P308 – Particle Interactions

Dr David Cussans

Mott Lecture TheatreMonday 11:10am

Tuesday,Wednesday: 10am

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”

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.)

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

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.

Advised texts – astrophysical applications:

• 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

Online Resources:

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

Outline and structure of the lectures:

• Lectures 1–4: – Introduction and scope of the course

– particle properties from the detector point of view

– particle glossary

– Kinematics

– cross-sections and decay rates.

Outline and structure of the lectures:

• Lectures 5–10:– Interactions of fast particles in a medium.

– Ionisation by charged particles

– Quantitative description of ionisation energy loss.

– Other energy loss processes

– Showering processes.

Outline and structure of the lectures:

• Lectures 11–12 (Information from detectors):– Position and timing measurement.

– Momentum, energy and velocity measurement.

– Measurement errors• counting fluctuations.

Outline and structure of the lectures:

• Lectures 13–18: – The general purpose detector.

– Some specific detector technologies.

– Technology choices for different applications.

What is a Particle?

e.g. EM radiation/photons:

Radio/microwave

Visible

X-ray/-ray

Ene

rgy

Wav

elen

gth Particle

behaviour becoming

more evident

FrequencyWavelength

EnergyMomentum

Wave Particle

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

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!)

-emitter

Relativity and QM

• Compton scattering of from electron:– Rest mass of = 0 eV

– Rest mass of electron = 511 keV

– Typical energy of ~ 10 MeV

– Need to use both relativity and QM

-emitter

e

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, , , e, ,

• Gauge bosons– , W , Z, gluons ( only is observed directly )

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

Particle glossary

• Most important particle properties from the detector point of view are:

– Mass

– Charge (electric, “strong”, “weak”)• Interactions ( EM, strong , weak )

– Lifetime

Stable particles

• Can be used as beam particles or for “low-energy physics”

• Decay prohibited by conservation laws– Photon ( )

– Neutrinos ( )

– Electron/positron

• Proton/antiproton

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, lepton

mc /

n: 3×1011m : 6km

1-10cm

50-200m

At high energy, 90% of detected particles from an hadronic interaction are charged pions!

,K,K0L: 5-50m

Very short-lived particles

• Detectable only by their decay products

• Electromagnetic decays to photons or lepton pairs– Includes 0 giving high-energy photons

• Strongly decaying “resonances” cm= 180nm

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.

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

Relativistic relations

• Special relativity applies to inertial ( ie. Not accelerating ) frames.

• Needed in most particle interaction physics.

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)

“Length” of a 4-vector

• Length of a 3-vector doesn’t change under rotations in (three-) space: x2 + y2 + z2 = x’2 + y’2 + z’2 = constant

• Lorentz 4-vectors are such that their “length” (magnitude) does not change under Lorentz transformation:

xx= x`x` = x02 – (x1

2+x22+x3

2) = constant

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 Contra+covariant pairs)

• ( Don’t worry about the terminology – included only for completeness.)

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.

bababababa ii

ic

3

033221100

bac

pcEpps222

/

minus sign comesfrom minus in spacecomponent of p

The Lorentz Transformation• Lorentz transformation:

))/(()/( βpcEγ=cE'

))/(( cEβpγ=p'

c/ 21/1

))(( ctβxγ=x' )( βxctγ=ct'

Energy, momentum and mass

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

Epc=β /

cmcp=E 42222

cγm=E 2 βγmc=p

2/ mcE

1 c

Useful Reference Frames• CM frame is Centre-of-Mass or

Centre-of-Momentum– “Rest frame” for a system of particles

– I.e. pi=0 ( where p is the usual 3-vector)

• LAB frame – may be:– Rest frame of some initial particle, or

– CM frame,or

– Neither

Invariant Quantities –Invariant Mass

• Lorentz invariant quantities exist for individual particles and systems.

• Invariant mass of a system:

))((..1..1

2

Ni

iNi

ippps

),(..1..1..1

NiNi

iNi

ipEp

2

..1

2

..1

Ni iNi

ipEs

Invariant Mass

• Invariant mass is equivalent to the CM frame energy for a particle system– If (pi)=0 then

• NB within a frame pi =constant

– (conservation of momentum)

2

..1

22

..1 ..1

Nii

Nii EpE

Nis

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

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 mtM,Ef

pmEp btb , pmEs btb22

mEmms tbbt 222 Em bt

s 2

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 :

mmmM

Et

btb

2

222

s

Mass of Short-lived Particle

• From invariant mass of its decay products, e.g. 2-body:

– How to measure ma? mc,Ec

ma

mb,Eb

bc

Two-body Decay

• Initial invariant mass s = ma2

• Final invariant mass =

• If Eb, Ec >> mc , mc then Eb, Ec~ pb, pc

• So,

ppEE cbcb 22

ppEEs cbcb .22

abcba CosEEm 122

q² for a Scattering Reaction

• For E,E` >> m• In elastic scattering, can use energy conservation

to get energy lost by incident particle ...

p`,E`

m,p,E

CosppEEmq '2'22 22

ppEEq ','

2'4 22 SinEEq

mt

Energy Loss in Elastic Scattering

• Energy transfer to target:

• Maximum energy transfer in scatter:

• Quoted without proof…

p`,E`

m,p,E

'22 EEmq t m

qEt2

2

mt

)/(2

22

max/21

2

mmmmm

Ttt

t

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

'

mcpc

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.

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)

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.

Cross-section and Interaction rate.

• For fixed target, with a target larger than the beam

• W=r L – W = interaction rate– r = rate of incoming particles– = number of target particles per unit volume– L = thickness of target– = cross-section for interaction

L

r

Cross-section and Interaction rate.

• For fixed target, in terms of particle flux, J• W=J n

– W = interaction rate– J = Flux: particles per unit area per unit time.– n = total number of particles in target.– = cross-section for interaction

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 I=nf

• Factors n1n2f/A normally called the Luminosity, L

LW

A

fnnW 21 Af

IIW 21

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 )

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:

beam

dd ),(

dd

dnJdW ),(..

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:

2

0

1

1

),(cos

d

ddd

21

2

dEdE

d

Decay Width,

• The lifetime of particles can tell us about the strength of the interaction in decay process (and about channels available)

• Decay rate, W=1/ (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

/~mE ~. tE

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

/ W

Ntot ....21

Decay Width Example

• Invariant mass of W = 80GeV

• Width=2.2GeV• Mean life

3.25 x 10-25s• c~10-16m

(I.e. less than “size” of proton )

/

End of Kinematics