We present a new measurement of the positive muon magnetic anomaly, a µ ≡ (gµ − 2)/2, from the Fermilab Muon g −2 Experiment based on data collected in 2019 and 2020. We have analyzed more than four 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 two due to better running conditions, a more stable beam, and improved knowledge of the magnetic field weighted by the muon distribution, ω′ p , and of the anomalous precession frequency corrected for beam dynamics effects, ωa. From the ratio ωa/ω ′ p , together with precisely determined external parameters, we determine a µ = 116 592 057(25) × 10 −11 (0.21 ppm). Combining this result with our previous result from the 2018 data, we obtain a µ (FNAL) = 116 592 055(24) × 10 −11 (0.20 ppm). The new experimental world average is aµ(Exp) = 116 592 059(22) × 10 −11 (0.19 ppm), which represents a factor of two improvement in precision.
The Twiss parameters provide a convenient description of beam optics in uncoupled linear beam lines. For coupled beam lines, a variety of approaches are possible for describing the linear optics; here, we propose an approach and notation that naturally generalizes the familiar Twiss parameters to the coupled case in 3 degrees of freedom. Our approach is based on an eigensystem analysis of the matrix of secondorder beam moments, or alternatively (in the case of a storage ring) on an eigensystem analysis of the linear single-turn map. The lattice functions that emerge from this approach have an interpretation that is conceptually very simple: in particular, the lattice functions directly relate the beam distribution in phasespace to the invariant emittances. To emphasize the physical significance of the coupled lattice functions, we develop the theory from first principles, using only the assumption of linear symplectic transport. We also give some examples of the application of this approach, demonstrating its advantages of conceptual and notational simplicity.
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