1. Parity Violation in Cosmic Muon Decay
Deanna Zapata, Alexander Ruckel, Joan Kim, and Stephen Armstrong
University of California Los Angeles
(Dated: December 9, 2015)
Muon decay was recorded through the use of plastic scintillators and an applied mag-
netic field with the goal of demonstrating parity violation in the electro-weak force. These
muons precessed along their axes with a frequency of 3.8±0.2 radians/µs and a lifetime
of 2.03±0.06µs and 2.28±0.07µs for the up and down decays, respectively.
Keywords: Parity Violation, Muon Decay
I. THEORY
Cosmic rays from our galaxy and beyond that enter
our atmosphere produce hadronic showers that approach
the planet’s surface at relativistic speeds. Charged and
neutral pions are produced in these hadronic showers: the
neutral pions decay to two photons, while the charged pi-
ons decay weakly into muons (π+
→ µ+
vµ; π−
→ µ−
¯vµ)
[1]. The muons then reach the surface and are able to be
detected. FIG 1 shows the measured spectrum of cosmic-
ray muons on Earth at sea level.
FIG. 1. Muon spectrum at sea level [1].
Because the muons are charged, they generate
light when passed through scintillators and this light
can then be detected by photomultiplier tubes. Due
to their heavier mass (relative to particles such as
electrons and positrons), muons experience very little
Brehmsstrahlung radiation, and can thus pass through
materials such as iron. This makes them ideal for
examining parity violation in weak decays.
Parity
The parity operator P ”flips” the spatial coordinates of
any observable it operates on. Some observables, such
as position and momentum, are odd under parity: e.g.
P(p) = −p. These are known as polar vectors. Other
observables, such as angular momentum and spin, are
even under parity: e.g. P(ˆs) = ˆs, and are known as axial
vectors. The dot product of an axial vector and a polar
vector (such as ˆs · p, otherwise known as helicity) is odd
under parity [2].
One way to test for parity violation is to start with an
ensemble of systems all in the same eigenstate of parity,
allow the ensemble to transition to a final state, then
take measurements of any observable that is odd under
parity (such as helicity). If the expectation value of the
odd observable in the final state is nonzero, then the
process has not conserved parity. This was used in 1956,
when C.S. Wu observed positive muons from pion decay
and found that their spins were aligned preferentially
in the direction of their momentum [5]. The muons
were stopped in material, preserving their polarization.
They then decayed to electrons, and the electrons had
a preferential direction with respect to the direction
of the entering muons. We use this same method in
our experiment to test whether or not the weak force
violates parity.
Parity Violation in Pion Decay
We first encounter parity violation in the decay of the
charged pion into a muon (FIG 2) - because helicity
(ˆs · p) is an odd observable, its expectation value must
be 0 in order for a process to converse parity [2]. Thus,
if the incoming muons have spins aligned preferentially
in the direction of their momenta, then the weak decay
(π+
→ µ+
vµ) has violated parity. By stopping incoming
muons in a copper target, placing them in a magnetic
field and observing the precession of their magnetic
moment as a function of time, we can determine
whether or not there is a preferential initial polarization.
The magnetic moment of the muons will precess at
the Larmor frequency around the axis of the applied
magnetic field, creating high-energy electrons which are
detected in the experiment as a periodic change in the
”up” and ”down” count rates as a function of time. A

2. 2
¡W+
¯d
u
µ+
vµ
FIG. 2. π+
→ µ+
vµ. This reaction takes place in the atmo-
sphere. The resulting muons then reach the planet’s surface
and enter our detector.
significant precession indicates that the stopped muons
have preferential spin polarization and nonzero helicity,
signaling parity violation.
Parity Violation in Muon Decay
Parity is violated for a second time in the weak decay
of the entering muons. As discussed above, the positive
muons that enter our detector have a preferential
spin polarization (experimentally, about 20% directed
upwards [1]). (Negative muons also enter our detector,
but their rate of capture in the target is larger than
their decay rate. Therefore, most negative muons will
be missed by our counters). Once they enter, they are
stopped in a copper target (p = 0), where they weakly
decay into a positron, electron neutrino, and muon
antineutrino, as shown in Figure 3. Because p = 0 and
s = 0 in the initial state, in order for parity to be
conserved, p must be 0 in the final state for parity to
be conserved.
