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

QCD 

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

Simela Aslanidou 

14.12.2006 

Simela Aslanidou 14.12.2006

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. 



• 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 



• 1950´s 

Robert Hofstadter performed elastic electronproton-scattering 

experiments to study the 

structure of protons 

protons are not pointlike 



• 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'. 



• 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. 



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. 



• 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-hadronscattering 

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 

dσ 

dΩ 

( ) 

Mott 

= 

4Z 

2 

α 

2 

( ) 

qc 

hc 

4 

2 

E′ 

2 

cos 

θ 

2 

Simela Aslanidou 14.12.2006 

2

Concepts 

The proton is an extended object, so the 

differential cross-section for the elastic scattering 

is given by the Rosenbluth-formula 

dσ 

⎛ dσ 

= 

⎞ 

⎜ ⎟ 

dΩ 

⎝dΩ⎠ 

Mott 

⎡ 

⎢G 

⎢ 

⎣ 

2 

M 

( 2 

Q ) 

2 

Q 

2M 

2 

tan 

θ G 

+ 

2 

( 2) 

2 

+ ( 2 

Q G Q ) 

1+ 

Q 

GM and GE are the magnetic and electric form 

factors respectively. 

2 

2 

E 

M 

⋅ 

Q 


2 

4M 

2 

2 

4M 

2 

⎤ 

⎥ 

⎥ 

⎦

Concepts 

• in electron-proton scattering experiments the 

electrons are relativistic and the process is 

e + 

p → e′ 

+ 

• for the kinematics we use the four-momentum 

and the energy-momentum relation leads to the 

invariant mass 

x 


Concepts 

invariant mass 

W² 

c² 

= P + q ² 

W 

= 

( ) 

invariant 

= 

mass 

M² 

c² 

• elastic scattering 

= W → 2M 

• inelastic scattering 

> 

M → 2 

; 

+ 

2Pq 

ν 

M ν − Q² 

= 

= 

W Mν 

− Q² 

> 

+ 

0 

q² 

Pq 

M 

0 

= 

M² 

c² 

+ 2Mν 

− 


Q²

Inelastic scattering 

The differential cross section is now given by 

2 

⎛ σ ⎞ ⎛ dσ 

⎞ 

⎜ 

⎟ = ⎜ ⎟ 

⎝ dΩdE′ 

⎠ ⎝ dΩ 

⎠ 

d 2 

Mott 

⎡ ( 2 ) ( 2 ) θ⎤ 

⎢ 

W2 

Q , ν + 2W1 

Q , ν tan 

⎥ 

⎣ 

2⎦ 

with W1, W2 the structure functions of the 

hadron and ν=Pq/M the energy transfer. 



Displaying the ratio 

d² 

σ 

dΩdE′ 

⎛ dσ 

⎞ 

⎜ ⎟ 

⎝ dΩ 

⎠ 

Mott 

as a function of the invariant mass shows an 

unexpected behaviour for a pointlike 

particle as there is only a small dependence in Q². 



Electron-proton-scattering: 

cross-sections for inelastic 

scattering for different 

invariant masses in 

comparison with elastic 

scattering 



d² 

σ 

dΩdE′ 

⎛ dσ 

⎞ 

⎟ 

⎝dΩ 

⎠ 

• the ratio ⎜ is independent from Q² 

lepton is scattered on a pointlike object 

• hadrons are extended objects 

they must have pointlike constituents 

Mott 


Bjorken scaling 

Structure-Function 

F2 as function of the 

Bjorken Variable x 

here called ξ for 

different Q² values. 

2 

x : = Q 2Mν 

is the additive term 

from the invariant 

mass 


Quarks 

Existence of Quarks 

there is empirical evidence since the momentum 

transfer realised at SLAC was much larger than 

the nucleon-mass 

Q²>>M² 

this result is interpreted at the following way: 

the nucleon must have a substructure of 

quasi-free, point-like particles. 


