CERN Accelerating science

Muon-decay positrons accumulated in \RunTwo and \RunThree after DQC. Positrons with \SI{1}{GeV}$$\SI{30}{\micro\second} after injection are shown. The \RunOne equivalent ($15.4\times10^9$) is shown for comparison.

Temperature of the calorimeter SiPMs (small dots) and the magnet yokes (thicker lines) across \RunOne, \RunTwo, and \RunThree. The two inserts show a box of four days with a temperature range of \SI{1}{\celsius}. The magnet thermal insulating blanket installed after \RunOne reduced the day-night oscillations of the magnet temperature. The upgraded air conditioning system greatly improved the long-term stability of both the calorimeters and magnet temperature after \RunTwo.

Muon loss time distribution L(t) for selected \RunOne (gray), \RunTwo (blue), and \RunThree (orange) data subsets showing the reduction in losses. The values here are normalized to the number of $e^+>1.7$ GeV in each dataset. The large modulation of the muon losses with the frequency $f_{CBO}$ is a reflection of the mechanism of the losses.

Azimuthally averaged muon beam distribution summed over $t>\SI{30}{\micro\second}$ ($_{\phi}$) from datasets from \RunTwo (2B) on the left and \RunThreeB (3O) on the right. The color represents the intensity, from low intensity in blue (outside) to high intensity in red (inside).

Azimuthally averaged muon beam distribution summed over $t>\SI{30}{\micro\second}$ ($_{\phi}$) from datasets from \RunTwo (2B) on the left and \RunThreeB (3O) on the right. The color represents the intensity, from low intensity in blue (outside) to high intensity in red (inside).

Representative example of the measure decay-asymmetry $A(E)$ versus the positron energy $E$ in the region $0.5-3.1$~GeV. In the A-method, each positron is weighted by $A(E)$ to achieve the greatest possible statistical power in the anomalous precession frequency measurement.

Illustration of the reconstructed pileup correction for the empirical method. The black curve is the raw energy distribution before the pileup correction. The dashed blue (dotted orange) curves show the reconstructed gain (loss) of positron events due to positron pileup. The agreement between the black curve and the blue curve in the energy region greater than the 3.1 GeV beam energy (vertical gray line) is an indication of the quality of the pileup correction.

Representative example of the discrete Fourier transform (FFT) of the fit residuals for a five-parameter fit (solid blue) and a multi-parameter fit (dotted orange) to the \RunThreeB dataset. The five-parameter Fourier transform indicates the presence of perturbations due to beam dynamics, muon losses, {\it etc}. The five-parameter fit shows peaks corresponding to radial beam oscillations ($f_{\text{CBO}}$, $2f_{\text{CBO}}$), vertical beam oscillations ($f_{\text{VW}}$, $f_y$), couplings between precession and radial frequencies ($f_{\text{CBO}} \pm f_{a}$), and radial and vertical frequencies ($f_{\text{VW}} - f_{\text{CBO}}$). Also evident at low frequencies are the effects of muon losses and other slow effects.

Plot of the results for the 19 analyses of the three different datasets. Note the muon-weighted magnetic field \ref{sec:field:muonWeighting} and beam dynamics corrections \ref{sec:bd:corr} are different for the three datasets. The plotted uncertainties are the statistical uncertainties from the multi-parameter fits to the associated time distributions. The allowed statistical and systematic differences between the results for a given dataset are discussed in \ref{ss:fitresults}.

Pulls between the 513 pairs of all $\omega_a$ measurements (top panel) and 45 pairs of A- and RA-method measurements the are used in the $\omega_a$ averaging (bottom panel). The pulls are defined as $( y_i - y_j ) / \sigma_{ij}$ where $y_i$, $y_j$ are the two measurements and $\sigma_{ij}$ is the estimated uncertainty on their difference. The values of $\sigma_{ij}$ are computed using the statistical and systematic uncertainties and their estimated correlations.

Pulls between the 513 pairs of all $\omega_a$ measurements (top panel) and 45 pairs of A- and RA-method measurements the are used in the $\omega_a$ averaging (bottom panel). The pulls are defined as $( y_i - y_j ) / \sigma_{ij}$ where $y_i$, $y_j$ are the two measurements and $\sigma_{ij}$ is the estimated uncertainty on their difference. The values of $\sigma_{ij}$ are computed using the statistical and systematic uncertainties and their estimated correlations.

A representative scan of the blinded R-value versus the fit start time for the Run-3a dataset and the asymmetry-weighted histogramming method. The black data points are the R-value fit results. The point-to-point values are highly correlated and the smooth blue curve is the $1$ allowed standard deviation band of any fit result from the canonical \SI{30.1}{\micro\second} fit start time. The allowed deviation band accounts for the statistical correlations between the \SI{30.1}{\micro\second} and $>\SI{30.1}{\micro\second}$ fit results. Note the vertical axis includes an analysis-dependent software blinding and cannot be compared to Fig.~\ref{f:comon-unblinded-R} and Table~\ref{t:comon-unblinded-R}.

A representative scan of the blinded R-value versus the calorimeter index for the \RunThreeA dataset and the asymmetry-weighted histogramming method. The black data points are the R-value fit results, and the solid blue line is a straight-line fit to the 24 individual calorimeter R-values. Note the vertical axis includes an analysis-dependent software blinding and cannot be compared to Fig.~\ref{f:comon-unblinded-R} and Table~\ref{t:comon-unblinded-R}.