¢W+
µ+
¯vµ
e+
ve
FIG. 3. µ+
→ vee+
¯vµ. This reaction takes place after the
muon is stopped in a copper target.
The charged positron emitted from the muon de-
cay travels either upwards or downwards from the
copper target. This momentum direction can be de-
tected with our setup, and an uneven distribution of
”up” and ”down” events indicates that p is nonzero,
signaling parity violation.
FIG. 4. Experimental setup for detection of µ+
decays.
II. EXPERIMENTAL METHODS
Detector
The cosmic ray telescope used for this experiment is
shown in FIG 4 [1]. The telescope is comprised of five
plastic scintillators with a light guide attached to each
one which attaches to a photomultipler tube (PMT),
which detects and amplifies the photons that are created
when the charged muons pass through.
These are arranged in three layers, with scintilla-
tion counters 2 and 3 in one layer, and counters 4 and 5
in another layer, as shown in FIG 5 [1]. Since muons are
relatively massive compared to the electron, they hardly
radiate and will maintain enough energy when passing
through a material. As a result of this, a 3 iron filter
is placed between the first and second layer of scintilla-
tors allows for muons to be selected with momenta 200
MeV/c2
and a target of copper is placed below the sec-
ond layer of scintillators which acts as both a stopper for
the muons and a magnetic field source when looking for
the Larmor precession of decay positrons. Muons with
momenta below this value will not be seen by the second
layer of scintillators and muons with momenta greater
than this will penetrate the copper target and the elec-
tronics will veto the event. The dimensions of counters
1, 2, and 5 are 20”x12”x3/4” and counters 3 and 4 are
19”x11”x5/16”. The copper stopper is 19”x11”x3/4” and
serves as a form for a coil of 351 turns made with #16
wire which has been wound such that the magnetic field
is along the 19” dimension [5].

3. 3
FIG. 5. Configuration of scintillation counters with an iron
filter and copper target.
The positively charged muons within the allowed
momentum bite will stop in the copper target, then
decay to a positron, an electron neutrino, and an
anti-muon neutrino. This experiment is sensitive to the
charged positron, as the neutrinos cant be detected with
our setup. The emitted positron is detected either above
or below the copper target, and these specific decays
are counted as either UP or DOWN events with our
electronics. The difference in time between the initial
muon passage and positron detection gives the muon
lifetime.
Electronics Setup
The setup of the experiments electronics modules is
shown in FIG 6. This includes an octal discriminator,
two quad logic (coincidence) units, a quad gate/delay
generator, a dual scalar unit and a high voltage power
supply.
FIG. 6. Laboratory setup of the NIM module electronics rack.
To convert the signal to meaningful data, the
output signals of the NIM module configuration are
connected to two time-to-digital converters (TDC),
each recognizing an up or down event and measures
the time interval between the start and stop incidents
then converts this into a digital output, and a SCSI bus
crate controller which transfers this data to a computer
readout. Also, not shown, is a power supply which
drives the current to the copper coil which produces a
magnetic field perpendicular to the beam axis.
The first set of discriminators (labelled DISCR A
in FIG 7) are used for each PMT output signal in order
to only pass through signals that are above the voltage
threshold in order to remove any circuit noise from being
counted as a muon event. The discriminator widths are
adjusted so that veto signals, or anti-counter incidences,
are considerably wider than the PMT signals (whose
widths are shortened by DISCR B discriminators) to
ensure all PMT signals overlap each other with enough
margin within the veto signal. Each logic unit is used to
signal an event once the detection logic is satisfied, in
this case for detection of a muon stopped in the copper
(labelled M in diagram) which starts the time-to-digital
converters (S) and two delay gates (G1 and G2),
detection of an upwards positron decay (U), detection
of a downwards positron decay (D), and the TDC stop
signals for up and down decays (UP and DN). The
gate/delay generators (G1, G2 and L/O) all delay the
output signal from the stopped muon by 20 µs. The G1
and G2 gate signals, in coincidence with a U or D signal
trigger a stop signal for the corresponding up or down
TDC to record as a decay time. The L/O gate signal is
a lockout for the M logic unit, so that once a muon is
stopped and detected, no other incoming particles will
be detected for a 20 µs period [1].
