Shell-like structures in an expanding quark-antiquark plasma.

The particle density distribution emerging from the solution of the Vlasov equation for relativistic, non-interacting particles with spherically symmetric initial conditions is shown to exhibit a shell-like structure for late fixed times in the center-of-mass (CM) frame of the system. A similar phenomenon was recently observed employing the test– particle method to solve the Vlasov equation for quarks with Nambu–Jona-Lasinio–type interactions, and was attributed to the attractive forces among the particles. Contrary to this claim, it is demonstrated here that this effect is of purely relativistic origin and is sensitive only to the mass of the particles and their initial phase–space distribution. e-mail address: [email protected] e-mail address: [email protected] One of the primary goals in present relativistic heavy–ion collision experiments at the AGS at Brookhaven, the SPS at CERN, and in future experiments at Brookhaven’s Relativistic Heavy Ion Collider (RHIC) and CERN’s Large Hadron Collider (LHC) is the temporary formation and subsequent observation of the so-called quark–gluon plasma (QGP), the deconfined, chirally restored phase of strongly interacting matter [1, 2]. In the common picture of chiral symmetry restoration [2], at sufficiently high temperatures or densities the massive and confined quasi-particle excitations, the constituent quarks, become bare, undressed quarks with current quark masses being much smaller than the constituent quark masses. Recently, Abada and Aichelin [3] studied the dynamical evolution of a collision-free quark– antiquark system by solving the relativistic Vlasov equation for the one–particle distribution function f(x,p, t), ∂f ∂t + p c E ·∇xf − ∇xE ·∇pf = 0 , (1) where the effective momentum–dependent force −∇xE = −c ∇x[p +M c (x, t) c] = − Mc(x, t) c 2 E c ∇xMc(x, t) (2) is generated by a self-consistent gap equation for the constituent quark massMc derived within the chiral quark model of Nambu and Jona-Lasinio [4, 5]. The initial condition was a sphere of radius R, homogeneously filled with quarks and antiquarks having a thermal momentum distribution corresponding to a temperature T well above the transition temperature Tc ∼ 140 MeV. Such “hot spots”, i.e., small–scale fluctuations with sufficiently high (and thermalized) energy density, are expected to occur in present heavy–ion experiments at the SPS [6] and also in future collider experiments at RHIC [7]. The striking observation of Abada and Aichelin was that, in the evolution of the system, most particles remain concentrated in an expanding shell, and that consequently the phase transition takes place at the inner and outer surface of that shell. Thus, the situation is fundamentally different from the naive expectation that the initial fireball expands by retaining its shape and consequently hadronizes from its surface only. For the interpretation of the results, Abada and Aichelin argued as follows: in the non-relativistic case the thermal velocity distribution of particles peaks around 〈v〉 = √ 2T/m, and in an interaction–free expansion, all the many particles with that velocity will stay together, leading to a corresponding peak in the density, i.e., the observed shell-like structure. For interacting quarks, the attractive forces in the Nambu–Jona-Lasinio model will enhance this effect by trying to keep the (bare) quarks in the region of high density. This conclusion was further motivated by a similar observation found in non-relativistic simulations of expanding nuclear matter, where a shell structure forms after the nucleons have frozen out and enter the liquid–gas transition regime [8]. In contrast, in this note we show that the observation by Abada and Aichelin [3], i.e., the shell-like structure in the density distribution, originates simply from ultrarelativistic kinematics. Moreover, contrary to their line of arguments, we show that in the non-relativistic limit, the density distribution does not exhibit such a structure. We will not justify the validity of the Vlasov equation for modelling the expansion of a quark–antiquark plasma.