The effects of beam-beam collision
dynamics on the luminosity spectrum
Final Project Report
3C00 Full Unit Project
Nicola Fancett
Project Supervisors:
Prof. D Miller & Dr. S Boogert
Acknowledgement
I would like to thank Dr. S Boogert and Prof. D Miller for their patience, time and
support throughout this project.
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Abstract
A linear collider is being planned which will accelerate bunches of electrons and
positrons to centre of mass energies between 200 GeV and 800GeV. Accurate and
precise measurements will be made of the masses of heaviest particles observed, the
top quark and W boson. These measurements will decrease the range of masses that
the Higgs boson can have. By comparing a measurement of the Higgs from the linear
collider and the predicted masses, the standard model can be tested.
This project studies the effects of beam-beam collision dynamics upon the luminosity
spectrum which is a distribution of the total centre of mass energies at collision.
GUINEA-PIG, a beam-beam collision simulation program is used to provide
simulation data for collisions between electrons-positrons. It has been shown that the
energy of a particle at the time of collision decreases the longer the time elapsed
before the collision occurs.
The effects of adding a Gaussian beam-spread to the beam-beam simulation causes
the centre of mass energy to be shifted a significant amount of 0.05%. In addition to
the beam-spread if dispersion is also added to the x, y and z components the
significant shift in the centre of mass energy are 0.05%, 0.08% and 0.09%
respectively.
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Contents
1. Introduction ………………………………………………………….
1.1 Why collide electrons and positrons? …………………………….
1.2 Why build a linear collider? ………………………………………
1.3 Project Outline/ Summary………………………………………....
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2. Effects within the bunches…………………………………………..
2.1 Sources of energy spread ………………………………………….
2.2 Disruption ………………………………………………………....
2.3 Dispersion …………………………………………………………
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3. GUINEA-PIG simulation program ………………………………...
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4. Skills and Tools ……………………………………………………...
4.1 Skills ……………………………………………………………...
4.2 Tools ……………………………………………………………...
4.3 Web Page – www.hep.ucl.ac.uk/~nmf …………………………..
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5. Initial Results………………………………………………………..
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6. Time dependence ……………………………………………………
7.1 Comparing the first and second halves of events …………………
7.2 Time dependence in more detail …………………………………
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7. Study of the time steps ……………………………………………...
7.1 Verification that the time steps are in chronological order ..……..
8.2 Why study the time steps …………………………………………
8.3 Changing the number of grid cells …….... ………………………
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8. Further study of the time dependence ……………………………..
8.1 The luminosity spectrum …………………………………………
8.2 Difference in beam energies ..…………………………………….
8.3 Angular distribution ………………………………………………
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9. Effects of increasing the number of grid cells ……………………..
9.1 Luminosity spectrum ……………………………………………..
9.2 Angular distribution ………………………………………………
9.3 Which simulation data to use ……………………………………..
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10. Introducing uncorrelated events to the analysis …………………
10.1 What is meant by an “uncorrelated event”? …………………….
10.2 The luminosity spectrum with uncorrelated events .…………….
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11. Introducing beam-spread into the GUINEA-PIG simulation ….
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12. Dispersion ………………………….. ………………………………
12.1 Result of introducing dispersion in x, y and z …………………..
12.2 Time-dependence …………………………………………………
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13. Conclusion …………………………………………………………
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References ………………………………………………………………….
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Appendix A ………………………………………………………………...
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1. Introduction
Within High Energy Physics groups worldwide, plans to design and build an electron
– positron linear collider are being developed. The linear collider will be used to
accelerate the electron – positron to higher energies than its predecessors. The
electron and positron bunches will be collided at centre of mass energies between
200Gev and 500GeV.
There are two main design proposals for the future linear collider. The first is called
TESLA which is a European collaboration. The second proposal is NLC/GLC from
the USA and Japan (both designs are very similar). Both the American and Japanese
propose to use the same copper technology to build the accelerating structures.
TESLA will use superconducting niobium accelerating structures. Figures 1.1 and 1.2
show the accelerating structures which are used by TESLA and NLC/GLC
respectively.
Figure 1.11: Superconducting niobium
accelerating structure which will be used
in TESLA.
Figure 1.22: Copper accelerating
structure for the NLC (cut open for
viewing).
Both collaborations, apart from the technology that will be used to accelerate the
particles, are otherwise fairly similar. Some parameters of both proposed linear
colliders are shown in Table 1.1 together with schematic layouts for each of the
proposed linear colliders in figures 1.3 and 1.4.
Parameter
Total Length (km)
Linear Accelerator Length (km)
Collision Energy (GeV)
Operating Temperature
Linear Collider Repetition Rate (Hz)
TESLA
33
2 x 15
500
2K
5
Table 1.14: Basic parameters for TESLA and NLC/GLC.
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NLC/GLC
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2 x 9.5
500
Room Temp.