To minimize the likelihood of counting false events,
which can be caused by PMT noise, the discriminator
thresholds must be adjusted as well as the high volt-
age values applied to the PMTs must be determined
empirically by two plateauing methods: single rate
and coincidence. For a fixed discriminator setting (this
was 20mV for this experiment), a PMT should reach
a constant triggering efficiency, the plateau, within a
range of high voltage settings. To determine the plateau
high voltage values for each PMT using the single rate
method, the output signal from the PMT was connected
to the input of the scalar unit to count each event and
the rate was recorded by timing these events for each
voltage setting. These rates were plotted against the
high voltage values in order to show the plateau effect,
as shown in FIG 9. These single rate plateau voltage
settings serve as an approximate value to set each PMT
at for the coincidence method.

4. 4
FIG. 7. Schematic diagram of our circuit logic used to detect stopped muons and their resulting decay time. Signals flow from
left to right across the electrical connections. Each PMT signal inputs into a discriminator, N2 Phil 705, which then passes
on to either a second stage discriminator, N4 Phil 705, or an initial logic unit, N6 Phil 754. The signals then pass through a
trigger, N8 Phil 794, and a final logic unit, N1 Phil 754, before recorded in a two-channel time-to-digital converter, C6 or C7
TDC.
FIG. 8. Wiring diagram for coincidence plateau measure-
ments when holding counters C1 and C3 at a constant voltage
and varying the voltage of C2.
The coincidence plateau method requires two
PMTs to be set at a constant voltage, using the single
rate plateau approximation to determine the voltage,and
a third PMT to be varied over a range of high voltage
values to optimize its operating value. The ratio of coinci-
dences, as shown in Equation 1, will approach a constant,
or plateau, at some high voltage value, which signifies
when the scintillation counter becomes fully efficient at
detecting muons.
R =
C1 ⊗ C2 ⊗ C3
C1 ⊗ C3
(1)
Figure 8 shows the wiring diagram for the coincidence
plateau method, if the voltage for counters C1 and C3
are held constant and the voltage for C2 is varied.

5. 5
FIG. 9. Single rate plateau method graphs. Plotted values of rate of incident particles vs. the high voltage value.

6. 6
FIG. 10. Coincidence plateau method graphs. Plotted values of the ratio of coincidences (Equation 1) vs. the high voltage value.

7. 7
Counter
Coin.
Counter
1
Coin.
Counter 1
Voltage
Coin.
Counter
2
Coin.
Counter 2
Voltage
Operating
High Voltage
C1 C2 -1450 V C3 -1500 V -1500 V
C2 C1 -1500 V C4 -1450 V -1450 V
C3 C1 -1500 V C4 -1450 V -1500 V
C4 C3 -1500 V C5 -1500 V -1450 V
C5 C1 -1500 V C4 -1450 V -1500 V
TABLE I. High voltage operating values used while tak-
ing data for each counter. Voltage values used for each
counter while taking measurements for the coincidence
plateau method.
Figure 10 shows the ratios plotted against high volt-
age in order to show the plateau effect at the optimized
high voltage value for each counter. These values are
shown in Table 1.
To ensure the detector counts the incident muons
with the preferred momenta, the signals from each
counter must coincide in time before being input into
the logic unit which determines whether or not the event
is counted. An oscilloscope was used to look at all input
signals going into each logic unit in the time domain. If
the signals did not coincide, longer cables were used as
a way to delay signals until they all triggered within the
same time range.
Each TDC occupies a slot in the SCSI bus crate con-
troller which determines if an up or down event should
be recorded. Lifetime data for an up or down event from
the TDC is written to a text file through the SCSI bus
crate controller. A program written in C++ was used to
select events in which a decay occurred within the data
and plotted that data in a histogram using the ROOT
analysis framework. Data contents were also stored in a
ROOT file for further processing and data analysis.
