We perform a new calculation of the hadronic contributions, a(Hadronic), to the anomalous magnetic moment of the muon, $a_{\mu }.$ For the low-energy contributions of order ${\alpha }^2$ we carry over an analysis of the pion form factor $F_{\pi }\mathrm{}\left(t\right)\mathrm{}$ using recent data both on $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ and ${\tau }^{+}\to {\overline{\nu }}_{\tau }{\pi }^{+}{\pi }^0.$ In this analysis we take into account that the phase of the form factor is equal to that of ππ scattering. This allows us to profit fully from analyticity properties so we can also use experimental information on $F_{\pi }\mathrm{}\left(t\right)\mathrm{}$ at spacelike t. At higher energy we use QCD to supplement experimental data, including the recent measurements of $e^{+}{e}^{-}\to \mathrm{hadrons}$ both around 1 GeV and near the $\overline{c}c$ threshold. This yields a precise determination of the $O\left({\alpha }^2\right)$ and $O\left({\alpha }^2\right)+O\left({\alpha }^3\right)$ hadronic part of the photon vacuum polarization ${10}^{11}\times a^{\mathrm{}\left(2\right)\mathrm{}}(\mathrm{h}.\mathrm{v}.\mathrm{p}.\mathrm{})=6909\mathrm{}\pm 64;\hspace{0.5em}{10}^{11}\times a^{\mathrm{}(2+3)\mathrm{}}(\mathrm{h}.\mathrm{v}.\mathrm{p}.\mathrm{})=7002\mathrm{}\pm 66.$ As by-products we also get the masses and widths of the ${\rho }^0,{\rho }^{+},$ and very accurate values for the charge radius and second coefficient of the pion. Adding the remaining order ${\alpha }^3$ hadronic contributions we find ${10}^{11}\times a^{\mathrm{theory}}\left(\mathrm{hadronic}\right)=6993\mathrm{}\pm 69\hspace{0.5em}{(e}^{+}{e}^{-}+\tau +\mathrm{spacel}).\mathrm{}$ The error above includes statistical, systematic, and estimated theoretical errors. The figures given are obtained including τ decay data; if we restrict ourselves to $e^{+}{e}^{-}$ data, slightly lower values and somewhat higher errors are found. This is to be compared with the figure obtained by subtracting pure electroweak contributions from the recent experimental value, obtained from measurements of the muon gyromagnetic ratio $\left(g-2\right),$ which reads ${10}^{11}\times a^{\mathrm{expt}}\left(\mathrm{hadronic}\right)=7174\mathrm{}\pm 150.$

• Received 29 June 2001

DOI:https://doi.org/10.1103/PhysRevD.65.093001