Download Particle Physics Design Group Studies Worksheet Introduction

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Particle Physics Design Group Studies Worksheet
Introduction
In this worksheet a number of topics are introduced which are relevant to the design study you
will be performing in group studies. Some subjects are reminders of material you have met in
lectures, others are topics which you may not have met in any detail in your lectures so far.
In each case only a brief introduction is given; you are expected to follow up each subject in
textbooks or other sources before you will be able to complete the problems given. Pointers
to appropriate reference sources are given in each section; you may have to follow up further
references within those sources, or elsewhere, to gain in-depth knowledge that you will need for
the project work. There is a list of all the references at the end, plus some additional possible
starting points.
Collaborative work and discussions within the group to address these issues and problems
are very much expected and encouraged. However, written and numerical answers to questions
should be your own individual work, as always.
Note that the convention h/2π = c = 1 is normally used in particle physics, including in
this worksheet.
Particle Physics Experiments: Colliders vs Fixed Targets
In particle physics there are two basic kinds of experiments. One uses a beam hitting a stationary
target in the laboratory, a so-called fixed target experiment. The second collides two beams,
usually “head on”, and often with equal and opposite momentum; this is a particle collider.
The instantaneous interaction rate in a collider is usually much less than that in a fixed target
experiment, and so colliding-beam machines are often built as storage rings to allow the beams
to circulate and collide over several hours.
Q1. (5 points) Colliders
Consider the interaction of a high-energy proton beam (p) hitting protons (p) which are essentially at rest in a liquid hydrogen fixed target. Calculate relativistically, from first principles, the
proton energy needed to create a Higgs boson (H) with mass mH = 125 GeV via the reaction
pp → ppH. What is the minimum proton beam energy needed to produce a Higgs boson with
the above mass in a pp colliding-beam experiment? Compare this value with the beam energy
of the LHC. Comment briefly on why the LHC beam energy is so much different.
Event Rates and Luminosity
At a particle collider the number of events produced per second dN/dt(s−1 ) for a process with
cross section σ(m2 ) depends only on the luminosity L(m−2 s−1 ), where dN/dt = Lσ. Rare
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processes have low cross sections and so it is essential to keep the collider luminosity high to
observe them. This is achieved by having large numbers of particles in each beam, colliding
these as frequently as possible and making the beam sizes small. Of course these quantities
do vary and so L is the instantaneous luminosity. In particle and nuclear physics the barn is
frequently used as the unit of cross section and this is equal to 10−28 m2 . The cross sections for
many interesting processes at high-energy colliders are much smaller than this, e.g. picobarns,
pb (10−40 m2 ). The instantaneous luminosity is usually quoted in units of cm−2 s−1 , but may
also be quoted in units of pb−1 s−1 . Depending on the magnitude of the cross section it may
sometimes be more convenient to work with e.g. femtobarns, fb (10−43 m2 ), or nanobarns, nb
(10−37 m2 ), rather than pb.
When running an experiment over a long period the instantaneous luminosity will vary and
the total number of events recorded will depend on the average instantaneous luminosity multiplied by the total time of the experiment (t). This product is known as the integrated luminosity
(LINT ) and is frequently used in describing the sensitivity of an experiment. It is often quoted
in inverse picobarns (pb−1 ). We need high LINT and this is achieved by maximising both the
instantaneous luminosity L and the running time t as far as possible.
An experiment with an integrated luminosity of 500 pb−1 would expect to observe, on average, 500 events of a process with a cross section of 1pb, if it were fully efficient. As the events
occur at random, this is an average expectation and the actual numbers observed in a series of
independent experiments of this type follow a Poisson distribution with this mean. Unless this
number N is very small,
√it is reasonable to approximate the spread (standard deviation) in the
number of events to be N .
Q2. (15 points) Event rates
The LHC will
with a centre-of-mass
√ deliver proton-proton collisions to the LHCb experiment
33
−2 −1
energy of s = 14 TeV, an instantaneous luminosity of 3 × 10 cm s and with proton
bunches crossing in the experimental interaction region every 25 ns. The total cross section
for pp interactions at this energy is 120 mb, and the inclusive bb̄ production cross section,
pp → bb̄X, is 1 mb.