A total of 582 hours of data with the magnetic field
on (at a current of 3.84 A) was recorded over five
separate data runs. A total of 115 hours of data with the
magnetic field off was recorded over three separate data
runs. And a total of 47 hours of data with the copper
target removed was recorded over a single data run. In
order to subtract this background data from the signal
data with the magnetic field on, the target removed data
was scaled by a factor of 12.28 to correlate the datasets.
Detection Logic
For this experiment, incident cosmic-ray muons with
momenta 200 MeV/c hit the detector and stop in
the copper target then decay. The decay particle of
interest is the positron, which either decays upwards
or downwards. If the positron decays upwards, it will
be detected by the counters above the target, and if it
decays downwards, it will be detected by the counters
below the target. Consequently, the muon lifetime is
measured as the time it took the muon to pass through
the first counter and decay into a positron detected
in the secondary counter. If parity is conserved, the
number of up decays and down decays should be equal
and the emitted positron decays symmetrically in each
direction.
An event begins when there is a simultaneous
coincidence in counters C1 × C2 × C3 × ¯C4. The output
of this starts the up and down TDCs and two 20 µ
s gates (G1 and G2 in the diagram) which ensure
that only decay positrons are counted. The output
of this also triggers a lockout gate (L/O) that lasts
for 20 µ s which suppresses the M logic unit from
counting any more particles during the lockout time. An
upward positron creates coincidence signals in counters
¯C1 ⊗ C2 ⊗ C3 ⊗ ¯C4, while a downward positron creates
coincidence signals in counters ¯C1 ⊗ C3 ⊗ C4 ⊗ ¯C5.
The outputs of these logic units trigger up or down
(U or D) positron events, respectively. If one of these
events coincide with the G1 or G2gate (within 20 µ s
of when the incident muon was stopped), the TDC is
stopped and the event is recorded. This ensures that
only muon decays are counted. If there is no stop signal
generated, both up and down TDCs overflow and the
data acquisition does not write any data.
ANALYSIS AND RESULTS
Muon Count Rate Estimation
As adapted from Sullivan 1971, the coincidence count
rate is formally determined by:
S
ˆr · σ
Ω
dω
∞
0
dpJ(p, ω) (p) (2)
with integration over telescopic area S, total solid an-
gle Ω, and momentum p, where spectral intensity J is
a function of momentum and solid angle and detection
efficiency . If we assume the flux is isotropic from above
and the efficiency is approximately ideal, or ∼1, the count
rate simplifies to
C = ΓI0 (3)
where Γ is the gathering power of the telescope and
I0 is momentum-integrated flux, both calculated from
computer-assisted numerical methods.
Telescopic Gathering Power
A Monte Carlo simulation produces random muon
tracks which propagate through our computer generated
scintillating telescope. The telescopes gathering power
is then related to the gathering power through the
aperture, G = πA, by the equation:
Γ =
number of tracks detected in instrument
total number of generated tracks
× G (4)
where A is the aperture area. This simulation is entirely
dependant on the specific geometry of our detector

8. 8
array, which is 61×30cm2
with a 24 cm spacing between
the top of C1 and the bottom of C4.
The complete distance between the incident C1-
plane and bottom C4-plane was used due to its influence
on detection triggering and vetoing. To better account
for solid angle, distance from the center of C4 was
measured to the outer corner of C1 as about 70 cm.
With this experimental configuration, the simulation
calculates a gathering power value of Γ = 592.373cm2
sr.
Measurements of muon decays were first made with
the magnetic field turned off, and later with both the
magnet and 4 copper target removed from the telescope.
Counts produced while the magnetic field was off were
subtracted to correct for background noise, while counts
with the magnet are target removed were subtracted to
account for background in muons which stopped outside
of the target material.
