Title:

Scale dependence and renormaloninspired resummations for some QCD observables

Since the advent of Quantum Field Theory (QFT) in the late 1940's, perturbation theory has become one of the most successful means of extracting phenomenologically useful information from QFT. In the everincreasing enthusiasm for new phenomenological predictions, the mechanics of perturbation theory itself have taken a back seat. It is in this light that this thesis aims to investigate some of the more fundamental properties of perturbation theory. In the first part of this thesis, we develop the idea, suggested by C.J.Maxwell, that at any given order of Feynman diagram calculation for a QCD observable all renormalization group (RG)predictable terms should be resummed to allorders. This "complete" RGimprovement (CORGI) serves to separate the perturbation series into infinite subsets of terms which when summed are renormalization scheme (RS)invariant. Crucially all ultraviolet logarithms involving the dimensionful parameter, Q, on which the observable depends are resummed, thereby building the correct Qdependence. We extend this idea, and show for moments of leptoproduction structure functions that all dependence on the renormahzation and factorization scales disappears provided that all the ultraviolet logarithms involving the physical energy scale Q are completely resummed. The approach is closely related to Grunberg's method of Effective Charges. In the second part, we perform an allorders resummation of the QCD Adler Dfunction for the vector correlator, in which the portion of perturbative coefficients containing the leading power of b, the first betafunction coefficient, is resummed to allorders. To avoid a renormalization scale dependence when we match the resummation to the exactly known nexttoleading order (NLO), and nextNLO (NNLO) results, we employ the Complete Renormalization Group Improvement (CORGI) approach , removing all dependence on the renormalization scale. We can also obtain fixedorder CORGI results. Including suitable weightfunctions we can numerically integrate these results for the Dfunction in the complex energy plane to obtain socalled "contourimproved" results for the ratio R and its tau decay analogue Rr. We use the difference between the allorders and fixedorder (NNLO) results to estimate the uncertainty in αs(M2/z) extracted from Rr measurements, and find αs(M2/z) = 0.120±0.002. We also estimate the corresponding uncertainty in a{Ml) arising from hadronic corrections by considering the uncertainty in R(s), in the lowenergy region, and compare with other estimates. Analogous resummations are also given for the scalar correlator. As an adjunct to these studies we show how fixedorder contourimproved results can be obtained analytically in closed form at the twoloop level in terms of the Lambert Wfunction and hypergeometric functions.