150
Figure 1.33: Schematic layout of TESLA
Figure 1.44: Schematic layout of NLC
1.1 Why collide electrons and positrons?
In a linear collider electrons will collide with their anti-particle the positron. Both
these particles are ‘point like’. When they collide they can annihilate forming a Z or a
photon which decays, possibly to W bosons or top quarks. Due to ‘point like’ nature
of the electron and positron the initial conditions are known. When the electron and
positron have collided there will not be any remaining fragments of the initial
colliding particles. This allows the detectors within the future linear collider to detect
clear signals and hence precise measurements can be made. Also the beam energy is
known and so the mass of the final state can be determined.
1.2 Why build a linear collider?
A linear collider will allow particles to be accelerated without energy loss due to
synchrotron radiation. In circular, high energy electron-positron colliders the beam
particles emit synchrotron radiation proportional to γ4/r, where γ = E/m and r is the
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accelerator radius. If a linear collider is used instead of a circular collider then in
theory there will be no synchrotron radiation losses. However, there is a disadvantage
when using a linear collider, as the accelerated particles only pass the accelerating
structures the once and it is hard to achieve the high energies required.
A linear collider which can reach energies of 500GeV and higher would provide the
means to create the heaviest particles. The aim is to determine the mass of heavy
particles such as the top quark and the W boson to high degrees of accuracy and
precision. The higher the degree of accuracy these masses are measured to allows the
possible masses of the Higgs boson to be restricted to a smaller range. By reducing
the range of the energies that the Higgs boson can possess and comparing this with a
direct measurement of the Higgs mass the standard model can be tested.
Furthermore the top is the heaviest particle yet observed and mass measurements are
of particular interest. Figure 1.1 also shows the top – anti-top threshold at
approximately 350GeV. The shape of figure 1.6 can be explained by the Feynman
diagram in figure 1.5. If the centre of mass energy is greater than twice the top mass
then they can be produced. Otherwise it is not possible to produce pairs of top – antitop. The threshold shape is used to find the top mass but if the beam energies are not
well known then this will introduce an error. So the average beam energy and its
distribution must be well known. The distribution of the centre of mass energies is the
luminosity spectrum.
γ
Figure 1.5: Feynman diagram showing the annihilation of an electron and positron to
produce a particle and anti-particle pair.
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Figure 1.65: Graph demonstrating the effects of energy loss.
1.3 Project Outline/Summary
During this project the effects of beam-beam collision dynamics on the luminosity
spectrum have been studied. The effects were studied using a computer program that
simulates the electron – positron collisions and interactions that would occur in a
linear collider. By introducing different effects to the electron and positron bunches
that are collided in the simulation the effects upon the luminosity spectrum can be
investigated. The effects upon the angular distributions of the particles can also be
studied.
The time dependence of the energies of the particles at the time of collision was
studied. If there is a time-dependent relationship this will also be seen in the
luminosity spectrum. The energy of the electron and positron at the time of collision
is important. The total centre of mass energy at the time of collision affects the
possible products of the electron-positron annihilation. The aim was to determine
whether there is a significant difference in the energies of the particles that collide at
the beginning or end of the simulation.
The reason for studying these areas is that there are many beam effects that can
develop within an accelerator. It is important that the beam effects that could occur in
a linear collider are fully understood. With this knowledge outputs from accelerators
will be analysed correctly while the beam-beam effects are occurring.
2. Effects within the bunches
There are a number of effects which can affect the particles within the bunches before
each electron – positron collision. The bunches of particles can be effected by
disruption, energy spread and dispersion. These factors are important, they effect the
behaviour of the bunches during the simulation.
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2.1 Sources of energy spread
As the two bunches of electrons and positrons are collided there are three processes
by which the energy can be spread from the nominal value. These processes are initial
state radiation (ISR), beamstrahlung, and beam spread. Figure 5.1 shows the effect
upon the luminosity spectrum of adding ISR, beamstrahlung and beam spread to an
electron - positron beam-beam collision simulation.
Beamstrahlung
Initial State Radiation
Beamstrahlung
Beam
spread
s/2Eb 
Figure
2.16:
Graph showing sources of energy spread within the bunches.
ISR is inevitable. It occurs when the electrons or positrons spontaneously emit
photons. However, this effect is calculable to a high level of precision and accuracy.
Beamstrahlung is synchrotron radiation which is emitted by individual particles. It is
caused by the strong electromagnetic fields created by the opposite charge of the
opposing bunch of particles.
Beam spread will introduced by the accelerator itself in the linear collider. All the
particles will not see exactly the same accelerating fields and so the momentum will
be spread over the bunch. This will introduce a small effect in spreading the beam
energy. In the GUINEA-PIG beam-beam collision simulation beam-spread can be
added.
2.2 Disruption
Disruption is the effect that occurs as a particle oscillates violently whilst it travels
through the opposing bunch of particles. This motion is due to the strong
electromagnetic fields that are generated due to the opposite charge of the opposing
bunch.
Figure 2.27: Diagram to show the motion of a particle through the opposing bunch
due to disruption.
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Disruption is the root cause of beamstrahlung. The particles are accelerated in the
strong electromagnetic fields generated by the opposing bunch. Radiation is emitted
from the individual particles as the particles are accelerated.
