history, quark model and the su(3)-symmetry

Simela Aslanidou 14.12.2006

QCDHistory, quark model and the SU(3)-symmetry

Simela Aslanidou14.12.2006


Contents

• development and motivation for the foundation of QCD

• concepts

• the SU(3)-colour group


History• up to the beginning of the 20th century the

common view in physics was that the Coulomb-force is responsible for the formation of atoms.

• with the discovery of the neutron it became clear, that there must be an unknown interaction “gluing” the nucleons in the core together.

• Quantumchromodynamics (QCD) was established much later but this was the beginning.


History

• with the development of particle accelerators in the 1940´s and 1950´s a large number of particles was discovered, called the “particle-zoo”.

• great efforts have been made in order to classify the “zoo” according to simple principles


History

• 1950´sRobert Hofstadter performed elastic electron-proton-scattering experiments to study the structure of protons

protons are not pointlike


History

• Murray Gell-Mann and Yuval Ne´eman postulated in 1964 that hadrons can be constructed from a number of fundamental particles called 'quarks'.

• almost at the same time George Zweig proposed the same idea and he coined the fundamental particles 'aces'.


History

• both, Gell-Mann and Zweig, chose the SU(3)-symmetry (flavour)group theory to organise the increasing number of discovered particles.

• they made their predictions on the basis of the known baryon and meson octets.

• in their theory of SU(3)-symmetry (flavor) they could organize all known hadrons as states of three different constituents.


History

In the 1960´s experiments were performed at SLAC in order to obtain information on the structure and the substructure of hadrons assuming that they are made of constituents and not fundamental particles.


History• the method of the experiments at SLAC was

the same as Rutherford had used in his experiments to explore the structure of an atom. In case of hadrons lepton-hadron-scattering was a suitable probe.

• the idea was to apply a beam of structureless particles (leptons) at high energy and thus high momentum transfer to obtain a high space-time resolution.

• this process is called the deep-inelastic electron-proton scattering.


Concepts

• to probe the structure of a particle one has to measure the cross-section. If we want to explore the structure of a hadron we use structureless pointlike projectiles like leptons.

• if the recoil effect is neglected and the target particle is assumed to be pointlike the differential cross-section is given by the Mott-Formula

( ) ( )2

coscq

EcZ4dd 24

2222

Mott

θ′α=Ωσ h


Concepts

The proton is an extended object, so thedifferential cross-section for the elastic scatteringis given by the Rosenbluth-formula

GM and GE are the magnetic and electric formfactors respectively.

( )( ) ( )

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

+

⋅++θ

⎟⎠⎞

⎜⎝⎛

Ωσ=

Ωσ

2

2

2

222

M22

E22

222

MMott

M4Q1

M4QQGQG

2tan

M2QQG

dd

dd


Concepts

• in electron-proton scattering experiments the electrons are relativistic and the process is

• for the kinematics we use the four-momentum and the energy-momentum relation leads to the invariant mass

xepe +′→+


Concepts

invariant mass

• elastic scattering

• inelastic scattering

( )

MPq ; massinvariant W

²QM2²c²M²qPq2²c²M²qP²c²W

=ν=

−ν+=++=+=

0²QM2WM =−ν→=

0²QM2MW >−ν→>


Inelastic scattering

The differential cross section is now given by

with W1, W2 the structure functions of thehadron and ν=Pq/M the energy transfer.

( ) ( ) ⎥⎦⎤

⎢⎣⎡ θν+ν⎟

⎠⎞

⎜⎝⎛

Ωσ=⎟⎟

⎠

⎞⎜⎜⎝

⎛′Ω

σ2

tan,QW2,QWdd

Eddd 22

12

2Mott

2



Displaying the ratio

as a function of the invariant mass shows anunexpected behaviour for a pointlikeparticle as there is only a small dependence in Q².

Mottdd

Edd²d

⎟⎠⎞

⎜⎝⎛

Ωσ

′Ωσ



Electron-proton-scattering:cross-sections for inelastic scattering for different invariant masses in comparison with elastic scattering



• the ratio is independent from Q²

lepton is scattered on a pointlike object

• hadrons are extended objects

they must have pointlike constituents

Mottdd

Eddd

⎟⎠⎞

⎜⎝⎛

Ω′Ωσσ²


Bjorken scaling

Structure-Function F2 as function of the Bjorken Variable x here called ξ for different Q² values.

is the additive term from the invariant mass

ν= M2Q:x 2


Quarks

Existence of Quarksthere is empirical evidence since the momentum transfer realised at SLAC was much larger than the nucleon-mass

Q²>>M²this result is interpreted at the following way:the nucleon must have a substructure ofquasi-free, point-like particles.


Quark model

• quarks come in 6 flavoursup, down, strange, charm, bottom top, these are the six different kinds of quarks

• according to the regularity of the leptons quarks show the same family structure

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛bt

sc

du

, , ( )GeV 1.53.174m GeV 4.4 to 1.4m

GeV 35.1 to 15.1m MeV 130 to 80mMeV 8 to 4m MeV 4 to 5,1m

tb

cs

du

±======


Quark model

Problems• non-observation of isolated quarks

• discrepancy between predicted and experimental data on cross-sections

• problem in constructing baryon wave functions(violation of the Pauli principle)


Quark model

Most of the difficulties can be resolved byintroduction of a new quantum number called colour.


