Course Outline for Physics 880.05

II. C. Hard Processes in Nucleon-Nucleon Collisions (Ch. 4)

We will substitute most of the details given in Wong with more up-to-date descriptions. Therefore in preparation, read the Introduction to 4, sections 4.2, 4.3, and 4.5 but don't worry about the supplemental discussions in those sections.

1. Anatomy of a Heavy Ion Collision
• Formation: from the initial collision to the thermalization (0.1-0.3 fm)
• Expansion: cooling as different particles come to chemical equilibrium (1-5 fm)
• Hadronization: breaking up into color singlets which can be considered thermalized hadrons that collectively interact. (20-40 fm)
• Freeze-out: Measurement.

This lecture will concentrate on the formation stage which includes the hard processes talked about in Wong. The latter three stages will be discussed more in future classes. Although the shortest in duration, the formation sets the initial conditions for the entire evolution.

• Before the messy nucleon on nucleon collision, consider point like probe on one nucleon.
2. Deep Inelastic Scattering (DIS): carried out at SLAC in 1972 (Nobel Prize in 1990) between an electron and a proton.
• Why "deep" (i.e. high energy) collisions needed? Like a microscope or the ocean, need smaller wavelengths to distinguish smaller distances.
• Why inelastic? We know the proton is not pointlike due to the form factor having a momentum dependence. But restricting the collisions to be elastic demands all constituents of the proton to act together, so cannot see inside proton.
• Note: the photon which is exchanged has four-momentum q^2 < 0 since two particles can only exchange space-like signals. Furthermore, for elastic collisions, -q^2 := Q^2 = 2M\nu with \nu=E_{initial electron} - E_{final electron}
• The constituents are often called "partons" and we know there have to be an odd number of fermions in the partons (to make a composite fermion). Is that all there is?
• To make things simpler:
• accelerate the proton to large velocities so that the partons don't interact (if exactly on light-cone, can't interact).
• so fast that transverse momentum can be neglected and say parton 'i' carries a fraction of the initial proton momentum: p_i = x_i P, with x_i between 0 and 1. This is the FEYNMAN x.
• note, these particles are off shell, since can't square above four-momentum to get the parton mass (too large!)
• at the scale of the parton-photon collision, assume elastic, this then says x=Q^2/(2M\nu) which is called BJORKEN's x. Note it is equal to the Feynman x but not exactly the light cone x_+ used in Wong. (See handout from previous class.) To distinguish, we will explicitly call the Bjorken x = x_B and the Wong x will remain x below.
• If partons are now considered point-like and the masses are small enough to be neglected, then there is no dimensional scale in the collision and the form factors describing the collision should only depend on the dimensionless variable, x_B. This hypothesis is called scaling and was shown to hold in experiment.
• In fact, the cross section is just the Rutherford cross-section for point-like particles times a function of x_B which is interpreted as the mean square charge distribution of the partons carrying four-momentum x_B P: a structure function, specifying the structure of the quarks in the proton. This is a fundamental property of the hadron and does not depend on the collision that occurs.
3. Beyond Deep Inelastic Scattering: Scaling violations
• Notice in the figure that not all x are scale independent. Works best for "large" x (x > 0.3 as discussed in last lecture) but even for x near 1 the scaling starts to break down. What is going on?
• This depended on the assumption that the transverse momentum of the parton is much less than the longitunidal, but a more accurate limit is (p_T)_i^2 < Q^2, which can be derived from requiring the interaction time to be much less than the lifetime of the parton.
• As increase the probe momentum, Q^2, the number of partons that can be in the hierarchy increases: Q^2 >> (p_T)_1^2 >> (p_T)_2^2 >> ...
• This means that as Q^2 increases the number of partons will increase so the structure function increases. And for large Q^2 and large x, the number of partons are enough that the probability that they can have large x decreases, so the structure function will decrease here. This explains the violation of scaling.
• Next thing to consider is hadron-hadron collisions, but in heavy ion collisions, we do not have the luxury of choosing x about equal to 1 (remember this means 0.3 < x < 1) so we need to also look at semi-hard processes.
• Plus, we know the parton can branch into two (or more) partons. Although the coupling is small for the momentum transfers we are talking about, the phase space is large and this leads to unit probability of this branching to occur.
• This means that at smaller and smaller x, the momentum of the parton is small and the probability increases even faster to produce more partons.
• Defining the number of partons with momentum fraction x that are able to absorb a photon of momentum q^2 as F(x,q^2), the parton density is proportional to R_hadron^2/F(x,q^2). If this is too large, the partons will start to recombine into more fundamental partons.
4. There are two theoretical approaches which are presently used to answer the following question:

