Improved Measurement of the Positive Muon Lifetime and Determination of
  the Fermi Constant
Improved Measurement of the Positive Muon Lifetime and
Determination of the Fermi Constant
D.B. Chitwood,1 T.I. Banks,2 M.J. Barnes,3 S. Battu,4 R.M. Carey,5 S. Cheekatmalla,4 S.M. Clayton,1 J. Crnkovic,1
K.M. Crowe,2 P.T. Debevec,1 S. Dhamija,4 W. Earle,5 A. Gafarov,5 K. Giovanetti,6 T.P. Gorringe,4 F.E. Gray,1, 2
M. Hance,5 D.W. Hertzog,1 M.F. Hare,5 P. Kammel,1 B. Kiburg,1 J. Kunkle,1 B. Lauss,2 I. Logashenko,5 K.R. Lynch,5
R. McNabb,1 J.P. Miller,5 F. Mulhauser,1 C.J.G. Onderwater,1, 7 C.S. Özben,1 Q. Peng,5 C.C. Polly,1 S. Rath,4
B.L. Roberts,5 V. Tishchenko,4 G.D. Wait,3 J. Wasserman,5 D.M. Webber,1 P. Winter,1 and P.A. Żo lnierczuk4
(MuLan Collaboration)
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Department of Physics, University of California, Berkeley, CA 94720, USA
TRIUMF, Vancouver, BC, V6T 2A3, Canada
Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA
Department of Physics, Boston University, Boston, MA 02215, USA
Department of Physics, James Madison University, Harrisonburg, VA 22807, USA
Kernfysisch Versneller Instituut, Groningen University, NL 9747 AA Groningen, The Netherlands
The mean life of the positive muon has been measured to a precision of 11 ppm using a low-
energy, pulsed muon beam stopped in a ferromagnetic target, which was surrounded by a scintillator
detector array. The result, τµ = 2.197 013(24) µs, is in excellent agreement with the previous world
average. The new world average τµ = 2.197 019(21) µs determines the Fermi constant GF =
1.166 371(6)× 10−5 GeV−2 (5 ppm). Additionally, the precision measurement of the positive muon
lifetime is needed to determine the nucleon pseudoscalar coupling gP .
The predictive power of the standard model relies on
precision measurements of its input parameters. Impres-
sive examples include the fine-structure constant [1], the
Z mass [2], and the Fermi constant [3], having relative
precisions of 0.0007, 23, and 9 ppm, respectively.
The Fermi constant GF is related [3] to the electroweak
gauge coupling g by
(1 + ∆r) , (1)
where ∆r represents the weak-boson-mediated tree-level
and radiative corrections, which have been computed to
second order [4]. Comparison of the Fermi constant ex-
tracted from various measurements stringently tests the
universality of the weak interaction’s strength [5].
The most precise determination of GF is based on the
mean life of the positive muon, τµ. It has long been
known that in the Fermi theory, the QED radiative cor-
rections are finite to first order in GF and to all orders in
the electromagnetic coupling constant, α [6]. This pro-
vides a framework [3] for extracting GF from τµ,
192π3
(1 + ∆q) , (2)
where ∆q is the sum of phase space and both QED and
hadronic radiative corrections, which have been known
in lowest-order since the 1950s [7]. Relation 2 does not
depend on the specifics of the underlying electroweak
model.
Until recently, the uncertainty on extracting GF from
τµ was limited by the uncertainty in higher-order QED
corrections, rather than by measurement. In 1999, van
Ritbergen and Stuart’s calculation of the second-order
QED corrections [3] reduced the relative theoretical un-
certainty in the determination of GF to less than 0.3 ppm.
The dominant uncertainty is currently from τµ, which
motivates this work. While the final goal of our experi-
ment is a 1 ppm uncertainty on τµ, we report here a re-
sult having a precision of 11 ppm—2.5 times better than
any previous measurement [8]—based on data obtained
in 2004, the commissioning run period.
The experimental design is conceptually simple. Lon-
gitudinally polarized muons from the πE3 beamline at
the Paul Scherrer Institute are stopped in a thin ferro-
magnetic target. A fast-switching kicker imposes a cycle
on the continuous beam, consisting of a 5 µs “beam-on”
period of stopped-muon accumulation, TA, followed by a
22 µs “beam-off” measurement period, TM . The muon
decay positrons are detected in a scintillator array which
surrounds the target. A decay time spectrum for a sub-
set of the detectors is shown in Fig. 1. The background
level is indicative of the “extinction” of the beam during
TM , caused by the kicker.
The beamline is tuned to transport 28.8 MeV/c muons
from pions that decay at rest near the surface of the
production target. Two opposing 60◦ bends raise the
beam from ground level to the experimental area, where
it is directed parallel to the optic axis through an ~E × ~B
velocity separator that removes the e+ contamination.
