particle physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · introduction to...

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1 Particle Physics Dr M.A. Thomson Made on 30-Aug-2000 17:24:02 by konstant with DALI_F1. Filename: DC056698_007455_000830_1723.PS DALI Run=56698 Evt=7455 ALEPH Z0<5 D0<2 RO TPC 0 -1cm 1cm X" 0 -1cm 1cm Y" (φ -175)*SIN(θ) θ=180 θ=0 x x x - x - x x - - - - - - x x - - - x x x x x - - - x x x x - x x - - x - - x - - x - x - - x x - - x o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 15 GeV 3 Gev EC 6 Gev HC Part II, Lent Term 2004 HANDOUT I Comments/Corrections to : [email protected] see : http://www.hep.phy.cam.ac.uk/ thomson/particles/ Dr M.A. Thomson Lent 2004

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Page 1: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

1

Particle Physics

Dr M.A. Thomson

Made on 30-A

ug-2000 17:24:02 by konstant with D

AL

I_F1.Filenam

e: DC

056698_007455_000830_1723.PS

DALI

Run=56698 Evt=7455 ALEPH

Z0<5 D0<2 RO TPC

0 −1cm 1cm X"0

−1cm 1cm

Y"

(φ−175)*SIN(θ)

θ=180 θ=0

x

x

x

x

x

x

x

x

x

x

xx

x

x

x

x−

x

x

xx

x

x

x

−x

−x

x −

−xx

− −

x

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o

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o

o

oo

oo

o

ooo

oo

o

o

o

o

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o

oo

o

o

o

15 GeV

3 Gev EC6 Gev HC

Part II, Lent Term 2004HANDOUT I

Comments/Corrections to : [email protected] : http://www.hep.phy.cam.ac.uk/thomson/particles/

Dr M.A. Thomson Lent 2004

Page 2: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

2

Course Synopsis

Overview:

❶ Introduction to the Standard Model

❷ Quantum Electrodynamics QED

❸ Strong Interaction QCD

❹ The Quark Model

❺ Relativistic Quantum Mechanics

❻ Weak Interaction

❼ Electroweak Unification

❽ Beyond the Standard Model

Recommended Reading:

Particle Physics : Martin B.R. and G. Shaw

(2nd Edition, 1997)

Introduction to High Energy Physics :

D.H.Perkins (4th Edition, 2000)

Dr M.A. Thomson Lent 2004

Page 3: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

3

Introduction to theStandard Model

Particle Physics is the study of

MATTER: the fundamental constituents that make

up the universe - the elementary particles

FORCE: the basic forces in nature i.e. the

interactions between the elementary

particles

Try to categorize the PARTICLES and FORCES in

as simple and fundamental manner as possible.

Current understanding embodied in the

STANDARD MODEL

Explains all current experimental

observations.

Forces described by particle exchange.

It is not the ultimate theory - many mysteries.

Dr M.A. Thomson Lent 2004

Page 4: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

4

The Briefest History ofParticle Physics

the Greek View

400 B.C : Democritus : concept of matter comprisedof indivisible “atoms”.

“Fundamental Elements” : air, earth, water, fire

Newton’s Definition

1704 : matter comprised of “primitive particles ...incomparably harder than any porous Bodiescompounded of them, even so very hard, as never towear out or break in pieces.”

A good definition - e.g. kinetic theory of gases.

CHEMISTRY

Fundamental particles : “elements” Patterns 1869 Mendeleev’s Periodic Table

sub-structure Explained by atomic shell model

ATOMIC PHYSICS

Bohr Model Fundamental particles : electrons orbiting the atomic

nucleus

Dr M.A. Thomson Lent 2004

Page 5: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

5

NUCLEAR PHYSICS Patterns in nuclear structure - e.g. magic numbers in

the shell model sub-structure Fundamental particles :

proton,neutron,electron,neutrino Fundamental forces :

ELECTROMAGNETIC : atomic structureSTRONG: nuclear bindingWEAK: -decay

Nuclear physics is complicated : not dealing withfundamental particles/forces

1960s PARTICLE PHYSICS Fundamental particles ? : far too many !

