Extrapolating Structure Functions to Very Small x

We review small x contributions to perturbative evolution equations for parton distributions, and their resummation. We emphasize in particular the resummation technique recently developed in order to deal with the apparent instability of naive small x evolution kernels and understand the empirical sucess of fixed–order perturbation theory. We give predictions for the gluon distribution and the structure functions F2(x,Q ) and FL(x,Q ) in an extended kinematic region, such as would be relevant for THERA or LEP+LHC ep colliders. to be published in the THERA book On leave from INFN, Sezione di Torino, Italy Measurements of the inclusive structure functions F2(x,Q ) and FL(x,Q ) at HERA have shown that the scaling violations of structure functions are in extremely good agreement with the perturbative next-to-leading order (NLO) QCD prediction, down to the smallest values of x, and for all Q > ∼ 1GeV [1]. This agreement is surprising in that it is known that perturbative corrections beyond NLO in αs are enhanced by powers of ξ ≡ ln(1/x), and thus one would expect higher order corrections to be sizable whenever αs(Q )ξ > ∼ 1, i.e. in most of the HERA kinematic region. Whereas techniques for the inclusion of small x contributions to leading twist evolution equations have been known for some time [2,3], only recently did a consistent picture of the general structure of these contributions and their resummation emerge. Indeed, considerable theoretical progress has been spurred by the determination [4] of next-to-leading corrections to the BFKL kernel, which allows the computation of the next-to-leading log(1/x) (NLLx) contributions to anomalous dimensions to all orders in αs. Specifically, it is now understood that the inclusion of NLLx contributions leads to instability [5] of perturbative evolution, unless it is suitably combined with a resummation of the collinear singularities [6–8] which are resummed order by order in the standard QCD evolution equations. Furthermore, the NLLx perturbative corrections give rise to increasingly large contributions to high orders of perturbation theory [9, 10] that make a nonsense of the perturbative expansion and call for an all-order resummation of the small-x behaviour of the anomalous dimensions [11, 12]. Practical methods to deal with these issues have been developed recently [7, 13], and lead to a resummation prescription which is amenable to numerical treatment and direct comparison with the data. It then appears that the observed smallness of perturbative higher order corrections at small x can be accommodated within the current knowledge of the general structure of anomalous dimensions, but it poses very stringent constraints on the form of the unknown higher order terms. Furthermore, even when these constraints are respected, so that, as required by the data, deviations of the behaviour of the observable structure functions from the fixed next-to-leading order prediction are very small, still non–negligible modifications of the fitted parton distributions at small x are found. This, because of ambiguities in the resummation procedure, entails larger uncertainties on parton distributions at small x. Likewise, these corrections have a sizable impact on the extraction of αs from small x data, both on the central value and the estimates of overall theoretical uncertainties [13, 14]. In the wider kinematic region available at THERA the small differences between resummed and fixed–order predictions could be put to more stringent tests. This would allow one to pin down more precisely the ambiguities in the resummation procedure, thereby reducing the uncertainty on parton distributions at small x and on precision determinations of αs at small x. Also, the possibility of reaching smaller values of x for given Q would allow a test of resummed perturbation theory in a region where the relevant resummation parameter αsξ is large, and also to see whether the perturbative description of scaling violations remains satisfactory or starts to break down, as is often suggested [15]. Here we briefly review our current understanding of resummed perturbation theory at small x. We then give predictions for the gluon distribution and the structure functions F2 and FL in two different resummation scenarios, and compare these to fixed next– to–leading order results in the kinematic range which is relevant for THERA. This is