2.3 Dispersion
Dispersion is when the momentum of the bunches is not constant throughout the
bunch. Dispersion could occur in a future linear collider due to the individual particles
seeing different accelerating forces. This creates a dispersion effect with the
momentum being spread over the bunch of particles.
p-
p+
Figure 2.38: Diagram showing the principle of dispersion in relation to the bunches of
particles.
Dispersion can be introduced into the GUINEA-PIG beam-beam simulation for
investigation.
3. GUINEA-PIG Simulation Program
GUINEA-PIG is a beam-beam simulation program. It simulates the collisions
between two simple Gaussian bunches of particles. A bunch of electrons and another
of positrons is modelled in this case (there are other options to simulate the collisions
of different particles). Each of these bunches contains approximately 1010 particles
and has the initial dimensions of 554nm x 5nm x 300µm in the x, y and z components
respectively.
e+
e-
Figure 3.1: Diagram illustrating the orientation of the two bunches of particles before
collision and their relative momenta.
The GUINEA-PIG simulation uses methods that are mainly used in plasma physics
for simulations of many body problems. The particles within the bunches are
simulated by a reduced number of macro-particles, typically a few ten thousands. For
a given longitudinal (z) position the electromagnetic fields only depend upon particles
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within this plane. In calculating the field at a given position the bunch can be divided
into planes or slices. This allows the bunches of particles to be divided into the slices
shown below in figure 3.1. The dimensions of the grid are determined by nine input
variables: beam size (σ), the number of cells (n) and total grid size inside which the
simulation occurs (c), each of which are defined in the x, y and z directions.
cy
-
e
cx
e+
cz
Figure 3.29: Diagram showing the grid inside which the GUINEA-PIG simulation
occurs, where cx, y, z are the dimensions of the grid.
The particles are modelled such that there are no effects due to its own beam.
Interactions occur when two macro-particles are at the same z position. The forces on
each of the particles are determined by using the Lorentz Law (3.1). Finally the
particles are moved accordingly to interact with the next slice of the beam.
FL = q (E + v x B)
(3.1)
The output of GUINEA-PIG which has been used during this project is a file that
contains the event number, the nominal energy of the beams (GeV), the energies of
the particles at the time of collision (GeV), the x, y positions of the collision (nm), the
z position of the collision (μm) and the angles with respect to the x and y axes for
both beams (radians). This data is provided for all of the collision events that occur
during the GUINEA-PIG simulation. Later on in the project GUINEA-PIG Dr. S
Boogert was able to modify the GUINEA-PIG output to give a time variable for each
collision.
4. Skills & Tools
Throughout the duration of this project a number of new skills had to be learnt and
new tools had to be used. Figure 4.1 below summarises the skills learn and the tools
used.
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UNIX
Operating
system
Output
Histograms,
Data
Guinea-Pig
Beam-beam
collision simulation
program
ROOT
Statistical Analysis
Tool
EMACS
Text
editing
program
C / C++
Programming
Language
Figure 4.1: Flow chart showing the different skills that were developed and the tools
that were used during this project.
4.1 Skills
Initially it was important to learn how to use the UNIX operating system. The UNIX
operating system was used to run programs which analysed the data produced by the
GUINEA-PIG simulation of the beam-beam collisions. All the data produced for this
project was also stored by the UNIX operating system which meant that commands to
access and edit the files were learnt.
It was necessary to take some time to learn some of the C and C++ programming
languages. By using the C/C++ language code was developed and written to analyse
output files from the GUINEA-PIG program.
EMACS is a text editor, this program was used extensively during the project and had
to be clearly understood before it was used. EMACS is the program used to edit the
C/C++ codes of the program used to analyse the GUINEA-PIG output.
4.2 Tools
ROOT is a statistical analysis tool. Knowledge of the properties of ROOT had to be
established before it could be used to analyse the data from the GUINEA-PIG
simulations. ROOT can be used to analyse data in many ways. The main functions
used during this project were the histogram plotting functions. ROOT was used to plot
a variety of one, two and three dimensional plots. In order to present the histograms
produced by ROOT effectively, a macro was used to add titles, labels, legends and
colour. The macro is written in C/C++ within EMACS. ROOT executes the macro to
produce the final histograms.
4.3 Web Page- www.hep.ucl.ac.uk/~nmf
In order to produce a web page to summarise the work carried out for project some of
the html language had to be learnt. The basic tags to change headings, fonts, colours,
add images, links and bullet points were all understood. A pop-up window was also
added to the page.
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5. Initial Results
To begin with simple plots were made for a GUINEA-PIG beam- beam collision
output file that was received. This was to ensure that the basic aspects of the data
could be verified. The confirmation of the properties of the data was carried out by
using an analysis program to plot histograms. A histogram displaying the luminosity
spectrum is shown in figure 5.1. The histogram shows, as expected, some of the
particles have lost energy by emitting beamstrahlung radiation. Neither ISR or beam
spread are included in this simulation.
Figure 5.1: Graph showing the luminosity spectrum.
The next step was to check that both of the bunches were consistent with one another.
Both of the beams should display the same behaviour. In order to check the
consistency of the bunches, the difference in the energies of the particles at the time of
collision was plotted, see figure 5.2.
