Resumming QCD perturbation series

Abstract

Since the advent of Quantum Field Theory (QFT) in the late 1940's, perturbation theory has become one of the most developed and successful means of extracting phenomenologically useful information from a QFT. In the ever- increasing enthusiasm for new phenomenological predictions, the mechanics of perturbation theory itself have often taken a back seat. It is in this light that this thesis aims to investigate some of the more fundamental properties of perturbation theory. The benefits of resumming perturbative series are highlighted by the explicit calculation of the three-jet rate in e+e- annihilation, resummed to all orders in leading and next-to-leading large logarithms. It is found that the result can be expressed simply in terms of exponentials and error functions. In general it is found that perturbative expansions in QED and QCD diverge at large orders. The nature of these divergences has been explored and found to come from two sources. The first are instanton singularities, which correspond to the combinatoric factors involved in counting Feynman diagrams at large orders. The second are renormalon singularities, which are closely linked to non-perturbative effects through the operator product expansion (OPE).By using Borel transform techniques, the singularity structure in the Borel plane for the QCD vacuum polarization is studied in detail. The renormalon singularity structure is as expected from OPE considerations. These results and existing exact large-A^/ results for the QCD Adler D-function and Deep Inelastic Scattering sum rules are used to resum to all orders the portion of the QCD perturbative coefficients which is leading in b, the first coefficient of the QCD beta-function. This part is expected asymptotically to dominate the coefficients in a large-Nj expansion. Resummed results are also obtained for the e+e- R-ratio and the r-lepton decay ratio. The renormalization scheme dependence of these resummed results is discussed in some detail.