xi strong interaction - uniudcobal/lezione_11.pdf · 4 single gluon exchange confinment . m. cobal,...
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M. Cobal, PIF 2006/7
Strong Interactions
M. Cobal, PIF 2006/7
• 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
Cx χχψ )(
=Ψ
Colour
M. Cobal, PIF 2006/7
Just a new quantum number..
M. Cobal, PIF 2006/7
Colour charge
M. Cobal, PIF 2006/7
• 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
=
=
=
100
,010
,001
bgr
M. Cobal, PIF 2006/7
• 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 Ic
3 and Yc are eigenvalues of corresponding operators • Colour hypercharge Yc and colour isospin Ic
3 charge are additive quantum numbers, having opposite sign for quark and antiquark. Confinement condition for the total colour charges of a hadron: Ic
3 = Yc = 0
M. Cobal, PIF 2006/7
Gluons
M. Cobal, PIF 2006/7
QCD Colour transformations
M. Cobal, PIF 2006/7
Local colour transformation
M. Cobal, PIF 2006/7
Self Interaction
M. Cobal, PIF 2006/7
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:
222qEqQ −=
)( 1−= rOq
M. Cobal, PIF 2006/7
Colour charge strenght
M. Cobal, PIF 2006/7
- 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()233(12
22 Λ−=
QN fs
πα
M. Cobal, PIF 2006/7
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
( ) opK π+Λ→Σ→+− 13850
( ) γ+Λ→Σ 11920
1001010 2
1
23
19≅
≅
−
−
αα s
14
2
≅=π
α ssg
M. Cobal, PIF 2006/7
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
M. Cobal, PIF 2006/7
- 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
M. Cobal, PIF 2006/7
• 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
krr
V ss +−=
α34
Single gluon exchange Confinment
M. Cobal, PIF 2006/7
Free Quarks
M. Cobal, PIF 2006/7
Quark confinement
M. Cobal, PIF 2006/7
Hadronization
M. Cobal, PIF 2006/7
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.
hadronsee →→+ −+ *γ
M. Cobal, PIF 2006/7
e-
e+ q
EMα Sα
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
M. Cobal, PIF 2006/7
−+−+ +→→+ µµγ *ee 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)
)cos1(2
)(cos
22
2θ
παµµ
θσ
+=→ −+−+
Qee
dd
)(cos
3)(cos
2 −+−+−+ →=→ µµθ
σθ
σ eeddeqqee
dd
q
M. Cobal, PIF 2006/7
Angular distribution of the quark jet in e+e- annihilation, compared with models - Experimentally measured angular dependence is clearly proportional to (1+cos2θ) ⇒jets are aligned with spin-1/2 quarks
M. Cobal, PIF 2006/7
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
M. Cobal, PIF 2006/7
- 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 distances of 1 fm between partons.
M. Cobal, PIF 2006/7
Zweig Rule