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We present a new measurement of the positive muon magnetic anomaly, \\(a_\\mu \\equiv (g_\\mu - 2)/2\\), from the Fermilab Muon \\(g\\!-\\!2\\) Experiment using data collected in 2019 and 2020. We have analyzed more than 4 times the number of positrons from muon decay than in our previous result from 2018 data. The systematic error is reduced by more than a factor of 2 due to better running conditions, a more stable beam, and improved knowledge of the magnetic field weighted by the muon distribution, \\(\\tilde{\\omega}'^{}_p\\), and of the anomalous precession frequency corrected for beam dynamics effects, \\(\\omega_a\\). From the ratio \\(\\omega_a / \\tilde{\\omega}'^{}_p\\), together with precisely determined external parameters, we determine \\(a_\\mu = 116\\,592\\,057(25) \\times 10^{-11}\\) (0.21 ppm). Combining this result with our previous result from the 2018 data, we obtain \\(a_\\mu\\text{(FNAL)} = 116\\,592\\,055(24) \\times 10^{-11}\\) (0.20 ppm). The new experimental world average is \\(a_\\mu (\\text{Exp}) = 116\\,592\\,059(22)\\times 10^{-11}\\) (0.19 ppm), which represents a factor of 2 improvement in precision.

The Fermi National Accelerator Laboratory has measured the anomalous precession frequency \\(a^_ = (g^_-2)/2\\) of the muon to a combined precision of 0.46 parts per million with data collected during its first physics run in 2018. This paper documents the measurement of the magnetic field in the muon storage ring. The magnetic field is monitored by nuclear magnetic resonance systems and calibrated in terms of the equivalent proton spin precession frequency in a spherical water sample at 34.7\\(^\\)C. The measured field is weighted by the muon distribution resulting in \\('^_p\\), the denominator in the ratio \\(^_a\\)/\\('^_p\\) that together with known fundamental constants yields \\(a^_\\). The reported uncertainty on \\('^_p\\) for the Run-1 data set is 114 ppb consisting of uncertainty contributions from frequency extraction, calibration, mapping, tracking, and averaging of 56 ppb, and contributions from fast transient fields of 99 ppb.

A measurement of the CKM angle \\(\\) and related strong-phase parameters is performed using a novel, model-independent approach in \\(B^ D( K^0_ S h^+h^-) h^\\) decays, where \\(h^() K\\). The analysis uses a joint data sample of electron-positron collisions collected by the BESIII experiment at the Beijing Electron-Positron Collider II during 2010--2011 and 2021--2022, corresponding to an integrated luminosity of 8 fb\\(^-1\\), and proton-proton collisions collected by the LHCb experiment at the Large Hadron Collider during 2011--2018, corresponding to an integrated luminosity of 9 fb\\(^-1\\). The two datasets are analyzed simultaneously by applying per-event weights based on the amplitude variation over the \\(D\\)-decay phase space to enhance the sensitivity to \\(C\\!P\\)-violating observables. The CKM angle \\(\\) is determined to be \\( = (71.3 5.0)^\\), which constitutes the most precise single measurement to date.

A measurement of the CKM angle \\(\\) is performed by applying a novel, unbinned, model-independent approach to datasets of electron-positron collisions collected by the BESIII experiment and proton-proton collisions by the LHCb experiment, corresponding to integrated luminosities of 8 fb\\(^-1\\) and 9 fb\\(^-1\\), respectively. The \\(C\\!P\\)-violating phase \\(\\) is determined from \\(B^ D( K_ S^0 h^+h^-) h^\\) decays in LHCb data, where \\(h^()\\) is either a pion or kaon, while the corresponding strong-phase parameters are measured using doubly tagged \\(D K_ S/L^0 h^+ h^-\\) decays in the quantum-correlated \\(DD\\) system present in BESIII data. A joint fit to both datasets, which allows for a simultaneous determination of the associated \\(C\\!P\\)-violating observables and strong-phase parameters, yields \\( = (71.3 5.0)^\\). The result is the most precise to date and consistent with previous measurements and world averages.

