Calculating the hadronic vacuum polarization and leading hadronic contribution to the muon anomalous magnetic moment with improved staggered quarks

We present a lattice calculation of the hadronic vacuum polarization and the lowest order hadronic contribution (HLO) to the muon anomalous magnetic moment, $a_{\mu }=(g-2)/2$, using $2+1$ flavors of improved staggered fermions. A precise fit to the low-$q^2$ region of the vacuum polarization is necessary to accurately extract the muon $g-2$. To obtain this fit, we use staggered chiral perturbation theory, including a model to incorporate the vector particles as resonances, and compare these to polynomial fits to the lattice data. We discuss the fit results and associated systematic uncertainties, paying particular attention to the relative contributions of the pions and vector mesons. Using a single lattice spacing ensemble generated by the MILC Collaboration ($a=0.086\mathrm{fm}$), light quark masses as small as roughly one-tenth the strange quark mass, and volumes as large as $(3.4\mathrm{fm}{)}^3$, we find $a_{\mu }^{\mathrm{HLO}}=(713\pm 15)\times {10}^{-10}$ and $(748\pm 21)\times {10}^{-10}$ where the error is statistical only and the two values correspond to linear and quadratic extrapolations in the light quark mass, respectively. Considering various systematic uncertainties not eliminated in this study (including a model of vector resonances used to fit the lattice data and the omission of disconnected quark contractions in the vector-vector correlation function), we view this as agreement with the current best calculations using the experimental cross section for $e^{+}{e}^{-}$ annihilation to hadrons, $(692.4\pm 5.9\pm 2.4)\times {10}^{-10}$, and including the experimental decay rate of the tau lepton to hadrons, $(711.0\pm 5.0\pm 0.8\pm 2.8)\times {10}^{-10}$. We discuss several ways to improve the current lattice calculation.