Fast-rotation signal from Run-2 data, showing individual turns around the storage ring over short time scales (top) and broader decoherence envelope over long time scales (bottom).

Fast-rotation signal from Run-2 data, showing individual turns around the storage ring over short time scales (top) and broader decoherence envelope over long time scales (bottom).

Fractional momentum distributions from the fast-rotation $\chi^2$ method, the tracking analysis method (data from the straw tracking detector at $180^\circ$), and the corrected Fourier analysis for the data subset 3F.

Joint distribution from the fast-rotation $\chi^2$ method of revolution frequency and injection time determined by the direct fit method for the data subset 3N, first bunch in the beam pulse sequence.

Electric-field corrections $C_E$ by data subset obtained from the tracking analysis method and the fast-rotation $\chi^2$ method. The final values for Run-2, Run-3a, and Run-3b are shown in color, which come from the combination of the calorimeter and tracker-based analyses.

Comparison between method-1 and method-2 of the pitch correction, $C_p$, results for all data subsets available in Run-2 and Run-3. The errors in the two methods are dominated by the tracking uncertainty.

Average radial coordinate $\langle x\rangle$ of the beam distribution per momentum offset at injection, from a \texttt{gm2ringsim} tracking simulation of stored muons. In this example, a nominal configuration of the injection parameters is implemented in the simulation. The $dx/d\delta$ correlations to quantify $C_{dd}^{p\text{-}x}$ are obtained from these tracking simulation results.

Momentum-phase distribution from the momentum-time distribution for one bunch in data subset 2C. The gray markers are the averaged relative spin phases per fractional momentum, exhibiting the correlation that drives $C_{dd}^{p\mathrm{-}t_0}$.

Momentum-time differential decay correction $C_{dd}^{p\text{-}t_0}$ per data subset (black). In gray crosses, correction predictions where the ratio between $p\text{-}t_{0}$ correlations and kicker timing offsets relative to beam injection, based on \texttt{gm2ringsim} beam tracking simulations, is scaled in proportion to the per-data-subset kicker timing offsets.

Simulated azimuthally averaged phase maps for the asymmetry-weighted analysis. The coupling between the overall quadratic-like detected phase acceptance in the vertical direction and the in-fill reduction in vertical beam width is the most significant effect on $C_{pa}$.

Calculation of $\phi_{\mathrm{pa}}$ for calorimeter 13 in data subset 1D (gray) and data subset 2C (blue) using data from the tracker station at 180$^{\circ}$. The shown fit function is of the form $\phi + \Delta\phi \cdot e^{{(-t/\tau_{\phi})}}$.

The relative (Rel.) dipole $m_1$ coefficient as a function of azimuth for three field maps with respect to its azimuthal average. A) is from April \nth{8} 2019, the beginning of \RunTwo, B) is from June \nth{20}, 2019, the end of \RunTwo, and C) from March \nth{11}, the end of \RunThree. The peak-to-peak amplitudes are \SI{76}{ppm}, \SI{108}{ppm}, and \SI{93}{ppm}, respectively, with RMSs of \SI{14.6}{ppm}, \SI{20.5}{ppm}, and \SI{15.8}{ppm}.

Top: Tracking offset (inability to track field) as a function of azimuth (azi.) around a yoke boundary. Different colors indicate different times after the magnet ramp. Bottom: Amplitude of effect at \SI{45}{\degree} as a function of time after magnet ramp. The x show the azimuthally averaged values scaled up by a factor of x10. A dedicated campaign of back-to-back trolley runs was performed in Run-6 to study this effect.

Top: Tracking offset (inability to track field) as a function of azimuth (azi.) around a yoke boundary. Different colors indicate different times after the magnet ramp. Bottom: Amplitude of effect at \SI{45}{\degree} as a function of time after magnet ramp. The x show the azimuthally averaged values scaled up by a factor of x10. A dedicated campaign of back-to-back trolley runs was performed in Run-6 to study this effect.

The relative muon-weighted magnetic field ($\tilde\omega_{p'}$) as a function of time for the \RunTwo (left side) and \RunThreeA and \RunThreeB (right side). The dipole $m_1$ contribution alone is shown in gray below. On this scale, they barely differ. The lower two plots show the tracked $m_2$ and $m_3$ moments.

Magnetic field transient induced by kicker magnets measured by the optical fiber magnetometer in summer 2021 and summer 2022.

Top) The transient magnetic field from the vibration caused by the ESQ pulsing for all times as a function of azimuth in the storage ring. Bottom) The transient magnetic field as a function of time at one specific location (\SI{-17}{deg}). The times during which muons are stored are highlighted by gray bands. The shown field transients are scaled up to the ESQ operation voltage.

$R^{\prime}_{\mu}(T_{r})$ versus data subset. The fit line has a $\chi^2$/ndf$ = 19.31/19$ with a p-value of \SI{44}{\percent}.

From top to bottom: experimental values of \amu from BNL E821, the FNAL 2021 measurement (FNAL Run-1), this measurement (FNAL Run-2/3), the FNAL combined measurement (FNAL Run-1 + 2/3), and the combined experimental average (Exp. average). The inner tick marks indicate the statistical contribution to the total uncertainties.

Top: Trolley calibration constants per trolley probe for \RunTwo (blue) and \RunThree (orange) and the combination (black). Predictions from \texttt{COMSOL} simulations (gray) with simplified geometry, only considers the trolley shell, show qualitative consistency. Bottom: The difference of \RunTwo and \RunThree calibration constants with respect to the combined value that are used for this analysis.