# Course Outline for Physics 880.05

## II. B. Nucleon-Nucleon Collisions (Ch. 3)

1. Overview
• Particle production in discussed in detail in chapter 4 (hard processes) and chapters 5-8 (soft processes). This chapter is an overview with mostly empirical facts about NN collisions.
• We want to bring in the concepts here that we'll need for nucleus-nucleus collisions, as well as to illustrate the kinematic variables introduced in Chap. 2. Plan: Organize the discussion by looking at each of the figures in turn (plus two from Perkins) and asking what is learned from each.
• Any behavior noted for nucleus-nucleus collisions should be calibrated to NN collisions.
• An important question: To what extent is the large longitudinal kinetic energy dissipated by the collision, with energy going into other degrees of freedom?
2. Figure 4.15 from Perkins
• Proton-proton total cross section for (COM energy) 3 GeV < \sqrt{s} < 100 GeV is about 40 mb.
• Note that p_lab=1 to p_lab=2000 GeV/c is roughly the range above.
• total cross section is found from incident flux and scattering rates. More precisely, it is the transition rate (# per unit time) divided by incident flux.
• 10 mb = 1 fm^2
• Order of magnitude estimate of cross section: If nucleons only interact strongly if within 1 fm or 2 fm, then should be of same magnitude as hard sphere scattering ("geometric cross section"). So
\sigma \approx \pi R^2 \approx \pi (1 fm)^2 to \pi (2 fm)^2
\approx 3-12 fm^2 \approx 30-120 mb ,
so we're in the ball park.
• If we assumed a totally absorbing black disk of well-defined radius R, the elastic and inelastic cross sections are each \pi R^2. In reality, the ratio of elastic to total cross section is much less than half with increasing energy. [Recall elastic vs. inelastic cross sections.] Inelastic cross section is around 30 mb.
• Inelastic processes dominated by projectile energy loss and particle production. That is, the colliding particles lose lots of kinetic energy into other degrees of freedom.
• There are empirical parameterizations of pp total and elastic (and therefore inelastic) cross sections.
3. Figure 4.16 from Perkins
• Geometric cross section estimate gives order of magnitude for all.
• \bar p p has most energy dependence.
• Can you explain why \pi+ p is below \pi- p?
4. Figure 3.1 from Wong
• Inelastic processes are dominated by multiple production of secondary mesons and baryon-antibaryon pairs.
• Colliding nucleons typically lose a large fraction of energy, which goes into producing particles
• Diffractive dissociation
• small number of particles produced
• of order 10% of inelastic cross section
• not a large fraction of energy lost
• we'll focus on nondiffractive inelastic collision processes
• Note that the bottom axis in the figure is center-of-mass energy while the top is lab momentum for the pp process.
• The graphs show the average number of charged particles (called the "charged multiplicity") produced in a collision at a given energy.
• The multiplicity (number of particles produced) also includes uncharged particles, but these are often undetected, in which case only the charged multiplicity is given. One can infer the full multiplicity from the charged multiplicity with additional assumptions, such as that mostly pions are produced [in which case we'd multiply the charged multiplicity by 3/2 to account for the undetected \pi_0's].
• Mostly pions are produced, since they are the lightest hadrons. The rest are kaons, baryon-antibaryon pairs, and so on.
• The two curves are for e+e- annihilation and pp scattering. The e+e- curve is higher because it has more energy available to create particles (the "leading" particles in pp have half the light-cone momentum fraction).
• The multiplicity increases as the logarithm of \sqrt{s}.
5. Figure 3.2 from Wong
• Rapidity (pseudorapidity) distributions in pp reactions show in the figure for three different energies
• E.g., for the top figure roughly two particles are produced per unit rapidity in the central rapidity region.
• Are the curves consistent with Figure 3.1?
• Characteristic shapes
• bell-shaped at low energies (lower picture)
• at higher energies a plateau around the central rapidity region
• peak is at y \approx \eta \approx 0 in the center of mass or \eta \approx \eta_b/2 in the lab (if fixed target)
• The small dip in the center comes from plotting dN/d\eta instead of dN/dy (see the equations in Wong).
• Note that the data in the top two graphs are reflected around \eta=0. Why?
• Wong describes some parameterizations in terms of x+ and x-.
6. Figure 3.3 from Wong
• Differential invariant cross section for producing \pi^+'s (positively charged pions) in proton-proton collisions, as a function of the transverse momentum p_T of the detected pion.
• The cross section is invariant because the volume element of three-momentum d^3p/E is a Lorentz invariant.
• We're really interested in seeing the p_T distribution in the pion. The plot is shown at a center-of-mass (com) angle of close to 90 degrees, which isolates the variation of the cross section with p_T. So the plot, in effect, shows the probability of producing pions with different p_T's.
• A p_T around 1 GeV/c roughly separates the momenta into "soft" (below 1 GeV/c) and "hard" (above 1 GeV/c), which in turn correspond to nonperturbative or perturbative domains of QCD.