(a) Calculate the average number of pp interactions per bunch crossing, and the average number
of bb̄ pairs produced each second.
(b) The LHCb Collaboration wants to study the rare decay mode Bs0 → φγ, which has a branching fraction of 3.6 × 10−5 (i.e. this fraction of all Bs0 mesons decay in this way). Assume that
10% of all the b̄ produced form Bs0 mesons, and that the overall efficiency of the LHCb experiment to detect and reconstruct Bs0 → φγ decays is 1%.
For how long will the LHCb experiment need collect data, running at this luminosity, in order
to expect to reconstruct an average of 10 Bs0 → φγ decays. What is the probability that less
than 5 Bs0 → φγ decays are reconstructed in this data sample?
(c) The LHCb Collaboration is also searching for another, previously unobserved, very rare decay mode of the Bs0 meson with a predicted branching fraction of 10−10 . Assuming the same
detection and reconstruction efficiency as in part (b), and that the LHC can continue to deliver
proton collisions at an average instantaneous luminosity of 3 × 1033 cm−2 s−1 with an operational efficiency of 25%, for how long would the experiment need to run, in order to expect to
reconstruct an average of one of these very rare decays? What is the probability that none of
these very rare decays would be reconstructed in such a data sample? For how long would the
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LHCb experiment need to run under these conditions, so that the probability to reconstruct none
of these very rare decays is less than 5%.
Q3. (10 points) Statistics of discovery
The LHCb experiment is used to search for the production of new particles among the decay
products of known, heavier and copiously produced particles. Consider the search for a new
particle that is produced in the decay of the Λb baryon in a data sample corresponding to an
integrated luminosity of 3 fb−1 . Assuming the visible production cross section of the Λb to be
1 nb, and the branching fraction for decays into the new particle to be 10−6 :
a: What is the probability that at least one of these new particles has been produced within this
data sample?
b: Assume that after further data-taking, 30 fb−1 of data are collected. What is the probability
that no new particle has been produced?
c: A measure of the statistical significance of a signal may be given in a counting experiment by
the number of signal events divided by the uncertainty on the number of expected background
events. The Standard Model background in the mass range relevant to the above new particle has an effective branching fraction of 10−5 . Estimate how much integrated luminosity the
experiment requires to observe a significance of 3σ?
Accelerator Physics
The motion of a particle with charge e and velocity v passing through a volume containing a
magnetic field B and an electric field E is determined by the Lorentz force
F = e(v × B + E) .
(1)
The magnetic force acts in a direction perpendicular both to the direction of the particle and
the direction of the magnetic field, whereas the electric force acts along the direction of the
electric field. In the absence of an electric field, a particle travelling at right angles to a constant
magnetic field will describe a circular trajectory. The formula that relates the radius R of the
orbit to the momentum p of the particle and the magnetic field is
p = BeR .
(2)
An electric field can be used to accelerate the particle along its direction of motion. In a fixed
magnetic field, the radius R will then increase in proportion to the momentum p as in the cyclotron. Alternatively, the radius may be kept constant by increasing the magnetic field in
proportion to the momentum as the particle accelerates, and this is the principle of the synchrotron. The advantage of the synchrotron is that the volume of the magnet is greatly reduced
as the field is only needed along a fixed trajectory, resulting in an annular ring of magnets for a
practical device.
High-energy protons and electrons are accelerated using synchrotrons. If, after acceleration
to a given momentum, the magnetic field is held fixed and the particles are allowed to circulate
at constant momentum, the synchrotron becomes a storage ring. A colliding-beam machine
can be formed by allowing two beams of particles to circulate in opposite directions along
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approximately the same trajectory and causing them to collide at specific locations at which the
beams are focussed to a small size to increase the rate of collision (L · σ). This can only be
done for particles of opposite charge, the same energy but opposite momentum, e.g. electrons
and positrons in LEP at CERN, protons and antiprotons in the Tevatron at Fermilab. In order to
collide two particle beams of the same charge, two intersecting rings of magnets are required
with fields in opposite directions. The LHC at CERN is of this type.
