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Elementary Particle Physics

David Milstead

milstead@physto.seA4:1021

tel: 5537 8663/0768727608

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Format● 19 lecture sessions● 2 räkneövningnar● Homepage http://www.physto.se/~milstead/fk7003/course.html● Course book

Particle Physics (Martin and Shaw,3rd edition, Wiley) ● Earlier editions can be used – handouts to be provided where appropriate.

● Supplementary books which may be useful but which are not essential Introduction to Elementary Particles (Griffiths, Wiley) Subatomic Physics (Henley and Garcia, World Scientific) Particles and Nuclei (Povh, Rith, Scholz and Zetsche, Springer) Quarks and Leptons (Halzen and Martin, Wiley)

● Assessment 2 x inlämningsuppgifter tenta

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Lecture outlineLecture Topic Martin and

Shaw (2nd edition)

Martin and Shaw (3rd edition)

Extra info

1 Antiparticles, Klein-Gordon and Dirac equations, Feynman diagrams, em and weak forces

1 1 Handout

2 Units, fundamental particles and forces, Charged leptons and neutrino oscillations

2 2 Handout

3 Quarks and hadrons, multiplets, resonances 2,5 3

4 Räkneövning 1

5 Symmetries: Noether’s theorem, C, P and T 4 5

6 Symmetries: C, P, CP violation, CPT 10 10

7 Hadrons: isospin and symmetries 5 6

8 Hadrons: bound states, quarkonia 6 6

9 Quantum chromodynamics: asymptotic freedom, jets, elastic lepton-nucleon scattering

7 7

10 Räkneövning 2

11 Relativistic kinematics: four-vectors, cross section Appendix B Appendix B Handouts

12 Deep-inelastic lepton-nucleon scattering: quark parton model, structure functions, scaling violations, parton density functions

7 7 Handouts

13 Weak interaction: charged and neutral currents, Caibbo theory 8 8

14 Standard Model: renormalisation, Electroweak unification, Higgs 9 9

15 Beyond the Standard Model: hierarchy problems, dark matter, supersymmetry, grand unified theories

11 11

16 Accelerators – synchrotron, cylcotron + LHC 3 4 Handouts

17 Detectors: calorimeter, tracking, LHC detectors, particle interactions in matter

3 4 Handouts

18 Revision lecture 1

19 Revision lecture 2

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Particle physics is frontier research of fundamental importance.

particle physics research

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The aim of this course

● Survey the elementary constituents in nature Identification and classification

of the fundamental particles Theory of the forces which

govern them over short distances

● Experimental techniques Accelerator Particle detectors

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Lecture 1 Basic concepts

Particles and antiparticles

Klein-Gordon and Dirac equations

Feynman diagrams

Electromagnetic force

Weak force

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Going beyond the Schrödinger equation

Classical mechanics

Quantum mechanics (Schrödingers equation)

Relativistic mechanics

Quantum field theory (Dirac, Klein-Gordon equations,QED, weak, QCD)

2 2 2 2 4

Collider experiments typically involve energies of several hundred GeV.

Eg a proton (mass 1 GeV) with 50 GeV energy

- relativity effects can't be ignored.

Dirac, Klein-Gordon equati

E p c m c

E pc

ons and quantum field theory necessary.

small

fast

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Implications of introducing special relativity

2 2 2 4

2 2 2 4 2 2 2 4

Consider a particle of charge , mass with momentum moving along the -axis

What is its energy ?

Special relativity gives us a choice: (1.1)

(1.2)

q m p x

E p c m c

E p c m c E p c m c

(1.3)

Surely the negative energy solution is unphysical and daft.

Can't we just ignore it ?

No - from quantum mechanics, every observable must have a complete

set of eigenstates. The negative energy states are needed to form

that complete set.

They must mean something....

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Negative energy states

22 2,

Plane wave corresponding to momentum along -direction.

Positive energy solution: (1.4)

(1.5) (moves to the right)

Negative energy solution:

px E ti

p x

x t Ne E pc mc

Epx E t x t

p

x

, 0 (1.6)

(1.7)

Negative energy state moving forwards in time is equivalent to a

positive energy state moving backwards in time.

px E ti

t Ne E E

EEx t t

p p

t

x

t

x

0E

, 0E E

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What does a particle moving backwards in time look like ?