Necessity of the colour

Quarks are required to be fermions with spin ½and according to the Pauli principle they cannot occupy the same state.

The quantum state has to be antisymmetric with respect to the exchange of all quantum numbers of the quarks.


Necessity of the colour

Example : pion-proton resonance Δ++

charge Q=2, isospin I and strangeness S=0

is symmetric under exchange in contradictionto the Pauli principle

23=

↑↑↑==Δ ++ uuuJ23,

3


Colour

• in 1965 O.W.Greenberg introduced the property of colour charge.

• arbitrary denomination to assign an additional quantum number for which the state is antisymmetric:

εijk is the total antisymmetric tensor and the indices represent the colour

↑↑↑ε==Δ ++ kjiijk3 uuu

23J,


Colour

• idea:hadrons are colour-neutral. Each quark carries colour (red, blue, green) such that their combination gives white

• in the case of the mesons made of quark-antiquark-pair, the antiquark carries the complementary colour


Colour

The additivecolour model


Quantum numbers of Quarks

0

0

0

+1

0

0

charm

r, g, b0-1/20-1/3ebottom

r, g, b0+1/20+2/3etop

r, g, b+1-1/20-1/3estrange

r, g, b0+1/20+2/3echarm

r, g, b0-1/21/2-1/3edown

r, g, b0+1/21/2+2/3eup

colourstrangenessIzIsospin IChargeName


Quark model

• why are there three flavours (at least in the beginning) and three colours?

π0 2γ decayse+ + e- - annihilation}The factor of three appears


Quark model

e+e--AnnihilationAt high energies electrons and positrons annihilate into hadrons

e+ + e- hadrons


Quark model

Cross section

More common

( ) ∑πα=→σ −+

fN

iic²QN

s3²4hadronsee

( ) ²QNs34

hadronseeRfN

iic∑=

πα→σ=

−+

31Q

32Q

31Q

32Q

31Q

32Q

quarks of chargeQcolours ofnumber N

4e

bottom

top

strange

charm

down

up

i

i

2

−=

=

−=

=

−=

=

==

π=α


Ratio of the cross section


Ratio of the cross section

Comparison of R with experimental dataEnergy=<3Gev

R=24GeV<Energy<9GeV

R=10/310GeV<Energy

R=11/3Substitution in the formula leads to Nc=3


Gluons

another experimental result from deep-inelastic electron-proton scattering was, that quarks only carry 50% of the momentum in the proton

there must be other constituents called “gluons”


Gluons

• gluons are the gauge bosons of the strong interaction

• gluons are the force carriers between quarks.

• gluons are responsible for the quark confinement.


Gluons

gluons carry colour and anticolour at the sametime

gluons interact with each other (in contrast to photons)

a) a quark radiates a gluon ; b) a gluon splits into a quark-antiquark pair c) three-gluon-interaction ; d) four-gluon-interaction


SU(N)

• group of the non-abelian Lie-Algebra.

• group of unitary NxN matrices with detU=±1 and N²-1 parameters.

• the N²-1 dimensional space is formed by the N²-1 generators of the Algebra.


SU(3)

The algebra is generated by the 8 Gell-Mann matrices λi, i=1,...,8.Define

The algebra satisfy the commutation relations

ii 21T λ=

[ ] kijkji TifT,T =


SU(3)

Why SU(3)-flavour?With the SU(3)-flavour it is possible to constructall hadron states from the fundamental triplet.


Ladder Operators

Define the ladder operators T±, U±, V±

(± stands for step-up/step-down operator)

Applying the operators to the hadron states stepsup or steps down the states and the multiplet canbe constructed.

765421 FiFUFiFVFiFT ±=±=±= ±±± ; ;


The Baryon-Oktet

Start from thefundamental triplet andapplying the ladderoperators leads to thestates of a multiplet as it isshown in the figure for theexample of the baryon octet


SU(3)c

• strong interaction is governed by the colour and not by the flavour.

• two fundamental tripletscolour ci , I=1,2,3 ; complementary colour , i=1,2,3ic


References

T. Muta, World Scientific Lectures Notes in Physics-Vol 57„Foundations of quantumchromodynamics“W. Greiner, Theoretische Physik Band 6„Symmetrien“W. Greiner, Theoretische Physik Band 10„Quantenchromodynamik“B. Povh, K. Rith, C. Scholz, F. Zetsche„Teilchen und Kerne“G. Musiol, J. Ranft, R. Reif, D. Seeliger„Kern- und Elementarteilchenphysik“G. Zweig, Cern-Libraries, Geneva„An SU(3) Model of strong interaction symmetry and ist breaking“http://www.personal.uni-jena.deM. E. Peskin, PiTP Summer School, July 2005


THE END

history, quark model and the su(3)-symmetry

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