Q4. (2 points) Bending power of LHC dipole magnets
One of the factors limiting the maximum beam energy at the LHC is the strength of the dipole
magnets. Calculate the radius of an “ideal LHC”, using only dipole magnets all the way around
a circular circumference. Use typical numbers for the magnetic field, B = 8 T, and the beam
momentum, p = 7 TeV. How does this result compare to the actual radius of the LHC?
The ultimate momentum (or energy) that can be achieved in a synchrotron is obviously proportional to the product of the magnetic field and the radius of the orbit. However, at sufficiently
high energy, or more strictly γ = E/mc2 , particles that are accelerated emit a significant amount
of radiation. The transverse acceleration in a synchrotron gives rise to synchrotron radiation
which, at a fixed radius, increases as the fourth power of the particle energy and also as the
inverse of the fourth power of the particle mass, making it much more serious for electrons. In a
colliding-beam machine such as LEP, the energy lost in each revolution due to synchrotron radiation has to be replaced by re-accelerating the electrons and positrons with a radio-frequency
electric field. Otherwise the particles would spiral inwards and be lost. This effect limits the
maximum energy at which it is practical to operate a circular collider.
The instantaneous luminosity, L, in a colliding-beam experiment can be calculated from
several equivalent expressions. One useful form for circular colliders, assuming that the particle
densities are Gaussian transverse to the beam direction, is the following:
L = I1 I2 /(e2 kfrev 4πσx σy ) ,
(3)
where frev is the revolution frequency, k is the number of bunches of particles per beam, σx and
σy are the RMS beam sizes in the transverse direction, and I1 , I2 are the two beam currents.
To achieve very high luminosities some present and future machines have many bunches to
overcome space-charge limitations. LEP was operated with 4, 8 and 12 bunches per beam, and
the LHC is designed for ∼2800 bunches spaced apart by just 25 ns. To increase the luminosity,
it is important to keep the transverse beam size small at the interaction point; this is done by a
process known as squeezing the beams.
More information on accelerator physics can be found in the books by Tigner and Chao,
Wilson, and Wille.
Q5. (8 points) Accelerator luminosities
a) Convert the above equation for L into an expression containing instead N , the number of
particles per bunch. Calculate the minimum value for N at a 500 GeV (250 GeV on 250 GeV)
collider (k = 1) to produce an average of 10 events per day for a process with a cross section
of 100 fb, if the beam dimensions are σx = σy = 1 mm, and frev is 103 Hz. Convert your value
for N into a beam current (A) and an instantaneous luminosity (in units of cm−2 s−1 ).
b) Redo the calculations assuming σx = σy = 0.01 mm(10 µm). This question should make
you appreciate some of the real problems in studying these rare reactions (tiny beam sizes with
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lots of particles in them). At LEP, typical instantaneous luminosities were 1031 cm−2 s−1 . At
LHC the design is 1034 cm−2 s−1 .
Interaction of Radiation with Matter: Particle Detectors
The detection of particles in experiments depends on the characteristics of the interactions of
the particles with the detector material. Several different types of interactions are important,
the main ones being: ionisation and excitation of atoms by charged particles; electromagnetic
interactions in nuclear electric fields; and hadronic interactions of strongly-interacting particles
with nuclei.
Many standard particle and nuclear physics textbooks have an introduction to the interactions of radiation with matter, and particle detectors. Good, more advanced, textbooks are those
by Green and by Kleinknecht. The “Review of Particle Physics” contains many useful formulae.
Energetic charged particles cause ionisation and excitation of atoms as they move through
material. The average energy loss of charged particles per unit length, dE/dx, through ionisation and excitation is described by the Bethe-Bloch equation. The energy spectrum of ionised
electrons falls steeply with increasing energy, and energetic primary ionisation electrons may
produce secondary ionisation. The total number of ions produced per unit length when a particle passes through the material depends both on the energy loss (given by the Bethe-Bloch
equation, on average) and on the specific properties of the material, particularly the average
energy needed to create each ion. This is as high as 40 eV for helium, but effectively as low as
3.6 eV in silicon, where the charge is carried by electron-hole pairs rather than electrons and
ions.
