Lattice Calculation of the Lowest-Order Hadronic Contribution to the Muon Anomalous Magnetic Moment

We present a quenched lattice calculation of the lowest order [$\mathcal{O}({\alpha }^2)$] hadronic contribution to the anomalous magnetic moment of the muon which arises from the hadronic vacuum polarization. A general method is presented for computing entirely in Euclidean space, obviating the need for the usual dispersive treatment which relies on experimental data for $e^{+}{e}^{-}$ annihilation to hadrons. While the result is not yet of comparable precision to those state-of-the-art calculations, systematic improvement of the quenched lattice computation to this level is straightforward and well within the reach of present computers. Including the effects of dynamical quarks is conceptually trivial; the computer resources required are not.

Figure 1

The lowest order hadronic contribution to the muon anomalous magnetic moment [4]. The muon has outgoing momentum $p+q/2$ after scattering from a photon with momentum $q$. The loop momentum is $k$. The blob represents the nonperturbative hadronic vacuum polarization.

Figure 2

The unsubtracted vacuum polarization for the $a^{-1}=1.3\mathrm{G}\mathrm{e}\mathrm{V}$ lattice. The two volumes described in the text are shown, $8^3$ (squares) and ${16}^3$ (circles). The solid line denotes a three-loop perturbation theory calculation [19], which has been shifted by a constant for comparison, and the dashed line the fit of the larger volume data points.