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\begin{document}
\title{Minerva Lab Experiment}
\author{L.A.Lewis}
\maketitle
\begin{abstract}
This experiment consists of a general introduction of the ATLAS experiment
at the LHC at CERN. Students will be using MINERVA to visualise collisions
from the proton-proton beam. Intially, students will be asked to identify
specific events such as Z and W decays, leading onto more difficult
decays from $t\bar{t}$. Using the information gathered from MINERVA
the mass and lifetime of the particles can be determined. Using the
values of the mass and width of the Z the number of neutrinos that
exist can be investigated. As an extension students can try looking
further into the $t\bar{t}$ decays.
\end{abstract}
\section{Introduction}
In this experiment you will be visualizing events similar to those
that will be seen in ATLAS \cite{1}, which is a particle physics
experiment at the Large Hadron Collider at CERN. The ATLAS detector
will search for new discoveries in the head-on collisions of protons
of extraordinarily high energy. ATLAS will search for new particles
and learn about the basic forces that have shaped our Universe since
the beginning of time and that will determine its fate. Among the
possible unknowns that will be probed are the origin of mass, extra
dimensions of space, microscopic black holes, and evidence for dark
matter candidates in the Universe.
MINERVA is a software package that visualizes the information recorded
by ATLAS after each collision. The same package is used by physicists
working on the experiment. You will need to familiarize yourself with
using MINERVA; the primary goals of the program are the visual investigation
and the understanding of the physics of single collisions (events)\cite{2}.
You will be learning how measurements of different types of particles
are made; and investigate how the mass and lifetime of short-lived
particles are determined.
\section{MINERVA}
See Separate sheet. Quick Guide to MINERVA for labs.
\section{Identifying Events}
The W and Z bosons were discovered in 1983 at CERN. Their properties
are very well known, so why are you being asked to learn how to detect
them? Investigating W and Z events at the LHC will provide additional
tests of the standard model of particle physics. Another reason to
study these particles is the search for the Higgs boson. Physicists
have theorized a Higgs field that is made of Higgs bosons. Particles
acquire mass by interacting with this field. The Higgs boson may be
discovered by its decay into WW or ZZ. The Z boson is also a common
background process for other searches for new particles and must be
understood well.
The Z boson is a massive, neutral particle that decays into any lepton-anti-lepton
pair or a quark antiquark pair. If at rest the decay products are produced
back-to-back to conserve momentum.
The W boson is a massive, charged particle that decays into a lepton
and neutrino or a quark antiquark pair (u-d or s-c). The presence of the
neutrino can be deduced by using momentum conservation. The branching
ratio is the fraction of particles that decay by individual decay
paths. Write down all the different decay paths of the W boson, remembering
that quarks each have 3 colours. Hence, calculate the branching ratios
for the W boson.
The events from W and Z decays chosen for this experiment will not
include those decaying to quarks. It is more difficult to study quarks
as they are detected as a collection of hadrons travelling close together,
called a jet. There are many jets produced in proton-proton collisions.
\paragraph{Exercise 1}
Read the Quick Guide to MINERVA. Open five of the sample sets, this
will give you 100 events from proton-proton collisions. Determine
which events are W decays, Z decays or Di-jet. Record how many of
each type of event there are. Measure the W/Z ratio. Comment on your
result.
(If continuing onto exercise 2 record for each Z event the momentum
components of the two decay leptons).
If continuing onto exercise 3 record for each W event the transverse
momentum components(Px and Py) of the charged lepton track and missing energy.
\section{Determining Rest Mass of Z and W Bosons}
The invariant mass, M is the same as the rest mass of a particle.
\[
M^{2}=(\sum E)^{2}-(\sum\bar{p})^{2}\]
$\sum E$ is the sum of all the energies of the decay particles. $\sum\bar{p}$
is the vector sum of the momenta of the decay particles (including
both magnitude and direction of the momenta).
For electrons and muons you can assume that the magnitude of momentum is equal
to the energy of the particle since the mass is negligible for energetic
particles.
\paragraph{Exercise 2}
Ask your demonstrator for the file containing only Z events. Read
the Guide to create histograms using MS excel%
\footnote{If continuing onto exercise 4 you may save time by using a different
program than excel.%
}. Plot a graph of the invariant mass of the Z boson.
What can the shape of the graph can tell you about the properties
of the particle?
\paragraph{Exercise 3}
For the W boson, only the transverse momentum can be measured because
of the undetectable neutrino. Transverse momentum is the momentum
of a particle in the x-y plane. Only the transverse component of the
missing momentum (neutrino momentum) can be measured because other
particles may escape along the beam pipe (z-direction), therefore,
the measurement of missing momentum along the z direction would be inaccurate.
Using the transverse momentum of the lepton and transverse missing
energy, find the transverse invariant mass of the W events. Plot this
as a histogram; can you explain the shape of the plot?
The transverse invariant mass does not take into account any momentum
in the beam direction. Since the W and Z bosons have similar mass,
it may be possible to find the actual invariant mass by calculating
a mass correction factor. Investigate this; find the Z transverse mass, using the transverse
momentum of the electrons and muons and find the ratio Z mass/ Z transverse
mass . Apply this correction factor to the W transverse mass results
and compare the W 'corrected' mass result with the known mass from the
Particle Data Group \cite{3}.