In addition to interaction with atomic electrons, charged particles may also interact in the
strong electric fields around nuclei. All charged particles are affected by Coulomb scattering, which gives rise to multiple scattering when passing through matter. Electrons may give
up a significant fraction of their energy in inelastic collisions with nuclei, via the so-called
bremsstrahlung (radiation loss) process, where the energy is radiated off as a photon. Incident photons with sufficient energy can also interact in the nuclear electric field to produce
electron-positron pairs, so-called conversions. High-energy electrons or photons impacting on
dense materials thus produce electromagnetic showers of electrons and photons via multiple
bremsstrahlung and conversion processes. Eventually all the energy of an incoming particle
may be absorbed, if the material is thick enough. The length scale of these electromagnetic
processes is the radiation length, denoted X0 .
Strongly interacting particles (hadrons) can also interact via collisions with atomic nuclei;
typically for high-energy particles several collision products will be produced, the charged
ones going on to ionise surrounding material. Analogously to the electron/photon case, this
in turn gives rise to an hadronic shower of particles, which may be absorbed if the material is
thick enough. In this case, the typical length scale is characterised by the nuclear interaction
length λI .
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Tracking Detectors
The basic idea of a tracking detector is to detect the passage of electrically charged particles by
ionisation energy loss, but to include only a little material so that the processes of showering do
not start, and to minimise multiple scattering. In this way, particle trajectories can be followed
over distances of metres or more. Most particle physics experiments employ magnetic fields
in the region around the interaction point so that moving charged particles are bent by the
Lorentz force on them. In colliding-beam experiments this is typically provided by a solenoid
wound around the beam direction, providing a uniform magnetic field along the beam axis.
The measurement of several points along a particle trajectory with tracking detectors allows the
curvature of a particle track to be measured, and hence the momentum can be deduced.
Typical tracking detectors in use today are constructed with large gas volumes used for
detection of ionisation. Ions drift over distances varying from a centimetre or two to several
metres. Best established are drift chambers, where the ionisation drifts a few centimetres
before being amplified at wire planes. Another technology, the time-projection chamber,
uses much longer drift distances of up to a few metres. An alternative technology for tracking
detectors is to use thin silicon sensors, and to measure electron-hole pairs produced by passing
particles. These devices may be made very precisely, as sensor tracks may be etched at the
level of tens of microns. They are particularly useful for measuring particle trajectories close
to the primary interaction point, to look for particles which travel only short distances (tens of
microns or more) before decaying.
Q6. (10 points) Momentum resolution
A tracking detector surrounding a colliding-beam interaction point makes 40 equally spaced
measurements of the particle trajectories in the coordinate plane perpendicular to the beam
direction, over a distance of 1.5 m. The precision of each measurement is 200 µm in the
azimuthal direction (i.e. perpendicular both to the beam direction and to a line drawn from the
measurement to the beam axis). The detector is immersed in a uniform magnetic field along the
beam direction of 2 T.
a) Ignoring multiple scattering effects, what is the expected momentum resolution for muons
with 1 GeV, 100 GeV and 1 TeV momentum, respectively, travelling out at 90◦ to the beam
direction?
b) If the detector is filled with argon at 20◦ C and atmospheric pressure, estimate approximately
the size of the error on the momentum measurements from multiple scattering, for the three
momentum values given in a). How does this compare with the errors from the intrinsic point
resolution as a function of momentum?
Calorimetric Detectors
Tracking detectors have two disadvantages for measuring the particles produced in a collision;
they see only electrically charged products, and also the resolution for very energetic particles,
which are bent little by magnetic fields, is quite poor. The measurements may therefore be
complemented by calorimeters, which are large detectors typically made out of dense solid
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materials such as lead and iron. Calorimeters are designed to absorb the energy of particles hitting them via the showering mechanism, and to be thick enough that the full shower is contained
inside them.
Lead and iron in themselves are not sufficient, as although incident particles shower inside
them there is no way to see the resultant energy deposition. A standard technique to overcome
this is to place thin layers of a detector material, such as scintillator slabs or gas detectors,
between thick layers of the dense absorber. In this way, the charged particles in the shower are
sampled at various points in the shower. Alternative techniques exist by using a dense active
medium, such as crystal calorimeters, lead-glass, etc.
