Chemical equilibration volume: Measuring the degree of thermalization

The chemical equilibration volume: measuring the degree of thermalization A. Majumder and V. Koch Nuclear Science Division, Lawrence Berkeley National Laboratory, 1 Cyclotron road, Berkeley, CA 94720. (Dated: June 16, 2006) We address the issue of the degree of equilibrium achieved in a high energy heavy-ion collision. Speciﬁcally, we explore the consequences of incomplete strangeness chemical equilibrium. This is achieved over a volume V of the order of the strangeness correlation length and is assumed to be smaller than the freeze-out volume V f . Probability distributions of strange hadrons emanating from the system are computed for varying sizes of V and simple experimental observables based on these are proposed. Measurements of such observables may be used to estimate V and as a result the degree of strangeness chemical equilibration achieved. This sets a lower bound on the degree of kinetic equilibrium. We also point out that a determination of two-body correlations or second moments of the distributions are not suﬃcient for this estimation. I. INTRODUCTION arXiv:nucl-th/0305047 v1 16 May 2003 Experiments are presently underway at the relativistic heavy-ion collider (RHIC) at BNL and the SPS at CERN to collide large nuclei at ultra-relativistic energies. The aim is to create energy densities high enough for the production of a state of deconﬁned quarks and gluons: the quark gluon plasma (QGP). Inherent in this statement is the assumption that, in such a collision, we have created a system close to kinetic and possibly chemical equilibrium. The QGP is rather short lived and soon hadronizes, leading one to observe a plethora of mesons and baryons at the detectors. One has to surmise by indirect means as to whether a QGP or at least a thermalized system was created in the history of a given collision. On confronting experimental results, two possibilities of what may have occurred in such collisions emerge. Even though most energetic partons are expected to undergo small angle (soft) scattering, it is possible that a sizable number of partons undergo large angle (hard) scattering followed by multiple rescattering in the central region. This may lead to the appearance of suﬃcient energy densities in the central region of such collisions that quarks and gluons that are usually bound in nuclei become deconﬁned over a region much larger than the size of a nucleon. One obtains a thermalized Quark Gluon Plasma (QGP) [1, 2]. Such a state will undoubtably be short lived and is expected to quickly expand and hadronize into a hot plasma of mesons and baryons. The system will continue to expand and cool, ﬁnally interactions freeze out and the hadrons free stream to the detectors. The above paragraph represents a rather optimistic possibility. The other extreme is to view the above collision as a superposition of a series of nucleon nucleon collisions. Each such collision produces jets of hard hadrons and some soft hadrons as is the case in a proton-proton (or proton-antiproton) collision. The hadrons mostly emerge without interaction from the central region. There is no multiple partonic rescattering, and as a result no widespread thermalization entails and no quark gluon plasma is formed. Another equivalent statement, would be that that the quarks and gluons that eventually lead to the formation of the hadrons are not deconﬁned over a region much larger than the volume of a nucleon. It is reasonable to assume that actual heavy-ion experiments lie somewhere in between these two extremes. To obtain complete thermalization one needs an inﬁnite amount of time, not available in a heavy-ion collision. On the other hand, it is rather unlikely that every hadron produced in each of the nucleon nucleon collisions will emerge untouched from a region with a density many times that of normal nuclear matter. Undoubtedly, there is some thermalization: we picture a scenario where an impinging nucleon has a probability distribution to undergo n hard or semi-hard scatterings; each of these n points becomes a source of particles (see e.g., Ref. [3]). There is also a probability distribution of the number of particles produced at these n points, greater particle production leads to a greater probability of ﬁnal state multiple scattering. Multiple scattering leads to the onset of kinetic followed by chemical equilibrium. As a result, one obtains multiple small domains in the larger volume, each, very close to being in chemical and kinetic equilibrium. The size of the chemically equilibrated volume is assumed to be smaller than the kinetically equilibrated volume. The particles in these domains can undergo further rescattering with particles in surrounding domains leading to an increase in the domain size. If very little rescattering occurs then the size of these domains is closer to the size of a nucleon. Extensive rescattering leads to domain sizes of the order of the size of the system. We will concentrate on the size of a chemically equilibrated domain. Whether the size of such domains is of the order of the system size or the size of a nucleon will be the primary question addressed in this article. The primary observables in such reactions are the populations of the various species of hadrons [4, 5, 6, 7] and their momentum distributions [8, 9, 10]. Interestingly, the broad characteristics of these observables fail to distinguish between the above two scenarios [11]. The reason behind this strife is the observation that the abundances of various