Again Patterns emerged:

n p

Σ- Σ0 Σ+

ΛΞ- Ξ+

Σ*

Ξ*

Ω-

B+S

I3

sub-structure - explained by QUARK model : u,d,c,s

TODAY Simple/Elegant description of the fundamental

particles/forces

These lectures will describe our current understandingand most recent experimental results

Dr M.A. Thomson Lent 2004

Page 6: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

6

Modern Particle Physics

Our current theory is embodied in the

Standard Model

which accurately describes all data.

MATTER: made of spin-

FERMIONS of whichthere are two types. LEPTONS: e.g. e, QUARKS: e.g. up quark and down quark

proton - (u,u,d) + ANTIMATTER: e.g. positron e,

anti-proton - (,,)

FORCES: forces between quarks and leptonsmediated by the exchange of spin-1 bosons - theGAUGE BOSONS.

e-

e-

γe-

e-

Electromagnetic Photon Strong Gluon Weak W and Z

Gravity Graviton

Dr M.A. Thomson Lent 2004

Page 7: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

7

Matter: Generation

Almost all phenomena you will have encounteredcan be described by the interactions of FOURspin-half particles : “the First Generation”

particle symbol type chargeElectron lepton -1Neutrino lepton 0Up Quark quark +2/3Down Quark quark -1/3

The proton and the neutron are the lowest energystates of a combination of three quarks:

Proton = (uud)

Neutron = (udd)

du u

ud d

e.g. beta-decay viewed in the quark picture

d dd uu u

e-

νe

n p

d→W-ue-νe

Dr M.A. Thomson Lent 2004

Page 8: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

8

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Page 9: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

9

The LEPTONSPARTICLES which do not interact via theSTRONG interaction - “colour charge” = 0.

spin 1/2 fermions 6 distinct FLAVOURS of leptons 3 charged leptons : ,,

Muon () - heavier version of the electron( )

3 neutral leptons : neutrinos

Gen. flavour Approx. Mass Electron -1 0.511 MeV/

Electron neutrino 0 massless ? Muon -1 105.7 MeV/

Muon neutrino 0 massless ? Tau -1 1777.0 MeV/

Tau neutrino 0 massless ?

+ antimatter partners,

Neutrinosstable and (almost ?) massless:

Mass 3 eV

Mass 0.19 MeV

Mass 18.2 MeV

Charged leptons only experience the

ELECTROMAGNETIC and WEAK and forces.

Neutrinos only experience the WEAK force.

Dr M.A. Thomson Lent 2004

Page 10: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

10

The Quarks spin 1/2 fermions fractional charge 6 distinct FLAVOURS of quarks

Generation flavour charge Approx. Mass down d -1/3 0.35 GeV/

up u +2/3 0.35 GeV/

strange s -1/3 0.5 GeV/

charm c +2/3 1.5 GeV/

bottom b -1/3 4.5 GeV/

top t +2/3 175 GeV/

Mass quoted in units of GeV/. To be comparedwith = 0.938 GeV/.

Quarks come in 3 “COLOURS”

“RED”, “GREEN”, “BLUE”

COLOUR is a label for the charge of the strong interaction.

Unlike the electric charge of an electron (), the strong

charge comes in three “orthogonal colours” RGB.

quarks confined within HADRONS

e.g. p (), ()

Quarks experience the ALL forces:ELECTROMAGNETIC, STRONG and WEAK(and of course gravity).

Dr M.A. Thomson Lent 2004

Page 11: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

11

HADRONS single free QUARKS are NEVER observed quarks always CONFINED in HADRONS

i.e. ONLY see bound states of or (qqq). HADRONS = MESONS,BARYONS

MESONS = qq

A meson is a bound state of a QUARK and an ANTI-QUARK All have INTEGER spin 0,1,2,... e.g.

charge, = =

is the ground state (L=0) of

there are other states, e.g. , ...

