New Trends in HERA Physics 1999 Proceedings of the Ringberg Workshop
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bstract. The status of the resummation of small x contributions to the unpolarized and polarized deep inelastic structure functions is reviewed.
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Introduction
The measurement of the nucleon structure functions in deep inelastic scattering provides important tests of the predictions of Quantum Chromodynamics (QCD) on the short–distance structure of nucleons. The experiments at the ep–collider HERA allowed to extend the kinematic region to very small values of x ∼ 10−4 at photon virtualities of Q2 ≥ 10GeV2 measuring the structure function F2 (x, Q2 ) at an accuracy of O(1%). These precise measurements allow dedicated tests of QCD. Also the polarized deep inelastic experiments approach smaller values of x with a higher accuracy. To obtain a description in this kinematic range potentially large contributions to the evolution kernels were studied during the last two decades and the resummations of ‘leading term’ contributions were performed. Here mainly two directions were followed. In one approach [1] a non–linear resummation of fan–diagrams of single ladder cascades is performed in the double logarithmic approximation [2]1 . Corrections of this type may ultimately become important at very small values of x to restore unitarity. Numerical studies of this equation were performed in Refs. [4]. One important assumption in the solution of this equation was that the nonperturbative input distributions for the N –ladder terms are given by the Nth power of the single gluon distribution at some starting scale. This would imply a strong constraint on the hierarchy of higher twist distributions. Later it was found [5] that the approximation [1] has to be supplemented by further color correlations even in the double logarithmic approximation, which cannot be cast into a non–linear equation anymore. As implied by the operator product expansion, the contributions due to different twist renormalize independently. The corresponding input distributions are likely to be unrelated between the different twists. Still saturation effects of the structure functions at very small x may be caused due to higher twist contributions. However, the detailed dynamics is yet unknown. 1
This approximation has to be considered as qualitative and leads often to an overestimate of the scaling violations, cf. [3].
G. Grindhammer, B.A. Kniehl, and G. Kramer (Eds.): LNP 546, pp. 42–57, 2000. c Springer-Verlag Berlin Heidelberg 2000
QCD Evolution of Structure Functions at Small x
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Ladder equations also form the basis of other approaches. In a physical gauge the emission of gluons along a single ladder–cascade describes2 in leading order (LO) the evolution of a parton density as predicted by the renormalization group equation if the emissions along the ladder are strongly ordered in the transverse momentum k⊥,1 ...k⊥,i k⊥,i+1 ... If these emissions are evaluated in the approximation x1 ...xi xi+1 ... instead, using effective vertices, one obtains the BFKL–resummation [6] in LO. This particular aspect led sometimes to the impression that these
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