# Problem Set #3

## Ground Rules for Problem Sets

• Due in class on Feb. 19.
• Will be handed back for corrections or revisions until we're happy.
• Answers to articulation problems are limited to three reasonable length sentences. You should imagine answering a question from a graduate student at a colloquium on the current topic. This is hard!
• Checkpoint questions should be fairly quick; if you are having a hard time, talk to classmates or the instructors.
• For any of the problems you can use any math tool such as maple or mathematica. Provide a printout of the session.

## Problem Set 3

### Articulation Questions

1. The phase diagram of hadronic matter on the Nuclear Science'' wall chart (there is a link from the web page) has temperature and mass density (kg/m$^3$) as the variables. Please comment.
2. What is the difference between a current quark and a constituent quark?
3. What keeps a bag in the bag model in equilibrium?
4. At a phase boundary between hadron matter and a quark-gluon plasma the chemical potentials of the phases are equal. Do the baryon densities have to be equal?

### Checkpoint Questions

1. Convert the estimates of the bag pressure $B$ given in the text to GeV/fm$^3$. Is the number reasonable?
2. Find the energies of the lowest two s-states in the bag model, using the value of $B$ in the text. Please comment.
3. Plot the running coupling in QCD as a function of $Q^2$. [Note: You'll need to look up a reference value; try the Particle Data Group web page.] For approximately what range of $Q^2$ is QCD perturbative?
4. The pions are often assumed massless in order to do the integral for the pion energy density analytically. Make a graph showing how good an approximation this is as a function of $m_\pi/T$. How does this affect the estimate for the transition temperature (at zero baryon density)? How much does the massless pion estimate change if the Boltzmann occupancy weight is used instead of the Bose-Einstein one?

### Project: Model of the Hadronic Matter Phase Diagram

Here you will model the hadronic and quark-gluon plasma phases of hadronic matter'' and study the thermodynamics of the phase transition between them. You will use a separate model for each phase and use thermodynamics to decide which phase is favored. You will need to evaluate the thermodynamic quantities numerically.

Model for quark-gluon plasma phase:

• Free gas of quarks and gluons plus bag pressure $B^{1/4} = 180\,$MeV [e.g., as in Eqs.~(9.9a) and (9.9b), only at nonzero chemical potential as well].
• Two flavors of massless quarks only (up and down).

• Baryons interacting via the exchange of neutral scalar and vector mesons (the sigma--omega'' model), plus a free gas of massive pions.
• The relevant equations for the energy density ${\cal E}$, pressure $p$, baryon density $\rho_{\scriptscriptstyle B}$, and the effective nucleon mass $M^*$ are given in a separate handout (except that you have to add the pion contributions). Use the experimental nucleon and pion masses. The only other parameters you need are the ratios given in Eq.~(3.64) of the handout.
• Note that $M^*$ must be determined {\it self-consistently\/}, which is another numerical problem for you!
1. At $T=0$, plot the saturation curves'' for each phase. This is a plot of the energy per baryon as a function of the baryon density (or the Fermi momentum $k_{\scriptscriptstyle F}$). (Note that at $T=0$ you can analytically eliminate the chemical potential $\mu$ in favor of the baryon density.) At what baryon density and binding energy per nucleon is the hadronic phase in equilibrium? How does this relate to the binding energy and density of ordinary nuclei? At what baryon density and binding energy per nucleon is the quark-gluon phase in equilibrium?