Renormalization Scheme Dependence in a QCD Cross Section

Abstract

The zero to four loop contribution to the cross section $R_{e^{+}{e}^{-}}$ for $e^{+}{e}^{-}⟶$ hadrons, when combined with the renormalization group equation, allows for summation of all leading-log ($LL$), next-to-leading-log $(NLL)\dots N^{3}LL$ 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 $N^{3}LL$ results.

Comments: 20 pages, 3 figures, LaTeX2e; accepted version to appear in Physical Review D