BARYONS= qq q

All have half-INTEGER spin ...

e.g.

e.g. Plus ANTI-BARYONS=

e.g. anti-proton

e.g. anti-neutron

Composite relatively complicated

Dr M.A. Thomson Lent 2004

Page 12: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

12

ForcesConsider ELECTROMAGNETISM and scattering of electronsfrom a proton:

Classical Picture

Electrons scatter in the static potential of the proton:

NEWTON : “...that one body can act upon another at adistance, through a vacuum, without the mediation ofanything else,..., is to me a great absurdity”

Modern Picture

Particles interact via the exchange of particles GAUGEBOSONS. The PHOTON is the gauge bosons ofelectromagnetic force.

Time

Spac

e

p p

e- e-

γ

Early next week we’ll learn how to calculate QuantumMechanical amplitudes for scattering via Gauge BosonExchange.

Dr M.A. Thomson Lent 2004

Page 13: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

13

All (known) particle interactions can be explained by 4fundamental forces:

Electromagnetic, Strong, Weak, Gravity

Relative strengths of the forces between two protons just incontact ( m):

du u

du u

Strong 1Electromagnetic

Weak

Gravity

At very small distances (high energies) - UNIFICATION

10-26

10-23

10-20

10-17

10-14

10-11

10-8

10-5

10-2

10

10 4

10 7

10-4

10-3

10-2

10-1

1 10 102

103

Distance (fm)

For

ce (

GeV

/fm

)

GravitationalForce

Weak Force

Strong Force(quarks)

Strong Force(hadrons)

ElectromagneticForce

Dr M.A. Thomson Lent 2004

Page 14: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

14

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Page 15: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

15

Practical Particle Physics

OR How do we study the particles and forces ?

Static Properties:MassSpin, Parity : J e.g.

Magnetic Moments (see Quark Model handout)

Particle Decays :Allowed/Forbidden decays conservation lawsParticle lifetimes

Force Typical Lifetime

STRONG s

E-M s

WEAK s

Accelerator physics - particle scattering:

Direct production of new massive particles inMatter/Antimatter ANNIHILATION

Study of particle interaction cross sections

FORCE Typical Cross section

STRONG mb

E-M mb

WEAK mb

Scattering and Annihilation in Quantum Electrodynamicsare the main topics of the next two lectures. Particle decaywill be covered later in the course.

Dr M.A. Thomson Lent 2004

Page 16: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

16

NATURAL UNITS

SI UNITS:kg m s

For everyday physics SI units are a natural

choice : (Widdecombe) kg

Not so good for particle physics:

proton kg

use a different basis - NATURAL UNITS

based on language of particle physics, i.e.

Quantum Mechanics and Relativityunit of action in QM (Js)velocity of light (ms)

Unit of energy : GeV = eV= - J

Natural UnitsGeV h c

Units becomeEnergy GeV Time (GeV/

Momentum GeV/c Length (GeV/c)

Mass GeV/c Area (GeV/c)

Dr M.A. Thomson Lent 2004

Page 17: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

17

Simplify (!) by choosing

All quantities expressed in powers of GeV

Energy GeV Time GeV

Momentum GeV Length GeV

Mass GeV Area GeV

Convert back to S.I. units by reintroducing

‘missing’ factors of and

EXAMPLE:

Heaviside-Lorentz units

Fine structure constant :

becomes

Dr M.A. Thomson Lent 2004

Page 18: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

18

Relativistic Kinematics

In Special Relativity and transformfrom frame-to-frame, BUT

---

are CONSTANT (invariant interval,invariant mass)Using natural units:

EXAMPLE: at rest.

π-

µ-

νµ

Conservation of Energy

Conservation of Momentum

Eπ = Eµ + Eν

0 = pµ + pν

(assume mν= 0)

but

!" #"

!"

"

"$

(see Question 1 on the problem sheet)Dr M.A. Thomson Lent 2004

Page 19: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

19

Colliders and

Consider the collision of two particles:

pµ pµp1(E1,p1) p2(E2,p2)

The invariant quantity

is the energy in the zero momentum frame.

It is the amount of energy available to interaction e.g.

the maximum energy/mass of a particle produced in

matter-antimatter annihilation.

