M. Cobal, PIF 2006/7
Strong Interactions
• Experimental data confirm predictions based on the assumption of symmetric wave functions
Problem: Δ++ is made out of 3 u quarks, and has spin J=3/2 (= 3 quarks of s= ½ in same state?) This is forbidden by Fermi
statistics (Pauli principle)!
Solution: there is a new internal degree of freedom (colour) which differentiate the quarks: Δ++=urugub
• This means that apart of space and spin degrees of freedom, quarks have yet another attribute
• In 1964-65, Greenberg and Nambu proposed the new property – the colour – with 3 possible states, and associated with the
corresponding wavefunction χ Ψ =ψ(x)χχC
Colour
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Just a new quantum number..
Colour charge
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• Conserved quantum numbers associated with χc are colour charges in strong interactions they play similar role to the electric charge in em interactions.
• A quark can carry one of the three colours (red, blue, green). An anti-quark one of the three anti-colours
• All the observable particles are “white” (they do not carry colour)
• Quarks have to be confined within the hadrons since non-zero colour states are forbidden.
• 3 independent colour wavefunctions are represented by colour spinor
Hadrons: neutral mix of r,g,b colours
Anti-hadrons: neutral mix of r,g,b anti-colours
Mesons: neutral mix of colours and anti-colours
=
=
=
1 0 0 ,
0 1 0 ,
0 0 1
b g
r
• These spinors are acted on by 8 independent “colour operators ” which are represented by a set of 3-dimensional matrices
(analogues of Pauli matrices)
• Colour charges Ic3 and Yc are eigenvalues of corresponding operators
• Colour hypercharge Yc and colour isospin Ic3 charge are additive quantum numbers, having opposite sign for quark and antiquark.
Confinement condition for the total colour charges of a hadron:
Ic = Yc = 0
M. Cobal, PIF 2006/7
Gluons
QCD Colour transformations
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Local colour transformation
Self Interaction
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Running of αs
The αs constant is the QCD analogue of αem and is a measure of the interaction strenght.
However αs is a “running constant”, increases with increase of r, becoming divergent at very big distances.
- At large distances, quarks are subject to the “confining potential”
which grows with r:
V(r) ~ λ r (r > 1 fm)
- Short distance interactions are associated with the large momentum transfer
Lorentz-invariant momentum transfer Q is defined as:
2 2
2 q Eq
Q = −
) ( −1
= O r q
Colour charge strenght
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- In the leading order of QCD, αs is given by:
Nf = number of allowed quark flavours
Λ ~ 0.2 GeV is the QCD scale parameter which has to be defined experimentally
) / ln(
) 2
33 (
12
2
2 Λ
= −
Q N f
s π
α
Strong Interactions
• Take place between quarks which make up the hadrons
• Magnitude of coupling can be estimated from decay probability (or width Γ) of unstable baryons.
• Consider:
Γ=36 MeV, τ = 10-23 s
If we compare this with the em decay: , τ = 10-19 s We get for the coupling of the strong charge
( ) o
p
K− + → Σ0 1385 → Λ +π
( )→ Λ +γ
Σ 11920
10 100
10 2
1 23 19
≅
≅ −
−
α αs
4 1
2
≅
= π αs gs
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QCD, Jets and gluons
• Quantum Chromodynamics (QCD): theory of strong interactions
Interactions are carried out by a massless spin-1 particle- gauge boson
In quantum electrodynamics (QED) gauge bosons are photons, in QCD, gluons
Gauge bosons couple to conserved charges: photons in QED- to conserved charges, and gluons in QCD – to colour charges.
Gluons do not have electric charge and couple to colour charges ⇒ strong
nteractions are flavour-independent
- Gluons can couple to other gluons
- Bound colourless states of gluons are called glueballs (not detected experimentally yet).
- Gluons are massless ⇒ long-range interaction Principle of asymptotic freedom
-At short distances, strong interactions are sufficiently weak
(lowest order diagrams) ⇒quarks and gluons are essentially free particles
-At large distances, higher-order diagrams dominate ⇒ interaction is very strong
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• For violent collisions (high q2), as < 1 and single gluon exchange is a good approximation.
• At low q2 (= larger distances) the coupling becomes large and the theory is not calculable. This large-distance behavior is linked with confinement of quarks and gluons inside hadrons.
• Potential between two quarks often taken as:
• Attempts to free a quark from a hadron results in production of new mesons. In the limit of high quark energies the confining
potential is responsible for the production of the so-called “jets
r kr Vs = − αs +
3 4
Single gluon exchange Confinment
Free Quarks
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Quark confinement
Hadronization
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QCD jets in e+e- collisions
- A clean laboratory to study QCD:
- At energies between 15 GeV and 40 GeV, e+e- annihilation produces a photon which converts into a quark-antiquark pair
- Quark and antiquark fragment into observable hadrons
- Since quark and antiquark momenta are equal and counterparallel, hadrons are produced in two opposite jets of equal energies
- Direction of a jet reflects direction of a corresponding quarks.
hadrons e
e+ + − →γ * →
e-
e+ q
αEM αS
q
Colliding e+ and e- can give 2 quarks in final state. Then, they fragment in hadrons
2 collimated jets of hadrons travelling in opposite direction and following the momentum vectors of the original quarks
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− +
−
+ +e →γ* → µ +µ e
Comparison of the process with the reaction
must show the same angular distribution both for muons and jets
where θ is the production angle with respect to the initial electron direction in CM frame
For a quark-antiquark pair:
Where the fractional charge of a quark eq is taken into account and factor 3 arises from number of colours. If quarks have spin ½,
angular distribution goes like (1+cos2θ); if they have spin 0, like (1- cos2θ)
) cos
1 2 (
) cos (
2 2
2
πα θ µ
θ µ
σ + − → + − = +
e Q d e
d
) cos (
3 ) cos (
2 + − + −
−
+ → = → µ µ
θ σ θ
σ e e
d e d q
q e
d e d
q
Angular distribution of the quark jet in e+e- annihilation, compared with models
- Experimentally measured angular dependence is clearly proportional to (1+cos2
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If a high momentum (hard) gluon is emitted by the quark or the anti -quark, it fragments to a jet, leading to a 3-jet events
A 3-jet event seen in a e+e- annihilation at the DELPHI experiment
- In 3-jet events it is difficult to understand which jet come from the quarks and which from the gluon
- Observed rate of 3-jet and 2-jet events can be used to determine value of αs (probability for a quark to emit a gluon determined by αs) αs= 0.15 ± 0.03 for ECM = 30-40 GeV
Principal scheme of hadroproduction in e+e- hadronization begins at
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Zweig Rule