Up and Down Decay Distributions
Our first set of data was produced from muon decay
without an applied magnetic field. Decay times for both
up (left) and down (right) decaying muons are detailed
in FIG 11. A total of 15586 muons decayed upward ,
while 10421 muons decayed downward for a grand total
of 16007 decays observed over 115.0 hours. Each distri-
bution was fitted with an exponential function,
ni = ae−ΓTi
+ b (5)
from 1µs to 20µs, where Γ is the muon decay rate and Ti
is the measured decay time and parameters a, b, and Ti
are adjusted by the method of least squares, minimizing
Chi-squared,
χ2
= (ni − mi)2
/mi (6)
where mi is the observed number of events in the ith
time
interval. Because the data was plotted on a logarithmic
scale, the exponential decay appears as a negative linear
function. Counts below 1µs were ignored as these sig-
nals were likely created by captured µ−
particles which
decay in this shorter lifetime. Their capture indicates
why signals in this decay region are almost an order of
magnitude higher than observed decays just 0.2µs higher.
Decays above 12µs were still fit to, but are not displayed,
as the observed counts do not deviate beyond 12µs.
Once fitted, our measurements display mean muon
lifetimes of 2.03 ± 0.06µs and 2.28 ± 0.07µs for the up
and down distributions, respectively. These values were
found as the negative reciprocal of the fitted decay rate.
The observed upward distribution then differs from the
accepted 2.20µs mean lifetime by 0.17µs with 7.73% per-
cent error, while the observed downward distribution dif-
fers by 0.08µs with 3.64% percent error. A black dotted
line in each plot of FIG 11 represents the expected 2.20µs
mean lifetime. While both decay distributions do not
contain this value, the up decay lifetime is definitively
less accurate as its fitted exponential deviates further
from the theoretical function than the fitted downward
decay lifetime.
Error in each data point is purely statistical, as our
dependent measurements are numeric counts of a ran-
dom process. Therefore, each point has an error equal
to
√
N, where N is the count for each 0.25µs interval.
Because our data is displayed on a logarithmic scale,
the errors appear to increase with increasing decay time.
This effect makes the errors difficult to display visually as
higher counts produce greater error in the measurement.
Furthermore, because our data sets are displayed across
three orders of magnitude, the plots in FIG 1 do not have
the resolution to present error in the higher counts. Er-
ror in the decay time, or independent variable, is equal
to the bin width of the histogram. Measurements for de-
cay time were made within 20 ± 5ns, or 0.02 ± 0.005µs
intervals on the TDC, but were plotted within 0.25µs
wide bins. Therefore, any counts made within a 250 ns
interval, for this distribution, were grouped together and
become indistinguishable from one another. Each data
point was then plotted at the center of each bin, with
an error of ±0.125µs on either side. Because this error
is two orders of magnitude greater than the error associ-
ated with measurement, the binning error dominates the
decay time distribution.
The error in our calculated muon lifetime, στ , was
produced by propagating the error of our fitted decay
rate, σΓ, found by the chi-square minimizing fit in our
ROOT software. This error propagation took the form
of:
στ =
1
Γ − σΓ
−
1
Γ + σΓ
(7)
In effort to reduce this error, a background signal
was measured for decays which originate outside of our
copper target. These decays include muons which stop
in either the polystyrene scintillators or the aluminum
housing. Because our measurement only uses electric
signals created by incident particles upon the scintilla-
tors, there is no way to immediately determine where
the decayed particle originated. Therefore, a background
distribution was measured with both the copper target
and the solenoidal magnet removed from our detector.
This measurement was made over 47.4 hours, and our
results are displayed in FIG 12. Because this run was
made over a considerably shorter time interval, the
data set was scaled by a factor of 2.43 to equate to the
previous distributions.
As expected, the background displays the same ex-
ponential decay curve as our previous data set. More
notably, the upward decays appear approximately an
order of magnitude more often than the down decays.
This trend is consistent across decays up to 4µs, but is
most apparent for decays between 0.6µs and 0.8µs.

9. 9
FIG. 11. Up (cyan) and Down (red) decay distributions for detections found with copper target and solenoid magnet removed
from the detector. These decays originate in either the aluminum housing or plastic scintillators and provide constant background
noise to all measurements. A notable 37038 upward decays were observed, while only 160 downward decays were detected in the
experiment’s 47.4 hour run. The above histograms have been scaled by 2.43 to match the time-length of the initial magnet off
data set.