Dr M.A. Thomson Lent 2004

Page 20: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

20

pµ pµp1(E,p) p2(m2,0)Fixed Target Collision

e.g. 100 GeV proton hitting a proton at rest:

pµ pµp1(E,p) p2(E,-p)Collider Experiment

Now consider two protons colliding head-on.

If then and

e.g. 100 GeV proton colliding with a 100 GeV proton :

In a fixed target experiment most of the proton’s

energy is wasted - providing momentum to the C.O.Msystem rather than being available for the interaction.

(NOTE: UNITS G = Giga =10, M = Mega =10)

Dr M.A. Thomson Lent 2004

Page 21: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

21

Summary

FERMIONS : spin

Chargeleptons

e-

νe

µ-

νµ

τ-

ντ

-10( ) ( ) ( )

quarks e.g. proton ( )uud

ud

cs

tb

+-23- -13

( ) ( ) ( )+ anti-particles

Fermion interactions = exchange of spin 1 bosons

e-

e-

γe-

e-

BOSONS : spin 1

GluonZ-bosonW-bosonPhoton

gZ0W±γ 0

91.2 GeV

80.3 GeV

0

Electromagnetic

Weak (CC)

Weak (NC)

Strong (QCD)

ForceMass

Dr M.A. Thomson Lent 2004

Page 22: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

22

Theoretical

Framework

e-

e+

e-

e+

νe

νe

γe

Z

e-

e+

e-

e+

νe

νe

Z eZ

e-

e+

e-

e+

νe

νe

γe

W

e-

e+

e-

e+

νe

νe

Z eW

e-

e+

e-

e+

νe

νe

γe

Z

e-

e+

e-

e+

νe

νe

Z eZ

e-

e+e-

e+

νe

νe

γe

W

e-

e+e-

e+

νe

νe

Z eW

e-

e+e-

e+

νe

νe

ZW

e-

e+e-

e+

νe

νe

ZZ

e-

e+

e-

e+

νe

νe

ZW

e-

e+e-

e+

νe

νe

ZZ

e-

e+

e-

e+

νe

νe

γ

W

W

e-

e+

e-

e+

νe

νe

Z

W

W

e-

e+

e-

e+

νe

νeγ

e Z

e-

e+

e-

e+

νe

νeZ

e Z

e-

e+

e-

e+

νe

νe

γ

e W

e-

e+

e-

e+

νe

νe

Z

e W

e-

e+

e-

e+

νe

νe

γ

eZ

e-

e+

e-

e+

νe

νe

Z

eZ

e-

e+

e-

e+

νe

νe

γe

W

e-

e+

e-

e+

νe

νe

Z

e

W

e-

e+

e-

e+

νe

νe

Zνe

W

e-

e+

e-

e+

νe

νe

Zνe

Z

e-

e+

e-

e+

νe

νeZ

νe

W

e-

e+

e-

e+

νe

νeZνe

Z

e-

e+

e-

e+

νe

νe

γ

W

W

e-

e+

e-

e+

νe

νe

Z

W

W

e-

e+

e-

e+

νe

νeW

νe

W

e-

e+

e-

e+

νe

νe

W

νe

Z

e-

e+

e-

e+

νe

νe

Wνe

W

e-

e+

e-

e+

νe

νe

Wνe

Z

e-

e+

e-

e+

νe

νeW

e

γ

e-

e+

e-

e+

νe

νeW

eZ

e-

e+

e-

e+

νe

νeWe

γ

e-

e+

e-

e+

νe

νeW

e

Z

e-

e+

e-

e+

νe

νe

W

γW

e-

e+

e-

e+

νe

νe

W

Z

W

e-

e+

e-

e+

νe

νe

W

W

Z

e-

e+

e-

e+

νe

νe

W

W

γ

e-

e+

e-

e+

νe

νeγe Z

e-

e+

e-

e+

νe

νeZ

e Z

e-

e+

e-

e+

νe

νeγe W

e-

e+

e-

e+

νe

νeZ

e W

e-

e+

e-

e+

νe

νe

W

νe

W

e-

e+

e-

e+

νe

νe

W

νe

Z

e-

e+

e-

e+

νe

νe

e

γ

Z

e-

e+

e-

e+

νe

νe

e

Z

Z

e-

e+

e-

e+

νe

νeνe

W

W

e-

e+

e-

e+

νe

νe

e

Z

γ

e-

e+

e-

e+

νe

νe

e

Z

Z

e-

e+

e-

e+

νe

νe

γe

W

e-

e+

e-

e+

νe

νe

Ze

W

e-

e+

e-

e+

νe

νeWνe

Z

e-

e+

e-

e+

νe

νeeZ

γ

e-

e+

e-

e+

νe

νeeZ

Z

produced by GRACEFIG

(Lowest order Feynman diagrams for )

