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Nuclear Transparency in Heavy Ion Collisions at 14.6 GeV/nucleon

The probability of a projectile nucleon to traverse a target nucleus without interaction is calculated for central Si-Pb collisions and compared to the data of E814. The calculations are performed in two independent ways, via Glauber theory and using the transport code UrQMD. For central collisions Glauber predictions are about 30 to 50% higher than experiment, while the output of UrQMD does not show the experimental peak of beam rapidity particles.

The dynamics of a heavy ion collision is a complicated many-body problem. It is the task of appropriately designed experiments to isolate one particular aspect of the dynamics and elucidate its physics. Wounded nucleons are one of the open problems. In a heavy ion collision a nucleon may undergo a sequence of collisions, which follow so rapidly one after another, that the nucleon is no more in its ground state, even not necessarily in any definite excited baryonic resonance (like ∆ or N * ). We will speak of a wounded nucleon. In a next encounter with another nucleon this wounded nucleon will not interact with the free space NN cross section σ N N in but with an effective one σ eff . Is it possible to determine σ eff from experiment?
The E814 collaboration has designed an experiment which may answer this question [1]. At the energy of 14.6 GeV/nucleon, the projectile 28 Si collides with Al, Cu and Pb and the beam rapidity nucleons are studied as a function of the centrality of the reaction (controlled by a measurement of the transverse energy). The beam-rapidity nucleons belong to the projectile and have not lost any energy in the reaction. Of course, in a peripheral reaction one always has the so called spectator nucleons, which never hit the target nucleus. They are not interesting for our purpose. However, in a central event, e.g., for Si on Pb, one still sees beam rapidity nucleons, i.e., nucleons of Si which go through the target nucleus without energy loss. The target nucleus is transparent for these projectile nucleons. We expect the transparency of a heavy nucleus like Pb to be small. Indeed, the "survival probability" S for a projectile nucleon to pass through the target without any inelastic interaction has been measured to S exp = 3.5 × 10 −3 for central Si-Pb collisions [1]. Does this result contain information about wounded nucleons and effective NN cross sections?
We proceed in the following ways: The transparency is calculated without assuming any exotic phenomena and then the quality of agreement with experiment is judged. We use two complementary methods of calculation • Glauber theory, in which the physics and formalism is simple and transparent, though not always realistic; • Cascade calculation (UrQMD), which represents present-day state-of-theart and may be compared to earlier calculations (Bass et al. [2]).
The survival probability S for a projectile nucleon to pass through the target without inelastic interaction is calculated in a Glauber-type approximation (straight trajectories, frozen nucleons) as where b is the impact parameter of the nucleus-nucleus collision, T p (b) and T t (b) are the thickness functions (T (b) = dzρ(b, z), d 3 xρ(x) = 1) of projectile and target, respectively. The straight line geometry is certainly justified for the through-going nucleons. The use of "frozen" nucleons and the neglect of any other degrees of freedom, like mesons, need justification. We consider a target nucleon and estimate the time ∆t in its rest system which has elapsed between the arrival times of the first and the last projectile nucleons: ∆t ≤ 2R p /γ p where 2R p ≃ 7 fm is the diameter of the Si projectile and γ p ≃ 16 is the Lorenz factor for the experiment under consideration. The time ∆t ≈ 0.5 fm/c is rather short: (i) The target nucleon has not moved significantly in space (frozen approximation is good). (ii) According to the uncertainty principle the intrinsic excitation energy is uncertain The value of the transparency S in Eq. (1) depends on the effective cross section σ eff . Using Saxon-Woods parameterizations for the densities ρ t and ρ p with the surface thickness a = 0.52 fm for all nuclei and the half-density radius r A such that the root-mean-square radius of the nucleus equals the charge radius [3], the survival probability S(b, σ eff ) is calculated for two values σ N N in = 30 and σ N N tot = 39 mb which correspond to the inelastic NN cross section and the total one at this energy, respectively. The results are displayed in Fig. 1. The experimental point, measured at transverse energy E c t = 15.5 GeV is also shown in the figure. It is obtained from the measured mean multiplicity M c of beam rapidity protons by where Z = 14 is the number of protons in Si. We have assumed -as the authors of the experiment do -that the highest E c t -bin corresponds to a central collision which is assigned a value b = 0. Then the experimental value is close to the curve σ eff = 30 mb. In fact the equation S G (b = 0, σ eff ) = S exp (E c t = 15.5 GeV) leads to a value σ eff = 31.1 ± 0.7 mb. This result reproduces a similar calculation using uniform density distributions by the E814 collaboration [1], who have concluded that σ eff = σ N N in within error bars and no anomaly being visible.
The crucial step in the argument is the assignment of b = 0 to the highest E t -bin. Actually, a given value of E t selects a distribution of impact parameters only within a band ∆b(E t ) which is quite large. We study the relation between E t and b and ∆b in the form of a probability distributionP (E t , b), which gives the probability that values of b contribute to events with a given value of E t . We normalize it as dE t P (E t , b) = 1. With this function and the differential inelastic heavy ion cross section d 2 σ in /d 2 b, one can obtain the dependence of the survival function S(E t , σ eff ) as a function of the transverse energy where We will use Eq. (4) to determine P (E t , b) from a comparison with the measured transverse energy distribution dσ in /dE t . With a reasonable Gaussian parameterization of P G (E t , b) for the Glauber calculation (or with P U (E t , b) from a UrQMD calculation, see below), S(E t , σ eff ) can be determined from Eqs. 1 and 3 without ambiguity. We discuss our parameterizations.