Figure 5.2: Graph showing the difference in energies of particles at the time of
collision.
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It was expected that the difference in the energy would be symmetric about zero if
both of the beams are behaving the same. Figure 5.2 shows a tendency for the energy
of the positron to be slightly greater than that of the electron. Since this is only a
simulation the bias must be created by a small bug within the GUINEA-PIG program.
The angular distribution of the particles at the time of collision was also studied for
each of the beams. Figure 5.3 below shows the horizontal and vertical angular
distributions for beams one and two.
Figure 5.3: Graph showing the horizontal (x) and the vertical (y) angular distributions
at the time of collision for beams one and two.
Both of the beams show very similar angular distributions and hence both of the
beams are consistent with one another here. The horizontal angular distribution will
always be larger than that in the vertical. This agrees with previous studies10.
Horizontal
Vertical
Focus
Wide
Narrow
Angles (Phase space)
Narrow
Wide
Angles (collision)
Wide
Narrow
Table 5.1: Summary of the distribution of angles that are seen after collision and the
angles that would be seen due to phase space alone.
6. Time dependence
After checking the basic properties of the two beams of particles, the next step was to
study the time dependence of the collisions. It is thought that the particles that collide
early on in the simulation will have a higher energy than those colliding towards the
end of the simulation. This is because particles colliding early have little time to emit
beamstrahlung. The longer the time elapsed before the collision of a particle the
greater the probability of beamstrahlung having been emitted and hence the particles
energy will be lower.
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6.1 Comparing first and second halves of events
To begin the investigation into the time dependence the collisions, the events were
divided up into two groups. The collisions were divided according to their event
number, i.e. the first and second halves of the particles to collide. In order to give each
collision a timing (which is not included in the GUINEA-PIG output file) it was
assumed that the events were recorded in chronological order. This meant that the
event number from the GUINEA-PIG output file was used as a guide to the timing of
the collisions.
As expected the first half of the electron - positron collisions has a greater mean
energy than those occurring in the latter half of the simulation. A test was carried out
to see whether there is a significant difference in the two energies.
Eb = 0.9780
Ee = 0.9693
Where Eb = mean energy of first half of events
Ee = mean energy of second half of events
The mean energies here are given as a fraction of the nominal beam energy
(250GeV).
To test the significance a statistical difference in means test was used. This tested a
null and an alternative hypothesis. The null hypothesis declared there was no
difference in the mean. While the alternative hypothesis is that there was a significant
difference in the mean energies of the two samples. Equation (7.1) was used to
determine the coefficient t which would be compared to statistical tables to determine
its significance.
tcalc =
Ē1 – Ē2 ___
(σ12/n1+ σ22/n2) ½
(7.1)
For the data that were being tested, tcalc = 44.60. (For data used to calculate t, see
Appendix A.1). This was compared to a value for a two sided rejection region of 99%
for which tsig = 2.576. Comparing the calculated tcalc and the significant value tsig, the
null hypothesis is rejected since tcalc > tsig.
The conclusion that can be drawn from this test is that there is a significant difference
in the mean energy of the particles at the beginning and at the end of the simulation.
6.2 Studying time dependence in more detail
The next step was to divide the collision events into smaller groups. It was decided to
split the collision events into ten equally weighted groups. This revealed more
information about the relationship between the timing of the collision and the energy
at which it occurs. It was expected that the mean energies of the groups would
decrease steadily with time due to the increasing probability of beamstrahlung
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radiation having being emitted by the individual particles. In order to display this, a
two dimensional plot was used.
Event No.
Fraction of nominal
energy
Figure 6.1: Graph showing the energy distributions varying with time.
The graph in figure 6.1 confirms the earlier result that the energy decreases as time
increases. The number of particles near to the nominal value decreases steadily over
time. This is due to the increasing probability of beamstrahlung having been emitted
from a particle as time increases.
7. Study of the time steps
It is important that the collision events are in chronological order. The grid that
GUINEA-PIG uses to simulate the beam-beam collisions are models the simulation
by using time steps.
7.1 Verification that time steps are in chronological order.
In order to verify that the event numbers that GUINEA-PIG was using were
chronological as assumed, a plot of the displacement of the particles in the x, y, and
directions and the event number was produced, see figure 8.1. This revealed as
anticipated a periodic movement through the grid cells for all the directions x, y, and
z. .
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Figure 7.1: Graph showing the event number and the displacements in the x, y, and z
components for a single time step.
Figure 7.1 shows how GUINEA-PIG moves through the grid cells to simulate the
beam-beam collisions. The program moves through the z component of the
displacement first as the bunch of particles’ z component is the largest. GUINEA-PIG
then works through the x component and lastly the y component as it has the smallest
dimensions.
Figure 7.1 provides a lot of information about the simulation. The number of grid
cells within the simulation was calculated to be approximately 240 per time step. By
counting up the number of periodic movements over all of the event numbers it was
found there were approximately 107 time steps. When counting the number of time
steps difficulties were encountered in the first and last few thousand events. This is
because there are relatively few events occurring. Using these values the total number
of grid cells can be calculated to approximately 26,000. This is consistent with the
expected number of grid cells being a few ten thousands.
