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Strong Interactions M. Cobal, PIF 2006/7 Colour • 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 Ψ = ψ ( x ) χχ C corresponding wavefunction χ 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) 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 • Quarks have to be confined within the hadrons since non-zero colour states are forbidden. 1 0 0 • 3 independent colour wavefunctions are represented by colour spinor M. Cobal, PIF 2006/7 r = 0 , 0 g = 1 , 0 b = 0 1 • 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: Ic3 = 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 q = O(r −1 ) 2 2 Lorentz-invariant momentum transfer Q is defined as: Q = q − E q 2 M. Cobal, PIF 2006/7 Colour charge strenght M. Cobal, PIF 2006/7 - In the leading order of QCD, αs is given by: αs = 12π (33 − 2 N f ) ln(Q 2 / Λ2 ) Nf = number of allowed quark flavours Λ ~ 0.2 GeV is the QCD scale parameter which has to be defined experimentally 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. − 0 o ( ) K + p → Σ 1385 → Λ + π • Consider: Γ=36 MeV, τ = 10-23 s If we compare this with the em decay: Σ 0 (1192) → Λ + γ , τ = 10-19 s g s2 ≅1 We get for the coupling of the strong charge α s = 4π 1 10 −19 2 αs ≅ 100 ≅ − 23 α 10 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: 4 αs Vs = − + kr 3 r Single gluon exchange Confinment • 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 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: e + + e − → γ * → hadrons - 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. M. Cobal, PIF 2006/7 e+ q αS α EM e- q 2 collimated jets of hadrons travelling in opposite direction and following the momentum vectors of the original quarks M. Cobal, PIF 2006/7 Colliding e+ and e- can give 2 quarks in final state. Then, they fragment in hadrons Comparison of the process with the reaction e+ + e− → γ * → µ + + µ − must show the same angular distribution both for muons and jets 2 dσ πα 2 (e + e − → µ + µ − ) = ( 1 + cos θ) 2 d cos θ 2Q where θ is the production angle with respect to the initial electron direction in CM frame For a quark-antiquark pair: dσ dσ (e + e − → qq ) = 3eq2 (e + e − → µ + µ − ) d cos θ d cos θ 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 (1cos2θ) 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 M. Cobal, PIF 2006/7