• At hard momenta, the QCD effective ("running") coupling is small enough ("asymptotic freedom") so that applying a perturbative QCD analysis is reliable.
• The graph shows that produced pions have mostly from soft p_T's at this energy according to the 1 GeV/c rule.
• But it may be that the region where a pQCD analysis can be applied in the form of a parton treatment extends lower. Furthermore, at higher energies the hard physics will certainly become more important.
• The curve is mostly straight, indicating an exponential dependence on p_T
• The detailed parameterization is a sum of two different exponentials.
• What is the average < p_T >?
• Approximating the curve as a straight line yields roughly < p_T > = 0.4 GeV/c (or less, depending on what part of the curve is used).
• Don't forget that d^3p \propto d(|p_T|^2)!
7. Figure 3.4 from Wong
• Differential invariant cross section for producing a variety of different hadrons in proton-proton collisions at a fixed center of mass energy and angle. The cross seciton is plotted as a function of m_T for the detected particles.
• Recall that m_T^2 = m^2 + p_T^2, where m is the mass of the particle.
• The curve is parameterized as proportional to e^{-m_T/T}/(m_T)^\lambda.
• The points lie essentially on top of each other, which implies very similar values for the exponential dependence on m_T (same T).
• This is known as "m_T scaling"
• As in 3.3, Wong claims mostly soft particle production
• Nonperturbative QCD unsolved for such processes
• Instead, phenomenological and qualitative models
• list of models or mechanisms:
• Schwinger mechanism
• QED_2
• preconfinement
• cluster fragmentation
• string fragmentation
• dual partons
• topological domains
• Summarized in review period (see schedule)
• Alternative discussion in terms of hard processes (especially for higher energies) next time.
8. Figure 3.5 from Wong
• Leading particles in NN inelastic collisions
• Here we focus not on produced particles but the remnants of the original beam and target particles.
• baryon number conservation says at least two baryons in products of collision
• one likely in projectile fragmentation region and other in target fragmentation region
• called "leading particles" in that case
• think of initial energetic baryons losing some of their light-cone momenta in inelastic collision and emerging as leading baryons.
• symmetric to target framentation process
• often use x to mean x+, where x+ is the ratio of p+ for leading baryon to the incident parent baryon
• so x \approx 1 if nothing (or not much) happens to its energy
• thus x characterizes degree of inelasticity
• Shape of transverse momentum distribution of leading proton shown in the figure.
• From figure 2.1, x+ > 0.3 is in region of projectile fragmentation, so we use x+ instead of y as the relevant variable.
• Shape depends on degree of energy loss
• exponential at x+=0.6
• gaussian at x+=0.4
• But approximately the same average p_T of 0.45-0.47 GeV/c! (The value for the exponential curve can be read off of the slope of the graph, with a factor of 2.)
• How can we explain p_T distributions? (Later!)
9. Figure 3.6 from Wong
• How is light-cone variable x=x+ for leading protons distributed?
• For x near 1, x and x_F are almost the same, so x distribution and x_F would look similar. [aside: Should plot x vs. x_F or the ratio or something like this to see how close they are for different x's.]
• Figure 3.6 shows that it is very flat. That is, we're just as likely to find any x from 0.2 to 0.9.
• It is nearly independent of energy as well. (Shown for pp beam momenta of 100 and 175 GeV/c. Actually sets in at lower energy.)
• This behavior is called "Feynman scaling"
• Interpretation of Feynman scaling
• Cross section measures some intrinsic properties of detected particle relative to parent
• Leads to concept of partons (chapter 4)
• detected particle originates from hard scattering of constituents and subsequent framentation or from direct fragmentation of parent particles
• near x=1, cross section for hard processes depends only on momentum distribution of parton ("structure function") in beam particle and on fragmentation function
• These should be properties of parent particle (how momentum is distributed among constituents) and fragmenting parton; it shouldn't depend on collision energy.
• Parameterization of d\sigma/dx
• equal probability to find product baryon in whole range of allowed x and < x > \approx 1/2
• so on average the product baryon carries 1/2 initial light-cone momentum, which means the other half is lost and used to produce particles.
• We can translate this into an average rapidity loss for the incident baryon.
• To find the average radidity of the leading proton, we need to find d\sigma/dy. But
d\sigma/dy = (d\sigma/dx)(dx/dy) = (d\sigma/dx)(m_cT/m_b) e^(y-y_b)
from the relationship of x+ and y. But the first term is constant, so d\sigma/dy is simply proportional to e^(y-y_b). Calculating the average:
< y > = [ \int_0^{y_b} y (d\sigma/dy) dy ] / [ \int_0^{y_b} (d\sigma/dy) dy ]
\approx y_b - 1
• So one unit of rapidity is lost on average in each nucleon-nucleon (NN) collision.
• In nucleus-nucleus collisions, many collisions of NN type can lead to large loss of incident energy/momentum => "stopping" if the incident momentum corresponds to a few units of rapidity.
• At higher incident energy, rapidity is still lost by the beam(s), but not enough to stop. In that case, the baryon density in the central rapidity region (where particles are produced) can be quite low.