Dr M.A. Thomson Lent 2004

Page 23: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

23

Theory : Road-map

Macroscopic MicroscopicSlow

Fastv → c

Classicalmechanics

Quantummechanics

SpecialRelativity

QuantumField Theory

To describe the fundamental interactions ofparticles we need a theory of RelativisticQuantum Mechanics.

OUTLINE: Klein-Gordon Equation

Anti-MatterYukawa Potential

Scattering in Quantum MechanicsBorn approximation (revision)

Quantum Electrodynamics2 Order Perturbation TheoryFeynman diagrams

Quantum Chromodynamicsthe theory of the STRONG interaction

Dirac Equationthe relativistic theory of spin-1/2 particles

Weak interaction Electro-weak Unification

Dr M.A. Thomson Lent 2004

Page 24: Particle Physics - hep.phy.cam.ac.ukthomson/particles/lectures/pp2004_intro.pdf · Introduction to High Energy Physics: D.H.Perkins ... ★ Accelerator physics - particle scattering:

24

The Klein-Gordon EquationSchrodinger Equation for a free particle can bewritten as

with energy and momentum operators:

giving ==:

which has plane wave solutions (=, =!)

"%%

Schrodinger Equation: 1st Order in time derivative () 2nd Order in space derivatives () Not Lorentz Invariant !!!!

Schrodinger Equation cannot be used to describethe physics of relativistic particles. We need arelativistic version of Quantum Mechanics. Ourfirst attempt is the Klein-Gordon equation.

Dr M.A. Thomson Lent 2004

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25

From Special Relativity (nat. units ):

from Quantum Mechanics ( ):

Combine to give the Klein-Gordon Equation:

Second order in both space and time derivatives -by construction Lorentz invariant.

Plane wave solutions %% solutions give

ALLOWS NEGATIVE ENERGY SOLUTIONS

Anti-Matter

Dr M.A. Thomson Lent 2004

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26

Static Solutions of K-G Equation

Consider a test particle near a mas-

sive source of bosons which mediate aforce. Solutions of the K-G equation in-terpreted as either boson wave-function

or as the potential

The bosons satisfy the Klein-Gordon Equation:

If we now consider the static case: K-G becomes

a solutions is

#

where the constant gives the strength of therelated force and is the mass of the bosons.

!

the Yukawa Potential - originally proposed byYukawa as the form of the nuclear potential.

NOTE: for we recover the Coulombpotential:

.

(the connection between the Yukawa Potential and a force

mediated by the exchange of massive bosons will soon

become apparent)Dr M.A. Thomson Lent 2004

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27

Anti-Particles

Negative energy solutions of the KG Equation ?

Dirac : vacuum corresponds to the state with all

states occupied by two electrons

of opposite spin.

Pauli exclusion principle

prevents ve energy electron

falling intove energy state

E

0

-mec2

+mec2

In this picture - a hole in the normally full set

of states corresponds to

Energy = i.e. $

Charge =%, i.e. % $

A hole in the negative energy particle electron

states corresponds to a positively charged,

positive energy anti-particle

Dr M.A. Thomson Lent 2004

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28

In the Dirac picture predict pair creation

and annihilation.

γ→e+e-

E

0

-mec2

+mec2

e+e-→γE

0

-mec2

+mec2

1931 : positron () was first observed.

Now favour Feynman interpretation: solutions represent negative energy

particle states traveling backward in time.

Interpreted as positive energy anti-particles,

of opposite charge, traveling forward in time.

Anti-particles have the same mass and equal

but opposite charge.