The characteristic length scales of electromagnetic and hadronic showers are different, but
a significant fraction of the products of collision events can be produced in the form of photons
and electrons. Calorimetry is therefore usually divided into two parts: a front part (nearest the
interaction point) is thin and highly instrumented to catch the electromagnetic showers, and it
will also catch the start of some of the hadronic showers; a thicker, less well-instrumented, back
part catches the rest of the hadronic showers. These are conventionally called electromagnetic
and hadronic calorimeters, although the functions actually overlap.
Q7. (20 points) Calorimeter resolutions
(a) Describe the main features of homogeneous and sampling electromagnetic calorimeters and
the differences between them. What are the advantages and disadvantages of each type of
calorimeter when used as part of a general-purpose particle physics detector?
(b) The energy resolution of a sampling electromagnetic calorimeter can be written in the form
σ(E)
b
a
= √ ⊕
⊕ c,
E
E
E
where the symbol ⊕ denotes addition in quadrature, E is the energy in GeV and
a, b and c are constants. Explain the physical origin of each of the three terms in this expression
and give typical values of a, b and c.
Multi-layer Detectors and Particle Identification
A typical particle physics experiment is a hybrid consisting of multiple layers: precise tracking
close to the interaction point; larger tracking detectors next; then electromagnetic and hadronic
calorimeters. On the outside of the detectors are placed extra tracking detectors called muon
detectors. Muon particles interact only minimally with the material of the detector and are,
therefore, more likely than other charged particles to travel unhindered to the outside of the
detector.
In addition to measuring particle momenta and energies, the information from the different
detectors may also be combined to give further information about the type of the particles, socalled particle identification. For example, by matching up electromagnetic calorimeter energy
deposits with charged particle tracks, it may be possible to identify individual neutral particles.
For charged particles, the rate of energy loss information, dE/dx, may allow protons, electrons
and charged pions, for example, to be separated.
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Further specialised detectors are used in some experiments to give extra information for the
purposes of particle identification. Examples are Cherenkov detectors and transition radiation detectors.
References - to get you started
D. H. Perkins, “Introduction to High Energy Physics”, 4th edition 2000, Camb. Univ. Press.
A. Das and T. Ferbel, “Introduction to Nuclear and Particle Physics”, 1993, Wiley.
G. Kane, “Modern Elementary Particle Physics: The Fundamental Particles and Forces”, 1993,
Perseus Books.
B. R. Martin and G. Shaw, “Particle Physics”, 2nd edition 1997, John Wiley.
I. Hughes, “Elementary Particles”, 3rd edition 1991, Camb. Univ. Press.
M. Tigner and A. Chao, “Handbook of Accelerator Physics and Engineering”, 2nd edition 2013,
World Scientific (online access to full text).
E. Wilson, “An Introduction to Particle Accelerators”, 2001, Oxford Univ. Press.
K. Wille, “The Physics of Particle Accelerators: an Introduction”, Oxford Univ. Press.
“Hands-on CERN”web pages:
http://hands-on-cern.physto.se/hoc v21en/index.html
(in particular the detectors part)
D. Green, “The Physics of Particle Detectors”, 2000, Camb. Univ. Press.
K. Kleinknecht, “Detectors for Particle Radiation”, 2nd edition 1998, Camb. Univ. Press.
“The Review of Particle Physics”, K.A. Olive et al. (Particle Data Group),
Chin. Phys. C, 38, 090001 (2014). http://pdg.lbl.gov/
W. R. Leo, “Techniques for Nuclear and Particle Physics Experiments - A How-to Approach”,
2nd edition 1994, Springer Verlag.
R. K. Bock and A. Vasilescu, “The Particle Detector BriefBook”,
http://rkb.home.cern.ch/rkb/titleD.html
R. K. Bock and W. Krischer, “The Data Analysis BriefBook”,
http://rkb.home.cern.ch/rkb/titleA.html
H. Murayama and M. Peskin, Ann. Rev. Nucl. Part. Sci. 46 (1996) 533-608.