XI strong interaction - Uniudcobal/lezione_11.pdf · 4 Single gluon exchange Confinment . M. Cobal,...

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

Cx χχψ )(

=Ψ

Colour


Just a new quantum number..


Colour charge


•  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


•  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


Gluons


QCD Colour transformations


Local colour transformation


Self Interaction


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


Colour charge strenght


- 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


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


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


•  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


Free Quarks


Quark confinement


Hadronization


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


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


−+−+ +→→+ 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


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


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 distances of 1 fm between partons.


Zweig Rule

XI strong interaction - Uniudcobal/lezione_11.pdf · 4 Single gluon exchange Confinment . M. Cobal,...

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