With background decays properly measured, a
magnetic field of 33.55 Gauss was then directed perpen-
dicular to the copper target and incident muon trajec-
tory. This field, created by 371 turns of copper wire with
with a current of 3.84 A, causes the muons to precess
along the magnetic field lines. The angular frequency of
this precession is theoretically given by, ω = geB/2mc,
where e is the electron charge, equivalent to the charge
of a muon, m is the mass of the muon, c is the speed of
light, B is the magnitude of our magnetic field, and g is a
numerical quantity approximately equal to 2 for a Dirac
particle [5]. This precession is observed in the Up minus
Down (U-D), distribution presented in FIG 13.

10. 10
FIG. 12. Up minus Down decay distribution with an applied magnetic field. The fitted exponential and cosine function (dotted
magenta line) indicates precession in the muon decays. Initial data points are ignored due to negative muon captures and a
correction for the stopping time.
Muons which decay within this field undergo a pre-
cession along the B-field lines proportional to the field’s
strength. Data for this precession was taken over 582
hours with background scaled by a factor of 12.28. A fit
of the form:
f = (A + B cos(ωT))e−ΓT
(8)
was applied to this distribution from 0.6µs to 4.0µs,
where A is accidental M (stopped) signals, and B is
the precession amplitude [3]. This fit, illustrated as the
dotted line in FIG 13, displays obvious precession in the
spin of the decayed muons. A decay rate of 0.7 ± 0.1
decays per µs, an accidental signal count of 550 ± 100
signals, a precession amplitude of 170 ± 80 decays,
and an angular frequency of 3.9 ± 0.2 radians per µs
were determined by ROOT’s Chi-Square minimizing
fit properties. Specifically, a Chi-Square value of 7.4
was determined for the 17 data points and 4 fitted
parameters. This value is slightly greater than half
the degrees of freedom, n=13, for this distribution.
Therefore, their ratio is neither excessively greater than
or less than 1, so the values obtained from the fit are
accurate to our actual observed values.
To correct for the actual stopping time of the muon,
T0, and captured µ−
particles, the fit itself began at
0.6µs. The T0 correction accounts for the fact that inci-
dent muons which trigger the M stop signal have a delay
before they actually stop in the Cu target. This delay be-
tween signals is the actual stopping time for the muon.
Failure to correct for this stop time results in a system-
atic error in determining when the precession originates
in muon decay. Regardless, T0 was found by increasing
the delay between the M stop pulse and the U logic unit.
Wires of lengths 16ns, 32ns, 48ns, 64ns, and 80ns were
connected between the output of M and the input of U.
Measurements of TDC units for only observed Upward
decay times were measured for each wire length over a
timescale of 10 minutes. This data is displayed graph-
ically in FIG 14. Because the delay was too short for
the smaller cable lengths, only the 64ns and 80ns cables
produced any measurable Upward decay times. These
two points where then fitted to a linear function (red)
to determine the slope and y intercept for TDC intervals
as a function of cable length. This fit produced a linear
function,
TDC = p0 + p1lcable (9)
where TDC is the measured TDC time interval, p0
is the TDC intercept for zero cable length, and p1
is the proportionality between TDC interval and cable
length defined in nanoseconds. This fit produced a p0
value of −1.25 ± 6.403 TDC intervals and a p1 value of
0.04437±0.08839 TDC intervals per ns cable length. Set-
ting TDC equal to zero, the x-intercept is algebraically
found at 28.17ns. In order to determine T0, this value
must then be added to the input-to-output delay time of
the Phillips 754 Logic Unit responsible for stopped M sig-
nals. According to the manufacter’s specification sheet,
this unit has a maximum delay of 8.5ns inherent to its
operation [5]. The true stopping time is then 36.67ns. Af-
ter T0, the muons have stopped in the copper target and
fully experience the magnetic field. Therefore, only de-
cay counts displayed after this time-length are attributed
to precession, as demonstrated by the fit in FIG 14.

