Renormalization scheme dependence in a QCD cross section

The zero to four loop contribution to the cross section $R_{e^{+}{e}^{-}}$ for $e^{+}{e}^{-}\to$ hadrons, when combined with the renormalization group equation, allows for summation of all leading-log, next-to-leading-log, …, next-to-next-to-next-to-leading-log perturbative contributions. It is 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 the first two schemes, $R_{e^{+}{e}^{-}}$ 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 renormalization group-summed next-to-next-to-next-to-leading-log results.

Figure 1

The $Q$ dependence of renormalization scale-independent $1+R_A^{(1)IV}$ and $1+R_A^{(2)IV}$ at $\mathrm{\Lambda }=m_{\tau }$ as compared to four-loop perturbative QCD and four-loop RG summed results for $1+R$ referenced at renormalization scale, $\mu =m_{\tau }$.

Figure 2

The $Q$ dependence of renormalization scale-independent $a_{(1)}^{IV}$ and $a_{(2)}^{IV}$.

Figure 3

The renormalization scale $\mu$ dependence of $1+R$ for four-loop perturbative QCD and four-loop RG summed results when $Q=m_{\tau }$.