P308 Particle Interactions

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P308 Particle Interactions

• Dr David Cussans
• Mott Lecture Theatre
• Monday 1110am
• Tuesday,Wednesday 10am

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Aims of the course

• To study the interaction of high energy particles
  with matter.
• To study the interaction of high energy particles
  with magnetic fields.
• To study the techniques developed to use these
  interactions to measure the particle properties.
• To look at how several different types of
  detector can be assembled into a general purpose
  detector

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Aims of the course

• (This course deals with particles as they are
  observed. We will try to be complementary to the
  material of the Quarks and Leptons course.)

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Advised texts Background on particles

• Everything you need to know about particles and
  more is in chapters 2 and 9 of Nuclear and
  Particle Physics, W.S.C.Williams, Oxford. If you
  want to know more, look at some general text on
  particles as advised for the Quarks and Leptons
  course, e.g. Particle Physics, Martin and Shaw,
  John Wiley

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Advised texts - Particle interaction with matter

• Single Particle Detection and Measurement R.
  Gilmore, Taylor and Francis.
• Detector for Particle Radiation K. Kleinknecht,
  Cambridge.
• The Physics of Particle Detectors, D. Green,
  Cambridge.

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Advised texts astrophysical applications

• High Energy Astrophysics (Vol. 1) M. Longair,
  Cambridge

Online Resources

• There are also some good subject reviews
  available online from the particle data group
• http//pdg.lbl.gov/2004/reviews/passagerpp.pdf
• passage of particles through matter
• http//pdg.lbl.gov/2004/reviews/pardetrpp.pdf
• particle detectors
• http//pdg.lbl.gov/2004/reviews/kinemarpp.pdf
• relativistic kinematics

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Outline and structure of the lectures

• Lectures 14
• Introduction and scope of the course
• particle properties from the detector point of
  view
• particle glossary
• Kinematics
• cross-sections and decay rates.

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Outline and structure of the lectures

• Lectures 510
• Interactions of fast particles in a medium.
• Ionisation by charged particles
• Quantitative description of ionisation energy
  loss.
• Other energy loss processes
• Showering processes.

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Outline and structure of the lectures

• Lectures 1112 (Information from detectors)
• Position and timing measurement.
• Momentum, energy and velocity measurement.
• Measurement errors
• counting fluctuations.

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Outline and structure of the lectures

• Lectures 1318
• The general purpose detector.
• Some specific detector technologies.
• Technology choices for different applications.

11
What is a Particle?
Wave
Particle
Frequency Wavelength
Energy Momentum
e.g. EM radiation/photons
Radio/microwave Visible X-ray/g-ray
Energy
Wavelength
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Relativity and QM

• Relativity describes particle behaviour at
• high speed ( close to speed of light)
• I.e. high energy (compared with particle rest
  mass)
• Quantum mechanics describes behaviour of waves
  (or fields)
• Probability interpretation for individual
  particles
• Often need both to analyse results of particle
  experiments

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Relativity and QM

• Alpha particle scattering from nuclei
• Rest mass of alpha 3.7 GeV
• Typical energy 10 MeV
• Can treat classically (fortunately for
  Rutherford!)

a-emitter
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Relativity and QM

• Compton scattering of g from electron
• Rest mass of g 0 eV
• Rest mass of electron 511 keV
• Typical energy of g 10 MeV
• Need to use both relativity and QM

g -emitter
g
e
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The Fundamental Particles

• Quarks
• u,c,t d,s,b
• We do not see free quarks, the particles actually
  observed are the traditional particles such as
  protons, neutrons and pions.
• Leptons
• e, m, t, ne, nm, nt
• Gauge bosons
• g , W , Z, gluons ( only g is observed directly )

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Types of Particle

• Particles divided into
• Fermions spin ½ , 3/2 , 5/2 etc.
• Bosons spin 0, 1, 2 etc.
• Hadrons made up of quarks
• Baryons and mesons
• Antiparticles
• appear to be a necessary consequence of quantum
  field theory

