1 Introduction to the Standard Model

8/13/2019 1 Introduction to the Standard Model

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Particle Physics

Dr M.A. Thomson

M a d e on 3 0 -A u g-2 0 0 0 1 7 : 2 4

: 0 2 b yk on s t a n t wi t h DAL I _F 1 .

F i l e n a m e : D C 0 5 6 6 9 8 _ 0 0 7 4 5 5 _ 0 0 0 8 3 0 _1 7 2 3 .P S

DALI Run=56698 Evt=7455ALEPH

Z0


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Course Synopsis

Overview:

Introduction to the Standard Model

Quantum Electrodynamics QED

Strong Interaction QCD

The Quark Model

Relativistic Quantum Mechanics

Weak Interaction

Electroweak Unication

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


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Introduction to the

Standard 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.



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The Briefest History of

Particle Physicsthe Greek View

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

Fundamental Elements : air, earth, water, re

Newtons Denition 1704 : matter comprised of primitive particles ...

incomparably harder than any porous Bodiescompounded of them, even so very hard, as never to

wear out or break in pieces. A good denition - e.g. kinetic theory of gases.

CHEMISTRY Fundamental particles : elements Patterns 1869 Mendeleevs Periodic Table

sub-structure Explained by atomic shell model

ATOMIC PHYSICS Bohr Model Fundamental particles : electrons orbiting the atomic

nucleus



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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

- 0 +

- +

*

*

-

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



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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 quarkproton - ( 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-

e-

e-Electromagnetic Photon Strong Gluon Weak W and Z

Gravity Graviton



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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-

e

n p

d W -ue

-

eDr M.A. Thomson Lent 2004


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GE N E R A T I ON S

N a t ur ei sn o t q ui t e t h a t si m pl e.

T h er e ar e 3 GE N E R A T I ON S of f un d am en t al f er mi on s.

F i r s t G en er a t i on

S e c on d

G en er a t i on

T h i r d G en er a t i on

E l e c t r on

M u on

T a u

E l e c t r onN e u t r i n o

M u onN e u t r i n o

T a uN e u t r i n o

U p Q u ar k

C h ar m Q u ar k

T o p Q u ar k

D ow

n Q u ar k

S t r an g e Q u ar k

B o t t om Q u ar k

E a ch g en er a t i on e. g.

i s an ex a c t c o p y of

.

T h e onl y d i f f er en c ei s t h em a s s of t h e p ar t i cl e s- wi t h r s t

g en er a t i on t h el i gh t e s t an d t h i r d

g en er a t i onh e avi e s t .

C l e ar s ymm e t r y- or i gi n of 3 g en er a t i on si sN OT

U N D E R S T O OD


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The LEPTONSPARTICLES which do not interact via the

STRONG 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 : neutrinosGen. avour 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.



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The Quarks spin 1/2 fermions

fractional charge 6 distinct FLAVOURS of quarks

Generation avour 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 conned within HADRONS

e.g. p ( ), ( )

Quarks experience the ALL forces:

ELECTROMAGNETIC , STRONG and WEAK(and of course gravity).



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HADRONS single free QUARKS are NEVER observed

quarks always CONFINED in HADRONSi.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



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ForcesConsider ELECTROMAGNETISM and scattering of electrons

from 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

S p a c e

p p

e- e-

Early next week well learn how to calculate QuantumMechanical amplitudes for scattering via Gauge BosonExchange.



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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

10-1710-14

10-11

10-8

10-5

10-2

10

10 410 7

10 -4

10 -3

10 -2

10 -1

1 10 102

103

Distance (fm)

F o r c e

( G e V

/ f m )

GravitationalForce

Weak Force

Strong Force(quarks)

Strong Force(hadrons)

ElectromagneticForce



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T h e G a u g eB o s on s

GA U GE B O S ON S

m e d i a t e t h ef un d am en t al f or c e s

S pi n- 1

p ar t i cl e s ( i . e.V E C T OR B O

S ON S )

T h em ann er i nwh i ch t h e GA U GE

B O S ON S i n t er a c t wi t h t h e

L E P T ON S an d Q U A R K S d e t er mi n e s t h en a t ur e of t h ef un d am en t al

f o

r c e s.

