Transverse radial expansion in nuclear collisions and two particle correlations

At the very first stage of an ultra-relativistic nucleus-nucleus collision new particles are produced in individual nucleon-nucleon collisions. In the transverse plane, all particles from a single $NN$ collision are initially located at the same position. The subsequent thermalization and transverse radial expansion of the system create strong position-momentum correlations and lead to characteristic rapidity, transverse momentum, and azimuthal correlations among the produced particles.

many individual nucleon-nucleon collision happen. The strings stretched and new particles are produced (partons are freed; the exact mechanism of the particle production is not important for the presentation of the main idea of this paper). Due to a rapid thermalization, the newly produced particles become "frozen in" the medium, preserving all the correlations between them, e.g. the correlations due to the conservation laws of electric and/or baryon charges, strangeness, etc. The relative particle position (in the transverse plane) could be only modified by particle diffusion during the system evolution before the final freeze-out (for a more detailed discussion of the role of the thermalization time and diffusion see later in the paper). Then the transverse expansion of the system would create strong position-momentum correlations in the transverse plane: in general, farther from the center axis of the system a particle is produced initially, on average the larger (transverse, in radial outward direction) push it receives from other particles during the system evolution. It is common [3] that the radial expansion collective velocity is parameterized by monotonically increasing function of the radial distance, though some deviations from that could be possible at very large distances due to low density at the periphery of the system. All particles produced from the same string (from the same N N collision) have initially the same spatial position in the transverse plane. Consequently, all particle from the same string gets on average the same push and thus become correlated. Correlated pairs of particles, being produced at the same place, for example containing s− ands−quarks, would be boosted in the same direction and become azimuthally correlated, etc.. Note that in rapidity space the correlations due to described mechanism could extend up to the length of the entire string. For simplicity, in the most of the discussion below we assume the radial expansion of the system to be boost invariant. The 'push' is in the transverse direction, and on average does not affects the longitudinal momentum component.
Based on the above mentioned ideas it appears a simple picture of AA collisions, which could be considered in some sense as a next order approximation to a simple superposition of independent N N collisions often use as a base line in correlation studies of AA collisions. In this picture, the particles produced in each independent collision are boosted in the radial direction depending to the location of the collision in the transverse plane and the transverse expansion velocity profile.
It corresponds to non-zero space-momentum correlations in the transverse plane, x i , ∆p t,i = 0 ( see [6] as an example of the role of space-momentum correlations in the development of directed flow). As discussed below this picture leads to many distinctive phenomena, most of which can be studied by means of two (and many-) particle correlations.
Two particle transverse momentum correlations. The single particle spectra are affected by radial flow in such a way that the effective slope as well as the mean transverse momentum are mostly sensitive to the average expansion velocity squared (the validity of this approximation is discussed in more detail below) and to much lesser extend to the actual velocity profile (dependence of the expansion velocity on the radial distance from the center axis of the system). The two-particle transverse momentum correlations [7], p t,1 p t,2 − p t,1 p t,2 ≡ δp t,1 δp t,2 , would be sensitive mostly to the variance in collective transverse expansion velocity, and thus are more sensitive to the actual velocity profile. Simple estimates show that the corresponding contribution could be comparable in magnitude to the primordial correlations existed in N N collisions. For a rough estimate one can use a relation where the coefficient α is of the order of typical particle mass, and v being the expansion velocity.
Then the correlation between transverse momenta of two particles would be The factor D, takes into account the dilution of the correlations due to a mixture of particles from N coll uncorrelated N N collisions, and that in an individual N N collision the mean number of particle pairs, n(n − 1) N N , on average is larger than the mean multiplicity squared, n 2 N N (see also [8,9]). For a linear velocity profile, σ 2 v 2 = v 2 2 /3. Taking into account that the increase in the average transverse momentum due to the radial expansion is of the order of 20-30%, one concludes that the two terms in Eq. 2 could be of the same order of magnitude: an increase in the relative correlations, could be of the order of 1-3%, similar to the value of δp t,1 δp t,2 N N / p t 2 ≈ (0.12) 2 , measured at ISR [10]. Thus, the correlations due to transverse expansion could be the major part in the observed centrality dependence of the mean p t fluctuations/correlations observed at the SPS and RHIC (for recent results, see [11]).
More accurate estimates can be obtained employing a thermal model presented in [4]. In this approach particles are produced by freeze-out of the thermalized matter at temperature T , approximated by a boosted Boltzmann distribution. Assuming boost-invariant longitudinal expansion and freeze-out at constant proper time, one finds, up to irrelevant constants, for single particle spectra: where ρ t is the transverse flow rapidity, φ b is the boost direction, α t = (p t /T ) sinh(ρ t ), and β t = (m t /T ) cosh(ρ t ). It also assumes a uniform matter density within a cylinder, r < R, and a power law transverse rapidity flow profile ρ t = ρ t,max (r/R) n .
In such a model, the mean transverse momentum is given by expression: The contribution of the transverse expansion to the two particle mean transverse momentum correlations (both particles are from the same N N collision, the dilution factor, Eq. 3, to be applied afterward) can be written as: δp t,1 δp t,2 = dρ t dφ b ρ 2/n−1 t dp t,1 dp t,2 (δp t,1 δp t,2 )m t,i K 1 (β t,1 )e α t,1 cos( This equation additionally assumes that during the expansion time (before the freeze-out) the particles originated from the same N N collision do not diffuse far one from another compared to the system size. We discuss this assumption in more detail later in this paper.