What has been recognised from figure 7.1 is that there are only two main grid cells in
the y component, see figure 7.2. It was thought there would need to be more than two
cells for GUINEA-PIG to be able to simulate accurately the behaviour of the particles
in the y direction.
Figure 7.2: Diagram showing the grid cells in the y direction. (Enlarged from figure
7.1).
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Another problem arose as a result of figure 7.2. It was noted that towards the end of
the simulations, the time steps appeared to change their orientation. It appeared that
the simulation was working backwards through the grid cells. This was found to be an
optical artefact and so concerns were dropped.
7.2 Why investigate the time steps?
When starting to look at the structure of the time steps used by GUINEA-PIG the aim
was to determine the event number at the start and end of each time step. Using this
information it would be possible to group the events that occurred at the same point in
time to make new histograms. By increasing the information on the time steps it is
hoped that the relationship between the energy of a collision and its timing would
become more clear.
However, once figure 7.1 had been plotted there was a bigger question that needed to
be addressed. This was in relation to the number of grid cells modelling the collisions
in the y direction (vertical).
7.3 Changing the number of grid cells
As was seen in figures 7.1 and 7.2 the vertical direction of the displacement is only
modelled by two main cells. The parameters that determine the dimensions of the
cells need to be changed to increase the number of cells within the grid in the y
direction. The parameter that determines the number of cells in the y direction was
increased by a factor of four. (For the full details of the parameters for simulations see
Appendix A.2.) All the other parameters that determine the beam size and the size of
the grid remained the same.
The new GUINEA-PIG simulation output was used to plot figure 7.3 showing the
displacement varying with event number
Figure 7.3: Graph showing the event number and the displacements in the x, y, and z
components for a single time step with the number of macro-particles in the vertical
increases by a factor of four.
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From the graph in figure 7.3 it is possible to see that the number of cells has been
successfully increased to a similar number to those in the x and z directions.
Figure 7.4: Diagram showing the macro-particles used to model the collisions in the
y component of the new data. (Enlarged from figure 7.3).
8. Further study of the time dependence
For this part of the project the GUINEA-PIG simulation output file had a new
parameter added to the file. The new parameter is a time variable that shows the time
step during which a collision occurred. This is a stage further than in section 7 where
the collisions are grouped by event number.
8.1 The luminosity spectrum
Figure 8.1: Graph showing the centre of mass energy varying with time for the initial
simulation where ny = 250 and the new simulation where ny = 1000.
The graph displayed in figure 8.1 shows how the centre of mass energy (√ S) varies
with time. It was predicted that there would be a decrease in the centre of mass energy
as time increased. This behaviour of the particles is reflected in figure 8.1. There is an
initial time where the centre of mass energy does not differ from the nominal value of
500GeV after which the centre of mass energy decreases slowly with time. In the last
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ten time steps are small number peaks in the centre of mass energy which are at the
nominal energy. This is possible as there is always a probability that the particles will
not have emitted energy as beamstrahlung by the final stages of the simulation.
8.2 Difference in beam energies
The difference in the two beams of particles energy is important. By plotting the
difference in their energies it can be checked that both the beams are behaving
similarly.
Figure 8.2: Graph showing the difference in the two beams energies varying with
time.
The graph in figure 8.2 shows how the difference of the two beams energy varies with
time. Initially there are very few particles that have lower energy than the nominal
beam energy. This is evident in figure 8.2 where at the early stages of the simulation
there is virtually no difference in the energy of the particles. The graph shows there is
some small oscillations in the difference. This can be interpreted as both of the beams
behaving in the same way, no beam has a dominant higher energy throughout the
simulation. Towards the latter stages of the simulation it is quite probable that some
of the particles will have emitted beamstrahlung more than once and so it is also more
probable that there will be larger differences between the energies of the particles.
This can be supported by seeing that in the latter stages figure 8.2 shows there are
differences of more than 20GeV.
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8.3 Angular Distribution
Figure 8.3: Graph showing the horizontal and vertical angular distributions varying
with time for beam one.
Figure 8.4: Graph showing the horizontal and vertical angular distributions varying
with time for beam two.
Figures 8.3 and 8.4 show that there is an oscillating tendency for both the horizontal
and the vertical angular distributions through out the simulation. In the very early
stages of the simulation (i.e. time steps < 20) and in the final stages of the simulation
(i.e. time steps > 80) the amplitudes of the oscillations are much greater because the
number of particles from the opposing bunch a particle will see are much fewer than
during the middle of the simulation. During the middle of the simulation the particles
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sees a greater number of particles from the opposing bunch. This causes the rapidly
oscillations that can be seen in the middle of the graphs in figures 13.3 and
9. Effects of increasing the number of grid cells
In this section the two sets of output data from the GUINEA-PIG simulation where
ny = 250 (original data) and when ny = 1000 (new data) are compared.
By increasing the number of cells in the vertical direction there may be effects on the
output data produced by GUINEA-PIG simulation. The effects will be due to
increased modelling, there will be a larger number of collision events simulated by
GUINEA-PIG.
