$W$-Boson Contribution to the Anomalous Magnetic Moment of the Muon

The contribution of the intermediate vector boson to the muon anomalous moment is calculated in the $\xi$-limiting formalism. We find $\Delta {\kappa }_{\mu }=\Delta \left[\frac{(g_{\mu }-2)}{2}\right]=\left(\frac{{g_W}^2}{8{\pi }^2}\right){\left(\frac{M_{\mu }}{M_W}\right)}^2\left[2\right(1-{\kappa }_W)\ln \xi +\frac{10}{3}],$ where $\sqrt{2}{\left(\frac{g_W}{M_W}\right)}^2=G_F$ is the Fermi weak coupling constant and ${\kappa }_W$ is the $W$ anomalous moment. For ${\kappa }_W=0\mathrm{}$, and $\xi$ chosen according to the prescription of Lee, this contribution is numerically $\Delta {\kappa }_{\mu }\cong -1.0\mathrm{}\times {10}^{-8\mathrm{}}\cong -0.8\mathrm{}{\left(\frac{\alpha }{\pi }\right)}^3.$