Hadronic contributions to the anomalous magnetic moment of the muon

We present a new, completely revised calculation of the muon anomalous magnetic moment $a_{\mu }=(g_{\mu }-2)/2$ comparing it with the more recent experimental determination of this quantity; this furnishes an important test of theories of strong, weak, and electromagnetic interactions. These theoretical determinations give the very precise numbers, ${10}^{11}\times a_{\mu }=116591806\pm 50\pm 10(\mathrm{rad})\pm 30(\ell \times \ell )[\mathrm{Theory},\mathrm{no}\tau ]$ and ${10}^{11}\times a_{\mu }=116591889\pm 49\pm 10(\mathrm{rad})\pm 30(\ell \times \ell )[\mathrm{Theory},\tau ]$ to be compared with the experimental number, ${10}^{11}\times a_{\mu }=116592080\pm 60$. In the theoretical evaluations, the first quantity does not, and the second one does, use information from $\tau$ decay. The first errors for the theoretical evaluations include statistical plus systematic errors; the other ones are the estimated errors due to incomplete treatment of radiative corrections and the estimated error in the light-by-light scattering contribution. We thus have a significant mismatch between theory and experiment. We also use part of the theoretical calculations to give a precise evaluation of the electromagnetic coupling on the $Z$, ${\overline{\alpha }}_{\mathrm{QED}}(M_Z^2)$, of the masses and widths of the (charged and neutral) rho resonances, of the scattering length and effective range for the $P$ wave in $\pi \pi$ scattering, and of the quadratic radius and second coefficient of the pion form factor.