9.1 Luminosity spectrum
The luminosity is changed by increasing the number of cells in the y direction. Figure
9.1 compares the luminosity spectra for data where ny = 250 and ny = 1000.
Figure 9.1: The normalised luminosity spectra for the data where ny = 250 and
ny = 1000.
From studying the luminosity spectra in figure 9.1 it can be seen that there is an effect
due to increasing the number of grid cells used by the simulation. There are fewer
events in the peak at 500GeV when ny = 1000 in the new data. This means there are a
greater number of particles losing energy due to beamstrahlung than when ny = 250.
The mean energies of the two samples were determined. A t test was used to
determine whether the difference in the means was significant.
As described in section 7.1 a t test was used by applying equation (7.1).
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The t value calculated for the difference in the mean total centre of mass energy for
the new and old data was found to be tcalc = 1.38. The value at which the difference
becomes significant is: tcrit = 2.576. The null hypothesis, there is no change in the
mean of the two set of data is therefore accepted.
The conclusion that can be drawn from this is that by increasing the number of cells
within the grid of the simulation, the mean energy of the particles at the time of
collision is not affected.
More importantly is the concern for what is happening near to the 500GeV peak of
the luminosity spectrum. To look at this area in more detail the fractions of the total
number of particles that have lost no energy, <0.01%, <0.1%, <1% and finally <5% of
their nominal energy were determined.
% Energy Loss
0
0.01
0.10
1.00
5.00
ny = 250
0.328624
0.396539
0.495235
0.716741
0.930031
ny = 1000
0.296275
0.360694
0.456214
0.679406
0.914427
Table 9.1: Fractions of particles which have lost energy.
The data shown in table 9.1 shows how the fractions of particles above certain
energies have changed by introducing a greater degree of modelling. The effect of
adding the extra macro-particles has been to reduce the number of particles at higher
energies. There is a significant difference in the fractions of energy loss from 0% until
<1% energy loss. This is because the increased modelling will have raise the amount
of disruption the particles are experiencing. Increasing the disruption increase the
amount of beamstrahlung radiation being emitted by the particles.
9.2 Angular Distribution
The effect upon the horizontal angular distribution on increasing the number of cells
in the y component can be seen in figure 9.3.
24
Figure 9.2: Graph comparing the horizontal (x) angular distribution of beam one for
the old and new data where ny = 250 and ny = 1000 respectively.
Figure 9.2 shows that the horizontal (x) angular distribution of the particles is not
effected by increasing the number of cells in the vertical direction. The effect upon
the vertical (y) angular distribution can be seen in figure 9.3.
Figure 9.3: Graph comparing the vertical angular distribution of beam one for the old
and new data where ny = 250 and ny = 1000 respectively.
The number of collisions which occur at zero radians in figure 9.4 has reduced by
(9.1 ± 0.1) % as a result of increasing the number of macro-particles in the vertical
direction. This is a significant effect upon the angular distribution, however the
distribution is still as anticipated symmetric about zero radians.
9.3 Which simulation data to use
25
From this point the simulation with the increased number of cells within the grid has
been used. This is because there are no significant unexplainable phenomena resulting
from increasing the number of cells within the simulation grid. By increasing the
number of cells the modelling is increasing and so can provide more information
about the effects that can be added to the beam.
10. Introducing uncorrelated luminosity events
10.1 What is an ‘uncorrelated luminosity event’
GUINEA-PIG uses an event number to order the beam-beam collisions. Each event
number has an energy for each of the particles at the time of collision. If events are
correlated the luminosity spectrum is created using both of the energies from the same
event. An uncorrelated event takes the energy of particle one and then randomly
selects, from all events, an event and uses this value to determine the total centre of
mass energy. bunch of particles. This process is carried out for all event numbers.
The reason for introducing this effect into the analysis is that the particles in the
accelerator collide with a particle at random. The particles in the accelerator may not
all see the same accelerating forces and so behave slightly different.
10.2 Uncorrelated luminosity spectrum
Figure 10.1: Graph showing the luminosity spectrum comparing uncorrelated and
correlated events.
The luminosity spectrum in figure 10.1 shows the comparison between uncorrelated
and correlated events. There is not a significant difference in the luminosity spectra. It
appears that the correlations do not affect the luminosity.
26
The difference in the means of the centre of mass energy at the time of collision was
tested to check its significance. The difference in the mean was found to be
insignificant. (See Appendix A.4 for data)
The fractions of energy loss were also compared for uncorrelated and correlated and
can be seen in table 10.1.
% Energy Loss
0
0.01
0.10
1.00
5.00
Correlated
0.296634
0.361277
0.454843
0.679954
0.913025
Uncorrelated
0.286254
0.351302
0.444843
0.674927
0.912780
Table 10.1: Fractions of particles which have lost energy.
From table 10.1 it can be shown that there is no significant change in the fractions of
particles losing energies of 0%, <0.01%, <0.1%, <1%, and <5%.
From this we can conclude that the uncorrelated events do not effect the overall
simulation.
11. Introducing beam-spread to the simulation
For this section of the project the GUINEA-PIG simulation has had a 0.1% Gaussian
beam-spread introduced. Beam-spread is an effect which is introduced in reality by
the accelerator. The individual particles see slightly different accelerating forces and
this results in the energy of the beam being spread. In a linear collider any beamspread present will most likely not be of a Gaussian distribution.
