QCD perturbative expansion fore+e−→hadrons

We study the perturbative QCD series for the hadronic width of the $Z$ boson. We sum a class of large "${\pi }^2$ terms" and reorganize the series so as to minimize "renormalon" effects. We also consider the renormalization scheme-scale ambiguity of the perturbative results. We find that, with three nontrivial known terms in the perturbative expansion, the treatment of the ${\pi }^2$ terms is quite important, while renormalon effects are less important. The measured hadronic width of the $Z$ is often used to determine the value of ${\alpha }_s\left(M_Z^2\right)$. A standard method is to use the perturbative expansion for the width truncated at order ${\alpha }_s^3$ in the $\overline{\mathrm{MS}}$ scheme with scale $\mu =M_Z$. We estimate that the determined value of ${\alpha }_s\left(M_Z^2\right)$ should be increased by 0.6% compared to the value extracted with this standard method. After this adjustment for ${\pi }^2$ and renormalon effects, we estimate that the uncertainty in ${\alpha }_s\left(M_Z^2\right)$ arising from QCD theory is about 0.4%. This is, of course, much less than the experimental uncertainty of about 5%.