# Problem Set #2

## Ground Rules for Problem Sets

• Due in class on Feb. 3.
• 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 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 2

### Articulation Questions

1. Why are "hard" processes in QCD easier to deal with than "soft" processes?
2. What is the difference between a string, a quark, and a parton?
3. Why is it reasonable to call the force between quarks called the "string tension"?
4. What is the empirical (experimental) evidence for a linear potential between quarks?

### Checkpoint Questions

1. We have seen graphs of $1/p_T (d \sigma)/(d p_T)$, $1/m_T (d \sigma)/(d m_T)$, and $1/m_T (d N)/(d m_T)$. How are they related to each other and which is most useful?
2. What would you plot to get straight Regge trajectories if the potential between quarks went like $r^2$, where $r$ is the distance between the quarks?
3. Look up the kaon resonances in the Particle Data Tables (it's all online at http://pdg.lbl.gov/). Identify a set of resonances that lie on a Regge trajectory and fit a least-squares straight line using Mathematica or Maple to extract the string tension.
4. Using the model for the thickness functions at the beginning of Section 12.3, estimate the average number of baryon-baryon collisions in a high-energy inelastic gold on gold collision.
5. Does the rapidity distribution in Fig. 13.2 imply that the collision is closer to the stopping'' region or the pure quark-gluon-plasma'' region?

### Project: Simulation of High-Energy Proton-Nucleus Scattering

Figure 12.1 in Wong shows differential cross sections for a detected proton from the collision of 100 GeV/c protons on various nuclear targets ranging from a proton to lead. The transverse momentum is fixed at p_T = 0.3 GeV/c and the cross section is given as a function of x.

We would like to reproduce, at least qualitatively, the features of this graph based on a numerical simulation (this means you're going to write a program!). We'd like to reproduce:

1. the scaling of the cross section with the number of nucleons A in the target
2. the concentration at low x for A >1
3. the increasing slope with increasing A
4. an estimate for the average number of baryon-baryon collisions for each target.

We'll apply the Glauber picture described in Chapter 12, with the following assumptions:

• The trajectory of an incident proton is a straight line at a fixed impact parameter b.
• The NN inelastic cross section is energy independent and given by \sigma_in = 30 mb = 3 fm^2.
• The incident nucleon thickness function and the baryon-baryon collision thickness function can be taken to be delta functions [as in the discussion above eq. (12.17)].
• Model the target thickness function as in section 12.3 [eq.(12.26) and after].

The basic procedure is:

1. Generate a distribution of protons with impact parameter b corresponding to a uniform beam.
2. For each proton and corresponding b, find out how many inelastic collisions it makes [distributed according to eq.(12.9)]. Keep track of the number of collisions (if greater than 0) and compute the average number at the end.
3. Compute the final value of x for each proton based on the number of collisions, according to the distribution D^(n)(x_n) in Supplement 13.1.
4. Collect statistics for bins in x to generate relative cross sections and discuss your results.