Simela Aslanidou 14.12.2006
QCDHistory, quark model and the SU(3)-symmetry
Simela Aslanidou14.12.2006
Simela Aslanidou 14.12.2006
Contents
• development and motivation for the foundation of QCD
• concepts
• the SU(3)-colour group
Simela Aslanidou 14.12.2006
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.
Simela Aslanidou 14.12.2006
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
Simela Aslanidou 14.12.2006
History
• 1950´sRobert Hofstadter performed elastic electron-proton-scattering experiments to study the structure of protons
protons are not pointlike
Simela Aslanidou 14.12.2006
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'.
Simela Aslanidou 14.12.2006
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.
Simela Aslanidou 14.12.2006
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.
Simela Aslanidou 14.12.2006
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.
Simela Aslanidou 14.12.2006
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
Simela Aslanidou 14.12.2006
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
Simela Aslanidou 14.12.2006
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 +′→+
Simela Aslanidou 14.12.2006
Concepts
invariant mass
• elastic scattering
• inelastic scattering
( )
MPq ; massinvariant W
²QM2²c²M²qPq2²c²M²qP²c²W
=ν=
−ν+=++=+=
0²QM2WM =−ν→=
0²QM2MW >−ν→>
Simela Aslanidou 14.12.2006
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
Simela Aslanidou 14.12.2006
Inelastic scattering
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
Simela Aslanidou 14.12.2006
Inelastic scattering
Electron-proton-scattering:cross-sections for inelastic scattering for different invariant masses in comparison with elastic scattering
Simela Aslanidou 14.12.2006
Inelastic 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
Simela Aslanidou 14.12.2006
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
Simela Aslanidou 14.12.2006
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.
Simela Aslanidou 14.12.2006
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
±======
Simela Aslanidou 14.12.2006
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)
Simela Aslanidou 14.12.2006
Quark model
Most of the difficulties can be resolved byintroduction of a new quantum number called colour.
Simela Aslanidou 14.12.2006
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.
Simela Aslanidou 14.12.2006
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
Simela Aslanidou 14.12.2006
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,
Simela Aslanidou 14.12.2006
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
Simela Aslanidou 14.12.2006
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
Simela Aslanidou 14.12.2006
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
Simela Aslanidou 14.12.2006
Quark model
e+e--AnnihilationAt high energies electrons and positrons annihilate into hadrons
e+ + e- hadrons
Simela Aslanidou 14.12.2006
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
Simela Aslanidou 14.12.2006
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
Simela Aslanidou 14.12.2006
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”
Simela Aslanidou 14.12.2006
Gluons
• gluons are the gauge bosons of the strong interaction
• gluons are the force carriers between quarks.
• gluons are responsible for the quark confinement.
Simela Aslanidou 14.12.2006
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
Simela Aslanidou 14.12.2006
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.
Simela Aslanidou 14.12.2006
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 =
Simela Aslanidou 14.12.2006
SU(3)
Why SU(3)-flavour?With the SU(3)-flavour it is possible to constructall hadron states from the fundamental triplet.
Simela Aslanidou 14.12.2006
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 ±=±=±= ±±± ; ;
Simela Aslanidou 14.12.2006
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
Simela Aslanidou 14.12.2006
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
Simela Aslanidou 14.12.2006
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