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Particle glossary

• Most important particle properties from the
  detector point of view are
• Mass
• Charge (electric, strong, weak)
• Interactions ( EM, strong , weak )
• Lifetime

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Stable particles

• Can be used as beam particles or for low-energy
  physics
• Decay prohibited by conservation laws
• Photon ( g )
• Neutrinos ( n )
• Electron/positron
• Proton/antiproton

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Weakly decaying particles

• Decay parameter
• Gives mean decay distance for 1GeV energy
• Neutron and muon
• Light quark mesons
• Strange baryons or Hyperons
• Heavy quark hadrons, t lepton

n 31011m
m 6km
p,K,K0L 5-50m
1-10cm
50-200mm
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Very short-lived particles

• Detectable only by their decay products
• Electromagnetic decays to photons or lepton pairs
• Includes p0 giving high-energy photons
• Strongly decaying resonances

p0 ct/m 180nm
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Very massive fundamental particles

• W,Z0
• top quark
• Higgs boson
• Super-symmetric particles,
• Decay indiscriminately to lighter known (and
  possibly unknown) objects leptons, quark jets
  (pions plus photons) etc.

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Relativity

• "Henceforth space by itself, and time by itself,
  are doomed to fade away into mere shadows, and
  only a kind of union of the two will preserve an
  independent reality."
• Hermann Minkowski,1908

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Relativistic relations

• Special relativity applies to inertial ( ie. Not
  accelerating ) frames.
• Needed in most particle interaction physics.

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Four-vectors

• Extension of normal 3-vector e.g.
• Position x? ( ct , x )
• Velocity ? ? ( ?c , ?? )
• Momentum p? m??
• ( ?mc , ?m? ) ( E/c , p )
• Have time-like component(scalar) and space-like
  component(vector)

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Length of a 4-vector

• Length of a 3-vector doesnt change under
  rotations in (three-) space
• x2 y2 z2 x2 y2 z2 constant
• Lorentz 4-vectors are such that their length
  (magnitude) does not change under Lorentz
  transformation
• x?x? x?x? x02 (x12x22x32) constant

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Four-vector terminology

• Contravariant vectors eg.
• x? ( ct , x )
• Covariant vectors eg.
• x? ( ct , -x )
• Contravariant and covariant differ in their
  behaviour under Lorentz transform (basically use
  them in Contracovariant pairs)
• ( Dont worry about the terminology included
  only for completeness.)

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Four-vector Operations

• dot-product for 4-vectors
• E.g. length of a 4-vector is the vector
  dotted with itself
• NB. The components of a 4-vector change under
  transformation, but its magnitude does not.

minus sign comes from minus in space component of
pm
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The Lorentz Transformation

• Lorentz transformation

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Energy, momentum and mass

• N.B. Will use natural units ,set
  and use units of eV for energy from now on.

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Useful Reference Frames

• CM frame is Centre-of-Mass or Centre-of-Momentum
• Rest frame for a system of particles
• I.e. ?pi0 ( where p is the usual 3-vector)
• LAB frame may be
• Rest frame of some initial particle, or
• CM frame,or
• Neither

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Invariant Quantities Invariant Mass

• Lorentz invariant quantities exist for individual
  particles and systems.
• Invariant mass of a system

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Invariant Mass

• Invariant mass is equivalent to the CM frame
  energy for a particle system
• If (?pi)0 then
• NB within a frame ?p?i constant
• (conservation of momentum)

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Four-momentum Transfer

• 4-momentum transfer is change in (E,p) between
  initial and final states
• q? p? - p?
• Its magnitude, q², is an invariant

p?
p?
k?
k?
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Total CM Energy in Fixed Target

• Fixed target experiment with a beam of
  particles, energy Eb, mass mb incident on a
  target of stationary particles, mass mt

mb,Eb
M,Ef
mt
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Threshold Energy for Particle Production

• If we want a fixed target experiment to have a CM
  energy, , higher than M then the beam energy
  Eb

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Mass of Short-lived Particle

• From invariant mass of its decay products, e.g.
  2-body
• How to measure ma?

mb,Eb
?bc
ma
mc,Ec
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Two-body Decay

