Analysis of Particle Decay Using Raw Data Acquired by a Detector
Liam Sadot1 , Binyamin Omer1
1
Raymond and Beverly Sackler, Faculty of Exact Sciences, School of Physics, Tel Aviv University
Particles
Abstract – In this study, we investigated a number of
theoretical phenomena, we used a simulation software
suitable for the study of high energy particle detectors. We
analyzed and characterized the measurements obtained
by injections of the two particles, κ0s and π 0 , in order to
calculate their mass and lifetime.
I. INTRODUCTION
A. Fundamentals of Physics and Nuclear Physics
Elementary particle physics investigates the basic
components of matter and the interactions between them. The
idea that all matter is fundamentally composed of elementary
particles dates back to at least the 6th century BC[1]. An
elementary particle is a subatomic particle whose properties
can be understood without assuming that it has an internal
structure. In the development of quantum physics, at the
early 20th century [2], subatomic particles that were initially
considered elementary were later proved to have an internal
structure. In 1897, the first subatomic particle was discovered
and identified as the Electron.
The study of the structure of matter at an ever-smaller
scale was done experimentally in collision processes at higherthan-ever energies, which is why the physics of elementary
particles is also known as the physics of high-energy. The
results of the experiment obtained, and the theoretical models
developed in the last decades of the twentieth century, were
synthesized in the standard model. According to the standard
model, elementary particles are quarks and leptons, together
with calibration bosons (which mediate electromagnetic, weak
and strong interactions) and the Higgs boson [3]. The graviton,
considered a mediator of gravitational interaction, remains
hypothetical for now and is not included in the standard model.
B. The Standard Model Theory
In elementary particle physics, the standard model
represents the consensus established at the end of the
20th century on the basic components of matter and
the fundamental forces that describe the interrelationships
between them.
The standard model includes a total of 61 particles that
are considered fundamental (without internal structure) and
3 of the 4 types of basic interactions (forces) [4]. The basic
particles are divided into two broad categories according to
a characteristic called spin. The two broad categories are
fermions and bosons. Fermions: Fermions are elementary
particles whose spin is a half or half integer. They create a
completely anti-symmetric quantum state for those particles.
They satisfy the Pauli principle according to which, unlike
bosons, no two particles can be in the same quantum
Leptons
Quarks
Flavour
eµτ
νe νµ ντ
uct
dsb
Charge [q/e]
-1
0
2/3
-1/3
Table 1 : The twelve fundamental particles of physics
according the Standard Model
state simultaneously. Bosons: A boson [3] is a particle
or quasi-particle with an integer value of spin. Bosons,
unlike fermions, obey Bose-Einstein statistics, which allow an
unlimited number of identical particles to be in one quantum
state. Some of the elementary bosons act as force carriers,
which give rise to forces between other particles, one example
is beta decay where a virtual W boson is emitted by a nucleon
and then decays to e± and an anti-neutrino.
The basic fermions [1] are divided into quarks and leptons:
•
•
Leptons: The leptons are charge carriers and are
considered to be fundamental particles. They do not
partake in strong interactions. A list of leptons is given
in Table 1.
Quarks: The quarks [5] are the massive elementary
fermions that strongly interact to form nuclear matter and
certain types of particles called hadrons.
C. Electromagnetic Interactions
A photon is a “packet” of electromagnetic energy [1],
when an electromagnetic interaction occurs in matter, the
particles exchange photons (a basic example is the electron
moving around the atomic nucleus, the electron maintains its
motion thanks to the electromagnetic force and exchanges
photons during its motion). There are several electromagnetic
interactions in the material encountered during the experiment
and they are: 1. The photoelectric effect, in this process
an energetic photon releases a bound electron. 2. Compton
scattering, this process describes the scattering of an electron,
which results from a photon collision with an electron (similar
to the scattering of free particles). 3. Pair production,
which is a process of particle and anti-particle formation
from a neutral boson. 4. Bremsstrahlung, which is the
electromagnetic radiation emitted from a charged particle due
to its acceleration.
