Measurement of the anomalous spin precession frequency $\omega_a$ in the Muon $g-2$ experiment at Fermilab

The muon anomaly, $a_\mu=(g_{\mu}-2)/2$, is a low-energy observable which can be both measured and computed to high precision, making it a sensitive test of the Standard Model (SM) and a probe for new physics. The current discrepancy between the experimental value and the Standard Model calculation from the Muon $g-2$ Theory Initiative~\cite{WhitePaper} is $a_{\mu}^{exp}-a_{\mu}^{SM}=(249\pm48)\cdot10^{-11}$, with a significance of $5.1\,\sigma$.
The Fermilab E989 experiment aims, with the full statistical power, to measure $a_{\mu}$ with a precision of $140\,$parts per billion (ppb), a four-fold improvement with respect to the previous measurement at the Brookhaven E821 experiment (1997-2001).
In April 2021 the FNAL E989 collaboration published their first result, based on the first year of data taking (2018 campaign)~\cite{Run-1}, and in August 2023 a new result was published, based on the datasets collected during Run-2 and Run-3 (2019 and 2020 campaigns)~\cite{Run-2/3}, in agreement with the first result and with the previous experiment at Brookhaven.\\
This paper will present details about the improvements and upgrades to the positron reconstruction and to the $\omega_a$ analysis since the Run-1 result, and it will describe the final statistical and systematic sources of uncertainty on $\omega_a$ in the Run-2 and Run-3 result.