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

TL;DR

This work develops a hybrid analytic–numerical framework to determine three-loop massive QED vacuum polarization with high precision and tests its impact on the four-loop muon anomalous magnetic moment. By combining $d$-dimensional recurrence-derived small-$q^2$ results, OS renormalization, and large-$q^2$ asymptotics with Padé approximants, the authors construct a reliable $\Pi(z)$ for all $z$ and insert it into the one-loop muon anomaly to estimate $A_4^{[1]}$ with $A_4^{[1]}=-0.230362(5)$. This result agrees, within uncertainties, with recent Monte Carlo evaluations but surpasses them in precision, arguing for the robustness of the analytic approach. The study also provides detailed analyses of OS/M$\overline{\rm MS}$ asymptotics and threshold behavior, reinforcing confidence in high-order QED predictions. The methods and internal consistency across multiple cross-checks position this technique as a powerful tool for precision electroweak calculations. $A_4^{[1]}=-0.230362(5)$.

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 Padé 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.