Pionic Contributions to the Magnetic Moment of the Muon

The anomalous magnetic moment $\kappa$ of the muon is affected in order ${\alpha }^2$ by vacuum polarization corrections to the photon propagator which involve strongly interacting particles. The contributions of the two- and three-pion systems to $\kappa$ are estimated using the present experimental information on the $J=1^{-}$ two-pion and three-pion resonances (the $\rho$ and $\omega$ mesons). The analysis is based on the use of dispersion relations for the muon vertex functions, and of the Lehmann-Källén representation for the photon propagator. If it is assumed that the $\rho$-meson resonance gives the dominant contribution to the electromagnetic form factor of the pion, the modifications of the photon propagator associated with the two-pion intermediate state lead to a change in $\kappa :{\left(\Delta \kappa \right)}_{2\pi }\sim \left(\frac{{\alpha }^2}{36\pi }\right){({m_{\rho }}^2-4\mathrm{}{m_{\pi }}^2)}^{\frac{3}{2}}{m_{\mu }}^2{m_{\rho }}^{-4\mathrm{}}{{\Gamma }_{\rho }}^{-1\mathrm{}}$, where ${\Gamma }_{\rho }$ is the width of the $\rho$ resonance in pion-pion scattering. In the limit of Gell-Mann's unitary symmetry for the meson-baryon interactions, the $\omega$-meson contributions to $\kappa$ are one-third of the $\rho$-meson contributions. With $m_{\rho }=750\mathrm{}$ MeV and the recent Yale value of 60 MeV for ${\Gamma }_{\rho }$, one obtains a combined result $\Delta \kappa \sim 1.1\times {10}^{-7\mathrm{}}$ which is an order of magnitude larger than (1) would be estimated using perturbation theory and (2) the nominal size of the sixth-order electrodynamic corrections; but which is still negligible in comparison with the quoted limits of error of ±5×${10}^{-6\mathrm{}}$ in the best measurement of $\kappa$, that of Charpak et al. The possibility of obtaining additional large contributions from $\rho -\pi$ and $\omega -\pi$ configurations in the three- and four-pion states is discussed briefly.