Course Outline for Physics 880.05

III. A. Thermodynamics and Phase Transitions Part 2 (Section 9.3)

Recap of the Bag Model

1. Summary of Complexity of the QCD vacuum: When we think of vacuums, we most likely think of an empty region void of any activity. Quantum corrections make for a more interesting picture due to virtual pairs, but in QED the coupling is small enough that this simple concept of a (perturbative) vacuum can still be used. However, in QCD, things are drastically different due to the large coupling constant. Virtual pairs created in the vacuum interact with each other to form vortices, non-vanishing expectation values of some operators (condensates), and even tunneling between degenerate vacua (instantons). This nonperturbative vacuum is the lowest energy state of QCD and the foundation for QCD calculations.
2. Definition of the Bag Constant and its interpretation: The Bag model's description of a nucleon (and similarly other hadrons) skirts this issue of the complicated vacuum by assuming the 3 quarks only exist in a bag with the simple, free (perturbative) vacuum present. Since the nonperturbative vacuum has lower energy, energy is required to sustain this bag. The energy required to clear away a volume dV is just defined to be B dV.
3. Energy in the Bag Model: The quarks are confined in the bag, and so this leads to a minimum energy given by the uncertainty principle (2.04/R, for a bag of radius R) and therefore the total energy of the bag is
``` E_bag = 2.04 N_q/R + (4 Pi R^3/3) B
```
which, when choosing R=0.8 fm gives B=200 MeV/fm^3 by minimizing the energy.
4. Deconfinement Transition: In general, the bag model has energy density and pressure given by
``` epsilon = epsilon_quarks + B
P = P_quarks - B
```
the minus sign in P is from the vacuum pushing in on the bag. This signifies that when the quark pressure increases beyond B the pressure will become positive and the quarks will break free of the bag and become deconfined. We would like to calculate this breakdown point, but first need to brush up on our relativistic statistical mechanics.

Calculation of Thermodynamic Quantities

``` NOTE: This section requires quite a few equations which I won't
even attempt to annotate on the web.  Just ask if you would like a
copy of my notes.
```
• Pressure: Starting with black body radiation as an example, I will show how the pressure relates to the free energy and energy density which leads us to a rederivation of the Stefan-Boltzmann Law:
``` energy density = constant T^4
```
for a system of massless bosons with temperature T. We also will derive the analogous relations for fermions and give the speed of sound in matter.

Deconfinement at High Temperatures

1. What temperature is needed to break out of the bag? We take the extreme case of rho_B=0. All we need is to compare the pressure from the QGP to the Bag constant. Taking the contribution from the gluons and assuming only the light quarks (up and down) contribute give 37 degrees of freedom in the QGP. The temperature where deconfinement occurs turns out to be about T_c=140 MeV.
2. Minimizing the Free Energy: We can also go about this by minimizing the free energy between the two systems (the hadrons and the QGP). For this we need a model for the hadronic side (the bag model only gives us a change in pressure between the two phases, not a direct pressure), so we just assume it is a gas of massless pions, which has 3 degrees of freedom. This gives T_c=145 MeV, which is not a big change.
3. However, what if we considered a more sophisticated model which took all the possible hadronic degrees of freedom into account (nucleons, higher mass mesons, ...)? This might change the result considerably. This is one of the things you are asked to do in your homework project.
4. Plot P vs. T^4 and e vs. T^4

Deconfinement at High Densities

1. Chemical Potential: OK, I cheated a little above to keep the discussion simple. There is also a chemical potential (fermi energy) for the distribution functions. The presence of one relates to the conservation of a charge. In this case it is baryon number which is conserved, so we needed to take the chemical potential into account above. However,
``` mu_quarks = - mu_anti-quarks
```
so the case of rho_B=0 is just the same as mu_q=0 and we were justified in doing it above (but only if rho_B=0!).
2. Now if we want to dial up the baryon density, we need more quarks than antiquarks, and this can be done by adjusting mu_q.
3. Let's assume the extreme case of zero temperature, then we can calculate the density required for the quarks to break out of the bag analytically, and it converts into a critical density rho_c=5 rho_0 (with rho_0=0.15/fm^3, the density of nuclear matter).
4. Plot schematic phase diagram of T vs. rho_B and show how it relates to the homework.