The muons continue undeflected through the (uncharged)
kicker and are focused on a 1.2-cm tall by 6.5-cm wide
aperture. The activated (charged) kicker induces a
36 mrad vertical deflection, which causes the beam to
http://arxiv.org/abs/0704.1981v2
Time Relative to Kicker Transition [ns]
-5000 0 5000 10000 15000 20000
Background Level
Measurement Period
Accumulation Period
Time [ct]
2000 4000 6000 8000 10000 12000 14000 16000
FIG. 1: Data from a subset of the MuLan detectors, illus-
trating the accumulation and measurement periods and the
background mainly caused by incomplete extinction. The fit
region used for the whole data set is indicated as a thick red
line. The residuals divided by their uncertainty (i.e, in stan-
dard deviations) are shown in the upper inset panel.
be blocked at the aperture. In 2004, the average un-
kicked muon rate was limited to 2 MHz; approximately
10 muons were accumulated per cycle, of which 4 re-
mained undecayed when TM began.
The kicker is described in detail in Ref. [9]. Briefly,
it consists of two pairs of electrode plates biased to pro-
duce a potential difference of up to VK = 25 kV, with
a virtual ground at the midplane. Modulators, using
series-connected MOSFETs operating in push-pull mode,
charge or discharge the plates. In 2004, a partial system
achieved an average beam extinction of ε = 260 with a
60 ns switching time [10]. During TM , VK changed by
less than 0.25 V. A time dependence of VK at this level,
together with a voltage dependent extinction, gives rise
to a 3.5 ppm systematic error on the muon lifetime.
The parity-violating correlation between the muon’s
spin orientation and the emission direction of its de-
cay positron can lead to a systematic shift in the ex-
tracted lifetime, for the following reasons: Suppose de-
tector A at position (θ, φ) counts positrons at the rate
~Pµ) exp(−t/τµ), where ~Pµ is the polarization of the
stopped muons. If ~Pµ varies with time because of relax-
ation or spin rotation caused by magnetic fields, so will
~Pµ). A temporal variation that is long compared to
τµ will manifest itself as an unobserved distortion to the
fitted lifetime of the detector. The spin-related system-
atic uncertainty in τµ is minimized through both detector
symmetry and target choice: the positron detectors are
arranged as a symmetric ball covering a large solid angle,
and every detector A at position (θ, φ) is mirrored by an-
other detector, Ā at (π − θ, φ + π). To the extent that
the detector pairs have the same geometrical acceptance
FIG. 2: Diagram of the experiment with several detector el-
ements removed. Muons enter through the beampipe vac-
uum window and traverse the EMC and a helium bag (not
shown) before stopping in the AK-3 target. Decay positrons
are recorded by the coincidence of inner and outer scintillators
in one triangular segment; the outer scintillators are visible.
Two example decay trajectories are shown.
and efficiency NA+Ā ≡ NA(~Pµ)+NĀ(~Pµ) is independent
of the value of ~Pµ, and therefore also its time variation.
Finally, a target possessing a high internal magnetic field
is used so that the muon spin precession period is ≪ τµ.
As depicted in Fig. 2, the muon beam exits its vac-
uum pipe through a 9.3-cm diameter, 76-µm-thick Mylar
window, then passes through a thin, high-rate multiwire
entrance muon chamber (EMC), which records the time
and position of muons entering the detector. Roughly 1
in 104 muons stop in the EMC. Their spins precess in
the field of a permanent magnet array, which has a mean
transverse field of 11 mT at the EMC center. The field
orientation was regularly reversed throughout data tak-
ing. The region between the exit of the EMC and the
target is spanned by a helium-filled balloon (instead of
air) to minimize muon stops and scattering.
The stopping target is a 0.5-mm thick, 50-cm diame-
ter disk of ArnokromeTM III (AK-3) [11], having an in-
ternal magnetic field of approximately 0.4 T, oriented
transverse to the muon spin axis. The field direction was
reversed at regular intervals. Dedicated µSR measure-
ments [12] on an AK-3 sample show an 18-ns oscillation
period with a large initial asymmetry that relaxes with
a time constant of 14 ns. These times are considerably
shorter than the accumulation period, TA. Using the dif-
ference spectrum of counts from mirrored detectors ver-
sus time; e.g., NA−Ā, the longer-term components are
shown to be negligible.
