# Problem Set #1

## Ground Rules for Problem Sets

• Due in class on Jan. 15.
• Will be handed back for corrections or revisions until we're happy.
• Suggestions and hints will become available through the 880.05 web page.
• Answers to articulation problems are limited to four 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 1

### Articulation Questions

1. Why is QCD asymptotically free but QED is not?
2. A ferromagnet minimizes its energy by having its spins aligned. So why do the spins become randomly aligned above the Curie temperature?
3. Why is the light-cone momentum fraction $x_+$ a useful variable?
4. Why is pseudorapidity often used instead of rapidity by experimentalists?
5. Why does the $e^+e^-$ curve in Figure 3.1 in Wong lie above the $pp$ curve?

### Checkpoint Questions

1. What is force in natural units? (Include the correct powers of $\hbar$ and c.)
2. What is the energy of a "leading" proton with transverse momentum 400 MeV/c and $x_+ = 0.7$, if it is the product of a collision of a 10 GeV proton beam and a fixed target?
3. Is the top graph in Figure 3.2 roughly consistent with Figure 3.1?
4. Estimate the average $p_T$ from Figure 3.3 in Wong. Is it what you expect?
5. Use Mathematica or Maple or some other computer math/graphics tool to compare graphically the rapidity $y$ and pseudorapidity $\eta$ for a pion with fixed $p_T = 300$ MeV/c as a function of the pion energy.

### Project: Rapidity and all that in heavy-ion collisions

At the CERN-SPS fixed target facility, beam energies can reach 200 GeV/nucleon in the laboratory frame. RHIC, the colliding beam facility that is being constructed at Brookhaven, is to have energies of 200 GeV/nucleon in the center of mass frame.

Assume that in a collision one-half of the center-of-mass energy goes into creating pions, whose density determines the temperature.

1. What is the maximum beam momentum at the CERN-SPS?
2. Compare the $\sqrt{s}$ center-of-mass energies of the two facilities.
3. What is the ratio of the longitudinal to transverse radii of the projectiles in the lab frame before collision for each facility?
4. Make a classical estimate of the corresponding density as a function of time. (Estimate the density profile at several relevant times.)
5. What temperature can be attained? Specify your assumptions.
6. Make a rough sketch of the rapidity distributions you expect for spectator nucleons (ones that don't interact) and produced hadrons. Indicate the beam rapidities quantitatively.
7. What measurements would be needed to determine the rapidity and pseudorapidity?