Transparency, Mixing and Reflection of Initial Flows in Relativistic Nuclear Collisions

We propose to use the hadron number fluctuations in the limited momentum regions to study the evolution of initial flows in high energy nuclear collisions. In this method by a proper preparation of a collision sample the projectile and target initial flows are marked in fluctuations in the number of colliding nucleons. We discuss three limiting cases of the evolution of flows, transparency, mixing and reflection, and present for them quantitative predictions obtained within several models. Finally, we apply the method to the NA49 results on fluctuations of the negatively charged hadron multiplicity in Pb+Pb interactions at 158A GeV and conclude that the data favor a hydrodynamical model with a significant degree of mixing of the initial flows at the early stage of collisions.

1. The main goal of investigations of high energy nucleus-nucleus (A+A) collisions is to uncover properties of strongly interacting matter at high energy densities and, in particular, to look for its hypothetical phases and transitions between them. Qualitative features of the rich experimental data collected thus far indicate that the produced matter experiences strong collective expansion and it is close to local equilibrium [1]. Moreover, the properties of the matter change rapidly at the low CERN SPS energies ( We are, however, far from a full understanding of the A+A dynamics. Many models based on different assumptions compete with each other and a consistent description of all aspects of the data within a single model is missing. The largest uncertainties concern the early stage of collisions. It is unclear how initial nuclear flows of energy and charges evolve. The majority of the dynamical models (e.g. the string-hadron transport approaches and the quark-gluon cascade models [5,6,7]) predict or assume that the colliding nuclear matter is transparent. The final longitudinal flows of the hadron production sources or the net baryon number related to the projectile and target follow the directions of the projectile and target, respectively. We call this class of models transparency (T-)models. Since the pioneering works of Fermi [8] and Landau [9] statistical and hydrodynamical approaches are successfully used to describe high energy nuclear collisions. Many models within this group, including the first Fermi formulation, assume full equilibration of the matter at the early stage of collisions.
The initial projectile and target flows of energy and charges are mixed. The approaches which predict or suppose the full mixing of the projectile and target flows we call the mixing (M-)models. Let us note that there are models which assume the mixing of hadron production sources (inelastic energy) whereas the transparency of baryon number flows, e.g. statistical model of the early stage [3] and the three-fluid hydrodynamical model [10]. Finally, one may even speculate that the initial flows are reflected in the collision process, i.e. the flows of matter related to the target and the projectile change their directions. This class of models we call the reflection (R-)models. The sketch of the rapidity distributions resulting from the T-, M-and R-models are shown in Fig. 1. The spectra related to the projectile and the target can be easily distinguished in the figure because they are marked in color and hatching the same way as the initial projectile and target nuclei. In this paper we propose a method to Thus, for the net baryon baryon number B one finds, for the fluctuations in the full phase space of participant nucleons. A factor 1/2 in the right hand side of Eq.
(1) appears because only a half of the total number of participants fluctuates.
Let us introduce ω p B and ω t B , where the superscripts p and t mark quantities measured in the projectile and target momentum hemispheres, respectively. By assumption, the mixing of the projectile and target participants is absent in T-and R-models. Therefore, in T-models, the net baryon number in the projectile hemisphere equals N proj p and does not fluctuate, i.e. ω p B (T ) = 0, whereas the net baryon number in the target hemisphere equals N targ p and fluctuates with ω t B (T ) = ω targ P . These relations are reversed in R-models. We introduce now a random mixing of baryons between the projectile and target hemispheres. Let α be a probability for (projectile) target participant to be detected in the (target) projectile hemisphere. It is easy to show that: Note that the term 1/2 of the right hand side of Eq. (4) is absent in (6).
Eventually were recently obtained by NA49 [11] for Pb+Pb collisions at 158A GeV. Note that at the CERN SPS and lower energies negatively charged hadrons are predominantly (more than 90%) π − mesons. In the following we consider T-, M-and R-scenarios within several approaches to particle production in high energy nuclear collisions. We suppose that a part of the initial projectile and target energy, the inelastic energy, is converted into hadron sources. Further on, the numbers of projectile and target related sources are taken to be proportional to the number of projectile and target participant nucleons, respectively. The physical meaning of a particle source depends upon the model under consideration, examples are wounded nucleons (see [13,15]), strings and resonances (see [5,6]), and volume cells of the expanding matter at the freeze-out in the hydrodynamical models. For the independent sources one finds regarding the scaled variance of i-th particle species (see e.g. [16]): where ω * i denotes the scaled variance for i-th hadron species (e.g., i may correspond to h − ) from a single source, n * i is the average multiplicity from a single source, and Ω * is the scaled variance for the fluctuation of the number of sources. Assuming that the number of hadron sources is proportional to the number of participating nucleons, N * = const · N P , one gets: where n i is the average multiplicity of the i-th species per participating nucleon. Thus the scaled variance (7) of the particle number multiplicity in the full phase space is: Consequently, the scaled variances of the i-th hadron multiplicity distribution in T-, M-and R-models read: Again two different versions of mixing with random rapidities (11) and fixed rapidities ( Note that Eqs. (10) and (13) are strictly valid provided that a source produces particles only in its hemisphere. Due of the finite width of the rapidity distribution resulting from the decay of a single source this condition is expected to be violated at least close to midrapidity.
