A PART-PER-MILLION MEASUREMENT OF THE
POSITIVE MUON LIFETIME AND DETERMINATION
OF THE FERMI CONSTANT
David M Webber
University of Illinois at Urbana-Champaign
(Now University of Wisconsin-Madison)
December 9, 2010
Outline
• Motivation
• Experiment Hardware
• Analysis
– Pulse Fitting
– Fit Results
• Systematic Uncertainties
– Gain Stability
– Pileup
– Spin Rotation
• Final Results
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Motivation
• m  gives the Fermi Constant to
unprecedented precision (actually Gm)
• m  needed for “reference” lifetime for
precision muon capture experiments
– MuCap
– MuSun
Capture rate from lifetime difference m- and m
The predictive power of the Standard Model
depends on well-measured input parameters
What are the fundamental electroweak parameters (need 3)?
a
GF
MZ
sin2qw
MW
0.00068 ppm
8.6 ppm
23 ppm
650 ppm
360 ppm
* circa 2000
Obtained from muon lifetime
Other input parameters include fermion masses, and mixing matrix elements:
CKM – quark mixing
PMNS – neutrino mixing
The Fermi constant is related to the
electroweak gauge coupling g by
Contains all weak interaction loop
corrections
In the Fermi theory, muon decay is a contact interaction
where Dq includes phase space, QED, hadronic and
radiative corrections
Dq
In 1999, van Ritbergen and Stuart completed full 2-loop QED corrections reducing the
uncertainty in GF from theory to < 0.3 ppm (it was the dominant error before)
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The push – pull of experiment and theory
• Lifetime now largest uncertainty leads to 2 new
experiments launched: MuLan & FAST
– Both @ PSI, but very different techniques
– Both aim at “ppm” level GF determinations
– Both published intermediate results on small data samples

Meanwhile, more theory updates !!
The lifetime difference between m and m- in
hydrogen leads to the singlet capture rate LS
m-  p  n  m
D m  - m -   0.16%
log(counts)
1.0 ppm MuLan
~10 ppm MuCap
MuCap nearly
complete
μ–
μ+
time
The singlet capture rate is used to determine gP and compare with theory
L S  L m - - L m   ( m - ) -1 - ( m  ) -1  gP
Experiment
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For 1ppm, need more than 1 trillion (1012) muons ...
πE3 Beamline,
Paul Scherrer Institut,
Villigen, Switzerland
The beamline transports ~107 “surface” muons
per second to the experimental area.
Parallel beam
Spatial focus
Momentum
Selection
Dp
p
 2%
Velocity separator
removes beam
positrons
A kicker is used to create the time structure.
counts arb.
5 ms
Trigger
Suppression
Accumulation
Period
22 ms
Extinction ~ 1000
Measuring Period
kicker
ARNOKROME™ III (AK-3) high-field target used in 2006
- Rapid precession of muon spin
- mSR studies show fast damping
The target was opened once per day
to view the beam profile.
Target rotates out of
beam
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In 2006, The ferromagnetic target dephases the
muons during accumulation.
•Arnokrome-3 (AK3) Target (~28% chromium, ~8% cobalt, ~64% iron)
•0.5 T internal magnetic field
•Muons arrive randomly during 5 ms accumulation period
•Muons precess by 0 to 350 revolutions
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2007 target: crystal quartz, surrounded by an external ~
135 G magnetic field
Installed
Halbach
Array
Quartz
• 90% muonium formation
– “Test” of lifetime in muonium vs. free
– Rapid spin precession not observable by us
• 10% “free” muons
– Precession noticeable and small longitudinal polarization exists
• Creates analysis challenges !
• Magnet ring “shadows” part of detector
The experimental concept…
Number (log scale)
Real dataKicker On
450 MHz
WaveForm
Digitization
(2006/07)
Measurement Period
time
Fill Period
MHTDC
(2004)
170 Inner/Outer
tile pairs
The detector is composed of 20 hexagon
and 10 pentagon sections,
forming a truncated icosahedron.
Each section contains either 6 or 5
tile elements
a
Each element is made from two independent
scintillator tiles with light guides and
photomultiplier tubes.