A new measurement of the magnetic anomaly \\(a_\\) of the positive muon is presented based on data taken from 2020 to 2023 by the Muon \\(g-2\\) Experiment at Fermi National Accelerator Laboratory (FNAL). This dataset contains over 2.5 times the total statistics of our previous results. From the ratio of the precession frequencies for muons and protons in our storage ring magnetic field, together with precisely known ratios of fundamental constants, we determine \\(a_ = 116\\,592\\,0710(162) 10^-12\\) (139 ppb) for the new datasets, and \\(a_ = 116\\,592\\,0705(148) 10^-12\\) (127 ppb) when combined with our previous results. The new experimental world average, dominated by the measurements at FNAL, is \\(a_(exp) =116\\,592\\,0715(145) 10^-12\\) (124 ppb). The measurements at FNAL have improved the precision on the world average by over a factor of four.

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 Muon g-2 Experiment at Fermi National Accelerator Laboratory (FNAL) has measured the muon anomalous precession frequency \\(\\omega_a\\) to an uncertainty of 434 parts per billion (ppb), statistical, and 56 ppb, systematic, with data collected in four storage ring configurations during its first physics run in 2018. When combined with a precision measurement of the magnetic field of the experiment's muon storage ring, the precession frequency measurement determines a muon magnetic anomaly of \\(a_{\\mu}({\\rm FNAL}) = 116\\,592\\,040(54) \\times 10^{-11}\\) (0.46 ppm). This article describes the multiple techniques employed in the reconstruction, analysis and fitting of the data to measure the precession frequency. It also presents the averaging of the results from the eleven separate determinations of \\omega_a, and the systematic uncertainties on the result.

We present details on a new measurement of the muon magnetic anomaly, \\(a_\\mu = (g_\\mu -2)/2\\). The result is based on positive muon data taken at Fermilab's Muon Campus during the 2019 and 2020 accelerator runs. The measurement uses \\(3.1\\) GeV\\(/c\\) polarized muons stored in a \\(7.1\\)-m-radius storage ring with a \\(1.45\\) T uniform magnetic field. The value of \\( a_{\\mu}\\) is determined from the measured difference between the muon spin precession frequency and its cyclotron frequency. This difference is normalized to the strength of the magnetic field, measured using Nuclear Magnetic Resonance (NMR). The ratio is then corrected for small contributions from beam motion, beam dispersion, and transient magnetic fields. We measure \\(a_\\mu = 116 592 057 (25) \\times 10^{-11}\\) (0.21 ppm). This is the world's most precise measurement of this quantity and represents a factor of \\(2.2\\) improvement over our previous result based on the 2018 dataset. In combination, the two datasets yield \\(a_\\mu(\\text{FNAL}) = 116 592 055 (24) \\times 10^{-11}\\) (0.20 ppm). Combining this with the measurements from Brookhaven National Laboratory for both positive and negative muons, the new world average is \\(a_\\mu\\)(exp) \\( = 116 592 059 (22) \\times 10^{-11}\\) (0.19 ppm).

The Muon \\(g-2\\) experiment, E989, is currently taking data at Fermilab with the aim of reducing the experimental error on the muon anomaly by a factor of four and possibly clarifying the current discrepancy with the theoretical prediction. A central component of this four-fold improvement in precision is the laser calibration system of the calorimeters, which has to monitor the gain variations of the photo-sensors with a 0.04\\% precision on the short-term (\\(\\sim 1\\,\\)ms). This is about one order of magnitude better than what has ever been achieved for the calibration of a particle physics calorimeter. The system is designed to monitor also long-term gain variations, mostly due to temperature effects, with a precision below the per mille level. This article reviews the design, the implementation and the performance of the Muon \\(g-2\\) laser calibration system, showing how the experimental requirements have been met.