Quark model 

• quarks come in 6 flavours 

up, 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 

⎛ 

⎜ 

⎝ 

u 

d 

⎞ 

⎟ 

⎠ 

, 

⎛ 

⎜ 

⎝ 

c 

s 

⎞ 

⎟ 

⎠ 

, 

⎛ t ⎞ 

⎜ ⎟ 

⎝b 

⎠ 

m 

m 

m 

u 

s 

b 

= 

1, 

5 to 4 MeV 

= 

= 

80 to130 

4. 

1 to 4. 

4 

MeV 

GeV 

m 

m 

m 

= 4 to 8 MeV 

= 1. 

15 to1. 

35 GeV 

= ( 174. 

3 ± 5. 

1)GeV 


d 

c 

t


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) 



Most of the difficulties can be resolved by 

introduction of a new quantum number called 

colour. 


Necessity of the colour 

Quarks are required to be fermions with spin ½ 

and 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 ++ 

3 

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

2 

3 

+ + 

Δ , J = 3 

2 

= u ↑ u ↑ u ↑ 

is symmetric under exchange in contradiction 

to the Pauli principle 


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: 

3 

2 

Δ = = ε ↑ ↑ 

+ + k j i 

, 

J3 

ijk u u u 

εijk is the total antisymmetric tensor and the indices 

represent the colour 

↑ 


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 quarkantiquark-pair, 

the antiquark carries the 

complementary colour 


Colour 

The additive 

colour model 


Name 

up 

down 

charm 

strange 

top 

bottom 

Quantum numbers of Quarks 

Charge 

+2/3e 

-1/3e 

+2/3e 

-1/3e 

+2/3e 

-1/3e 

Isospin I 

1/2 

1/2 

0 

0 

0 

0 

Iz 

+1/2 

-1/2 

+1/2 

-1/2 

+1/2 

-1/2 

charm 

0 

0 

+1 

0 

0 

0 

strangeness 

+1 


0 

0 

0 

0 

0 

colour 

r, g, b 

r, g, b 

r, g, b 

r, g, b 

r, g, b 

r, g, b


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

and three colours? 

π02γ decays 

e + + e- The factor of three appears 

- annihilation} 



e + e--Annihilation At high energies electrons and positrons 

annihilate into hadrons 

e + + e-hadrons Simela Aslanidou 14.12.2006



Cross section 

More common 

( ) ∑ 

πα 

= 

→ 

σ 

− 

+ 

f 

N 

i 

i 

c 

² 

Q 

N 

s 

3 

² 

4 

hadrons 

e 

e 

( ) 

² 

Q 

N 

s 

3 

4 

hadrons 

e 

e 

R 

f 

N 

i 

i 

c ∑ 

= 

πα 

→ 

σ 

= 

− 

+ 

3 

1 

Q 

3 

2 

Q 

3 

1 

Q 

3 

2 

Q 

3 

1 

Q 

3 

2 

Q 

quarks 

of 

charge 

Q 

colours 

of 

number 

N 

4 

e 

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 data 

Energy=

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 same 

time 

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 

1 

Ti = λ 

2 

i 

The algebra satisfy the commutation relations 

[ T ] i, 

Tj 

= 

ifijkTk 


SU(3) 

Why SU(3)-flavour? 

With the SU(3)-flavour it is possible to construct 

all hadron states from the fundamental triplet. 


Ladder Operators 

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

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

Fˆ Fˆ Fˆ Vˆ Fˆ Fˆ Tˆ = ± 

± 

= 1 ± i 2 ; 

± = 4 ± i 5 ; Û± 

Applying the operators to the hadron states steps 

up or steps down the states and the multiplet can 

be constructed. 


6 

Fˆ i 

7

The Baryon-Oktet 

Start from the 

fundamental triplet and 

applying the ladder 

operators leads to the 

states of a multiplet as it is 

shown in the figure for the 

example of the baryon octet 


SU(3)c 

• strong interaction is governed by the colour 

and not by the flavour. 

• two fundamental triplets 

colour ci , I=1,2,3 ; 

complementary colour c 

i , i=1,2,3 


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.de 

M. E. Peskin, PiTP Summer School, July 2005 


THE END

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

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