Thus, in order to be able to neglect the cross-talk of particles between the projectile and target hemispheres the width of the rapidity distribution of a single source, ∆y * , should be much smaller than the total width of the rapidity distribution.
Some comments concerning Eqs. (10-13) are appropriate. There is a general similarity of the expressions for produced particles and the corresponding expressions for baryons, Eqs. (3)(4)(5)(6). There are, however, two important differences. A single source produces particles in a probabilistic way with an average multiplicity n * i and a scaled variance ω * i . Consequently, it leads to an additional term, ω * i , in all expressions for ω p,t i , and an additional factor, n i , appears in terms related to the fluctuations of the number of sources. Following Eq. (8) the source number fluctuations can be substituted by ω targ P , and an average multiplicity, n * i , of a single source can be then transformed into an average multiplicity per participating nucleon, n i . The term, 1/2, in the r.h.s. of Eq. (11), as that in Eq. (4), is due to the random rapidity positions of the sources in M-models. In the hydrodynamical model particle production sources can be identified with the volume cells of the expanding matter at the freeze-out.
The source rapidities do not fluctuate and Eq. (11) is transformed then into Eq. (12).
We turn now to a discussion of multiplicity fluctuations of negatively charged hadrons in Pb+Pb collisions at 158A GeV. The value of N − / N P ≡ n − ≃ 2 was measured for the studied reactions [18]. For simplicity we assume ω * − ≃ 1, this is valid for the poissonian negatively charged particle multiplicity distribution from a single source. Note, in p+p interactions at SPS energies and in the limited acceptance of NA49 the measured distribution is in fact close to the Poisson one [11]. This gives: The dependence of ω p − on N proj P from Eqs. (14)(15)(16)(17) for T-, M-and R-models is presented in Fig. 3. .

5.
In a recent analysis [13] of d+Au interactions at √ s N N = 200 GeV [14] within the wounded nucleon model (WNM) [15] it was found that the wounded nucleon sources emit particles in a very broad rapidity interval which results in the mixing of particles from the target and projectile sources. In the following we consider predictions of this model with respect to multiplicity fluctuations.
The WNM assumes that i-th particle rapidity distribution in A+A collisions is presented where F t i (y) and F p i (y) are the contributions from a single wounded nucleon (identified with a particle source) of the target and projectile, respectively. The model also requires in the center of mass system of the collision. The mean number of particles in the rapidity interval ∆y for collisions with N proj P and N targ P is given by For interaction of identical heavy ions N targ P ≃ N proj P , and then Eq. (20) yields: Let us consider now fluctuations of N i (∆y) at a fixed N proj P . A contribution to the scaled variance of N i (∆y) (20) due to the fluctuations of N targ P reads: where As previously, for simplicity we assume that a single source emits particles according to the Poisson distribution, ω * i (∆y) = 1. This leads to a general expression on the scaled variance for a particle of i-th type: The parameter α t i quantifies the amount of mixing of the projectile and target contributions and can vary between 0 and 1. For full acceptance, ∆y = [−Y max , Y max ], Eq. (24) transforms to Eq. (9).
The T-, M-and R-limits of the WNM can be formulated in terms of the distribution functions of the single nucleon, The scaled variances in the projectile and target hemispheres, ω p i ≡ ω i (y ≥ 0) and ω t i ≡ ω i (y ≤ 0), can be found using Eq. (24). It follows that in T-and R-models, the ω p,t i coincide with those given by Eq. (10) and Eq. (13), respectively. For the M-models this gives the following result: which is identical to Eq. (12). In the WNM all projectile (target) sources are assumed to be identical and their positions are the same and fixed. Therefore, similar to the hydrodynamical model, the term, 1/2, in the r.h.s. of Eq. (11) is absent in Eq. (28). This is the M-model with the fixed rapidity positions of the sources. The mixing in the considered version of WNM results from a broad distribution of particles produced by a single source.