\section{Z Lifetime}
For particles with extremely short lifetimes, there will be a significant
uncertainty of the measured energy due to Heisenberg Uncertainty Principle,
\[
\Delta E\Delta t\geq\frac{\hbar}{2}\]
In the figure below gamma is the full width half maximum (FWHM). Gamma
relates to the lifetime of the particle by \[
\Delta E=\frac{\Gamma}{2}=\frac{\hbar}{2\tau}\]
Insert pic of z width{*}
\paragraph{Exercise 4}
Use the same Z events from exercise 2. For this exercise you will
need to use a program that can fit gaussian curves to a histogram
eg SCIDAVis (freeware) \cite{4}. Plot a histogram of the
invariant masses the mass and lifetime of the Z boson can be determined.
Use the Quick Guide to SCIDAVis create a histogram of the Z invariant
mass and fit a Gaussian curve to the plot. Find the width and peak position
of the fitted Gaussian curve. What is the lifetime of the Z boson?
\section{Z Line Shape}
Experiments at the large electron=positron collider (LEP)used the result
from the precisely measured Z mass distribution to show that there are only three types of neutrino
that the Z can decay into. The total width of the Z boson is made
up of partial widths of different decay modes. The Z can decay
into any lepton -antilepton or quark -antiquark pair, excluding $t\bar{t}$,
which is forbidden due to energy restrictions.
The decay width is \[ \Gamma_{Z}=\Gamma(z\rightarrow qq)+\Gamma(z\rightarrow ee)+\Gamma(z\rightarrow\mu\mu)+\Gamma(z\rightarrow\tau\tau)+N_{\nu}\Gamma(z\rightarrow\nu\nu)\]
where$N_{\nu}$is the number of neutrinos.
\paragraph{Optional}
It is possible to calculate this yourself using the partial Z decay widths $\Gamma_{f}$ which are given by\[ \Gamma_{f}=\frac{G_{f}M_{Z}^{3}}{6\pi\sqrt{2}}N_{c}\left[(C_{V}^{f})^{2}+(C_{A}^{f})^{2}\right]\]
Where Nc is number of colours (1 for leptons, 3 for quarks).
$C_{V}^{f}$ and $C_{A}^{f}$ are the Z couplings to fermions shown in the table below.
\noindent \begin{center} \begin{tabular}{|c|c|c|} \hline & $C_{V}^{f}$ & $C_{A}^{f}$\tabularnewline \hline \hline $\nu_{l}$ & $\frac{1}{2}$ & $\frac{1}{2}$\tabularnewline \hline $e\mu\tau$ & -$\frac{1}{2}+2sin^{2}\theta_{w}$ & $-\frac{1}{2}$\tabularnewline \hline uct & $\frac{1}{2}-\frac{4}{3}sin^{2}\theta_{w}$ & $\frac{1}{2}$\tabularnewline \hline dsb & -$\frac{1}{2}+\frac{2}{3}sin^{2}\theta_{w}$ & $-\frac{1}{2}$\tabularnewline \hline \end{tabular} \par\end{center}
\[ sin^{2}\theta_{w}=0.230\]
\[
\frac{G_{f}}{(\hbar c)^{3}}=1.6637\times10^{-5}GeV^{-2}\]
\section{Top Quark Mass}
In ATLAS a top quark and top antiquark pair are expected to be frequently
produced. The top quark is twice as heavy as the W and Z bosons, and
always decays into a high energy b quark and W boson. The W can be
detected as seen in the previous exercises or as two jets, whereas,
the b quark produces a jet including a B hadron which decays slightly
away from the collision point. The possible decay modes of the $t\bar{t}$
pair are:
$t\bar{t}\rightarrow b\bar{b}q\bar{q}'\bar{q}q'\qquad Fully\, Hadronic,\,6\, jets.$
$t\bar{t}\rightarrow b\bar{b}q\bar{q'}l^{-}\bar{\nu}_{l}\, or\, t\bar{t}\rightarrow b\bar{b}l^{+}\nu_{l}\bar{q}q'\qquad Semi\, Leptonic,\,4\, jets.$
$t\bar{t}\rightarrow b\bar{b}l^{+}\nu_{l}l^{-}\bar{\nu}_{l}\qquad Fully\, Leptonic,\,2\, jets.$
Using the W branching ratios, (1/9 for each lepton type and 2/3 for quark antiquark decays )calculate the branching ratios of the
$t\bar{t}$ decay being fully hadronic, semi leptonic or fully leptonic.
\noindent \begin{center}
%
\begin{figure}[H]
\noindent \begin{centering}
\includegraphics[scale=0.25]{ttdecays\lyxdot jpg}
\par\end{centering}
\end{figure}
\par\end{center}
\paragraph{Exercise 5}
Open the file containing the $t\bar{t}$ events. For each event identify
the lepton and the jets with the highest transverse momentum. The top quark
mass can be determined from finding the invariant mass of the three
jets comprising two quark jets and the b jet. Use the three jets with
the largest vector sum of transverse momentum for this calculation.
Plot a histogram of the top quark's invariant mass and compare your results
with the value of the top quark mass in the Particle Data Group (PDG).
\bibliographystyle{plain}
\bibliography{lab}
\end{document}