11. 11
FIG. 13. Stopping time, T0, calculation from measured Up decays in varied cable lengths between the stopping, M, signal and the
Upward decay, U, signal. Five lengths of cable were used with 16ns length intervals between them for a maximum of 80ns cable
length. A linear fit (red) was then used to extraoplate the delay time, 28.17ns, between the two logic units. The input-to-output
delay time, 8.5ns, inherent to the Phillips 754 M unit was then added to this value to produce the time which muons stopped in
the Cu target and began precession.
This fit starts, in TDS 20ns interval units, after the
nearest integer of T0
20ns , which is 40ns or 0.4 µs. The data
point at 0.5 µs was also ignored in the fit as this point,
with an order of magnitude higher count than nearby in-
tervals, was likely not created by precessing muons. Pre-
viously mentioned µ−
captures or after-pulsing in the Up
detector could have created this divergent count. Errors
in both measured counts and decay time are again at-
tributed to statistical error of a counted random process,√
N, and bin width, respectively. Bins in this histogram
have a width of 0.2 µs and therefore the decay times have
an error of ±0.1µs.
The Up and Down distributions can then sum to-
gether, U+D, to better display µ−
capture in our de-
tectors, as well as the overall decay rate for both decay
directions. This summed distribution is displayed in both
histograms in FIG 14. The top histograms was produced
by the raw data which our detector observed. This distri-
bution was again fitted to an exponential function, such
as Equation 5, which accounts for accidental detections
in the M signal, and, more importantly, produces a Chi-
Square minimizing value for the overall decay rate, Γ.
The fit, displayed as the magenta dotted line, has a de-
cay rate of 833 counts per µs with an accidental count of
15 counts. This decay rate corresponds to a mean muon
lifetime of 2.00 ± 0.05µs.
Negatively charged muon contamination is defini-
tively noticable in the first and third bins of the U+D
distributions. This contamination becomes more appar-
ent when the background decays are subtracted. The
bottom histogram in FIG 14 displays this correction. The
relative count distance between the counts at 0.1 µs, 0.2
µs, and 0.3 µs decreases as the non-target decays are
subtracted away.
To better observe the precession amongst in-target
decayed muons, our data was then plotted as the differ-
ence in Up and Down decay counts divided by their sum.
Symbolically, this distribution takes the form:
U − D
U + D
(10)
where U is the observed Up decay count, and D is
the observed Down decay count.
Once these counts were recorded, background de-
cays were scaled, again, by a factor of 12.28 to properly
correct our measured distribution. Because these decay-
ing µ+
particles were stopped outside of the target, they
experience no direct magnetic force other than from a
possible background field. Therefore, these non-target
muons display no precession in their spin. Without pre-
cession, this background is unable to affect the frequency
and amplitude of our actual precession signal, but does
produce an offset in the overall asymmetry. Specifically,
this offset is observed in the U+D distribution, which
acts as the denominator for the oscillating ratio seen in
FIG 15. Thus, background decays were subtracted from
the Up and Down distributions before their asymmetry
was plotted.
This overall asymmetric distribution was fitted to

12. 12
FIG. 14. Up plus Down distribution with a magnetic field applied to the stopping region within the Cu target. An exponential
fit (dotted magenta) is applied to this data set with clear background contamination at decay times between 0.1 µs and 0.3 µs.
The top histogram is without background decays subtracted, while the bottom histogram corrects this error in the counts. Their
difference indicates the prevalence of captured negative and non-target decayed muons.

13. 13
FIG. 15. Observed spin precession in relative decay counts for muons stopped in the Cu target under the influence of a magnetic
field. A fitted cosine function (dotted red) displays the observed precession with 11.875 Chi-Square value, indicative of an
accurate fit. Points before 0.6 µs were ignored due to corrected muon stopping time and negative muon contamination
the function:
A + B cos(ωT) (11)
where A is the offset for accidental triggers, B is the pre-
cession amplitude, and T is the decay time of the muon.
This fit, displayed in red, then produced an offset of
0.15 ± 0.02 with an amplitude of 0.04 ± 0.02 when fit-
ted between 0.6µs and 6µs. These parameters are unit-
less as they were found for a ratio of numeric counts.