• Initial invariant mass s ma2
• Final invariant mass
• If Eb, Ec gtgt mc , mc then Eb, Ec pb, pc
• So,

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q² for a Scattering Reaction

• For E,E gtgt m
• In elastic scattering, can use energy
  conservation to get energy lost by incident
  particle ...

p,E
?
m,p,E
mt
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Energy Loss in Elastic Scattering
p,E

• Energy transfer to target
• Maximum energy transfer in scatter
• Quoted without proof

?
m,p,E
mt
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Time Dilation and Decay Distance

• Often measure particle lifetime by distance
  between creation and decay.
• If mean life of particle is ? in its rest frame,
  in the lab frame the mean life is
• During this time it travels a distance ??c?
• Since p??m,
• mean decay distance in lab
• Decay length proportional to momentum

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Interaction Rates and Cross-sections

• Experiments measure rates of reactions these
  depend on both
• kinematics e.g. energy available to final state
  particles, and
• dynamics, e.g. strength of interaction,
  propagator factors etc.

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Cross section, ?

• Cross section incorporates
• Strength of underlying interaction (vertices)
• Propagators for virtual exchange factors
• Phase space factors (available energy)
• Does not depend on rate of incoming particles.
• Called the cross-section because it has units
  of area.
• Normally quoted in units of barns ( 10-28m2 )
• or multiples eg. Nanobarns (nb), picobarns
  (pb)

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Cross-Section physical interpretation.

• Can be thought of as an effective area centred on
  the target if the incident particle passes
  through this area an interaction occurs.
• Physical picture only realistic for short range
  interactions. (target behaves like a featureless
  extended ball)
• For long range interactions, like EM, integrated
  cross-section is infinite.
• Cross-section invariant under boost along
  incoming particle direction.

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Cross-section and Interaction rate.

• For fixed target, with a target larger than the
  beam
• Wr ? L ?
• W interaction rate
• r rate of incoming particles
• ? number of target
• particles per unit volume
• L thickness of target
• ? cross-section for interaction

r
L
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Cross-section and Interaction rate.

• For fixed target, in terms of particle flux, J
• WJ n ?
• W interaction rate
• J Flux particles per unit area per unit time.
• n total number of particles in target.
• ? cross-section for interaction

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Colliding Beam Interaction Rate

• In a colliding beam accelerator particles in each
  beam stored in bunches.
• Bunches pass through each other at interaction
  point, with a frequency f
• Have an effective overlap area, A
• Can express in terms of beam currents Inf
• Factors n1n2f/A normally called the Luminosity, L

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Differential Cross-section, d?

• We have just defined the total cross-section,?,
  related to the probability that an interaction of
  any kind occurred.
• Often interested in the probability of an
  interaction with a given outcome ( e.g. particle
  scatters through a given angle )

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Cross-section - Solid Angle

• Consider a particle scattering through ? , ?
• What is probability of
• scatter between (?, ?d?)
• and ( ? , ?d?) ?
• Element of solid angle d? d(cos?). d?
• Differential cross section
• For, e.g. fixed target

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Differential ?Total Cross-section

• To get from differential to total cross-section
• With unpolarized beams no dependence on ? -
  integrate to get d?(?)/d?
• If measuring some other variable ( e.g. final
  state energy, E) other differential
  cross-sections, e.g

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Decay Width, ?

• The lifetime of particles can tell us about the
  strength of the interaction in decay process (and
  about channels available)
• Decay rate, W1/? (in rest frame)
• ? - lifetime in rest frame
• For short lived particles reconstruct the mass,
  m, of the particle from decay products.
• Uncertainty principle
• ?t ? , so

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Decay Width

• Define the decay width, ?, to be the uncertainty
  in the mass, ?m
• For particle with several different modes of
  decay , can define partial widths, ?i
• Total width is the sum of all partial widths

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Decay Width Example

• Invariant mass of W 80GeV
• Width2.2GeV
• Mean life
• 3.25 x 10-25s
• c?10-16m
• (I.e. less than size of proton )

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End of Kinematics