Download Strong Interactions

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
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Local colour transformation
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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
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Free Quarks
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Quark confinement
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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