170 scintillator tile pairs readout using
450 MHz waveform digitizers.
2 Analog Pulses
Waveform
Digitizers
x2
1 clock tick = 2.2 ns
1/6 of system
Uncertainty on lifetime from gain stability:
0.25 ppm
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The clock was provided by an Agilent E4400B
Signal Generator, which was stable during the run
and found to be accurate to 0.025 ppm.
Agilent E4400 Function
Generator
f = 451.0
450.87649126
+/- 0.2 MHz
• Checked for consistency
throughout the run.
• Compared to Quartzlock A10-R
rubidium frequency standard.
• Compared to calibrated
frequency counter
• Different blinded frequencies in
2006 and 2007
Average difference = 0.025 ppm
MuLan collected two datasets, each
containing 1012 muon decays
Ferromagnetic Target, 2006
Quartz Target, 2007
• Two (very different) data sets
–
–
–
–
Different blinded clock frequencies used
Revealed only after all analyses of both data sets completed
Most systematic errors are common
Datasets agree to sub-ppm
Analysis
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Raw waveforms are fit with templates to find pulse
amplitudes and times
>2 x 1012 pulses in 2006 data set
>65 TBytes raw data
inner
A difficult fit
Normal Pulse
ADT
Template
outer
Two pulses close together
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Nearby pulses perturb the time of main pulses.
Studied with simulations
Fixed reference
perturbation
Dtavg
Dtavg
Estimated pull:
(0.03 ct )  (0.25% pileup )
 0.075 ppm
(1000 ct /  m )
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2006: Fit of 30,000 AK-3 pileup-corrected runs.
ppm m + Dsecret
22 ms
Relative
 (ppm)
R vs fit start time
Red band is the
set-subset allowed
variance
0
9 ms
2007: Quartz data fits well as a simple sum, exploiting the
symmetry of the detector. The mSR remnants vanish.
Systematics
Introduction
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Leading systematic considerations:
Systematics:
Gain Stability
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Gain is photomultiplier tube type
dependent
Artifact from start signal
Deviation at t=0
1 ADC = 0.004 V
Sag in tube response
0
10
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32
Gain variation vs. time is derived from the stability of
the peak (MPV) of the fit to pulse distribution
If MPV moves, implies greater or fewer hits will be over threshold
δN
 3.0 10 - 4
N
0
10
20 ms
Carefully studied over the summer. Gain correction is 0.5
ppm shift with 0.25 ppm uncertainty.
33
Systematics:
Pileup
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Leading order pileup to a ~5x10-4 effect
Fill i
•Statistically reconstruct
pileup time distribution
•Fit corrected distribution
Fill i+1
Measured  vs. Deadtime
Normal Time
Distribution
Pileup Time
Distribution
Raw Spectrum
Pileup
Corrected
Introducing higher-order pileup
A
B
C
D
E
hit
Inner tile
Artificial deadtime
F
G
triple
pileup
Artificial deadtime
time
Outer tile
Artificial deadtime
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Pileup to sub-ppm requires higher-order terms
•
12 ns deadtime, pileup has a 5 x 10-4 probability at our rates
•
Proof of procedure validated with detailed Monte Carlo simulation
– Left uncorrected, lifetime wrong by 100’s of ppm
Pileup terms
at different
orders …
R (ppm)
1 ppm
150 ns deadtime range
Artificial Deadtime (ct)
The pileup corrections were tested
with Monte-Carlo.
Monte-Carlo Simulation, 1012 events
agrees with truth to < 0.2 ppm
1.19 ppm statistical uncertainty
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Lifetime vs. artificially imposed deadtime
window is an important diagnostic
• A slope exists due to a pileup undercorrection
1 ppm
150 ns deadtime range
Extrapolation to 0 deadtime is correct answer
D. Correction
M. Webber
Pileup
Uncertainty: 0.2 ppm
39
Explanations of R vs. ADT slope
• Gain stability vs. Dt?
– No. Included in gain stability systematic uncertainty.
• Missed correction?
– Possibly
– Extrapolation to ADT=0 valid
• Beam fluctuations?
– Likely
– Fluctuations at 4% level in ion source exist
– Extrapolation to ADT=0 valid
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Systematics
Spin Precession
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The decay positron energy and angular
distributions are not uniform, resulting in position
dependant measurement rates.
