In this period we apply the Glauber approach from Chapter 12 to discuss the stopping power of nucleons in nucleon-nucleus and nucleus-nucleus collisions. The material corresponds to section 13.1 in Wong's text.

- Recall Figure 12.2.
- Inputs: NN inelastic cross section sigma_in and the density distributions of the nuclei and a nucleon. The probability of collisions is related to the probability of two nucleons colliding, summed over all places the nucleons can be in the two nuclei.
- The basic quantity is T(b)*sigma_in, which is the probability of a collision between a pair of nucleon when the nuclei are at impact parameter b. T(b) is the thickness function, which is conveniently parameterized as a gaussian.
- The probability of n collisions at b is given by a binomial distribution formula with T(b)*sigma_in as the probability. [Rewrite formula.]
- Probability of any collision defines:

d\sigma^{AB}_in/d\vec b = 1 - (1-T(b)\sigma_in)^(A*B)

(just the ratio of areas again) - The total inelastic cross section is found by integrating over impact parameters. It can be expressed as a finite sum if gaussians are used for the thickness function.
- Expressions for the average number of collisions at impact parameter b and for any b are given in the text.

- We've seen evidence that a substantial fraction of inital kinetic
energy of nucleons is lost by nucleon-nucleus collisions.
- It is deposited in the vicinity of the center of mass, implying regions of very high energy densities.
- We'll go through Bjorken's estimate of these energy densities.

- We expect two energy regions according to whether the nucleons in the
beam nucleus lose enough units of rapidity to be in the central rapidity
region or whether, despite losing energy, they continue along the z-axis
away from the region of high energy density.
- The baryon-free quark-gluon plasma region is expected for \sqrt{s} > 100 GeV/nucleon. So named since the baryon number recedes from the COM, leaving little net baryon content.
- The baryon-rich qgp region is around \sqrt{s} from 5 to 10 GeV/nucleon. Here the incident nucleons largely stop as the result of the collision.

- Let's apply our Glauber formalism to quantify the degree of stopping.

Supplement 13.1 discusses how to find the distribution expected in light-cone momentum fraction x after a given number n NN collisions in the nucleus-nucleus collision. Figure 12.1 shows the distribution for pA collisions and we've discussed qualitatively many of the features.

- Recall Feynman scaling result: independent of energy, the x value of a proton after a collision with another proton is equally probable to be anywhere from 0 to 1 (excluding the regions near the endpoints).
- What is the distribution after n collisions? We must
*compound*the results of each scattering to find the result. We can work through the results in the text, or discuss how to set this up for the problem set. We'll quote the former and concentrate on the latter. - So we restate the problem: Given the number of collisions n, use random numbers \gamma uniformly distributed between 0 and 1 to generate a final value for x that has the proper distribution.
- Before the first collision x_0=1.
- If after the i'th collision the value is x_i, then the value
of x_{i+1} is evenly distributed between 0 and x_i (not 0 and 1).
So we generate n random numbers \gamma_i, and take

x_i = \gamma_i * x_{i-1} for i=1 to n. - You can check (and should!) that the average value of x_n is (1/2)^n.
- Note that the actual distribution function derived in the text is quite complicated in comparison to this simple prescription.
- What if we generalize the distribution in x from a uniform
distribution? Wong introduces \alpha and postulates the distribution
probability:

w(x_0,x_1) = \alpha x_1^{\alpha-1} and

w(x_{n-1},x_n) = \alpha (x_n/x_{n-1})^{\alpha-1} 1/x_{n-1}- The key feature in the second line is x_n^{\alpha-1}. The rest is to normalize w properly, since x_n only runs from 0 to x_{n-1} [the integral over this region should be one, since it is the sum of all probabilities].
- Note that \alpha=1 reduces to the uniform case, while \alpha>1 means that larger values of x (toward 1) are more favored.
- Now we cannot simply scale are random numbers \gamma_i and expect to find this distribution for general \alpha.

- General rule: equate w(x_{n-1},x_n) dx_n to d\gamma_n and integrate
to find \gamma_n as a function of x_n. Then invert to find x_n as a
function of \gamma_n, which is what we want.
- Recall (normalized) distribution for impact parameters: 2\pi b db/(\pi b_max^2) = d\gamma. Integrating and inverting to find b yields b = b_max * sqrt{\gamma}, as before.
- Here we find x_n = x_{n-1} * \gamma_n^{1/\alpha}, so a pretty simple generalization!
- We can also make a hybrid rule, where \alpha=1 for the first collision but then \alpha=3 for subsequent collisions. Wong cites this as supported by some researchers.

- The general result is that <y_{n-1}> - <y_n> = 1/\alpha. (Note that you can calculate this from your program.) So we can relate the number of collisions to the net loss in rapidity.
- Since our result is independent of energy, we'll lose essentially the same number of units of rapidity at 10 GeV and 100 GeV. But in the first case, that will stop the beam (since beam and target rapidities are only a few units apart) and in the second case it will still have plenty of steam, despite having deposited a lot of energy. So we get the two regions: baryon-rich and baryon-free.

Consider Figure 13.1.

- Here we study likelihood for different amounts of energy deposited in a near-zero-degree detection for an O16 beam on various targets (C,Cu,Ag,Au).
- Particles within 0.3 degrees of the incident beam (so \eta>6) are
included. These are the ones that are
*not*stopped very much. - So "1-this" is a measure of stopping.
- Potentially 60*16 = 960 GeV is available. That ends up in the detector often on C12 but on Au the peak or average is most lower ==> much more stopping.

Next consider Figure 13.2.

- Rapidity distribution dN_p/dy of leading protons in Si on Al collision at 14.6*A GeV.
- It's claimed that one can tell the leading protons since those from B\bar B production can be ruled out by comparing \Lambda production at higher energy.
- Here y_b = 3.4 and y_t = 0. Thus y_cm is 1.7.
- There is almost a flat distribution in the central region.
- The average rapidity shift is about 1.5 units with a large distribution, implying lots of stopping (since the average takes a proton to y_cm).
- We argue that stopping should be much greater with heavier nuclei.
- At higher energies, like 100 GeV/nucleon => y_b-y_t = 10.7 units [quick: in the lab or com], so a loss of 2-4 units will leave the beam and target baryons far from the central rapidity region => low baryon density and a "baryon-free QGP".

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Copyright © 1997,1998 Richard Furnstahl and James Steele.