When two coherent parton wavefunctions collide, how does it initially form into a quark-gluon plasma?

The reason to have these models are actually two-fold: to decide what signals will determine if a quark-gluon plasma has been formed in heavy-ion collisions and to decide how to build the detectors to look for these signals.

Both rely on the idea of a Cascade: a scenario in which particles freely propagate and quantum mechanical cross sections are taken into account only when two particles are close enough together (like billiard balls or molecules in a box). The statistics of the particles are input through the Boltzmann transport equation which relates the propagation of the particles in question to the particle distribution function in phase space.

• QCD string breaking (Refs: NPA498 (89) 567c, PLB 289 (92) 67)
• These ideas were developed from the Lund model and are most notably used in a code called RQMD (Relativistic Quantum Molecular Dynamics, the name describes the fact that it is a cascade approach, see above)
• Already know that p+p collisions are not binary elastic collisions above 100 GeV momentum (although up to there works quite well). Therefore a naive hadron cascade doesn't work. Need to include parton effects.
• First: input string interactions (a string is the flux tube formed when two colored objects are pulled apart)
• the idea: when two nucleons collide, pushes one "parton" out, this then pulls a string with it from the wounded nucleon. If the density of these strings is not too large (number/area < 1 fm^{-2}), they can be considered non-interacting and fragment independently (in a proper time of about 1 fm). Works in AGS (Alternating Gradient Synchtron) collisions well.
• If strings interact, can still save the theory by considering color ropes where they all coalesce into a larger string which in the end fragments (this allows for more massive quark formation as well). Has shown some success at SPS energies, reproducing the Lambda distributions.
• The non-interacting string picture is nice because only use known parameters and no fits requires. Adding the interaction between strings is messier and more ad hoc. Some of these issues will be discussed again in the next class.
• In terms of a phase transition, the strings carry energy which is not kinetic energy and can be released in a relatively short time just like latent heat. This seems to suggest this model will give a 1st order phase transition although that has not been calculated directly.
• Parton Cascade (Ref: K. Geiger, Phys Rep 258 (95) 237)
• Normally, partons are considered coherent since they stay neatly tucked inside a hadron (with lifetimes much longer than the interaction time). However, an energetic probe can disturb this by kicking a parton loose (see Fig. 3 of Geiger, the reference given above).
• This parton cascade fully equilibrizes after 0.5 fm (!) with a temperature of about 325 MeV, therefore color correlations randomize fast enough upon impact to assume perturbative QCD between the partons is relevant
• Instead of using QCD's first principles which gets difficult, use the ideas of a evolution semiclassically in a parton cascade
• In the parton cascade, the Boltzmann equation is replaced by the DGLAP (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) equations for the evolution of the parton distributions.
• Problems with the parton cascade?
• Partons are looked at perturbatively, but there is a non-perturbative vacuum that must be addressed once the partons recombine into hadrons and this it is not clear how to do this.
• Until confinement is solved, the mechanism which clusters quarks into hadrons is still a qualitative picture.
• At finite temperature, partons develop a mass proportional to the temperature, but we assumed the partons to be massless. Simulations seem to show this is not a huge problem, but needs further investigation.

Comparing the two approaches, strings break down at large energies and parton cascade at low. Where does the AGS and CERN sit? How about RHIC and LHC? Where is the transition between the two and why?