D. Relativistic Particles In a Magnetic Field
The movement of a particle of charge q traveling in a
magnetic field with velocity v is influenced by the Lorentz
force [6] as follows:
⃗F = q ⃗v X ⃗B + ⃗E
(1)
c
In this experiment, particles under a magnetic field Bẑ will
be influenced so that a curve in plane XY is received, where
the curvature can be extracted from:
κ=
qB0
pT
(2)
Where κ is the curvature, q is the electric charge of the
particle, B0 is the magnitude and pT is the momentum.
A relativistic particle [6] is a particle which moves at a
relativistic speed; that is, a speed comparable to the speed
of light. In special relativity, the four-momentum is the
generalization of the classical three-dimensional momentum
to a four-dimensional space-time.
p = p0 , p1 , p2 , p3 = ( Ec , p1 , p2 , p3 )
We will use the four momentum norm, which is invariant
under the Lorentz transform:
2
PP = ηµν Pµ Pν = Pν Pν = − Ec2 + |P|2 = −m2 c2
(3)
Through the experiment we use equation (3) for mass
calculations of a parent particle using the masses of its two
daughter particles. We can identify parent particles using the
decay products’ trajectories, momentums and masses.
s
m
2
= m21 + m22 + 2
p2
p2
2
(m21 + 21 )(m22 + 22 ) − 2 P1 P2 cos (θ )
c
c
c
types of pions: π + , π − and π0 . Pions are the lightest mesons
and even the lightest hadrons. Along with other mesons, pions
are the carriers of nuclear power, which is the connecting
force between the nucleons in the atomic nucleus. pions are
generated in the atmosphere by nuclear collisions of incoming
cosmic photon rays.
kaons are a group that carries odd quantum numbers. The
neutral kaons are particles with complex properties that set
them apart from other known particles. There are two types of
neutral kaons: κs0 and κL0 . The short lived kaon (Ks0 ) consists
of two quarks - Ks0 = d s̄ (down and strange).
Particle
π0
Ks0
Dominant Decay Channel
τ[s]
2γ
π +π −
8.52 · 10−17
0.89 · 10−10
(5)
Where the mass distribution of a relativistic particle is
according to the Breit-Wigner formula [7]:
f (m) =
C
2
(m2 −m20 )
+m20 Γ2
(6)
Where m0 is the particle’s mass, and Γ is in energy units and
describes the energy spread through the decaying and equals
to Γ = h̄τ −1 while τ is the particle’s mean lifetime.
E. Hadrons
The hadron is a subatomic particle composed of quarks
[8], as well as their opposite counterparts (anti-particles). The
quarks are held together by gluons that carry the strong nuclear
force. The hadrons are usually classified according to the
number of quarks they include:
a) Barions which include three quarks each.
b) Mesons which are made up of one quark and one antiquark (of a different type). These include the pions, kaons and
other types of mesons.
The pion is a subatomic particle containing quark up and
anti-quark down, their antiparticles, or quark up or down
together with the anti-particles of each. Thus, there are three
134.97 ± 0.00 98.8
497.61 ± 0.01 66.6
F. Particle Lifetime
The lifetime of a relativistic, unstable particle with a resting
mass m0 that travels a certain distance L until its decay is given
by:
τ=
m0 L
p
(7)
The decay rate of an unstable particle is given by the
exponential distribution:
L
m2 = (P1 + P2 )2 = 2P1 P2 (1 − cos (θ ))
BR
Table 2: Theoretical values of lifetime and mass of particles
analyzed in this experiment
(4)
During the experiment we investigated particles that decay
into photons. A photon is a special particle since it is massless, and we can therefore argue that the energy of the photon
is equal to its momentum. We then get the following equation:
m0 [Mev]
N (t) = N (0) e− τ
(8)
II. THE DETECTOR
The particle decay analysis was performed using the
detector simulation software GEANT3 [10]. The simulated
detector consists of four different parts as will be discussed
below. The simulation allows the injection of selected particles
with known momentums into the detector, providing raw data
produced by the different parts of the detector for further
analysis.