-

,

What are the implications of moving backwards in time ?

Lorentz force on particle (charge ) in a -field travelling forwards in

time at a certain point in space and time and

(1.

q B

r t

F r t qv B

2

2

2 2

2 2

,

- .

, ,

8) (1.9)

Force on a particle with charge moving backwards in time:

(1.10) easily rearranged to (1.9)

d r drF r t m q B

dt dtq dt dt

d r dr d r drF r t m q B F r t m q B

dt dt dt dt

The equation of motion of charge q moving backwards in time in a magnetic field is the same as the equation of motion of a particle with charge -q moving backwards in time.

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Antiparticles Special relativity permits negative energy solutions and quantum

mechanics demands we find a use for them.

(1) The wave function of a particle with negative energy moving forwards in time is the same as the wave function of a particle with positive energy moving backwards in time.

Ok, the negative energy solutions must be used but we can convert them to positive energy states if we reverse the direction of time when considering their interactions.

(2) A particle with charge q moving backwards in time looks like a particle with charge –q moving forwards in time.

General argument that a particle with negative energy and charge q behaves like a particle with positive energy and charge -q.

We expect, for a given particle, to see the ”same particle” but with opposite charge: antiparticles.

Antiparticles can be considered to be particles moving backwards in time - Feynman and Stueckelberg.

Hole theory (not covered) provides an alternative, though more old fashioned way of thinking about antiparticles.

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Electron and the positron

1897 e- discovered by J.J. Thompson

1932 Anderson measured the track of a cosmic ray particle in a magnetic field.

Same mass as an electron but positive charge

The positron (e+ ) - anti-particle of the electron

Nobel prize 1936

Every particle has an antiparticle.Some particles, eg photon, are their own antiparticles. Special rules for writing particles and antiparticles, eg antiproton p, given in next lecture.

1.5 ( )

(to left)

B T out of page

F q v B

pr

eB

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Klein-Gordon equation

2

2

,

,, -

2

Start with Schrödinger equation:

Free particle: (1.11)

(1.12) , (1.13)

This is the quantum analogue to the non-relativistic conservati

i p r Et

r t Ne

r ti r t E i p i

t m t

2

222 2

22 2 2 2 2 4

2

1

2

,,

on of energy:

(1.14)

Try to build a relativistic wave equation (Oskar Klein, Walter Gordon 1927)

Based on (1.1)

(1.15) The Klein Gordon Equation

Two

E mv

E pc mc

r tc r t m c

t

22 2

*

22 2

,

,

plane wave solutions:

(1.16); Energy

(1.17)

Energy= -

Two possible solutions for a free particle. One has posi

i p r E t

i p r E t

r t Ne E p c mc

r t N e

i E E p c mct

tive energy and the other "negative energy"

eigen values. Schrödinger's equation only had one energy solution.

Special relativity demands antiparticles.

Klein-Gordon equation describes spin-0 bosons.

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The Dirac Equation

3

2 2

1

1

2

,, ,

ˆ

Dirac (1928):

Relativistic equation for spin particles (fermions).

Look for an equation based on form:

i (1.18)

Hamiltonian (1.19)

, co-effi

ii i

r tH r t r t

t

H i c mc c p mcx

2 21, 1, 0 0 ( )

,

cients constrained by need to satisfy the Klein-Gordon equation (1.15)

and (1.20)

are not numbers - represent as matrices

4-component vector :

i i i i j j i i j

1

2

3

4

,

,,

,

,

,

, ,

(1.21)

Plane wave solutions: (1.22)

are spinors

Four solutions:

Two positive energy energies corresponding to two spin st

p r Eti

r t

r tr t

r t

r t

r t u p e

u p r t

E E

1

21

2

ates of spin particles

Two negative energy solutions corresponding to two spin states of spin particlesE E E

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Implications of the Dirac Equation

1

21

2

Dirac equation implies that for every spin particle there is

(a) a corresponding spin antiparticle

(b) two spin states

Spin and antiparticles arises as a consequence of treating quantum mecha

22

2.002..

