The Constituent Quark Models

Outline 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)

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 Model1964 The model was proposed independently by Gell-Mann and Zweig Three fundamental building blocks 1960’s (p,n,) 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 =12

(uu - d d), k+= ds, ko= ds

baryons are bound state of 3 quarks:proton = (uud), neutron = (udd), = (uds)

anti-baryons are bound states of 3 anti-quarks:

p u u d n u d d u d s

Λ= (uds))( ud

QuarksThese 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 = I3 +1/2(S+B) = I3 +Y/2• Group Structure is SU(3)• 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 QuarksSuccesses 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. -)• 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. p/pp = 2/3)

Failures of the 1960's model:• No evidence for free quarks (fixed up by QCD)• Pauli principle violated (++= (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

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

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

Quantum Chromodynamics (QCD)

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

Quantum Chromodynamics (QCD)

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

Quark confinement

Spatial confinement Quarks cannot leave a certain region in space

String confinement The attractive( color singlet) quark-antiquark

Color confinement

What Models do we have?

Harmonic Potential Model (for N and N* states, mu=md=m)

2( )2ij ij

KV r r

1 2 3

1 2

1 2 3

1( )

31

( )2

1( 2 )

6

R r r r

r r

r r r

R

λ

ρ

3

21

23

01

1 1( ) ( ) ( )

2 2 2i

i ij ss iji ij iji

pH m V r V r

m

Solution of Harmonic Potential Model

2 2int 2 20

3 3

2 2 2 2

p pK KH

m m

0 0NE E N 0

3K

m

N N N L l l ( 1)l lP

1 2 2 2

1 2 2 2

3 4 (3 ) ( ) 200 2

(3 ) ( ) 211 3 2

3( )

3( )

Km

Kmx y

Kme

Kmi e

Spin-Spin Contact Interaction

34( ) ( )

9i i

ss i j si j

V q q xc m m

3

, 1

3 for S=1/2

3 for S=3/2i ii ji j

The three parametersms,d , αs|ψ(0)|2, ω0

are obtained by fitting to experimental data

32

3 2,

32

3 2,

43 (0) for N

9

43 (0) for

9

s

u d

ss

s

u d

c mM

c m

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

Start with a non-relativistic quark model with SU(3)xSU(2) spin-flavor

symmetry.

SU(3) flavor breaking via quark mass difference.

(mu,d is not equal to ms).

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

No spin-orbit interaction, comparing to shell model Spin-spin contact interaction acts when L is zero Tensor interaction acts when L is Nonzero

232

01

33

( ) ( )2 2 4

3( )( )2 8( )

3 3

i j hypi si ij ij

i i j i ji ij

i ij j ij i jhyp sij ij i j

i j ij

p KH m r V

m r

s r s r s sV r s s

m m r

Nstar Spectrum

M.I.T. Bag Model

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

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

The boundary condition generates discrete energy eigenvalues.

R

xnn

R - radius of the Bag

x1=2.04

BRRE

R

xNRE

pot

nqkin

3

3

4)(

)(

Nq = # of quarks inside the bag

B – bag constant that reflects the bag pressure

M.I.T. Bag Model

Minimizing E(R), one gets the equilibrium radius of the system

4133

41

43

4

4

nqn

nqn

xBNE

B

xNR

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

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?)

R

ME qs

X

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

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 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

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).