A. Magnetic Spectrometer
The magnetic spectrometer is the first layer of the
detector, whose purpose is to detect charged particles and to
provide their trajectories, allowing lifetime and momentum
calculations. The spectrometer consists of 10 drift chambers
under a uniform magnetic field B0 ẑ, and each chamber consists
of two planes of 5 anode and 6 cathode wires strung parallel
to the z-axis. In addition, each chamber is filled with a
mixture of gas and is enclosed by two sheets of mylar. The
wires allow the measurement of the y position needed for
curvature measurements. These provide hit positions and
enable the reconstruction of the particle’s track using a pattern
recognition algorithm.
B. Electromagnetic Calorimeter
The electromagnetic calorimeter is the second layer of
the detector, whose purpose is to detect and measure the
energy and location of electrons, positrons and photons. It
is simulated as an array of crystals, made of scintillating
material, positioned behind the spectrometer. The interaction
of photons, electrons and positrons with matter is mainly
through pair-production and bremsstrahlung. At high energies,
these interactions lead to showers that ionize the medium,
resulting in the release of photons of a constant energy. The
amount of emitted photons is therefore proportional to the
amount of deposited energy, and using a photomultiplier an
electric pulse is obtained. Using a reconstruction made by
the program, the ECAL provides the pulse height associated
into clusters (and cluster locations), which presents the energy
deposited in each crystal.
the measurements for each momentum were taken 5 times to
incorporate statistical errors. However, the statistical error
appeared negligible compared to the software’s outputted
resolution error for κ.
In order to calibrate the magnetic spectrometer, the
measurements above were taken for electrons and muons with
momentums of 10-100 GeV
C. Iron Yoke
2mm The iron yoke is the third layer of the detector. It is
100 [cm] long and is designed to block the passing of hadrons
using hadronic interactions.
D. Muon Detector
The muon detector is the fourth and final layer of the
detector, whose purpose is to measure the position of all
charged particles that managed to pass the iron yoke. It
is composed of two drift chambers, similar to those of
the magnetic spectrometer, and relies on the fact that the
probability of a pion not interacting before reaching it is slim.
III. SYSTEM CALIBRATION
In order to understand the correlation between the
simulation’s outputs and their physical meaning, the system
requires a calibration of the magnetic spectrometer and the
ECAL.
A. Magnetic Spectrometer Calibration
According to theory (2) the relationship between a
charged particle’s trajectory’s curvature κ, and its transverse
momentum PT , is given by:
κ=
qB0
2pT
(9)
Where κ is obtained by the simulator, and the 1/2 factor
is a result of a bug in the simulation. The angle of trajectory
can be extracted from the simulation’s output TANDIP, and the
transverse momentum of a particle with a selected momentum
p can be therefore given by:
pT = p cos(arctan(TANDIP))
(10)
The magnetic spectrometer’s calibration can be achieved by
injecting known charged particles with varying momentums
and measuring their trajectory’s curvature. Using (9) the
following linear relation can be obtained:
κ = a p1T + b
(11)
Where a = 21 qB0 and b is a free parameter.
In practice TANDIP measurements were very small and
therefore negligible, and pT ≈ p, where p’s error is set to
1% of its value by the simulation. Since the simulation uses
mathematical techniques to estimate the particle’s behavior,
Fig. 1. Linear fit for the trajectory’s curvature κ as a function of
the injected momentum pT for electrons and muons. The magnetic
spectrometer’s magnitude B0 can be extracted from the slope for
calibration.
Particle
a ± ∆a
∆a/a
2
χred
Pvalue
Electron
Muon
0.002000 ± 4.7 · 10−5
2.3%
1.7%
0.13
0.064
1.0
1.0
0.002000 ± 3.4 · 10−5
Table 3: linear fit results for the magnetic spectrometer’s
calibration
Both fits yield the following value for the magnitude:
B0 = 1.336 ± 0.027[T ]
B. Electromagnetic Calorimeter Calibration
As mentioned before, the ECAL provides an electric pulse
as a result of electrons, positrons and photons depositing their
energy in the medium. In order to make sense of the generated
pulse height, a calibration is needed. Such calibration
is achieved by injection of electrons and photons with
varying momentums, which participate in electromagnetic
interactions, thus disposing their energy in the calorimeter.