211

2

nics relativistically

Intrinsic magnetic moment of elementary particle: (1.23)

Prediction of Dirac equation for electrons:

Experiment

Precision experimental result:

eg S

mcg

g

g

12

12

59652180.7 0.3 10

21159652153.5 28 10

2Dirac prediction + quantum corrections for :

Quantum electrodynamics is "the best theory we've got!"

ge

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How particles interact – exchange forces Electromagnetic force

02

A photon is emitted - we don't know its momentum

and we don't know where it is

The "quantum path" between the start and end points

is not like a classical path.

The reaction can take place as

p xp

per the diagram.

- -

--

+

- -

+

+-

-

-

RepulsionAttraction

Easy to visualise but beware this is a

useful but limited "visual toy model"

for the quantum world.

photon

Particles carrying charge interact via the exchange of photons () mass=0, spin=1 (boson)

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Electromagnetic processes

2 2

1 1

( , )

( , )

Two possible interpretations

(1) A particle moves forward in time, emits two photons at and moves

back in time with negative energy to point where it scatters off a photon and

moves for

x t

x t

1 1

1 1

( , )

( , )

ward in time. There is only one particle moving through space and time.

(2) At point an antiparticle-particle pair is produced. The antiparticle moves forward

to point where it annihila

x t

x t tes with another particle producing to two photons.

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Feynman diagrams

Important mathematical tool for calculating

rates of processes - Feynman rules.

Qualitative treatment here but more detailed

treatment later in the course.

Represent any process by contributing diagrams.

Strategy:

(1) Build Feynman diagrams for electromagnetic processes

(2) Consider energy-momentum conservation/violation

(3) Consider how they can be used for simple rate estimates.

(4) Show Feynman diagram formalism for other fundamental forces.

One possible diagram for

e e e e

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(1) Electromagnetic processes

Convention - time flows to the right

The lines do not represent trajectories

of a particle.

Arrow for antiparticle goes "backward in time".

Lines should not be taken as "trajectories" of particles

Interac

2

0

1 1

4 137

tions occur at a vertex.

Rule of thumb: a vertex carries a factor associated

with the probability of that interaction taking place.

Probability

(1.24) Fine structure constant

em

e

c

A basic process.

e e

t

s

vertex

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( ) ( )

Consider all electromagnetic processes

built up from basic processes: to

The basic processes are never seen since

they violate energy conservation (next slide)

They can be combined to make observa

a h

( ) ( )

ble processes:

and

e e e e

e f

(1) Basic electromagnetic diagrams

t

s

( ) vacuum g e e ( ) vacuumh e e

((g) and (h) become clear soon)

vertex

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(2) Is energy conservation violated ?

1 1 1 1

1 1 1 1

1 1

0

0

(annihilation)

Electron and positron in centre-of-mass frame: , (1.24)

Annihilate to form photon , (1.25)

(according

e e e e

e e e e

e e

e e e e

E E p p

E E E p p p

E E E E

2 2 2 2 41 1

1 1

0 0

0!! 0

to conservation of momentum)

,

(1.26)

Nevertheless, the process happens. Two qualitative ways to interpret this

(a) Energy-momentum conservation

e e

E p c m c p m

E E E E

is violated for the short interaction time

Uncertainty principle: violation can happen over time (1.27)

(b) The mass of the photon is not zero (goes off mass-shell) for a short time

Mass chang

tE

t

2

2e (1.28) as permitted by the uncertainty principle (1.29)

Internal lines correspond to virtual particles (we can never see them)

External lines correspond to real particles which can

E cm t

c m

be observed and

always carry the expected mass.

Important: however we think about our diagrams, we measure energy and momentum

violation. The energy and momenta of the real particles always add up

never

1 1 2 2 1 1 2 2

:

(1.30)e e e e e e e e

E E E E p p p p

2 2,

e eE p

t

s

treal particle

virtual particle ()

1 1,

e eE p

1 1,

e eE p

,E p

2 2,

e eE p

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(3) Using Feynman diagrams

Even with a qualitative treatment it is possible to see how the Feynman diagram picture

agrees with observed reactions.

