Beam Dynamics simulations for the Fermilab E989 Muon $g-2$ Experiment*

The E989 Muon $g-2$ Experiment at Fermilab (FNAL) recently determined the anomalous magnetic moment of the positive muon to be $a_{\mu}(\mathrm{FNAL})=116592040(54) \times 10^{-11}(\qty{0.46}{ppm})$. This result combined with the previous Brookhaven National Laboratory E821 measurement sets a new experimental average of $a_{\mu}(\mathrm{Exp})=116592061(41) \times 10^{-11}(\qty{0.35}{ppm})$. This result is 4.2 standard deviations greater than the standard model prediction. For the Muon $g-2$ Experiment at Fermilab, \qty{3.094}{\GeV/ c} muons are injected into the storage ring through a magnetic inflector. The storage ring has a 1.45 T magnetic dipole field. Right after the injection, the beam is moved to the nominal equilibrium orbit by using magnetic kickers during the first turn, and the electric field generated on the quadrupole plates shift the beam radially and vertically to move the edges of the beam into the collimators. After this, the quadrupole plates provide a focusing electric field so the stored muons will travel inside the storage ring until they decay into positrons. These will be detected by calorimeters to reconstruct the anomalous frequency $\omega_a$. The described elements affect the motion of the muons in the storage ring and can bias the extraction of $\omega_a$. Detailed simulations have been implemented to study the dynamics of the beam and estimate the size and nature of these systematic effects. This talk will present some studies and strategies used in simulations.