Pionic Contributions to the Magnetic Moment of the Muon
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Abstract
The anomalous magnetic moment $\ensuremath{\kappa}$ of the muon is affected in order ${\ensuremath{\alpha}}^{2}$ by vacuum polarization corrections to the photon propagator which involve strongly interacting particles. The contributions of the two- and three-pion systems to $\ensuremath{\kappa}$ are estimated using the present experimental information on the $J={1}^{\ensuremath{-}}$ two-pion and three-pion resonances (the $\ensuremath{\rho}$ and $\ensuremath{\omega}$ mesons). The analysis is based on the use of dispersion relations for the muon vertex functions, and of the Lehmann-K\"all\'en representation for the photon propagator. If it is assumed that the $\ensuremath{\rho}$-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 $\ensuremath{\kappa}:{(\ensuremath{\Delta}\ensuremath{\kappa})}_{2\ensuremath{\pi}}\ensuremath{\sim}(\frac{{\ensuremath{\alpha}}^{2}}{36\ensuremath{\pi}}){({{m}_{\ensuremath{\rho}}}^{2}\ensuremath{-}4{{m}_{\ensuremath{\pi}}}^{2})}^{\frac{3}{2}}{{m}_{\ensuremath{\mu}}}^{2}{{m}_{\ensuremath{\rho}}}^{\ensuremath{-}4}{{\ensuremath{\Gamma}}_{\ensuremath{\rho}}}^{\ensuremath{-}1}$, where ${\ensuremath{\Gamma}}_{\ensuremath{\rho}}$ is the width of the $\ensuremath{\rho}$ resonance in pion-pion scattering. In the limit of Gell-Mann's unitary symmetry for the meson-baryon interactions, the $\ensuremath{\omega}$-meson contributions to $\ensuremath{\kappa}$ are one-third of the $\ensuremath{\rho}$-meson contributions. With ${m}_{\ensuremath{\rho}}=750$ MeV and the recent Yale value of 60 MeV for ${\ensuremath{\Gamma}}_{\ensuremath{\rho}}$, one obtains a combined result $\ensuremath{\Delta}\ensuremath{\kappa}\ensuremath{\sim}1.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ 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 \ifmmode\pm\else\textpm\fi{}5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$ in the best measurement of $\ensuremath{\kappa}$, that of Charpak et al. The possibility of obtaining additional large contributions from $\ensuremath{\rho}\ensuremath{-}\ensuremath{\pi}$ and $\ensuremath{\omega}\ensuremath{-}\ensuremath{\pi}$ configurations in the three- and four-pion states is discussed briefly.