Elimination of QCD Renormalization Scale and Scheme Ambiguities

We present results for the thrust distribution in the electron positron annihilation to the three jet process at NNLO in the perturbative conformal window of QCD, as a function of the number of flavors $N_f$. Given the existence of an infrared interacting fixed point in this region, we can compare the Conventional Scale Setting (CSS) and the Principle of Maximum Conformality (PMC${}_{\mathrm{\infty }}$) methods along the entire renormalization group flow from the highest energies to zero energy. We then consider also the QED thrust, obtained as the limit $N_c\to 0$ of the number of colors and we show analogous comparison. QED in the low energy regime develops an infrared non-interacting fixed point. Using these quantum field theory limits as theoretical laboratories, we arrive at interesting results showing new features of the PMC${}_{\mathrm{\infty }}$.