Not much more than noting (for a plane wave)

(more on this next lecture)

Dr M.A. Thomson Lent 2004

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29

Bubble chamber photograph of the process

. Only charged particles are visible -

i.e. only charged particles leave TRACKS

The full sequence of events is :

enters from bottom

Charge-exchange on a proton:

decays (lifetime s) to two photons:

Finally

(the other photon is not observed within the region of thephotograph)

Dr M.A. Thomson Lent 2004

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30

Rates and Cross SectionsFor reaction

The cross section, , is defined as the reaction rate

per target particle, , per unit incident flux,

where is given by Fermi’s Golden Rule.

Example: Consider a single particle of type traversing a beam (area) of particles of type ofnumber density .

v

v

δt

A

In time Æ traverses a region containing Æparticles of type .

σ

Interaction probability defined as ef-

fective cross sectional area occupied

by the Æ particles of type

&Æ '(

' &Æ (

Therefore the reaction rate is .

(see Question 2 on the problem sheet)Dr M.A. Thomson Lent 2004

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31

Scattering in Q.M.REVISION (see Dr Ritchie’s QMII transparencies 11.8-11.15)

NOTE: Natural Units used throughout

Consider a beam of particles scattering in Potential

pi

pf

θ

Scattering rate characterized by the interaction crosssection

number of particles scattered/unit time

incident flux

Use FERMI’S GOLDEN RULE for Transition rate, &:

& !

where ! is the Matrix Element and = density offinal states.

1st Order Perturbation Theory using plane wave

solutions of form " %%.

Require :

wave-function normalization matrix element in perturbation theory expression for flux expression for density of states

Dr M.A. Thomson Lent 2004

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32

Normalization: Normalize wave-functions to one particle ina box of side #

" #

" #

Matrix Element: this contains the physics of the interaction

! '(

!

'($%

!

"% % % "%%$%

!

#

%% % $%

where % % %

Incident Flux: Consider a “target” of area A and a beam ofparticles traveling at % towards the target. Anyincident particle with in a volume & will cross the targetarea every second. Flux = number of incident particlescrossing unit area per second :

)* &

&

where is number density of incident particles = per#

)* # #

Dr M.A. Thomson Lent 2004

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33

Density of states: for box of side# states are given byperiodic boundary conditions:

' #

#

#

#py

px

pz

Volume of single state in

momentum space:

Number of final states between $:

$" $$#

$"$ $#

In almost all scattering process considered in theselectures the final state particles have and to agood approximation .

$"

$$"

$

$

$

$#

$

Dr M.A. Thomson Lent 2004

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34

Putting all the separate bits together:

$

flux!

#

#

%% $%

#

$

$

$

%% $%

The normalization cancels and, in the limit where theincident particles have % and the out-going particleshave , arrive at a simpleexpression. Apply to the elastic scattering of a particle fromin a Yukawa potential.

Scattering from Yukawa Potential

!

pi

pf

θpi

pf p=pi-pf

θ

!

%% $%

!

!

%

+, ($$($

Where for the purposes of the integration, the z-axis is beendefined to lie in the direction of % and ( is the polar anglewith respect to this axis.

Dr M.A. Thomson Lent 2004

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35

!

!

%

+, ($($$

Integrate over $ and set ) $-+ (.

!

%$$)

*%

% % $

*%

% %$

*%

*%

*%

*%

*%

%

*

+

giving$

$

%

Scattering in the Yukawa potential introduces a term % in the denominator of the matrix element -this is known a the propagator.

Dr M.A. Thomson Lent 2004

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36

Rutherford Scattering

Let and replace !,

!

gives Coulomb potential: ,

Hence for elastic scattering in the Coulomb potential:

$

$

#,

%

pi

pf p=pi-pf

θ

$

$

#,

# +,

$

$

,

! +,

e.g. The upper points are theGieger and Marsden data (1911)

for the elastic scattering of particles as they traverse thingold and silver foils. The scatter-

ing rate, plotted versus

,

follows the Rutherford formula.

(note, plotted as log vs. log)

Dr M.A. Thomson Lent 2004