Course Outline for Physics 880.05

Soft Processes in Nucleon-Nucleon Collisions (Chs. 5-8)

Recap: Hard Processes

The previous class was ambitious and threw out a lot of ideas. Let's step back and see how hard scattering fits into the big picture so far developed in this class.

Overview: Chapters 5-8

Chapters 5 through 8 in Wong review a variety of phenomenological models and approaches used to describe soft QCD processes relevant for NN collisions. Much of this discussion is less relevant at the very high energies that will be typical at RHIC than was apparent several years ago when Wong wrote his text. However, we can understand many of the qualititative features of NN collisions that we have seen in the figures of Chapter 3 of Wong by considering strings.

These chapters contain much more detail than we have time to consider. So we will focus on some basic string physics from Chapters 5 and 7.

Chapter 5: Particle Production in a Strong Field

We've seen from Chapter 3 that we do produce lots of particles in high-energy NN (and nucleus-nucleus of course!) collisions. We observed various characteristics of the collisions:

  1. The transverse momentum distributions are typically exponential (or gaussian) with <p_T> of order 300 to 400 MeV/c.
  2. The multiplicity distribution in rapidity, dN/dy, has a characteristic plateau shape at large energies.

In this chapter we consider a mechanism for particle production originally due to Schwinger. The idea is to have a concrete model of how particles are produced and how a characteristic p_T distribution of the produced particles can arise.

  1. Basic idea: Between a quark and anti-quark being pulled apart will be a color electric field. A particle--anti-particle pair is produced when a particle tunnels from the negative-energy sea to a positive-energy state (both in the continuum).
  2. Confined color flux tube between quark and anti-quark
  3. Schwinger Particle Production Mechanism
  4. Flux tube picture (Figure 5.1)
  5. String tension \kappa.
  6. Basic picture of tunneling
  7. Tunneling calculation
  8. The end result is the tunneling probability:
    P = exp(-\pi m_T^2/\kappa) = exp(-\pi [p_T^2+m^2]/\kappa)
  9. How many kaons are produced compared to the number of pions?

Chapter 6: Particle Production in 2-D QED

In this chapter, a model field theory, QED in 1+1 dimensions (1 space plus 1 time dimension) is explored. The last chapter featured a static model; in actual collisions a quark and antiquark will separate at large velocity (close to c). We'd like a model of the production and subsequent hadronization (how the quarks form into hadrons).

Why is QED_2 of interest? Because of the analogs to QCD and because it can be solved!

  1. QED_2 with massless fermions can be solved exactly.
  2. The fermion and anti-fermion in the theory are a positive and negative charged electron. The electromagnetic interaction in 1+1 dimensions goes like r (compare 1/r in 3+1), so it is like the string potential we have talked about for QCD.
  3. As a consequent, the electrons are confined, while the interaction is also asymptotically free (becomes weak at short distances).
  4. A neutral boson exists as a nonperturbative bound state, like mesons in QCD.
  5. The model features vacuum polarization with charge screening as in QCD.

It can be shown that QED_2 with interacting fermions is equivalent to a free field theory of a boson field. So we can model the production process by mapping an initial condition of the fermions to simulate a quark-antiquark pair separating at high speed onto the boson field and then evolving it in time.

  1. We can define the initial charge density as delta functions at positions z=t and z=-t for t=0 to infinity.
    j^0_ext(z,t) = -e \delta(z+t) + e \delta(z-t)
    Note that the plus charge is at z=t.
  2. The bound state mass is m=e/sqrt(\pi), so we can choose e so that M=0.40 GeV, which is the average transvers mass for pions produced in high-energy processes.
  3. Figure 6.2 shows j^0_prod(z,t) [they use x^1 instead of z] at several fixed times t as a function of z (spatial distribution).
  4. The boson field \phi_pro(\tau)
  5. Figure 6.4 shows emergence of particles
  6. Rapidity distribution is predicted to be a plateau
  7. Can we understand this result qualitatively?

Chapter 7: Classical String Model

The important aspects here are the discussion of strings, which we have moved earlier under Chapter 5, and the overview of the Lund model, which we summarize below.

  1. Figure 7.1 shows meson-meson scattering in the string picture showing s-channel/t-channel duality (the string diagrams have the same topology hence the same amplitude) => dual resonance model.
  2. Mesons are described as string segments undergoing longitudinal expansion and contraction with superposed translational motion. There is a quark at one end and an antiquark at the other. "Yo-yo motion."
  3. Particle production is represented by the fragmentation of a stretching string as the quark and antiquark move apart.
  4. Lund model particle production indicated in Figure 7.6.
  5. The text describes the motivation for a splitting function f and a vertex probability function \rho, which combine to describe string fragmentation, and hence particle production.

Chapter 8: Dual Parton Model

This chapter extends the discussion of soft (and hard) particle production models. We'll skip it for now.

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Copyright © 1997,1998 Richard Furnstahl and James Steele.