The decay positrons are recorded by 170 detector el-
ements, each consisting of an inner and outer layer of
3-mm-thick, BC-404 plastic scintillator. Each triangle-
shaped scintillator is read edge-on using a lightguide
mounted at 90◦, which is coupled to a 29-mm photo-
multiplier tube (PMT). The 170 elements are organized
in groups of six and five to form the 20 hexagon and
10 pentagon faces of a truncated icosahedron (two pen-
tagons are omitted for the beam entry and symmetric
exit). The distance from target center to an inner scin-
tillator is 40.5 cm. The total acceptance is 64%, taking
into account the reduction in the geometrical coverage of
70% from positron range and annihilation in the target
and detector materials.
A clip line reshapes the natural PMT pulse width
to a full-width at 20% maximum of 9 ns. These sig-
nals are routed to leading-edge discriminators that have
10 ns output widths. On average, a throughgoing
positron gives a signal of 70 photoelectrons, producing
a 400-mV pulse height. The data taking was split al-
most evenly between periods of 80-mV and 200-mV dis-
criminator thresholds. The arrival time of a positron
is measured with respect to the kicker transition by
a CAEN V767 128-channel, multihit TDC. An Agi-
lent E4400B frequency synthesizer, operating at approx-
imately 190.2 MHz, serves as the master clock. Its abso-
lute frequency is accurate to 10−8 and its central value
did not change at this level over the course of the run.
A clock step-down and distribution system provides a
23.75 MHz square wave as the input clock for each TDC.
The master clock frequency was given a concealed off-
set within 250 ppm from 190.2 MHz. The analyzers
added a fitting offset to τµ when reporting intermedi-
ate results. Only after the analysis was complete was the
exact oscillator frequency revealed, and the fitting offset
removed, to obtain the lifetime.
The raw data consist of individual scintillator hit times
for each cycle. The kicker transition defines a common
(global) t = 0, using the 1.32 ns resolution provided by
the 32 subdivisions of the TDC input oscillator period.
To avoid problems with differential nonlinearities in the
TDC clock period division circuit, coincidence windows,
artificial deadtimes, and decay histogram bin widths were
always set at integer multiples of the undivided input
clock period (42 ns).
Events are missed if a positron passes through a detec-
tor during the electronic or software-imposed deadtime
following a recorded event in the same detector. With
peak rates in individual detectors of 7 kHz, the “pileup”
probability for a 42-ns deadtime is < 3 × 10−4. If un-
accounted for, this leads to a 67 ppm shift in the fit-
ted τµ. Pileup can be accommodated by including an
exp(−2t/τµ) term in the fit function, but this doubles
the uncertainty on the fitted τµ. Instead, an artificial
pileup spectrum, constructed from secondary hits occur-
ring in a fixed-width time window that is offset from a
primary hit, is added back to the raw spectrum, thus
restoring, on average, the missing hits. The procedure is
repeated using a wide range of artificial deadtime periods
and offsets, and the corrected spectra all give consistent
lifetimes. The systematic uncertainty from this proce-
dure is 2 ppm.
Preliminary fits to the decay time spectra using the
function N(t) = N0 exp(−t/τµ) + B showed a common
structure in the residuals at early times, independent
of experimental condition or detector. The structure is
caused by an intrinsic flaw in the TDC, which does not
lose events but can shift them in time by ±25 ps. This
behavior was characterized in extensive laboratory tests
using white-noise and fixed-frequency sources, together
with signals that simulate the kicker transitions. For a
fit start time of tstart = 1 µs, the TDC response settles
into a simple pattern that can be described well by a
modification of the decay time spectrum by a factor
ℑ(t′) = 1 + A cos(2πt′/T + δ) exp(−t′/τTDC), (3)
where t′ = t − tstart, and with typical values of A =
5 × 10−4, τTDC = 600 ns, and T = 370 ns. A spectrum
of 1011 white-noise events was fit to a constant function,
modified by ℑ(t′), achieving a good χ2 and structureless
residuals.
The function used to fit the decay time spectra is
N(t′) = ℑ(t′) · [N0 exp(−t′/τµ) + B]. (4)
Because the clock frequency was blinded during the anal-
ysis, T and τTDC in ℑ(t
′) could not be fixed in the fits.
With the clock frequency revealed, these parameters are
found to be consistent with the laboratory values. In an
important test of the appropriateness of Eq. 4, the fitted
τµ is found to be independent of the fit start time beyond
the minimal tstart = 1.05 µs. The systematic uncertainty
for the TDC response is 1 ppm.
Subsequent to the 2004 run, waveform digitizers
(WFDs) replaced the discriminator and TDC timing sys-
tem. The WFDs establish the stability of the PMT gain
versus time during TM . A gain change, together with a
fixed discriminator threshold (as in 2004), will appear as
a time-dependent efficiency. The analysis of the PMT
gain stability over a range of instantaneous rates indi-
cates a systematic effect of less than 1.8 ppm on τµ.
