Renormalization Scheme Dependence in a QCD Cross Section

TL;DR

This paper investigates how the cross section for electron-positron to hadrons in QCD depends on renormalization schemes and scales, demonstrating that the summed results are scheme and scale independent through renormalization group techniques.

Contribution

It introduces a method to sum all logarithmic contributions to the QCD cross section, showing scheme and scale independence of the summed results.

Findings

Summation of all leading and subleading logarithms yields scheme-independent results.

Explicit cancellation of renormalization scale dependence in the summed cross section.

Comparison of different renormalization schemes confirms scheme independence of the results.

Abstract

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.