Functional statistical inference of parton distributions

The parton model is applicable to large momentum transfer hadronic processes. Scaling in hadronic cross-sections is interpreted in the parton model as a consequence of the existence of charged pointlike constituents in hadrons. Parton model cross-sections are calculated by combining amplitudes for the scattering of these constituents with probability densities (parton distribution functions) for nding a given constituent in a hadron carrying a given fraction of the total momentum of the hadron. In perturbative QCD [1], scattering cross-sections can be given a parton model{inspired form, by invoking a factorization scale, f : Roughly, it is supposed that all internal lines with momenta o -shell by more than f are included in the amplitude factor of the cross-section, and the e ects of all lines with softer momenta are included in the parton distributions. (The precise meaning of f is incorporated in the operational de nition of parton distribution functions via factorization schemes, and will not be needed here.) The latter quantities cannot, at present, be derived from QCD, and must therefore be experimentally determined. There is an important theoretical constraint on these distributions: No physical quantity can depend on the choice of factorization scale, thus the calculable f dependence of the amplitude factors can be translated into the f dependence of the distribution functions, leading to the Gribov{Lipatov{Altarelli{Parisi (GLAP) evolution equations satis ed by the parton distribution functions. Knowledge of the parton distribution functions at some value of f therefore is input for the prediction of hadron scattering cross-sections at other energies. Thus the importance of inferring these parton distribution functions cannot be over-emphasized|the prospect of extracting new physics from hadron collider experiments requires the subtraction of known hadronic physics to a high degree of sprecision. Besides systematic experimental uncertainties, the determination of parton distribution functions is a di cult task because the tting procedure is of necessity somewhat circuitous [1]. Given the parton distribution functions at some momentum transfer, they must be evolved to other values using the GLAP equations. The experimental data is then t to cross-sections computed from the set of evolved parton distribution functions. Then the initial parton distribution functions are altered to improve the experimental t, and the whole set of steps is repeated. Furthermore, the functional form of the parton distributions is unknown|we do not know how to solve QCD. Thus a t to parameters in a postulated functional form is beset with the worry that the true functional form of the distribution is not the postulated one, in which case it is quite probable that the true distribution does not even lie on the manifold of parameters at all. That this is not a far-fetched scenario is pointed out by the ln x dependence that appears to smooth the f dependence of a; b in the original x (1 x) form [1]. The aim of this report is to show that a recent formulation of the problem of functional statistical inference due to Bialek, Callan and Strong [2] can be simply extended to provide a framework for parametrization independent inference of parton distribution functions. My attention was drawn to this problem by G. Sterman who emphasized the importance of a parametrizationindependent t. Further, after this work was completed, and presented at several seminars, M. Peskin drew my attention to the work of Giele and Keller [3]. These authors point out the utility of Bayesian methods in the inference of parton distributions [3], motivated by the problem of assigning error estimates to inferred parton distribution functions. It seems therefore that the present report contains material that complements [3], and lls a gap mentioned in the concluding section of [3]. I review brie y the elegant formulation of the problem of inferring a continuous probability distribution from a nite number of data points given by Bialek, Callan and Strong [2]. A small variation on their work, to take into account the fact that probability distributions are densities, was given in [4,5], but I shall not bother with such niceties here. I then extend the work of [2] by constructing an exact solution of the equations obtained in [2] for a nite data set, instead of the WKB type analysis given in [2]. With this material in hand, the parton distribution function problem involves just a few conceptual extensions. Ref. [2] used Bayes' rule to write the probability of the probability distribution Q; given the data fxig; as