Highest Energy
Energy Positrons
Positrons
Lowest
Ee = Emax
= 52.83 MeV
Positron energy distribution
•• Lowest
Highest energy
energy positron
positronwhen
when neutrinos
neutrinosare
are antiparallel.
Ee = 13.2 MeV
Ee = 26.4 MeV
x
Detection
threshold
Ee
Emax
parallel.
• Neutrino helicities cancel angular momentum.
• Neutrino helicities add so that they have angular
• Positron spin
must be in the same direction as muon
momentum
of 2.
spin.
q
• Positron spin must compensate to bring total to 1.
• Chiral limit dictates right handed positrons.
• Chiral suppression (not well justified at this energy)
•makes
Most probable
positron
direction
is same as muon
positron most
likely
right handed.
e+
spin
• Most probable positron direction is opposite muon
spin
mSR rotation results in an oscillation of the
measurement probability for a given detector.
eB
  gm
2
m
c
m
g 2
B

m
m
B = 34 G
This oscillation is easily detected
counts arb.
counts arb.
N (t )  N 0exp - t /  1  a Pcos t   
B=1G
This oscillation is not easily detected
and systematic errors may arise
counts arb.
The sum cancels muSR effects; the difference
accentuates the effect.
B
counts arb.
counts arb.

Sum
m
Difference/Sum
2006 target: AK3 ferromagnetic alloy with high internal
magnetic field
•Arnokrome-3 (AK3) Target (~28% chromium, ~8% cobalt, ~64% iron)
•0.4 T transverse field rotates muons with 18 ns period
•Muons arrive randomly during 5 ms accumulation period
Ensemble Averge Polarization
•Muons precess by 0 to 350 revolutions  DEPHASED  small ensemble avg. polarization
A small asymmetry exists front / back owing to residual longitudinal
polarization
85 Opposite Pairs
All 170 Detectors
pairs summed
OppositeLifetime
m
When front / back
opposite tile pairs are
added first, there is no
distortion
Front
“front-back folded”Back
2007 target: crystal quartz, surrounded by an external ~
135 G magnetic field
Installed
Halbach
Array
Quartz
• 90% muonium formation
– “Test” of lifetime in muonium vs. free
– Rapid spin precession not observable by us
• 10% “free” muons
– Precession noticeable and small longitudinal polarization exists
• Creates analysis challenges !
• Magnet ring “shadows” part of detector
We directly confront the mSR. Fit each detector for an “effective
lifetime.” Would be correct, except for remnant longitudinal
polarization relaxation.
Difference between Top of Ball and Bottom
of Ball to Sum, vs time-in-fill
Illustration of free muon precession in top/bottom detector differences
Longitudinal polarization distorts result in predictable manner
depending on location. The ensemble of lifetimes is fit to obtain
the actual lifetime. (Method robust in MC studies)
Relative effective
lifetime (ppm)
(+ blind offset)
Magnet-right data
2007: Consistency
against MANY special
runs, where we varied
target, magnet, ball
position, etc.
Start-time scan
Consistency Checks
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2006: Fit of 30,000 AK-3 pileup-corrected runs
ppm m + Dsecret
22 ms
Relative
 (ppm)
R vs fit start time
Red band is the
set-subset allowed
variance
0
9 ms
2006: AK-3 target consistent fits of individual detectors,
but opposite pairs – summed – is better
Difference of Individual lifetimes to average
All 170 Detectors
85 Opposite Pairs
2007: Quartz data fits well as a simple sum, exploiting the
symmetry of the detector. The mSR remnants vanish
Variations in m vs. fit start time are
within allowed statisical deviations
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Conclusions
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Final Errors and Numbers
ppm units
Effect
Kicker extinction stability
Residual polarization
Upstream muon stops
Overall gain stability:
Short time; after a pulse
Long time; during full fill
Electronic ped fluctuation
Unseen small pulses
Timing stability
Pileup correction
Clock stability
Total Systematic
Total Statistical
2006
0.20
0.10
2007
0.07
0.20
0.10
0.25
0.12
0.20
0.03
0.42
1.14
0.42
1.68
(R06) = 2 196 979.9 ± 2.5 ± 0.9 ps
(R07) = 2 196 981.2 ± 3.7 ± 0.9 ps
(Combined) = 2 196 980.3 ± 2.2 ps (1.0 ppm)
D(R07 – R06) = 1.3 ps
Comment
Voltage measurements of plates
Long relax; quartz spin cancelation
Upper limit from measurements
MPV vs time in fill; includes:
MPVs in next fill & laser studies
Different by PMT type
Bench-test supported
Uncorrected pileup effect  gain
Laser with external reference ctr.
