Transcript Slide 1

The Constituent Quark
Models
Outline
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The Quark Model
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Original Quark Model
Additions to the Original Quark Model
Color
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Harmonic Potential Model
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Isgur-Karl Model
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M.I.T. Bag Model
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Assumptions
Predictions
Constituent Quark Model
(Non-relativistic)
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Quasi–particles, have same quantum number like fundamental quarks
of QCD: electric charge, baryon number, color, flavor and spin.
Bare quark dressed by clouds of quark-antiquark pairs and gluons.
Mass is more than 300MeV, compared to bare quark about 10MeV.
Allow treatment similar to nuclear shell model
 Simpler: only three players ( for baryons ) while nuclei can have
many nucleons.
 Harder: more freedom,
 three colors, while nucleons are colorless
 three flavors, while nucleons only have neutrons and
protons.
Original Quark Model
1964 The model was proposed independently by Gell-Mann and Zweig
Three fundamental building blocks 1960’s (p,n,l)  1970’s (u,d,s)
mesons are bound states of a of quark and anti-quark:
Can make up "wave functions" by combining quarks:
+ = ud, - = du, o = 1 (uu - d d), k = ds, k = ds
2
+
o
baryons are bound state of 3 quarks:
proton = (uud), neutron = (udd), L= (uds)
anti-baryons are bound states of 3 anti-quarks:
p  uud
n  u d d L u d s
  (du )

Λ= (uds)
Quarks
These quark objects are:
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point like
•
spin 1/2 fermions
•
parity = +1 (-1 for anti-quarks)
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two quarks are in isospin doublet (u and d), s is an
iso-singlet (=0)
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Obey Q = I3 +1/2(S+B) = I3 +Y/2
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Group Structure is SU(3)
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For every quark there is an anti-quark
•
The anti-quark has opposite charge, baryon number and
strangeness
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Quarks feel all interactions (have mass, electric charge, etc)
Early 1960’s Quarks
Successes of 1960’s Quark Model:
• Classify all known (in the early 1960’s) particles in terms of
3 building blocks
• predict new particles (e.g. W-)
• explain why certain particles don’t exist (e.g. baryons with
spin 1)
• explain mass splitting between meson and baryons
• explain/predict magnetic moments of mesons and baryons
• explain/predict scattering cross sections (e.g. sp/spp = 2/3)
Failures of the 1960's model:
• No evidence for free quarks (fixed up by QCD)
• Pauli principle violated (D++= (uuu) wave function is totally
symmetric) (fixed up by color)
•
What holds quarks together in a proton ? (gluons! )
•
How many different types of quarks exist ? (6?)
Additions to the Original
Quark Model – Charm
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Another quark was needed to account for
some discrepancies between predictions of
the model and experimental results
Charm would be conserved in strong and
electromagnetic interactions, but not in weak
interactions
In 1974, a new meson, the J/Ψ was
discovered that was shown to be a charm
quark and charm antiquark pair
More Additions – Top and
Bottom
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Discovery led to the need for a more elaborate
quark model
This need led to the proposal of two new quarks
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t – top (or truth)
b – bottom (or beauty)
Added quantum numbers of topness and
bottomness
Verification
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b quark was found in a  meson in 1977
t quark was found in 1995 at Fermilab
Quantum Chromodynamics
(QCD)
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QCD gave a new theory of how quarks interact with
each other by means of color charge
The strong force between quarks is often called the
color force
The strong force between quarks is carried by
gluons
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Gluons are massless particles
There are 8 gluons, all with color charge
When a quark emits or absorbs a gluon, its color
changes
Quantum Chromodynamics (QCD)
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Asymptotic freedom
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Quark confinement
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Quarks move quasi-free inside the nucleon
Perturbation theoretical tools can be applied in
this regime
No single free quark has been observed in
experiments
Color force increases with increasing distance
Chiral symmetry
Quark confinement
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Spatial confinement
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String confinement
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Quarks cannot leave a certain region in space
The attractive( color singlet) quark-antiquark
Color confinement
What Models do we have?
Harmonic Potential Model
(for N and N* states, mu=md=m)
pi2
1
1
H 0   (mi 
)  V (rij )  Vss (rij )
2mi
2 ij
2 ij
i 1
3
1
R  (r1  r2  r3 )
3
1
l
(r1  r2 )
2
1

(r1  r2  2r3 )
6
2
K 2
V (rij )  rij
2
λ
1
ρ
R
3
Solution of Harmonic Potential Model
H
int
0
p2
3K 2 pl2 3K 2


