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This paper presents the beam dynamics systematic corrections and their uncertainties for the Run-1 data set of the Fermilab Muon g-2 Experiment. Two corrections to the measured muon precession frequency $omega_a^m$ are associated with well-known effects owing to the use of electrostatic quadrupole (ESQ) vertical focusing in the storage ring. An average vertically oriented motional magnetic field is felt by relativistic muons passing transversely through the radial electric field components created by the ESQ system. The correction depends on the stored momentum distribution and the tunes of the ring, which has relatively weak vertical focusing. Vertical betatron motions imply that the muons do not orbit the ring in a plane exactly orthogonal to the vertical magnetic field direction. A correction is necessary to account for an average pitch angle associated with their trajectories. A third small correction is necessary because muons that escape the ring during the storage time are slightly biased in initial spin phase compared to the parent distribution. Finally, because two high-voltage resistors in the ESQ network had longer than designed RC time constants, the vertical and horizontal centroids and envelopes of the stored muon beam drifted slightly, but coherently, during each storage ring fill. This led to the discovery of an important phase-acceptance relationship that requires a correction. The sum of the corrections to $omega_a^m$ is 0.50 $pm$ 0.09 ppm; the uncertainty is small compared to the 0.43 ppm statistical precision of $omega_a^m$.
The Neutrinos at the Main Injector (NuMI) facility at Fermilab began operations in late 2004. NuMI will deliver an intense muon neutrino beam of variable energy (2-20 GeV) directed into the Earth at 58 mrad for short (~1km) and long (~700-900 km) bas
After a brief self-contained introduction to the muon anomalous magnetic moment, (g-2), we review the status of lattice calculations of the hadronic vacuum polarization contribution and present first results from lattice QCD for the hadronic light-by
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