More importantly, the fit produced a angular frequency
of 3.8 ± 0.2 radians/µs. Error in this value, as well as
for the offset and amplitude were calculated by the Chi-
Square minimizing fit built into ROOT. This particular
fit produced a Chi-Square value of 11.875 for 28 data
points and 3 fitted parameters which produce 25 degrees
of freedom. A measure of the fit’s quality is produced by
the relation,
χ2
n
= 0.474 < 1 (12)
where χ2
is the Chi-Square value and n is the degrees
of freedom. This ratio is only an approximate factor
of two less than 1, and therefore a ”good” fit with
95% cumulative probability. Because this fit has a high
cumulative probability and a Chi-Square to degree-of-
freedom ratio neither much greater than, nor less than,
one, our fit parameters are close to the true values for
offset, amplitude, and angular frequency of our data set.
As a gauge to our frequency measurement, the ex-
pected angular frequency, when calculated through the
magnetic field value, has a value of 1.428 · g radians/µs,
where g is a numeric quantity close to 2 for a Dirac parti-
cle. Since the muon is in fact a Dirac particle, the actual
expected frequency for our precession is 2.856 radians/µs
[5]. This value differs from our observed frequency by
0.944 radians/µs with 33.05% percent error.
Our calculated angular frequency is then converted
to a muon lifetime, τ, by the equation,
2π
ω
= τ (13)
This equation then produced a lifetime of 1.7 ±
0.3µs, which is 0.5µs below the expected value, 0.27µs
below our Up distribution value, and 0.58µs below our
Down distribution value from non-magnetic field decays.
CONCLUSIONS
The high count that appear between 0.6µs and
0.8µs for our Up background distribution indicates a po-
tential error in our measurements. If after-pulsing caused
this increase, the Down background distribution would
display a similar count as the Up decays. But because a

14. 14
higher count appears only for the Upward distribution,
the theshold voltage was likely set too low for the second
discriminator connected to the C3 PMT. While both U
and D logic gates have input from this PMT, only U re-
ceives its C3 input after this second discriminator. Noise
fluctuations from the first stage discriminator may pass
through to the second which then go through and pro-
duce a false signal in the U logic gate.
Our measured mean muon lifetimes for Up and
Down decays, 2.03 ± 0.06µs and 2.28 ± 0.07µs, were just
outside of the accepted 2.20 µs lifetime value with respect
to their error bars. This difference is likely due to an in-
sufficient number of decay events, and can be improved
with a longer experiment run time. Our muon preces-
sion’s angular frequency, 3.8±0.2 radians/µs, was consid-
erably higher than the expected 2.856 radians/µs value.
Improper measurement of our magnetic field may likely
cause this large difference. While current was measured
and maintained at 3.84 A, imperfections in the solenoid
magnet could create a divergent magnetic field, and thus
cause this difference in precession frequency. A Gausome-
ter should then be implemented to test the uniformity of
our applied magnetic field.
Although our measured values for mean muon life-
time and precession frequency differ from the anticipated
results, the presence of the decay precession indicates
parity violation in the cosmic muons. Thus our experi-
ment succeeded in confirming this property of muon de-
cay. Our experiment could be expanded with magnetic
fields in different orientations to observe their effect on
the precessing muons.
REFERENCES
[1] Slater William E and Rene A. Ong, Laboratory
Manual for Physics 180F, UCLA, Version 5.4
(2015)
[2] Wikipedia contributors. Parity (physics).
Wikipedia, The Free Encyclopedia. Wikipedia,
The Free Encyclopedia, 20 Nov. 2015. Web. 10
Dec. 2015.
[3] Wu, Chien-Shiung, et al. Experimental test of par-
ity conservation in beta decay. Phys. Rev. 105,
1413 (1957)
[4] Physics 180F Software, UCLA, Version 2.2 (2012)
[5] Ticho H,180F Experiment: Muon Mean Life and
Magnetic Moment, UCLA High Energy Group
Notes, Memo No. 163 (1973)
[6] Scientific Phillips, Quad 300MHz Majority Logic