Three-loop QED Vacuum Polarization and the Four--loop Muon Anomalous Magnetic Moment

Abstract

Three--loop contributions to massive QED vacuum polarization are evaluated by a combination of analytical and numerical techniques. The first three Taylor coefficients, at small $q^2$, are obtained analytically, using $d$\/--dimensional recurrence relations. Combining these with analytical input at threshold, and at large $q^2$, an accurate Pad\'e approximation is obtained, for all $q^2$. Inserting this in the one--loop diagram for the muon anomalous magnetic moment, we find reasonable agreement with four--loop, single--electron--loop, muon--anomaly contributions, recently re--evaluated by Kinoshita, using 8--dimensional Monte--Carlo integration. We believe that our new method is at least two orders of magnitude more accurate than the Monte--Carlo approach, whose uncertainties appear to have been underestimated, by a factor of 6.