The particle’s energy is assumed to be approximately equal
to the momentum since the photons are massless and the
electrons’ mass (0.51 MeV) is negligible compared to the
selected momentums (∼ 10GeV ).
Since the pulse height is proportionate to the deposited
energy, the following linear relation is used for calibration:
E = a · P.H + b
(12)
Where E is the particle’s energy and equals the selected
momentum, and P.H is the pulse height [V] obtained by the
simulation with an error set to 5% of its value.
Similarly to the spectrometer’s calibration, the
measurements above were taken for electrons and photons
with momentums of 10-100 GeV, 5 times for each momentum.
The statistical error is again negligible compared to the given
resolution error.
Fig. 2. Linear fit for the Energy as a function of the pulse height P.H
[V] for electrons and photons. The constant of proportionality can be
extracted from the slope for calibration.
Particle
a ± ∆a
∆a/a
b ± ∆b
2
∆b/b χred
Pvalue
0.021000 ± 2.2 · 10−5
Electron
0.11% 0.458 ± 0.027 5.9% 0.002 1.0
Photons 0.021000 ± 3.5 · 10−5 0.17% 0.433 ± 0.042 9.6% 0.005 1.0
Table 4: Linear fit results for the electromagnetic
calorimeter’s calibration
Both fits yield the following relation between the energy
and the pulse height:
E = 0.021 · P.H + 0.4455 [GeV]
IV. MASS AND LIFETIME MEASUREMENTS
In this experiment, our goal was to calculate the mass and
lifetime of mesons using our knowledge of their most common
decay channels, and to find their branching ratio (BR).
A. π0 - The Neutral Pion
According to the PDG, the neutral pion’s most common
decay channel [11] is:
π0 → 2γ
In addition, its mean lifetime is known to be short, cτ =
25.3 nm [11], which leads to it decaying approximately at its
injection point (x=-10 cm) relative to the detector’s scale. As
a result, the mean lifetime of the pion couldn’t be measured in
this experiment.
Since photons, the pion’s main decay products, are neutral,
they show no track in the magnetic spectrometer and are only
detectable by the ECAL. Each photon was expected to leave
two clusters in the calorimeter, providing their pulse heights
and coordinates.
We therefore define an interesting event as:
1.
2.
No tracks detected by the spectrometer
Two clusters detected by the calorimeter
Using the ECAL’s calibration, each pulse height can be
translated into the photon’s energy and thus, taking advantage
of the fact that it’s massless, its momentum. The angle
between the photons’ tracks can be calculated using a scalar
multiplication:
cos(φ ) =
⃗r1 ·⃗r2
|⃗r1 ||⃗r2 |
According to the PDG, the short-lived neutral kaon’s most
common decay channels [11] are:
κ0s → π + π − (69%)
(13)
Finally,the pion’s mass can be calculated according to
equation (5). The neutral pion was injected 80 times with
momentum of 9.3 GeV, out of which 69 were deemed
“interesting” providing a BR of 86%, which is smaller than
the theoretical 99% [11]. Given the mass calculations for each
injection, we created a histogram fitted to the Breit-Wigner
distribution (6) such that:
f (m) =
B. κ0s - The Short Lived Neutral Kaon
a
(m2 +b2 )2 +b2 c2
κ0s → π0 π0 (31%)
In this experiment we focused on the most common decay
channel, κ0s → π + π − . Since the two pions are charged, they
leave tracks on the spectrometer, enabling their momentum
measurements. The short kaon travels approximately cτ =
2.68 [cm] before decaying into the two pions, which are
known to have a relatively long mean lifetime [11]. As a result,
this decay channel is expected to result in 0-2 clusters on the
calorimeter.
(14)
Where b is the pion’s invariant mass we are looking for, c
is the distribution width and a is a scaling factor. The mass
error was taken as the bin’s width, and the events’ error as the
square root of the amount of events per bin.
We therefore define an interesting event as:
1.
Two tracks detected by the spectrometer
2.
0-2 clusters detected by the calorimeter
Using the magnetic spectrometer’s calibration, each
trajectory can be translated into a pion’s momentum. The
angle between the two tracks is given by the simulation.
