Improved analytic theory of the muon anomalous magnetic moment

We consider the recent results of Kinoshita in which he improves the accuracy of the theoretical value for the anomalous magnetic moment of the muon. This is needed now that a new, more accurate experiment has been approved at Brookhaven National Laboratory. Kinoshita's results are completely numerical. Here we perform an independent check of his results in fourth and sixth order by analytical means, using expansions in the small mass ratios which occur in the computation. Our result for the fourth-order contribution is $a_{\mu }^{\mathrm{}\left(4\right)\mathrm{}}-a_e^{\mathrm{}\left(4\right)\mathrm{}}=5\mathrm{}904475.1\left(3\right)\mathrm{}\times {10}^{-12\mathrm{}}$. This is to be compared with Kinoshita's result $a_{\mu }^{\mathrm{}\left(4\right)\mathrm{}}-a_e^{\mathrm{}\left(4\right)\mathrm{}}=5\mathrm{}904485\times {10}^{-12\mathrm{}}$. For the sixth-order vacuum-polarization contribution we obtain $(a_{\mu }^{\mathrm{}\left(6\right)\mathrm{}}-a_e^{\mathrm{}\left(6\right)\mathrm{}})\left(\mathrm{vacuum}\mathrm{polarization}\right)=24\mathrm{}064.8\left(6\right)\mathrm{}\times {10}^{-12\mathrm{}}$. Kinoshita's result is $(a_{\mu }^{\mathrm{}\left(6\right)\mathrm{}}-a_e^{\mathrm{}\left(6\right)\mathrm{}})\left(\mathrm{vacuum}\mathrm{polarization}\right)=24\mathrm{}069\left(6\right)\mathrm{}\times {10}^{-12\mathrm{}}$. Our result for the total QED contribution is $a_{\mu }^{\mathrm{QED}}=1\mathrm{}165846943\left(28\right)\mathrm{}\left(27\right)\mathrm{}\times {10}^{-12\mathrm{}}$. This agrees with Kinoshita's result $a_{\mu }^{\mathrm{QED}}\mathrm{}\left(K\right)=1\mathrm{}165846961\left(44\right)\mathrm{}\left(28\right)\mathrm{}\times {10}^{-12\mathrm{}}$. Our final result for the muon anomaly is $a_{\mu }^{\mathrm{theory}}=116\mathrm{}591901\left(77\right)\mathrm{}\times {10}^{-11\mathrm{}}$. This should be compared with Kinoshita's result $a_{\mu }^{\mathrm{theory}}=116\mathrm{}591919\left(176\right)\mathrm{}\times {10}^{-11\mathrm{}}$ and the experimental value $a_{\mu }^{\mathrm{expt}}=1\mathrm{}165923\left(8.5\right)\mathrm{}\times {10}^{-9\mathrm{}}$. Our value makes use of a recent computation of the hadronic contribution, in which the error may be overly optimistic.