Course Outline for Physics 880.05
III. A. Thermodynamics and Phase Transitions Part 2
Recap of the Bag Model
- 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.
- 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.
- 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.
- 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.
Deconfinement at High Temperatures
- 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.
- 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.
- 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.
- Plot P vs. T^4 and e vs. T^4
Deconfinement at High Densities
- 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!).
- 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.
- 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).
- Plot schematic phase diagram of T vs. rho_B and show how it
relates to the homework.
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Copyright © 1997,1998 Richard Furnstahl and James Steele.