Photon-photon scattering contribution to the anomalous magnetic moment of the muon

The dominant contribution to the anomalous magnetic moment of the muon in sixth order is the photon-photon scattering contribution $a_{\mu }^{\mathrm{}\left(6\right)\mathrm{}}\left(\gamma \gamma \right)$. An accurate and reliable value is needed in order to properly compare theory with the new muon $g-2\mathrm{}$ experiment now underway at Brookhaven National Laboratory. Our result is $a_{\mu }^{\mathrm{}\left(6\right)\mathrm{}}\left(\gamma \gamma \right)=20.9469\mathrm{}\left(18\right)\mathrm{}{\left(\frac{\alpha }{\pi }\right)}^3$. This agrees extremely well with Kinoshita's result $a_{\mu }^{\mathrm{}\left(6\right)\mathrm{}}\left(\gamma \gamma \right)=20.9471\mathrm{}\left(29\right)\mathrm{}{\left(\frac{\alpha }{\pi }\right)}^3$ where the errors indicate the 90% C.L. This invalidates an old result (1975) of Samuel and Chlouber, for $a_{\mu }^{\mathrm{}\left(6\right)\mathrm{}}\left(\gamma \gamma \right)$, as well as that for the electron, Samuel and Chlouber (1978), and Samuel (1986).