F or c e

B o s on

M a s s

R an g e

( G eV /

)

( m )

E l e c t r om a g

n e t i c

P h o t on

m a s sl e s s

W e ak

8 0 / 9 0

1 0

S t r on

g

Gl u on

m a s sl e s s

/ 1 0


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Practical Particle PhysicsOR 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.



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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 Units

GeV h cUnits become

Energy GeV Time (GeV/

Momentum GeV/c Length (GeV/ c)

Mass GeV/c

Area (GeV/ c)



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Simplify ( !) by choosing

All quantities expressed in powers of GeVEnergy 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



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Relativistic Kinematics

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

-

-

-

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

EXAMPLE:

at rest.

-

-

Conservation of Energy

Conservation of Momentum

E = E + E

0 = p + p

(assume m = 0)

but

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


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Colliders and

Consider the collision of two particles:

p

p

p 1(E 1,p 1) p 2(E 2,p 2)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.



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p p p 1(E,p) p 2(m 2,0)Fixed Target Collision

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

p p p 1(E,p) p 2(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 xed target experiment most of the protonsenergy is wasted - providing momentum to the C.O.M

system rather than being available for the interaction.(NOTE: UNITS G = Giga =10 , M = Mega =10 )



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SummaryFERMIONS : spin

Chargeleptons

e-

e-

-

-10( ) ( ) ( )

quarks e.g. proton ( )uudud

cs

tb

+-23- -13( ) ( ) ( )

+ anti-particles

Fermion interactions = exchange of spin 1 bosons

e-e-

e-

e-

BOSONS : spin 1

GluonZ-bosonW-bosonPhoton

gZ 0W 0

91.2 GeV

80.3 GeV

0

Electromagnetic

Weak (CC)

Weak (NC)

Strong (QCD)

ForceMass



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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 Theory

Feynman diagrams Quantum Chromodynamics

the theory of the STRONG interaction Dirac Equation

the relativistic theory of spin-1/2 particles Weak interaction

Electro-weak Unication



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

with energy and momentum operators:

giving = = :

which has plane wave solutions ( = , = )

Schr odinger Equation: 1st Order in time derivative ( ) 2nd Order in space derivatives ( )

Not Lorentz Invariant !!!!

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



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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



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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-functionor 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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Anti-ParticlesNegative 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 into ve energy state

E

0-m ec

2

+m ec2

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



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In the Dirac picture predict pair creation

and annihilation.

e+

e-

E

0

-m ec2

+m ec2

+

e+

e-

E

0

-m ec2

+m ec2

+

1931 : positron ( ) was rst 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)



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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)



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Rates and Cross SectionsFor reaction

The cross section,

, is dened as the reaction rateper target particle, , per unit incident ux,

where is given by Fermis Golden Rule.

Example: Consider a single particle of type

traversing a beam (area

) of particles of type

ofnumber density

.

v

v

t

A

In time traverses a region containing

particles of type .

Interaction probability dened as ef-

fective cross sectional area occupied

by the

particles of type

Therefore the reaction rate is .

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


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Scattering in Q.M.REVISION (see Dr Ritchies QMII transparencies 11.8-11.15)

NOTE: Natural Units used throughout

Consider a beam of particles scattering in Potential

p i

p f

Scattering rate characterized by the interaction crosssection

number of particles scattered/unit time

incident uxUse FERMIS GOLDEN RULE for Transition rate, :

where is the Matrix Element and

= density ofnal states.

1st Order Perturbation Theory using plane wavesolutions of form .

Require :

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



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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



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

p y

p x

p z

Volume of single state inmomentum space:

Number of nal states between :

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



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Putting all the separate bits together:

ux

The normalization cancels and, in the limit where the

incident 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

p i

p f

p i

p f p=p i-p f

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



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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 .



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Rutherford Scattering

Let and replace

gives Coulomb potential:

Hence for elastic scattering in the Coulomb potential:

p i

p f p=p i-p f

e.g. The upper points are the

Gieger 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)

Documents

1 Introduction to the Standard Model