 

l
2m 2
2m 2
EN  E0  N0
L  l  ll
N  N  Nl
 00  (
l
3Km
 11  

2

34
) e
3Km
32
3K
0 
m
P  (1)
 (3 Km )1 2 (  2  l 2 ) 2
( l x  il y ) e
l ll
 (3 Km )1 2 (  2  l 2 ) 2
Spin-Spin Contact Interaction
s i s i
4
Vss (qi q j ) 
s
 ( x)
9 c
mi m j
3

4 3  s
2

3


(0)
for N

3
2
9 c mu ,d

DM ss  
3
s
4

2
 3 
 (0) for D
3
2

9 c mu ,d

 3 for S=1/2
s i s i  

 3 for S=3/2
i , j 1
3
i j
The three parameters
ms,d , αs|ψ(0)|2, ω0
are obtained by fitting to
experimental data
Spectrum of low lying N and N* states
ms,d = 360MeV , ω0 =500MeV
Non-relativistic quark model
with the salt of QCD
eg. Isgur-Karl Model
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Start with a non-relativistic quark model with SU(3)xSU(2)
spin-flavor symmetry.
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SU(3) flavor breaking via quark mass difference.
(mu,d is not equal to ms).
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Long range confining force independent of flavor and spin.
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Only one gluon exchange accounts for short range spin and
flavor dependent interaction.
(similar to electrodynamics of two slow moving fermions)
Isgur-Karl Model
pi2
K 2  s li  l j
H 0   (mi 
)   ( rij 
)  Vijhyp
2mi
4 rij
i 1
i j 2
i j
3
hyp
ij
V
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2 s

3mi m j
 8 3
3( si  rij )( s j  rij )  si  s j 
  (rij ) si  s j 

3
rij
 3

No spin-orbit interaction, comparing to shell model
Spin-spin contact interaction acts when L is zero
Tensor interaction acts when L is Nonzero
Nstar Spectrum
M.I.T. Bag Model
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Developed in 1974 at
Massachusetts Institute
of Technology
It models spatial
confinement only
• Quarks are forced by a fixed external pressure to move only
inside a given spatial region
• Quarks occupy single particle orbitals
• The shape of the bag is spherical, if all the quarks are in
ground state
M.I.T Bag Model
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Inside the bag, quarks are allowed to move
quasi-free.
An appropriate boundary condition at the bag
surface guarantees that no quark can leave
the bag
This implies that there are no quarks outside
the bag
M.I.T. Bag Model
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The boundary condition generates discrete
energy eigenvalues.
R - radius of the Bag
xn
n 
x =2.04
R
1
xn
Ekin ( R )  N q
R
4 3
E pot ( R )  R B
3
Nq = # of quarks inside the bag
B – bag constant that reflects the
bag pressure
M.I.T. Bag Model
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Minimizing E(R), one gets the equilibrium radius of
14
the system
 N q xn 

Rn  
 4B 
4
3 3
En 
4BNq xn
3


14
Fixing the only parameter of the model B, by
fitting the mass of the nucleon to 938MeV we
have first order predictions
One gluon exchange
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Model so far excluded all interactions between the
quarks
There should be some effective interaction that is
not contained in B( how do we know that?)
sM q
EX 
R
αs – the strong coupling constant
Mq depends on the quantum no. of
the coupled quarks
Predictions
The masses of N, Δ, Ω, ω
were used to fit the
parameters.
Conclusions
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The quark model
 classifies all known particles in terms of 6 building blocks
 Explains mass splitting between meson and baryons
 Explain/predict magnetic moments of mesons and baryons
 Explain/predict scattering cross sections
The MIT Bag Model
 predicts fairly accurate masses of the particles
 Explains color confinement
 Helps predict heavy quark spectrum
Simple models can give us a very good picture!
Bibliography
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Y. IWAMURA and Y. NOGAMI, IL NUOVO CIMENTO VOL. 89 A, N.
3(1985)
Peter HASENFRATZ and Julius KUTI, PHYSICS REPORTS (Section C
of Physics Letters) 40, No. 2 (1978) 75-179.
T. Barnes, arXiv:hep-ph/0406327v1
Carleton E. DeTar, John 12. Donoghue, Ann. Rev. Nucl. Part. Sci.
(1983)
E. Eichten et al. , Phys. Rev. D, 203 (1980)
E. Eichten et al. , Phys. Rev. Lett, 369 (1975)
Stephan Hartmann, Models and Stories in Hadron Physics
Theoretical papers
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N. Isgur and G. Karl, Phys. Rev. D 18, 4187 (1978); 20, 1191 (1979).
L. G. Landsberg, Phys. At. Nucl. 59, 2080 (1996).
J.W. Darewych, M. Horbatsch, and R. Koniuk, Phys. Rev. D 28,1125 (1983).
E. Kaxiras, E. J. Moniz, and M. Soyeur, Phys. Rev. D 32, 695 (1985).