In addition, the distance the kaon travelled before decaying
can be extracted from the coordinates of the two tracks’
intersection given by the simulation. Since the spectrometer’s
magnetic field was originally too small, causing the decay
products’ trajectories to cross, the simulation was altered such
that B0 = 4B0 . Finally, the kaon’s mass can be calculated
according to equation (4), and its lifetime can be calculated
according to equation (7).
The short-lived kaon was injected 100 times with
momentum of 9.3 GeV, out of which 59 were deemed
“interesting” providing a BR of 60%, which is somewhat
smaller than the theoretical 69% [11]. Similarly to the pion
measurements analysis, we created a histogram fitted to the
Breit-Wigner distribution (6).
Fig. 3. The mass histogram of π0 fitted to the Breit-Wigner
distribution. The pion’s invariant mass can be extracted from the
distribution.
Mass
[MeV]
137.8 ± 2.3
Distribution width Γ [GeV]
Nσ
Relative Error
0.0203 ± 0.0072
1.2
2.1%
Table 5: Results for the π0 mass histogram fitted to the
Breit-Wigner distribution
The extracted mass for the neutral pion according to these
results is 137.8 ± 2.3 [MeV], with a relative error of 2.1%
compared to the theoretical mass of 134.9768 ± 0.0005 [MeV]
[11]. This result deviates by 1.2 Standard deviations from
theory, which implies a good fit with the theoretical model.
Fig. 4. The mass histogram of κ0s fitted to the Breit-Wigner
distribution. The kaon’s invariant mass can be extracted from the
distribution.
Mass
[MeV]
229.2 ± 15.8
Distribution width Γ [GeV]
Nσ
Relative Error
0.119 ± 0.048
17
54%
κ0s
Table 6: Results for the mass histogram fitted to the
Breit-Wigner distribution
The extracted mass for the short-lived neutral kaon
according to these results is 229.2 ± 15.8 [MeV], with a
relative error of 54% compared to the theoretical mass
of 497.611 ± 0.013 [MeV] [11]. This result deviates
by 17 Standard deviations from theory, which implies an
unsatisfying fit with the theoretical model.
In addition, given the lifetime calculations for each
injection, we created a histogram fitted to the exponential
distribution as depicted in (8). The errors taken into account
for this fit are as described for the mass distribution.
and acquire information about its movement between the
different detector layers. We analyzed this data, and using the
appropriate distributions we calculated the mass of the neutral
pion, and the mass and lifetime of the short-lived kaon. The
results obtained for the pion’s mass and the kaon’s lifetime
calculations appeared to fit the theoretical values, deviating
1.2 and 0.12 standard deviations respectively. However,
the kaon’s mass calculation provided an unsatisfactory fit to
theory, deviating 17 standard deviations from the theoretical
value. Additional accuracy could be obtained by increasing
the amount of measurements taken, injecting each particle
more times. This could be performed by automatizing the
data collection, as opposed to manually extracting it from the
simulation.
In conclusion, it can be said that the parts of the
measurements resulted in a satisfactory fit with theory,
whereas others require further data collection and analysis.
REFERENCES
Fig. 5. The lifetime histogram of κ0s fitted to an exponential
distribution. The kaon’s mean life can be extracted from the
distribution.
Lifetime
[s]
0.929 · 10−10 ± 0.28 · 10−10
Nσ
Relative Error
0.12
3.7%
Table 7: Results for the κ0s lifetime histogram fitted to an
exponential distribution
The extracted lifetime for the short-lived neutral kaon
according to these results is 0.929 · 10−10 ± 0.28 · 10−10 [s],
with a relative error of 3.7% compared to the theoretical
mean life of 0.89564 · 10−10 ± 0.00033 · 10−10 [s] [11]. This
result deviates by 0.12 Standard deviations from theory, which
implies a relatively good fit with the theoretical model.
V. CONCLUSIONS
In this article we have discussed various phenomena of
physical particles. Under the standard model theory, we
investigated the decay and behavior of physical particles using
a computer software that simulates the atlas system. The
simulator allowed us to inject a particle into the detector
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