The results of the numerical calculations based on the above equations are presented in Figs. 1 and 2 as function of ρ 2 t = ρ t 2 (4n + 4)/(2 + n) 2 . The results are shown for two different velocity (transverse rapidity) profiles, n = 2, and n = 0.5. One observes that indeed for all the particle types presented, p t depends very weakly on the actual profile. On opposite, the correlations are drastically different for two cases presented.
Rapidity correlations. Charge balance functions. The transverse expansion 'push' consists of many individual collisions. It leads not only to the increase of the transverse momentum but also to the particle diffusion in the rapidity space. We do not consider the effect of such diffusion in this discussion concentrating only on the effect of the transverse radial push. We define the balance function as Very roughly, B ab (x b ; x a ) has a meaning of a distribution of the 'associated' particles b under condition of a 'trigger' particle a to be found at the location x a . (The charge balance function The width decreases as the mean transverse mass increases due to transverse expansion. This effect is consistent with the experimentally observed narrowing of the balance function with centrality [14]. Note that because the charge balance function is normalized to unity, the narrowing of the correlations means an increase in the magnitude of the two particle correlation function. In its turn it means an enhancement of the net charge multiplicity fluctuations if measured in a rapidity region compared or smaller than the correlation length (1-2 units of rapidity). This observation might be an explanation for the centrality dependence of the net charge fluctuations measured at RHIC [15].
Azimuthal correlations. As all particles from the same N N collisions are pushed in the same direction (radially in the transverse plane) they become correlated in azimuthal space. The corre-  lations can become really strong for large transverse flow as shown in Fig. 2 (again, for particles originated from the same spatial position in the transverse plane). Fig. 3 shows the strength of the azimuthal correlation in terms of the first two harmonics. cos(∆φ) reaches quite large values (even if one takes into account a dilution factor, D ∼ 1/150 for central Au+Au collisions). If the average would be taken over all particles including the particles from other N N collisions, the momentum conservation contribution [16] can not be neglected; it can greatly reduce the effect. Note, however, that the momentum conservation effects are expected to be small for the (azimuthal) charge balance function. In this case both correlated particles originate from the same N N collision and the results presented in Fig. 3 are valid. In our model cos(∆φ) should strongly depend on the particle mass and its transverse momentum. Identified particle correlation study would be of great interest in this respect.
The second harmonic in the azimuthal correlations generated by radial expansion is of a particular interest as it would contribute to the measurements of elliptic flow. The numbers from Fig. 3 corresponding to ρ 2 t ∼ 0.3 are comparable with the estimates of the strength of non-flow type azimuthal correlation estimates made in [17]. Thus the azimuthal correlations generated by transverse expansion could be a major contributor to the non-flow azimuthal correlations. More importantly, this contribution would depend on centrality (following the development of radial flow), unlike many other non-flow effects. Note, however, that the typical elliptic flow centrality dependence (rise and fall) is different from that of the correlations due to transverse radial expansion (continuous increase of N coll · cos(2∆φ) ).
Thermalization time. Diffusion. The magnitude of the correlations due to transverse expansion should be sensitive to the system thermalization time and the particle diffusion in the thermalized matter during the expansion; the azimuthal correlations would be the most interesting/useful for such a study. The dependence on the thermalization time comes from the following. Somewhat exaggerating, one can imagine that just after the collision, newly produced particles ('freed' partons) experience free streaming for some period of time before the thermalization happens. After that moment the particles become frozen into hydrodynamic type cell and an expansion starts with all the consequences discussed in the first part of this paper. The effect of the free streaming phase would be a diffusion in the transverse plane of the particles created initially at the same position (of the typical hadronic size). Such a diffusion would lead to a broadening of the azimuthal correlations. Many current estimates give the thermalization time of the order of or smaller than 1 fm/c [3]. Such a short thermalization time would be difficult to observe, but in any case the corresponding measurements would be of great interest giving an independent limit to this important parameter.
Single jet tomography? We could go even further with our speculations. Jet tomography of nuclear collisions is a popular subject based on the jet quenching phenomena. For such study it would be very useful to find an observable that is correlated to the space point where the hard collision occurred. It seems that the correlations due to transverse expansion provide such a possibility. One has to correlate the jet (high p t hadron) yield with mean transverse momentum of particles taken at different rapidity (but better at similar azimuthal angle). In the same N N collision where the high p t particle is emitted, the soft particles are produced as well. Those soft particles experience the transverse 'push' corresponding to the spatial position in the transverse plane where the original N N collision happens to be. Then the mean transverse momentum of the associated particles would provide the information on how close to the center of the system the collision occurred.
Summary. Expansion of the dense system created at the initial stage of a high energy nucleusnucleus collision leads to strong position-momentum correlations. As all particles from the same N N collision are produced at the same position in the transverse plane, they get a similar radial push during the expansion stage. It creates rapidity, azimuthal angle, and transverse momentum correlations. The correlations extend over wide rapidity range. The picture is not boost-invariant, as the initial geometry of the source and particle densities change with rapidity leading to a rapidity dependent radial expansion.
The above described picture of AA collisions has many interesting observable effects, only a few mentioned in this paper. The picture become even richer if one looks at the identified particle correlations. Many questions require a detailed model study, but the approach opens a potentially very interesting possibility to address the initial conditions and the subsequent evolution of the system created in an AA collision.