1. The Color Charge & Bag Model
Outline
Color in Quantum Chromodynamics (QCD)
Color Charges
Quark confinement
M.I.T. Bag Model
Assumptions
Bag Model of Quark Confinement

2. Color
Quarks have a physical property called color, it could be blue, green or red
Each color also has an anti-color
They are not really different colors, it is a property, like charge
Quarks cannot exist individually because the color force increases as they are pulled apart.

3. Color in QCD
Quantum Chromodynamics (QCD) is the theory of the strong interaction.
a) The interaction is governed by massless spin 1 objects called “gluons”.
b) Gluons couple only to objects that have “color”: quarks and gluons
c) There are three different charges (“colors”): red, green, blue.
(in QED there is only one charge (electric)).
gluon exchange can change the color of a quark but not its flavor.
e.g. a red u-quark can become a blue u-quark via gluon exchange. u g u
grb u u
e) Since gluons have color there are couplings involving gluons.

4. Color in QCD There are several interesting consequences of QCD:
a) Quarks are confined in space.
We can never “see” a quark the way we can an electron or proton.
Explains why there is no experimental evidence for “free” quarks.
b) All particles (mesons and baryons) are color singlets.
This “saves” the Pauli Principle.
In the quark model the D++ consists of 3 up quarks in a totally symmetric state.
Need something else to make the total
wave function anti-symmetric Þ color!
Asymptotic freedom.
The QCD coupling constant changes its value dramatically as function of energy.
As a result quarks can appear to be “free” when probed by high energy (virtual) g’s and yet be tightly bound into mesons and baryons (low energy).
d) In principle, the masses of mesons and baryons can be calculated using QCD.
But in reality, very difficult to calculate (almost) anything with QCD !

5. Color Charges Pauli exclusion principle?
two or more identical fermions may not exist in the same quantum state
what about the u quarks in D++ ?
Pauli principle violated (D++= (uuu) wave function is totally symmetric) (fixed up by color)
++ = u u u
It must be antisymmetric under Pauli principle!
More questions on the quark model 5

6. Color Charges ++ = uR uG uB
Another internal degree of freedom was needed “COLOR”
Postulates
quarks exist in three colors:
hadrons built from quarks have net zero color
(otherwise, color would be a measurable property)
We overcome the spin-statistics problem by dropping the concept of identical quarks; now distinguished by color
++ = uR uG uB 6

7. Color Force
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 7 7

8. The Gluon The strong force holds quarks together to form hadrons
Its carrier particles are called gluons
The strong force only acts on very short distances

9. Colored Quarks and Gluons
Each quark has one of the three color charges and each antiquark has one of the three anticolor charges
Baryons and mesons are color-neutral just as red-green-blue makes white light

10. Fundamental process: Electron-Positron Annihilation
If one of the quarks is pulled away from its neighbors, the color field stretches between that quark and its neighbors. Then New quark-antiquark pairs are created in the field

11. Color Force We have assigned a “hidden” color quantum # to quarks.
“hidden” because detectable particles are all “colorless”
It solves the embarrassment of fermion statistics problem for otherwise successful quark model.
Like colors repel and unlike colors attract
The color force between color-neutral hadrons (like a proton and a neutron) is negligible at large separations
The strong color force between the constituent quarks does not exactly cancel at small separations
This residual strong force is the nuclear force that binds the protons and neutrons to form nuclei 11

12. Quark Confinement
The only quark and anti-quark combinations for color confinement is given by
(Martin’s Book, Page-165):
For baryons are the states with p=1, n=0 and for mesons are states with p=0, n=1.
States such as qqqq and qq are forbidden by the singlet (confinement) requirement.
However, the following states are allowed:
What about bound
states of gluons?
“glueballs”
There have been many experimental searches for quark bound states other than the conventional mesons and baryons. To date there is only evidence for the existence of the conventional mesons and baryons.

13. QCD & Color Asymptotic freedom Quark confinement
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 (will be discussed Later)

14. M.I.T. Bag Model
Developed in 1974 at Massachusetts Institute of Technology (MIT)
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 14

15. 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 15

16. Bag Model of Quark Confinement
In dealing with the nature of quark confinement, one visualization is that of an elastic bag which allows the quark to move freely around, as long as you don't try to pull them further apart. But if you try to pull a quark out, the bag stretches and resists.
                                                                                                                   
The quarks of a proton are free to move within the
proton volume
If you try to pull one of the quarks out, the energy required is on the order of 1 GeV per fermi, like
stretching an elastic bag.
The energy required to produce a separation far exceeds the pair production energy of a quark-antiquark pair, so instead of pulling out an isolated quark, you produce mesons as the produced quark-antiquark pairs combine.

17. Bag Model of Quark Confinement
The models of quark confinement help in understanding why we have not seen isolated quarks. If one of the constituent quarks of a particle is given enough energy, it can create a jet of mesons as the energy imparted to the quark is used to produce quark-antiquark pairs.
Experiments show that the forces containing the quarks get weaker as the quarks get closer together, so that within the confines of a baryon or hadron, they are essentially free to move about. This condition is referred to as "asymptotic freedom".

18. Bag Model of Quark Confinement
Asymptotic Freedom
As the quarks within a meson or baryon get closer together, the force of containment gets weaker so that it asymptotically approaches zero for close confinement.
The implication is that the quarks in close confinement are completely free to move about.
Part of the nature of quark confinement is that the further you try to force the quarks apart, the greater the force of containment. This is often visualized in terms of the "bag model" of quark confinement.

19. M.I.T. Bag Model
A potential function which has been successfully used to describe some quark systems is of the form:
The boundary condition generates discrete energy eigenvalues.
R - radius of the Bag
x1=2.04
Nq = # of quarks inside the bag
B – bag constant that reflects the bag pressure

20. M.I.T. Bag Model
Minimizing E(R), one gets the equilibrium radius of the system
Fixing the only parameter of the model B, by fitting the mass of the nucleon to 938MeV we have first order predictions