Estimate of the Sixth-Order Contribution to the Anomalous Magnetic Moment of the Electron

Abstract

An improvement of the estimate by Drell and Pagels of the anomalous magnetic moment of the electron is given. Use is made of a sidewise dispersion relation in which the mass $W^2$ of one of the external electron lines is analytically continued off its mass shell. Only one-electron, one-photon states are retained in the absorptive amplitude, but this is sufficient to obtain all terms in the absorptive part of the anomalous moment proportional to ($W^2-m^2$) and ${(W^2-m^2)}^2$ as $W^2\to m^2$. This calculation relates $\frac{1}{2}\mathrm{}(g-2\mathrm{})\mathrm{}$ to the Compton amplitude in its exact threshold region. An expansion is made in powers of the photon energy in the Compton amplitude rather than a perturbation calculation in powers of $\alpha \approx \frac{1}{137}$. The Schwinger term $\frac{\alpha }{2\pi }$ is reproduced exactly. A cutoff is chosen so that the fourth-order term is $-\frac{0.328{\alpha }^2}{{\pi }^2}$ approximately reproduced. This cutoff leads to an estimate of the sixth-order term of $\approx +\frac{0.13{\alpha }^3}{{\pi }^3}$.