

The Quark Model



Original Quark Model
Additions to the Original Quark Model
Color
Harmonic Potential Model
 Isgur-Karl Model
 M.I.T. Bag Model
 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
2
(uu - d d), k
+
= d s, k o
= ds baryons are bound state of 3 quarks: proton = (uud), neutron = (udd),
L
= (uds) anti-baryons are bound states of 3 anti-quarks: p
 u u d n
 u d d
L u d s
  
( d u )
Λ = (uds)

These quark objects are:
• point like
• spin 1/2 fermions
• parity = +1 (-1 for anti-quarks)
• two quarks are in isospin doublet (u and d), s is an iso-singlet (=0)
•
Obey Q = I
3
+1/2(S+B) = I
• Group Structure is SU(3)
3
+Y/2
• For every quark there is an anti-quark
•
The anti-quark has opposite charge, baryon number and strangeness
•
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
/ s pp
= 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 ?)

 Another quark was needed to account for some discrepancies between predictions of the model and experimental results
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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

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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

 t
– top (or truth) b – bottom (or beauty)
Added quantum numbers of topness and bottomness
Verification
 b quark was found in a
 meson in 1977
 t quark was found in 1995 at Fermilab

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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
 Gluons are massless particles
 There are 8 gluons, all with color charge
When a quark emits or absorbs a gluon, its color changes

 Asymptotic freedom


Quarks move quasi-free inside the nucleon
Perturbation theoretical tools can be applied in this regime
 Quark confinement
 No single free quark has been observed in experiments
 Color force increases with increasing distance
 Chiral symmetry

 Spatial confinement
 Quarks cannot leave a certain region in space
 String confinement
 The attractive( color singlet) quark-antiquark
 Color confinement

(for N and N* states, m u
=m d
=m)
H
0
 i
3 

1
( m i
 p i
2
2 m i
)

1
2
 ij
V r ij

1
2
 ij
V r ss ij
V r ij

K r ij
2
2
R

1
3
( r
1 r r
3
) l 
1
2
( r
1
 r
2
)
 
1
6
( r
1 r r
R
2 λ
ρ
1
3

Solution of Harmonic Potential Model
H
0 int  p
2
 
2 m
3 K
2

2  p
2 l 
2 m
3 K
2 l
2
E
N

E
0

N

0
 
0
3 K m
N

N


N l
L
  l l
P ( 1) l 
 l l

00

(
3 Km

2
)
3 4 e

(3 Km )
1 2
(
2 2
) 2
 l
11
 
3 Km

3 2
( l x
 i l y
) e

(3 Km )
1 2
(
 l 2
) 2

V ( q q ss i j
)

4

9
3
 s s i
 s c m m i j i

3 
,

1 s s i
 i



3 for S=1/2

3 for S=3/2
D
M ss


 
4

9
4

9
3
 s
3 c m
2
3
 s
3 c m
2
 2
(0) for N
 2
(0) for
D
The three parameters m s,d
, α s
| ψ(0)| 2 , ω
0 are obtained by fitting to experimental data

Spectrum of low lying N and N* states m s,d
= 360MeV , ω
0
=500MeV

Non-relativistic quark model with the salt of QCD eg. Isgur-Karl Model
 Start with a non-relativistic quark model with SU(3)xSU(2) spin-flavor symmetry.
 SU(3) flavor breaking via quark mass difference.

(m u,d is not equal to m s
).
Long range confining force independent of flavor and spin.
 Only one gluon exchange accounts for short range spin and flavor dependent interaction.
(similar to electrodynamics of two slow moving fermions)

Isgur-Karl Model
H
0
 i
3 

1
( m i
 p i
2
2 m i
)
 
(
K
2
V ij hyp 
2
 s
3 m m i j
 


8
3
 
3
( ) ij i j r ij
2 
 s
4 l l i
 j
)
 
V ij hyp r ij
3(
 i ij
)( s
 r j ij
)
  i j r ij
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

 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

 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

 The boundary condition generates discrete energy eigenvalues.
 n
 x n
R
R - radius of the Bag x
1
=2.04
E kin
( R )

N q x n
R
N q
= # of quarks inside the bag
E pot
( R )

4
3

R
3
B
B – bag constant that reflects the bag pressure

 Minimizing E(R), one gets the equilibrium radius of the system
R n



N q x n
4

B


1 4
E n

4
3

4

BN q
3 x n
3
1

4
Fixing the only parameter of the model B, by fitting the mass of the nucleon to 938MeV we have first order predictions

 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?)
E
X

 s
M q
R
α s
– the strong coupling constant
M q depends on the quantum no. of the coupled quarks

The masses of N, Δ, Ω, ω were used to fit the parameters.



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!

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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).