Extrapolation to zero ADT
Calibration and measurement
Highly correlated for 2006/2007
GF & m precision has improved by
~4 orders of magnitude over 60 years.
Achieved!
Lifetime “history”
FAST
The most precise particle or nuclear or (we believe)
atomic lifetime ever measured
New GF
GF(MuLan) = 1.166 378 8(7) x 10-5 GeV-2 (0.6 ppm)
The lifetime difference between m and m- in
hydrogen leads to the singlet capture rate LS
m-  p  n  m
D m  - m -   0.16%
log(counts)
1.0 ppm MuLan
~10 ppm MuCap
MuCap nearly
complete
μ–
μ+
time
The singlet capture rate is used to determine gP and compare with theory
L S  L m - - L m   ( m - ) -1 - ( m  ) -1  gP
In hydrogen: 1/m-)-(1/m+) = LS gP
now in even better agreement with ChPT*
Using previous m world average
Using new MuLan m average
Shifts the MuCap result
*Chiral
Perturbation Theory
64
Conclusions
• MuLan has finished
– PRL accepted and in press. (see also arxiv:1010.0991)
– 1.0 ppm final error achieved, as proposed
• Most precise lifetime
– Most precise Fermi constant
– “Modest” check of muonium versus free muon
• Influence on muon capture
– Shift moves gP to better agreement with theory
– “Eliminates” the error from the positive muon lifetime, needed in
future MuCap and MuSun capture determinations
(R06) = 2 196 979.9 ± 2.5 ± 0.9 ps
(R07) = 2 196 981.2 ± 3.7 ± 0.9 ps
(Combined) = 2 196 980.3 ± 2.2 ps (1.0 ppm)
D(R07 – R06) = 1.3 ps
MuLan Collaborators
2004
Institutions:
University of Illinois at Urbana-Champaign
University of California, Berkeley
TRIUMF
University of Kentucky
Boston University
James Madison University
Groningen University
Kentucky Wesleyan College
2006
2007
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D. M. Webber
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Backup Slides
What is gP?
gP is the pseudoscalar form
factor of the proton
D. M. Webber
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Muon capture
At a fundamental level, the leptonic and quark currents possess the
simple V−A structure characteristic of the weak interaction.
GFVud
L
2
ν
d
μ
u
  a (1-  5 )m d a (1-  5 )u
D. M. Webber
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Muon capture
In reality, the QCD substructure of the nucleon complicates the weak
interaction physics. These effects are encapsulated in the nucleonic
charged current’s four “induced form factors”:
ν
n
μ
p
n ( a ) gV  (i a q  ) g M  ( a  5 ) g A  (qa  5 ) g P p
Return
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Miscellaneous
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The decay positron energy and angular
distributions are not uniform, resulting in position
dependant measurement rates.
Highest Energy
Energy Positrons
Positrons
Lowest
Ee = Emax
= 52.83 MeV
Positron energy distribution
•• Lowest
Highest energy
energy positron
positron when
when neutrinos
neutrinos are
are
Ee = 13.2 MeV
Ee = 26.4 MeV
x
Detection
threshold
Ee
Emax
anti-parallel.
parallel.
•• Neutrino
so angular
that theymomentum.
have
Neutrino helicities
helicities add
cancel
angular momentum of 2.
q
• Positron spin must be in the same direction as
•muon
Positron
spin.spin must compensate to bring total
to 1.
• Chiral limit dictates right handed positrons.
e+
• Chiral suppression (not well justified at this
•energy)
Most probable
positronmost
direction
same as
makes positron
likelyisright
muon
spin
handed.
Effect of h on GF
• In the Standard Model, h=0,
• General form of h
• Drop second-order nonstandard couplings
• Effect on GF
return
D. M. Webber
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