k⊥-factorization and perturbative invariants at small x

I review the basic idea of k⊥-factorization and its relation to collinear factorization. Theoretical results in resummed perturbation theory are summarized and the example of the heavy-flavour structure functions is explicitly considered. Using these results one can investigate the small-x behaviour of quantities that are independent of the non-perturbative parton densities. In particular, one can introduce physical anomalous dimensions that relate the scaling violations in different hadronic observables. DFF 254-7-96 July 1996 ∗Talk given at International Workshop on Deep Inelastic Scattering and Related Phenomena, DIS 96, Rome, Italy, 15-19 April 1996. A shorter version will appear in the Proceedings. 1 k⊥-factorization in hadron collisions at high energy Hadron collisions at large transferred momentum Q (Q ≫ Λ, Λ being the QCD scale) can be studied in QCD perturbation theory by computing the corresponding cross sections as power-series expansions in αS(Q ). In the high-energy regime √ S ≫ Q ( √ S the centreof-mass energy) or, equivalently, at small values of the ratio x = Q/S, the coefficients of these power-series expansions contain logarithmically-enhanced contributions of the type ln x. As soon as αS ln 1/x ∼ 1, the fixed-order expansion in αS is no longer reliable. The higher-order contributions (αS ln x) n have to be evaluated and, possibly, resummed to all orders in perturbation theory. The basic theoretical input for the resummation is provided by the BFKL equation [1] for the gluon distribution F(x, k⊥;Q0). This distribution describes the evolution of an initial-state gluon with momentum p (p = Q0 ∼ 1GeV) into an off-shell gluon with momentum k = xp + k⊥ μ (−k2 = −k⊥ ≫ Q0). The evolution process is obtained by radiating final-state partons with momenta k⊥i (k⊥ = − ∑ i k⊥i). The BFKL equation thus resums (αS ln x) n terms due to gluon evolution over the large rapidity gap y = ln 1/x. Since these terms are produced by emission of partons with any transverse momentum k⊥i, no k⊥-ordering is embodied in the BFKL equation. Gluons are not directly observable in scattering processes. Having at our disposal the BFKL equation, we still have to relate it to physical cross sections. The relation is provided by the k⊥-factorization theorem [2, 3, 4]. In the case of processes involving a single incoming hadron, like, for instance, in deep-inelastic lepton-hadron scattering (DIS), the cross section is written as follows σ(x,Q) = ∫ 1