Renormalization group and Pade applications to perturbative and non-perturbative quantum field theory

Pade approximants (PA) have been widely applied in practically all areas of physics. This thesis focuses on developing PA as tools for both perturbative and non-perturbative quantum field theory (QFT).; In perturbative QFT, we systematically estimate higher (unknown) loop terms via the asymptotic formula devised by Samuel et al. This algorithm, generally denoted as the asymptotic Pade approximation procedure (APAP), has greatly enhanced scope when it is applied to renormalization-group-(RG-) invariant quantities. A presently-unknown higher-loop quantity can then be matched with the approximant over the entire momentum region of phenomenological interest. Furthermore, the predicted value of the RG coefficients can be compared with the RG-accessible coefficients (at the higher-loop order), allowing a clearer indication of the accuracy of the predicted RG-inaccessible term. This methodology is applied to hadronic Higgs decay rates (H → bb¯ and H → gg, both within the Standard Model and its MSSM extension), Higgs-sector cross-sections ($W+LW-L\to ZLZL$), inclusive semileptonic b → u decays (leading to reduced theoretical uncertainties in the extraction of |V_ub|), QCD (Quantum Chromodynamics) correlation functions (scalar-fermionic, scalar-gluonic and vector correlators) and the QCD static potential. APAP is also applied directly to RG β- and γ-functions in massive &phis;4 theory.; In non-perturbative QFT we use Pade summation methods to probe the large coupling regions of QCD. In analysing all the possible Pade-approximants to truncated β-function for QCD, we are able to probe the singularity structure corresponding to the all orders β-function. Noting the consistent ordering of poles and roots for such approximants (regardless of the next unknown higher-loop contribution), we conclude that these approximants are free of defective (pole) behaviour and hence we can safely draw physical conclusions from them. QCD is shown to have a flavour threshold (6 ≤ n_f ≤ 8) for any infrared-stable fixed points (IRFP) to occur. Moreover, the behaviour below this threshold is seen to be analogous to supersymmetric gluodynamics, in which the coupling at low energy approaches an infrared attractor rather than an IRFP.