Aim: compare rates of and

Start and consider possib

e e e e

e e

2

le contributing diagrams. Use the simplest possible

diagrams (leading order) - in this case diagrams with two vertices.

Diagrams with two vertices : probability of process occuring

Life gets much easier if you don't think

about things going back in time. Instead, take your

diagram, think of the lines as "rubber" and see how they

can be bent in such a way as to change the time order of

the vertices. Two possibilities here :

(a) emits a photon and goes on to annihilate with

a leading to a photon.

(b) emits a photon and goes on to annihilate with

a leading to a photon

e

e

e

e

.

Usually only one such diagram is shown and the others

implied.Negative energy solutions –antiparticles.QM insists we use them!

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(3) Using Feynman diagrams

3

3

2

2

3

0.7 10

10

Three vertices probability

(1.31)

Observed

Qualitative Feynman diagram picture gives

suppression with (very) rough accuracy.

e e

Rate e eR

Rate e e

R

Full QED calculation gives correct rates.+ 5 other contributions

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QuestionFor the interaction draw three Feynman diagrams which would be

suppressed wrt to those we studied earlier.

e e

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(3) Using Feynman diagrams

6

4

e-

e-

e-

e-

e-

e-

e-

e-

e-

e-

e-

e-

11

137

Two electrons are observed to repell each other:

Many different indistinguishable processes,

eg one-photon, two-photon exchange,

can contribute to the scattering

Coupling is weak

e e e e

higher order processes contribute less and less to the

calculation and can be safely be neglected in any approximate

solution.

+

2

+

….+

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QuestionFor the interaction draw all six possible time ordered Feynman diagrams for

the leading order (3 vertices) processes

e e

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Understanding forces

,A AE p

,X AE p

2 ,0AM c

2

1/22 2 2 4

.

0,

,

Go beyond EM force

Generic force between particles and via exchange of particle

Inititally particle at rest: (1.32)

After vertex:

Particle A: (1.33)

Parti

A

A A A

A B X

A p E m c

p p E E p c m c

1/22 2 2 4

2

2

,

2 ( )

( 0)

0

cle : (1.34)

Energy difference between initial and final state:

(1.35)

(1.36)

(apparent energ

A X X

X A A

X

X p p E E p c m c

E E E m c pc p

M c p

E

max 2

y conservation violation!)

Energy violation can only persist time:

minimum energy violation corresponds to longest time

(1.37)

Max speed

Range of force (1.38)

Electrom

X

X

tE

tM c

v c

RM c

agnetic range due to massless photon.

R

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e- e-e-

W-

ee+

0 2 2

18,

91.2 , 80.4

2 10

Use same formalism as for electromagnetic force

Very brief overview:

Exchange of 3 spin-1 particles: (mass= GeV/c ) , (mass= GeV/c )

range m (tiny - proton "radius" 1W ZW

Z W W

RM c

15

2 2

0

2

4 4

1

4

0 m)

Define coupling constant analagous to fine structure constant

(1.39) (1.24)

analagous to electric charge (very few talk of "weak charge")

WW

W

WW

g e

c c

g

g

c

1

, , .240 137

(1.41) the weak force is only weak due to massesW Z

The weak force

(neutrinos – next lecture)(decay)

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The fundamental forcesDifferent exchange particles mediate the forces:

strongelectromagnetic

weak

Interaction Relative strength

Range Exchange Mass (GeV)

Charge Spin

Strong 1 Short ( fm)

Gluon 0 0 1

Electromagnetic 1/137 Long (1/r2) Photon 0 0 1

Weak 10-9 Short ( 10-3 fm)

W+ W-,Z 80.4,80.4, 91.2

+e,-e,0 1

Gravitational 10-38 Long (1/r2) Graviton ? 0 0 2No quantum field theory yet for gravity

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Summary Antiparticles and spin states are predicted when

when relativity and quantum mechanics meet up! Antiparticles correspond to negative energy states

moving backwards in time. Feynman diagram formalism developed and used

for (very basic) rate estimation Generic approach for all forces Weak force is weak because of the mass of the

exchanged particles.