by ex- changing gluons, in all possible ways as it is shown in Fig. 1. The full system of n gluon lines has to be in the forward direction in order to contribute to the struc- ture function. Any subsystem, however, where two