Last period we considered the idea of strings in QCD as color electric flux tubes between quarks. Summary of basic points:

- The flux tube implies that the energy is linearly proportional
to the length of the tube (E = \kappa L), or that the force is constant.
- The constant of proportionality \kappa is called the "string tension".
- By looking at plots of hadron spins versus their mass^2, we see that they lie on "Regge trajectories" with close to the same slope. The slope is directly related to the "string tension" (we did the calculation last time), so \kappa has a universal value, which is found to be about 1 GeV/fm. [That is, J = const. + (1/2\pi\kappa)*M^2.]

- The flux tube between a quark and anti-quark is analogous to the
electric field in a capacitor. For large enough electric fields
in the capacitor, electron-positron pairs are produced. The mechanism
was described by Schwinger.
- What happens is tunneling of negative-energy ("dirac sea") particles to positive energy states, made possible by the linearly decreasing potential between the plates.
- The tunneling probability depends exponentially on the transverse mass^2 of the produced particle (a quark or quark-antiquark pair in this case). So smaller m_T particles are produced preferentially, and the distribution is predicted to be a gaussian: P = e^{-\pi*m_T^2/\kappa}.
- The scale in the exponential is set by the string tension,
which explains the observed
distribution for pions we saw from Chapter 3 figures. - Also sets the scale for the minimum string lengths to produce quarks: L_{min} = 2m_T/\kappa, which is about 0.7 fm for p_T=0 and 1.0 fm for p_T around 350 MeV/c.
- We can also estimate the ratio of produced kaons to pions, which also gives a reasonable number compared to experiment (see notes from last period). The basic idea is that a (constituent) strange quark is more massive than a (constituent) up or down quark, so the production of kaons relative to pions will be suppressed according to the gaussian probability.

In summary, we understand (qualitatively at least) why pions are produced the most and why the transverse momentum (p_T) distributions decrease rapidly with increasing p_T, with an average p_T around 3-400 MeV/c. Other aspects such as the plateau in the multiplicity distributions can also be explained by models. We'll briefly discuss Chapter 6 and the other pictures on the handout from last time.

Now we turn from pp scattering to protons on nuclei (pA) and nucleus-nucleus scattering.

Let's think about nucleus-nucleus collisions now. From Chapter 3 and the figures we learned that a nucleon colliding with another at high energy loses a significant fraction of its energy, which goes into creating other particles. We can imagine that a nucleus-nucleus collision will involve many nucleon-nucleon collisions, so that we can deposit a tremendous amount of energy in the region of the collision. This is our proposed method for reaching high temperatures or densities, with the hope of reaching the QGP phase transition.

Let's start with proton-nucleus collisions to see the patterns. Consider figure 12.1. This shows the cross section for protons on a variety of targets ranging from a proton to nuclei from carbon (A=12) to lead (A=208). The cross sections are all at fixed transverse momentum p_T=0.3 GeV/c and are plotted as a function of x. Recall that "x" here is a shorthand for x+. (How is dx d^p_T related to the invariant cross section?) Some observations:

- For pp collisions, we see that the x distribution is flat and energy
independent ("Feynman scaling") in the projectile fragmentation region
(x > 0.2). What does this tell us?
- A proton comes in with x=1 (by definition) and can leave equally likely with any x from 0 (really the lower kinematic limit) to 1.
- The average <x> is 1/2. So the average proton loses half of its light cone momentum in the collision.

- Where are the protons that get through having lost little energy? Near x=1. So less and less likely as the target gets heavier (=thicker).
- The cross sections are all approximately straight lines in the region shown. The average value of x is considerably less than 1/2, however.
- Curves for heavier nuclei lie higher => greater cross section. How does it scale with A? Could it be a geometric factor like the radius of a nucleus [A^(1/3)], or the surface area [A^(2/3)], or the volume [A]?
- The
*slopes*of the curves increase as A increases.- So <x> decreases.
- Or, we're more likely to have found that the proton slows down more as it passes through the target as the target A increases.

How can we understand what is happening in Fig. 12.1 and, more generally, how much energy is lost in a nucleus-nucleus collision? If we know how many collisions the incident nucleon makes, and we can model how x changes with each collision, then we should be able to reproduce (at least semi-quantitatively) Fig. 12.1 and generalize to nucleus-nucleus collisions. We'll use the Glauber model of multiple-collision processes, which is a high-energy scattering approximation.

- As a nucleon in the projectile suffers a collision with a target nucleon, the resulting baryon-like object (it may not be a nucleon) can still be treated as a nucleon continuing on the same path to make further collisions.
- We take the basic baryon-baryon cross section to be the same throughout (and energy independent), equal to about 30 mb = 3 fm^2. We also assume that the same cross section is appropriate even if the nucleon becomes an excited baryon in the midst of its collisions.
- The Glauber model uses the notion of a mean free path. We relate the probability of a collision between a nucleon in the beam nucleus and a nucleon in the target nucleus to the basic cross section and the basic geometry of the collision.

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Copyright © 1997,1998 Richard Furnstahl and James Steele.