27
Figure 11.1: Graph comparing the luminosity spectrum of a normal simulation and a
simulation with 0.1% Gaussian beam spread added.
From figure 11.1 it can been seen that the luminosity spectrum is affected by the
addition of beam-spread to the simulation. The change in the luminosity spectrum is
significant near to the peak at 500 GeV, see table 11.1.
% Energy Loss
No beam-spread
0
0.01
0.10
1.00
5.00
0.296634
0.361277
0.454843
0.679954
0.913025
0.1%
Gaussian
beam-spread
n/a
0.232772
0.41949
0.679993
0.912853
Table 11.1: Table showing the fraction of particles which have retained certain values
of energy.
The fraction of particles which have retained 100% of their energy does not apply to
the simulation which includes beam-spread due to the nature of the beam-spread. The
effect of the beam-spread is seen near to the 500GeV peak of the luminosity
spectrum. There is a considerable smaller number of particles that have retained
99.99% and 99.9% of their energies at the time of collision. This is as was expected.
The beam-spread added is of a value of 0.1% of the nominal beam energy. Therefore
a significant number of the particles are going to be shifted away from the peak to an
energy of less than the 99.9%. When the energy loss is of a greater order than the
beam-spread, (i.e. >0.1%) then the effect of the added beam-spread becomes
insignificant.
The effect upon the beam that is significant is determined by setting a target error.
The errors are:
δMt < 100 MeV
δexp < 45 MeV
( Error on the mass of top quark)
(Experimental error)
This means that approximately 2.5 in 104 is a significant change in the centre of mass
energy.
The centre of mass energy was shifted by 0.05% which is a significant value.
12. Dispersion
In this section energy-displacement correlations were added to the GUINEA-PIG
simulations of the beam-beam collisions. All of the simulations containing dispersion
also have a Gaussian beam-spread of 0.1% of the nominal beam energy.
28
The dispersion added to the bunches causes the energy of the particles to be
dependent upon their position within the bunch. The affect upon the distribution of the
energy within the bunches is that energy changes steadily from the maximum energy
on the top edge of the bunch to the minimum energy at the bottom edge of the bunch.
The function below is used to create the dispersion within the bunches.
ΔE = E(U) – E0 = K( U – U0) = K ΔU
10.1
Where the position of the particle, U = x, y, z. E0 is the nominal beam energy

and K  E is the factor used to scale the energy distribution.
U
12.1 Result of introducing dispersion in x, y, and z
Figure 12.1: Graph showing the luminosity spectra which result as an effect of
correlating the energy of the particles with displacement.
The luminosity spectra are very similar for all of the simulations containing dispersion
regardless of the orientation x, y, or z. The effects upon the luminosity spectrum due
to the dispersion are summarised in table
29
Figure 12.3: Graph showing the horizontal and vertical angular distributions when
there are energy-displacement correlations in the simulation.
The graphs plotted in figures 12.1-12.3 show that the luminosity spectrum and the
angular distributions are as expected. There is no great difference caused by adding
the dispersion to the GUINEA-PIG beam-beam simulation to the overall distributions.
However, more importantly there are changes in the mean of the centre of mass that
are significant. The results are summarised in table 12.1.
Data Set
Average
Δ(√s ) %
X 0.1% BS
0.05
Y 0.1% BS
0.08
Z 0.1% BS
0.09
Table 12.1: Summary of the effects due to added dispersion in the x, y, and z
directions.
Again comparing the shift in the luminosity spectrum with the target errors it can be
seen that this shifts due to adding the dispersion is significant in x, y and z.
The values in table 12.1 tell us that the effect of adding dispersion in the x direction is
lower than either of the y or z directions. This is because a greater amount of
disruption occurs in the x direction which will be cancelling out some of the effects
due to the added dispersion.
30
12.2 Time-dependence
Figure 12.4: Graph showing the luminosity spectrum with energy-displacement
correlations varying with time.
Figure 12.5: Graph showing the vertical angular distribution varying with time when
there are energy-displacement correlations in the simulation varying.
31
Figure 12.6: Graph showing the horizontal angular distribution when there are
energy-displacement correlations in the simulation, varying with time.
Figures 8.4, 8.5, 8.6 display the same properties as the earlier plots (figures 8.1,8.2,
and 8.3) of the luminosity spectrum and the angular distributions varying with time.
Further studies into these plots will be required in order to see if there is any
significant effect introduced by the correlations and beam-spread which have been
added to the simulations.
14. Conclusion
Initially the GUINEA-PIG simulation output data was used to plot the luminosity
spectrum. This luminosity spectrum was as expected, it showed energy losses due to
beamstrahlung radiation. The angular distribution of the particles was also shown to
be as described in earlier studies. The horizontal angular distribution always has a
greater range than the vertical. Particles are oscillating much quicker in the vertical
than in the horizontal, so they are less able to be deflected at large angles.
