Physically Motivated Approximation to a Parton Distribution Function in QCD

It has been suggested that parton distributions in coordinate space, so called Ioffe-time distributions, provide a more natural object for non-perturbative methods compared to the usual momentum distributions. In this paper we argue that the shape of experimentally determined Ioffe-time distributions of quarks in a nucleon target clearly indicates separation of longitudinal scales, which is not easily recognizable in terms of conventional longitudinal momentum space considerations. We demonstrate how to use this observation to determine parton distributions, using non-perturbative information about the first few moments and the Regge asymptotics at small x. Published in Phys.Lett. B380, 134 (1996) Work supported in part by BMBF On leave of absence from N. Copernicus Astronomical Center, Polish Academy of Science, ul. Bartycka 18, PL–00-716 Warsaw (Poland) Deep inelastic scattering provides one of the cleanest applications of perturbative QCD. According to factorization theorems [1], the entire Q dependence of the cross section can be calculated perturbatively, while all dynamical effects of large distances can be parametrised by a set of one-particle parton distribution functions given at a certain reference scale. Although these parton distributions are determined experimentally with a good accuracy, their calculation from first principles remains a challenge for non-perturbative QCD methods. In the past decade a remarkable progress has been made at the experimental side, and apart from the region of very small Bjorken x, there is now not much controversy regarding the existing parametrizations of parton distributions [2]. The theoretical progress has been much slower. Apart from several quark-model or MIT bag model calculations, there have been relatively few attempts to determine parton distributions from QCD. The problem has proved to be very difficult for the theory. QCD sum rules calculations of properties of parton distributions have been moderately successful [3, 4]. The stateof-art lattice QCD calculation of the lowest moments of quark distribution functions has appeared just recently [5, 6]. In the present paper we propose an approximate scheme to compute bulk of momentum space quark distributions starting from relatively modest theoretical information. The approximation can be systematically improved when new information becomes available. Note that we are not primarily concerned here with an effective mathematical method of reconstruction of a parton distribution from its (many) moments. Such methods exists [7, 8, 9] and are very useful in the perturbative QCD analysis of experimental data. In this paper we consider a different problem of computation of structure function from a non-perturbative theoretical input. Hence, anticipating technical difficulties in calculation of higher moments from QCD, we have been looking for a method which can give a satisfactory description using a minimal amount of theoretical information. Parton distribution functions arise from long-distance QCD physics which is still perhaps the least understood domain of strong interactions. Given the intrinsic complexity of the problem it is clear that an approximation scheme is necessary. The main problem is to understand what information is required to compute a parton distribution function in QCD with a reasonable accuracy. Usually modern parametrizations of parton densities [10, 11, 12] rely on input distributions assumed to be valid at some low normalization scale. Parton distributions at any larger scale can be uniquely determined thanks to the QCD evolution equations. As very little is understood at present about the low scale input distributions, they are usually assumed to follow a simple shape which is adjusted iteratively to reproduce the experimental data. Alternatively one can calculate them in a certain QCD motivated model, but the significance of such calculation is not obvious. Thus, the question we address in this paper can be formulated as follows: how to determine input distributions making the best use of the information available from state-of-art non-perturbative QCD calculations. It is a textbook statement that twist-2 parton distribution functions describe probability densities of partons longitudinal momenta. Furthermore it is usually assumed that