Infrared behaviour of QCD observables
The perturbation series created through the method of Feynman diagrams can be used to calculate experimental quantities to a good level of accuracy in many cases. The method of obtaining such a series and relating it to an observable for a QFT such as QCD is outlined. This is followed by an introduction into some of the complications of QFT's such as renormalisation, which leads to scale dependence for dimensionless quantities, and the factorial growth of perturbation series. The problem, due to the Landau pole, of defining the QCD perturbation series for the ratio Re+e- in the infrared is approached through a contour improved or analytic perturbation theory (CIPT or APT) series. For a fixed-order truncation such a series will smoothly freeze to the value 2/6 where b = (33- 2Nf)/6 is the first beta function coefficient. This is extended to considering all-orders perturbation theory through Borel summation which is used to evaluate the series factorial growth. This, along with the non-perturbative operator product expansion (OPE), is shown to be well-defined and finite for all values of the energy scale. The perturbative component again freezes to 2/b. A phenomenological comparison with low energy data is performed using a smeared Re+e- and a good agreement is found. A similar approach is developed for the ratio Rr. This case has an added complication due to a non-freezing ambiguous term that arises when one tries to perform CIPT/APT. The apparent ambiguity is fixed through Borel summation. A comparison with experimental data allows the QCD parameter Λ-MS to be evaluated for three flavours of quark, with the result Λ-MS = 382±1820 MeV obtained from a fixed-order calculation and Λ-MS = 340+12-13 MeV from an all-orders calculation.