A powerful consistency test is performed by grouping
data from detectors having a common azimuthal angle φ
(see Fig. 2), fitting each group independently, and sorting
the lifetime results by cos θ, where θ is the polar angle.
For the AK-3 data, the lifetime distribution is flat over
cos θ (χ2/dof = 17.9/19). For data taken with a 20-cm
diameter sulfur target, surrounded by a permanent mag-
net array, the same distribution is not flat (χ2/dof = 8.0).
The cause in the latter case is a higher fraction of muons
that miss the target and stop downstream along inner
detector walls. The SRIM program [13] finds that 0.55%
of the muons miss the smaller sulfur target, while only
0.07% miss the AK-3 target. In both cases, ≈ 0.11% suf-
fer large-angle scatters in the EMC or backscatter from
the target, stopping in upstream detector walls. Sim-
ulated decay spectra for all “errant” muons—using the
muon stopping distribution predicted by SRIM and in-
cluding detector element acceptance, initial muon po-
larization, and relaxation—were used to determine the
expected distortion to the lifetime. The procedure was
tested against a special data set in which all incoming
muons were stopped in a plastic plate placed midway
between the EMC and the target. The distribution of
lifetimes with cos θ was successfully reproduced. For
the AK-3 target, a distortion as large as 1 ppm could
be expected. However, a negative shift in the range of
4 − 12 ppm is implied for sulfur, principally owing to
the ∼ 8 times larger fraction of downstream muon stops.
The uncertainty on a correction of this magnitude could
be as large as 100%, exceeding the 15 ppm statistical
precision of the sulfur data sample. Therefore, we chose
not to use the sulfur data in our reported results—a deci-
sion made prior to unblinding the clock frequency—even
though such a correction would bring the fitted sulfur life-
time into excellent agreement with that found for AK-3.
We conservatively assign a 2 ppm systematic uncertainty
to this procedure for the AK-3 data set.
Other small systematic uncertainties are listed in Ta-
ble I. Data integrity checks indicate a small fraction of
hits (6 × 10−6) that may be duplicates of earlier hits.
When the duplicates are removed, τµ shifts upwards by
2 ppm. Since the status of these hits is uncertain, a
correction of +1 ppm is applied to τµ with a systematic
uncertainty of 1 ppm. A systematic error from queuing
losses in the TDC single-channel buffer is calculated to
be less than 0.7 ppm. The systematic error from timing
shifts induced by previous hits in a channel during the
measurement period is less than 0.8 ppm.
The final result is based on a fit using Eq. 4 to the
1.8 × 1010 events in the summed AK-3 spectra, giving
τµ(MuLan) = 2.197 013(21)(11) µs (11.0 ppm)
with a χ2/dof = 452.5/484. The first error is statistical
and the second is the quadrature sum of the systematic
uncertainties in Table I. Figure 1 indicates the range of
the fitted region and the inset displays the residuals di-
vided by their uncertainty in units of standard deviations
from the fit. The consistency of τµ was checked against
experimental conditions, including detector, threshold,
target and magnet orientation, extinction factor, kicker
voltage, and run number. Only a sub-group of runs at the
beginning of the production period exhibited an anoma-
lous lifetime compared to the sum. When all runs, in
groups of 10, are fit to a constant, a χ2/dof = 108/102 is
obtained, suggestive that the sub-group fluctuation was
statistical.
Our fitted lifetime is in excellent agreement with the
world average, 2.197 03(4) µs, which is based on four pre-
TABLE I: Systematic uncertainties.
Source Size (ppm)
Extinction stability 3.5
Deadtime correction 2.0
TDC response 1.0
Gain stability 1.8
Errant muon stops 2.0
Duplicate words (+1 ppm shift) 1.0
Queuing loss 0.7
Multiple hit timing shifts 0.8
Total 5.2
vious measurements [8]. The improved world average is
τµ(W.A.) = 2.197 019(21) µs (9.6 ppm).
Assuming the standard model value of the Michel pa-
rameter η = 0, and light neutrinos, determines the Fermi
constant
GF = 1.166 371(6)× 10−5 GeV−2 (5 ppm).
In a companion Letter [14], a new determination of the
induced pseudoscalar coupling gP is reported. It depends
mainly on a comparison of negative and positive muon
lifetimes, the latter quantity being reported here.
We acknowledge the generous support from PSI and
the assistance of its accelerator and detector groups.
We thank W. Bertl, J. Blackburn, K. Gabathuler,
K. Dieters, J. Doornbos, J. Egger, W.J. Marciano,
D. Renker, U. Rohrer, R. Scheuermann, R.G. Stuart
and E. Thorsland for discussions and assistance. This
work was supported in part by the U.S. Department of
Energy, the U.S. National Science Foundation, and the
John Simon Guggenheim Foundation (DWH).
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