The next section involved determining whether the timing of the collision has any
effect upon the energy of the particle at the time of collision. The collision events
were split up into two groups and the energies plotted. The mean energy of the first
half of the collisions was found to be significantly higher than the second half of the
collisions. To see how the time effects the collision energy of a particle in more detail
the events were split up into ten equal groups and plotted in a two dimensional
histogram. From this it was clear that the energy of the particles at collision decreased
steadily with time. This is occurring because the particles colliding further into the
simulation has a longer period of time during which they can emit beamstrahlung. To
complete this section of the project, GUINEA-PIG was modified by Dr. S Boogert to
include a time variable in the output file. By using this new variable, the change in the
luminosity spectrum over time can be seen very clearly. It shows in the early stages of
the beam-beam collision that there are very few energy losses. As time increases the
total centre of mass energy decreases steadily with time.
32
By using this new time variable in the GUINEA-PIG output file the time dependence
of the angular distributions can shown. The results of this study show that there is an
oscillating tendency for both the vertical and horizontal angular distributions. In the
early and final stages of the simulation the amplitude of the oscillations are much
greater than during the middle. This is because the particles are seeing fewer particles
from the opposing bunch and there are therefore fewer interactions between the
particles causing the angles to oscillate less rapidly.
There was also investigation into the time steps that the simulation uses. It was
assumed in plotting the energies against time that the event numbers allocated to the
collisions are in chronological order. A plot showing the displacement against event
number revealed that the event numbers are in chronological order as assumed and
there are definite time steps used in the simulation. It was by using this plot that the
time variable could be added to the GUINEA-PIG output file.
It could also be seen from the plot of displacement against event number that there
were only two grid cells modelling the vertical direction of the simulation. The
number of grid cells was increased to allow y to have a similar number to those seen
in the x and z directions. The effects of this increased modelling did affect the
luminosity spectrum. It was decided to continue the remainder of the project using
this data because the y direction may not have been modelled significantly to show
the effects that would later be added to the bunches of particles.
Uncorrelated events were created by using an analysis program. The effects of
removing the correlations from the collisions were insignificant.
Gaussian beam-spread of 0.1% was added to the GUINEA-PIG beam-beam collision
simulation. The luminosity spectrum was affected by the addition of beam spread.
The mean centre of mass energy was shifted by 0.05% which is a significantly large
shift according to error targets upon the mass of the top quark and experimental error.
Dispersion was also added, in addition to the Gaussian beam-spread, to the GUINEAPIG beam-beam collision simulation. The luminosity spectrum was affected by the
addition of dispersion in all the directions x, y, and z.. The mean centre of mass
energy was shifted by 0.05% for x, 0.08% for y, and 0.09% in z all of which are
significantly large shifts according to error targets upon the mass of the top quark and
experimental error. The shift in energy for the dispersion in the x direction is much les
than the shift in either the y or the z direction. This is because there is greater
disruption in the x plane than in either the y or z plane. The disruption will be
cancelling out some of the effects due to the dispersion being added to the simulation.
Finally, if this project were to be continued it would be interesting to see the effects of
adding beam-spread to the simulations that are not Gaussian. It is most probable that
any beam spread in a linear collider will not be Gaussian. Therefore, this would be a
highly relevant area to study. It would also be interesting to study simulations that are
not of Gaussian bunches of particles to see the effect that this has upon the luminosity
spectrum.
33
References
1: http://tesla.desy.de/
2: http://www-project.slac.stanford.edu/nlc/components.html
3: http://tesla.desy.de/
4: http://www-project.slac.stanford.edu/nlc/components.html
5: Dr. S Boogert, Graph demonstrating the effects of energy loss.
6: Talk given by Dr. S Boogert and Prof. D Miller, Luminosity measurement
questions: beam related, slide 3.
7: Talk given by Dr. S Boogert and Prof. D Miller, Luminosity measurement
questions: beam related, slide 6.
8: Talk given by Dr. S Boogert and Prof. D Miller, Luminosity measurement
questions: beam related, slide 6.
9: Grid inside which the GUINEA-PIG beam0beam collision simulation occurs, Dr. S
Boogert.
10: Beam-beam phenomena in linear colliders, Kaoru Yokaya
11, 12: Study of electromagnetic and hadronic background in the interaction region of
the tesla collider. Daniel Schulte, 1996.
34
Appendix A
A.1
Eb
Mean, Ē
Variance, σ
No. of events, n
0.9780
0.03773
99999
Ee
0.9693
0.04824
97689
Table A1: Table showing the data used to test the significance of the difference of the
mean energy of the particles at the beginning and those at the end of the simulation.
A.2
Parameters
No of macroparticles
Cut off
(dimensions of the
grid)
Beam dimensions
Original Data
32
250
12
3
100
3
554 nm
5 nm
300 µm
Nx
Ny
Nz
Cx
Cy
Cz
σx
σy
σz
New Data
32
1000
12
3
100
3
554 nm
5 nm
300 µm
Table A2. Table showing the parameters that are used to determine the number of
macro-particles, the dimensions of the beam and the size of the grid that the
simulation occurs in. Highlighted is the only change that was implemented to increase
the number of cells in the y component of displacement.
35