The inelastic cross section d 2 σ in /d 2 b is taken from the folding model The parameterization of P (E t , b) is more model dependent, and we calculate it in two ways: (i) in the formalism of Glauber theory, and call it P G (E t , b), and (ii) deduce it from results of the cascade calculation, P U (E t , b).
To calculate P G we assume a Gaussian parameterization which satisfies the normalization condition. We make the usual assump-  tions [4,5] for the functions E t (b) and σ t (b) Here N(b) is calculated in the "collision model" [4] which equals the mean number of NN collisions in a projectile-target interaction with impact parameter b. The proportionality constant ǫ 0 between the observed transverse energy E t and N(b) depends on the dynamics of hadron production, but also on the experimental set up (chosen rapidity interval and acceptance of counter). It will be a fit parameter. The ansatz σ 2 t ∝ N(b) corresponds to the hypothesis that the fluctuations in N(b) are of statistical origin. It is known, however, that the proportionality constant ω in Eq. (8) depends strongly on the rapidity interval of the accepted particles [4]. The reason is not clear [6]. We take ω as a free parameter.
Using expressions (5) to (8) we have calculated dσ in /dE t and have varied the parameters ǫ 0 and ω until the data for Si-Pb are fitted. The result is shown in Fig. 2 with σ N N in = 30 mb, ǫ 0 = 0.067 GeV, ω = 7.8. A calculation of N(b) in Eqs. (7)(8) within the wounded-nucleon model [5] gives a similarly good fit to the data for dσ in /dE t .
According to Eq. (3), P (E t , b) determines the integration region in impact parameter b, which contributes to the integral for a given value of E t . Fig. 1 shows the distribution bP G (E t , b) for a central collision (E c t = 15.5 GeV) derived in the Glauber formalism, and the corresponding one, bP U (E c t , b), calculated (see below) with the help of the code UrQMD [8]. Both distributions agree within statistics, with a peak around 1.7 fm with a half width of about 2.5 fm. Using the functions S G (b, σ eff ) and P G (E c t , b) as shown in Fig. 1, the calculated value of S G (E c t , σ eff ) gets contributions from a considerable range of values of b. We find The calculated transparency is too large by a factor of three.. Should one rather use σ eff = σ N N tot instead? We think yes: Although elastic scattering changes the rapidity of the projectile nucleon only marginally, it increases the transverse momentum of a nucleon, and an elastically scattered nucleon is not accepted within the experimental window p t < 0.3 GeV with a large probability. The results of the Glauber-type calculations are compared with experiment in Fig. 4. We discuss this figure after we have described the second approach.
The UrQMD calculations presented in this paper are based on the standard version 1.1 [8]. As mentioned in [1], due to leakage in the TCAL [7], there is less than half of the E t visible. This has two consequences: (i) The E t scale of the data cannot be directly compared with that of the cascade calculation. (ii) The data have additional fluctuations, visible as a flatter slope of the E814 E t -distribution beyond the knee. Simulating the leakage in the Monte Carlo by simply selecting only 37% of the particles hitting the TCAL fixes both problems. The resulting E t -distribution agrees quite well with the data (Fig. 2). In order to compare to the E t -selected proton rapidity distributions of E814, we have used the absolute cross section given there -593, 102 and 6.9 mb -for the three centrality bins shown in Fig. 3. The mean impact parameters for these bins are 3.6, 2.1, and 1.6 fm, resp., in UrQMD. (hatched area). The two histograms are rather similar in the region of beam rapidity (y B = 3.44). While calculations agree with the data in the interval 6 < E t < 11 GeV, there is a clear discrepancy in the region around beam rapidity for the two other E t intervals (middle and lower picture): The calculations do not show any peak, while the data do. five smaller for the second method. This reflects the fact already observed in Fig. 3 that UrQMD does not produce a peak at beam rapidity for the more central values of E t . It is therefore not meaningful to compare data and results from UrQMD for E t > 10 GeV. For values of E t < 10 GeV calculation and data agree to better than a factor of two -the calculations are above the data for E t = 5 GeV and below the data at E t = 10 GeV. The result of the Glauber calculations (solid curve for σ eff = σ tot = 39 mb, dashed curve σ eff = σ in = 30 mb) are nearly always above the data. The solid curve agrees within 30-50% for central values of E t , but agreement worsens towards smaller values of E t . At E t = 5 GeV the Glauber calculation is a factor 3 higher than the data. Note that in this region also the results of UrQMD lie above the data by a factor of 1.5. These discrepancies may partly have the following origin: Small values of E t correspond to peripheral reactions with several nucleons being spectator particles. All spectator nucleons are included in the calculations; spectator nucleons bound in deuterons, alpha particles etc., however, are not contained in the data.
In summary, the two theoretical approaches, Glauber theory and UrQMD, applied to the transverse energy dependence of the nuclear transparency, have yielded only a semi-quantitative understanding of the data. At small values of E t (peripheral reactions) calculations are systematically above the data. In the interesting regime of central collisions, Glauber theory predicts values of 30 to 50% above the data, while UrQMD shows clearly no peak of beam-rapidity particles so that the results are only an upper limit.
Although the theoretical situation is still far from being satisfactory the analysis of beam-rapidity particles in heavy ion collisions may potentially be a good way to investigate the question of effective cross sections of wounded nucleons.