Renormalization Scheme Dependence in a QCD Cross Section

The zero to four loop contribution to the cross section Re+e− for e +e− −→ hadrons, when combined with the renormalization group equation, allows for summation of all leading-log (LL), next-to-leading-log (NLL) . . . N3LL perturbative contributions. It is shown how all logarithmic contributions to Re+e− can be summed and that Re+e− can be expressed in terms of the log independent contributions, and once this is done the running coupling a is evaluated at a point independent of the renormalization scale μ. All explicit dependence of Re+e− on μ cancels against its implicit dependence on μ through the running coupling a so that the ambiguity associated with the value of μ is shown to disappear. The renormalization scheme dependency of the “summed” cross section Re+e− is examined in three distinct renormalization schemes. In the first two schemes, Re+e− is expressible in terms of renormalization scheme independent parameters τi and is explicitly and implicitly independent of the renormalization scale μ. Two of the forms are then compared graphically both with each other and with the purely perturbative results and the RG-summed N3LL results. 1