Renormalization Scheme Dependence in a QCD Cross Section

The zero to four loop contribution to the cross section $R_{e^{+}e^{-}}$ for $e^{+}e^{-} \longrightarrow$ hadrons, when combined with the renormalization group equation, allows for summation of all leading-log ($LL$), next-to-leading-log $(NLL) \ldots N^3LL$ perturbative contributions. It is also shown how all logarithmic contributions to $R_{e^{+}e^{-}}$ can be summed and that $R_{e^{+}e^{-}}$ 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 $\mu$. All explicit dependence of $R_{e^{+}e^{-}}$ on $\mu$ cancels against its implicit dependence on $\mu$ through the running coupling $a$ so that the ambiguity associated with the value of $\mu$ is shown to disappear. The renormalization scheme dependency of the "summed" cross section $R_{e^{+}e^{-}}$ is examined in three distinct renormalization schemes. In each case, $R_{e^{+}e^{-}}$ is expressible in terms of renormalization scheme independent parameters $\tau_i$ and is explicitly and implicitly independent of the renormalization scale $\mu$. Two of the forms are then compared graphically both with each other and with the purely perturbative results and the $RG$-summed $N^3LL$ results.