Detailed report of the MuLan measurement of the positive muon lifetime and determination of the Fermi constant

V. Tishchenko et al. (MuLan Collaboration)

Phys. Rev. D 87, 052003 – Published 4 March 2013

Abstract

We present a detailed report of the method, setup, analysis, and results of a precision measurement of the positive muon lifetime. The experiment was conducted at the Paul Scherrer Institute using a time-structured, nearly 100% polarized surface muon beam and a segmented, fast-timing plastic scintillator array. The measurement employed two target arrangements: a magnetized ferromagnetic target with a $\sim 4 kG$ internal magnetic field and a crystal quartz target in a 130 G external magnetic field. Approximately $1.6 \times 10^{12}$ positrons were accumulated and together the data yield a muon lifetime of $τ_{μ} (MuLan) = 2196980.3 (2.2) ps$ (1.0 ppm), 30 times more precise than previous generations of lifetime experiments. The lifetime measurement yields the most accurate value of the Fermi constant $G_{F} (MuLan) = 1.1663787 (6) \times 10^{- 5} {GeV}^{- 2}$ (0.5 ppm). It also enables new precision studies of weak interactions via lifetime measurements of muonic atoms.

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Received 5 November 2012

DOI:https://doi.org/10.1103/PhysRevD.87.052003

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Vol. 87, Iss. 5 — 1 March 2013