A complete mixing in the WNM means according to Eq. (26) that projectile and target source functions become identical.
6. Let us consider fluctuations in limited phase-space domains in which only fractions, q p,t , of all particles in the projectile or target hemispheres are accepted. Then the scaled variances in the acceptance, ω p acc and ω t acc , will be different from the ω p and ω t . We start with the scaled variances ω p B and ω t B of the net baryon number fluctuations. Assuming that inside the projectile and target hemispheres the baryon rapidities are not correlated one gets (see e.g., [16,19]): It can be shown that the scaled variance of the produced particles in the limited momentum acceptance within the M-model with fixed source rapidities (12) reads: where N i acc is a mean multiplicity of a particle of i-th type in the acceptance. The formula above assumes that the produced particles are uncorrelated in the momentum space, i.e. it neglects effects of motional conservation laws and resonance decays. The scaled variance in a limited rapidity acceptance, ∆y, within WNM can be directly obtained from Eq. (24). It coincides with that of Eq. (30).
Let us consider now as an example the NA49 acceptance, which is located in the projectile hemisphere about one and half rapidity units from mid-rapidity, ∆y = [1.1; 2.6] in the c.m.s.
The acceptance probability was measured to be q p ≃ 0.4 [11,17] (i.e. about 40% of negatively charged particles in the projectile hemisphere are accepted). In the limiting case of the fixed rapidities of the sources, this is assumed to be valid for both the hydrodynamical and WN models, one finds: The relations, n t − (∆y; T ) = 0; n t − (∆y; M) = 0.5 q p n − ; n t − (∆y; R) = q p n − , with q p ≃ 0.4 and n − ≃ 2 have been used in Eqs. (31-33). Note that in the limit q p → 0 one finds ω p −,acc ≃ 1 for all type of models. The predictions given by Eqs. We now discuss an effect of a limited acceptance for the approaches with randomly fluctuating source rapidities. In this case the scaled variances in the projectile and target hemispheres are given by Eqs. (10)(11)(12)(13), provided a width of a rapidity spectrum of particles emitted from a single source is narrow. In a general case, when a rapidity width of the source particles and an acceptance window are comparable in size, it is difficult to make analytical estimates. The problem can be solved in the limit of very narrow sources (a source width, ∆y * , is much smaller than the experimental acceptance interval, ∆y). The hadrons created by "narrow" sources have correlated rapidities, but Eq. (29) can be used for the scaled variances of the number of sources assuming that the source rapidities in the projectile hemisphere are not correlated. In this case our Eqs. (10-13) yield: ω p −,acc (R; rr) ≃ 1 + n − 1 − q t + q t · ω targ P ≃ 2.2 + 0.8 ω targ P .
As before, we use q p ≃ 0.4 and n − ≃ 2 in Eqs. (34-36). The corresponding curves are plotted in Fig. 4. The experimental points for Pb+Pb collisions at 158A GeV clearly exclude transparency and reflection approaches discussed here. The mixing model with random fluctuations of a source rapidity and a narrow width also strongly disagree with the data. A reasonable agreement is observed only for the mixing-hydrodynamical and WNM models. We remind that a large degree of mixing was found previously in the analysis of the pseudo-rapidity spectra of charged hadrons produced in d+Au interactions at √ s N N = 200 GeV [14] within WNM [13].
Note however that the WNM model, used here as a simple example to illustrate the MinF method, can not reproduce many observables connected to collective behavior of matter created in high energy nuclear collisions, like radial and anisotropic flows and the strangeness enhancement. On the other hand, these effects are at least qualitatively described by statistical and hydrodynamical approaches. . This alone may result in additional multiplicity fluctuations. We estimated that for the NA49 data discussed above the corresponding increase of the scaled variance is smaller than 5%.
The MinF-method can be used independently of the degrees of freedom relevant at an early stage of collisions (e.g. hadrons at a low collision energy or quark and gluons at a high energy). This is because the concepts of the spectators and the participants as well as hadron multiplicity fluctuations are valid at all relativistic energies and for all collision scenarios. In the case of collisions of non-identical nuclei (different baryon numbers and/or electric charge to baryon ratios) one can trace flows of the conserved charges by looking at their inclusive final state distributions (see e.g. [20]). An interesting information can be extracted from collisions of two nuclei with different atomic numbers (see [13]).
In the case of identical nuclei only the MinF method can be used. It gives a unique possibility to investigate the flows of both the net baryon number and particle production sources.