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Figure 1
Feynman diagrams for ordinary muon decay in Fermi theory. The upper graph shows the tree-level diagram for the ordinary decay amplitude while the lower graph shows a one-loop QED correction to the ordinary decay amplitude. Fermion mass renormalization and soft bremsstrahlung diagrams also contribute at this order.
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Figure 2
Standard Model tree-level diagram of ordinary muon decay showing the $W$ boson mediating the electroweak interaction between the weak leptonic currents.
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Figure 3
The positron energy distribution $N (E)$ and energy-dependent positron asymmetry $A (E)$ in ordinary muon decay $μ^{+} \to e^{+} ν_{e} {\bar{ν}}_{μ}$ . $E_{m}$ is the end-point energy of the ordinary decay spectrum.
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Figure 4
Depiction of typical time spectra from TF $μ SR$ (upper panel) and LF $μ SR$ (lower panel). The paired-solid lines represent the time distributions of outgoing positrons in geometrically opposite detectors. The single-dashed line represents a positron distribution without $μ SR$ effects. The TF $μ SR$ spectrum shows both spin precession and spin relaxation whereas the LF $μ SR$ spectrum shows a relaxation signal but no precession signal. Note that these figures greatly exaggerate the observed effects of $μ SR$ in this experiment.
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Figure 5
Schematic view of the $π E 3$ beamline including the E target station, opposing vertical bending magnets (D), quadrupole focusing magnets (denoted Q), slit systems (S), electrostatic kicker, $\vec{E} \times \vec{B}$ velocity separator, and the location of the experiment. See text for details.
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Figure 6
Plot of the muon arrival times (upper panel) and decay positron times (lower panel) that are produced by the pulsed beam technique. The fill cycle comprises the $T_{A} = 5 μ s$ beam-on accumulation period and the $T_{M} = 22 μ s$ beam-off measurement period.
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Figure 7
Cutaway drawing of the experimental setup showing the vacuum beam pipe (1), stopping target (2), Halbach magnet (3), scintillator array (4), and the beam monitor (5) at the downstream window of the beamline. Note that the Halbach magnet was not installed for the AK-3 data taking and the stopping target was rotated out of the beam path for the beam monitoring.
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Figure 8
Hit distribution for the 170 tile pairs versus the angle $θ$ relative to the beamline axis for all of the AK-3/quartz data sets. The upper panel is the hit distribution for the AK-3 setup without the Halbach magnet and the lower panel is the hit distribution for the quartz setup with the Halbach magnet. The vertical groupings with common $θ$ values are tile pairs with different values of angle $ϕ$ .
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Figure 9
Examples of the digitized waveforms and the corresponding fit results for a single pulse (upper panel) and two overlapping pulses (lower panel). The waveforms are indicated by the histogram and the fit function by the curve.
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Figure 10
Representative pulse amplitude distribution for an individual scintillator tile. The histogram represents the measured data and the smooth curve the fit function (see text for details). The fit-function minimum defines the normal amplitude threshold $A_{thr}$ for each scintillator tile.
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Figure 11
Representative time-difference distribution for a single inner-outer pair. The histogram is the measured distribution and the curve is the Gaussian fit. The observed timing jitter between inner-outer pairs is typically 0.55 ns ( $1 σ$ ).
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Figure 12
Representative time distributions for the trigger hits before the pileup correction (curve 1) and corrections accounting for leading-order pileup (curve 2), extended pileup (curve 3), accidental coincidences (curve 4), and second-order pileup (curve 5). These histograms correspond to an $ADT = 6 c . t .$ (13.3 ns).
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Figure 13
Fitted muon lifetime versus artificial deadtime obtained from an analysis of a Monte Carlo simulation of pileup effects, accidental coincidences, and timing jitter. The simulated data are analyzed using the same software and methods as measured data. The fitted lifetime is ADT-independent to better than 0.1 ppm and in statistical agreement with the “true” lifetime depicted at the location $ADT = 0$ . The error bars on the $ADT > 0$ points correspond to the statistical variations allowed by the ADT correction. The error bar on the $ADT = 0$ point corresponds to the statistical variation allowed between the “true” lifetime that omits the pulse pileup, accidental coincidences, and timing jitter, and the $ADT > 0$ lifetimes that correct for pulse pileup, accidental coincidences, and timing jitter.
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Figure 14
Fractional MPV change versus the hit time in the measurement period. The solid circles correspond to the amplitude variation of tiles with Photonis PMTs and the open triangles correspond to the amplitude variation of tiles with Electron Tube PMTs.
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Figure 15
Time dependencies of the Landau MPV versus the time interval after a preceding pulse for representative groups of Electron Tube PMTs (upper panel) and Photonis PMTs (lower panel). The filled data points correspond to time intervals between two hits in the same tile pair and the same fill cycle. The open data points correspond to time intervals between two hits in the same tile pair but two neighboring fill cycles. The open points show no evidence of longer-term effects from gain variations between neighboring fills.
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Figure 16
Three-parameter fit to the entire AK-3 data set. The upper panel shows both the measured data and the fit function and the lower panel shows the normalized residuals between the data points and the fit function.
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Figure 17
Effective muon lifetime $τ_{eff}$ versus tile coordinate $θ_{B}$ relative to the $B$ -field axis. The data points are the 170 individual effective lifetime measurements and the solid curve is the effective lifetime fit from Eq. (14). The small irregularities in the solid curve arise from the $ϕ_{B}$ dependence of the asymmetry parameter $A$ .
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Figure 18
The distribution of $χ^{2} / dof$ values from the 170 fits to the individual time histograms for the R07-A data set. The fit function [Eq. (14)] incorporates the TF $μ SR$ signal.
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Figure 19
Plots of the fitted lifetimes from the individual time histograms grouped by the detector angle $θ$ relative to the beam axis. The upper panel represents the AK-3 production data with the magnetization axis at 90 degrees to the beam axis. The lower panel represents the AK-3 supplemental data with the magnetization axis at 45 degrees to the beam axis. The four-fold increase in the $θ$ dependence of the fitted lifetime in the lower plot reflects the increase in the longitudinal polarization for the 45 degree data set. Note that the vertical scales on the two plots are the same.
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Figure 20
Ratio $R_{U D} (t)$ of the normalized difference between the time distributions in the upstream/downstream hemispheres of the positron detector. The upper panel corresponds to production data with an accumulation period $T_{A} = 5.0 μ s$ and the lower panel corresponds to supplemental data with a shortened accumulation period $T_{A} = 0.15 μ s$ (note the different scales of the vertical axes). The plots highlight the contribution of the TF $μ SR$ signal in the quartz data.
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Figure 21
Ratio of hit-time distributions from the beam-left hemisphere [ $R_{L} (t)$ ] and the beam-right hemisphere [ $R_{R} (t)$ ] to the detector sum. The upper panel corresponds to production data with a longitudinal polarization $P_{1} \sim 0.01$ and the lower panel corresponds to supplemental data with a longitudinal polarization $P_{1} \sim 0.25$ . The gradual slopes in $R_{L / R} (t)$ are the effect of the LF $μ SR$ in the quartz data.
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Figure 22
Muon lifetime versus fit start-time for the AK-3 data (top) and the quartz data (bottom). The data points and gray band indicate the lifetime results and corresponding uncertainties from individual fits. The solid envelopes indicate the permissible statistical deviations ( $1 σ$ ) from the benchmark fits with a start time $t_{start} = 0.2 μ s$ . The $1 σ$ deviations are calculated with appropriate accounting for data set correlations between different fit ranges.
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Figure 23
The distribution $(τ_{i} - τ_{μ}) / σ_{i}$ of the normalized deviations of the fitted lifetimes $τ_{i}$ from the individual runs from the final result $τ_{μ}$ . The data points are the run-by-run results and the solid curve is a least squares fit to a Gaussian distribution. The left panel is the AK-3 data set and the right panel is the quartz data set.
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Figure 24
The upper panel shows the lifetime results from opposite tile-pair sums grouped by detector coordinate $θ$ for the entire AK-3 data set. The lower panel shows the lifetime results from opposite tile-pair sums grouped by detector coordinate $θ_{B}$ for the entire quartz data set. The plots demonstrate the cancellation of $μ SR$ distortions from target stops in opposite tile pairs.
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Figure 25
Fitted lifetime for quartz data sets with different magnetic field orientations and positron-detector offsets. Entries R07-A and R07-B correspond to the quartz production data with the $B$ -field orientation to the beam-left and beam-right, respectively. Entries R07-C through R07-F correspond to quartz production data with different detector positions and entries $θ = - 9.0 °$ through $θ = 20.0 °$ correspond to quartz systematics data with different magnet tilt angles. The different detector positions and magnet angles change the magnitudes and axes of the LF/TF polarizations and their corresponding $μ SR$ signals.
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Figure 26
Plot of $τ_{μ}$ versus ADT before the pileup correction (open circles) and after the pileup correction (filled circles). The statistical uncertainties associated with the pileup correction are substantially smaller than the statistical uncertainty on the muon lifetime and not visible on this scale. The solid lines through data points are only to guide the eye.
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Figure 27
Examples of precession frequency $ω$ (upper panel) and relaxation constant $T_{2}$ (lower panel) versus tile coordinate $θ_{B}$ obtained from seven-parameter fits using Eq. (14). The solid horizontal lines indicate the fixed “best-fit” values of $ω$ and $T_{2}$ that are used in the fits of the tile-dependent effective lifetimes.
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Figure 28
Plot of post-1970 measurements of the positive muon lifetime. The data points represent the individual measurements and the vertical band corresponds to the combined R06 and R07 result. The result of Duclos et al. is not shown in the figure.
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