STRANGENESS IN RELATIVISTIC HEAVY ION COLLISIONS P. KOCH, B. MULLER and J. RAFELSKI Institut für Theoretische Physik der Universitdt Frankfurt, Postfach 111932, 6000 Frankfurt 11, W. Germany Institute of Theoretical Physics and Astrophysics, University of Cape Town, Rondebosch 7700, Cape, R.S.A. and Gesellschaft für Schwerionenforschung (GSI), Postfach 110541, 6100 Darmstadt 11, W. Germany I NORTH-HOLLAND - AMSTERDAM PHYSICS REPORTS (Review Section of Physics Letters) 142, No. 4 (1986) 167—262. North-Holland, Amsterdam STRANGENESS IN RELATIVISTIC HEAVY ION COLLISIONS P. KOCH, B. MULLER and J. RAFELSKI Institut für Theoretische Physik der Universitdt Frankfurt, Postfach 111932, 6000 Frankfurt 11, W Germany Institute of Theoretical Physics and Astrophysics, University of Cape Town, Rondebosch 7700, Cape, R. S. A. and Gesellschaft fur Schwerionenforschung (GSI), Postfach 110541, 6100 Darmstadt ii, W. Germany Received March 1986 Contents: 1. Introduction 169 5.1. Relevant reactions 217 1.1. Overview 169 5.2. Rate equations for the production of strange particles 219 1.2. Bagged QCD 172 5.3. Approach to equilibrium of strangeness abundance in 1.3. Ideal gas of quarks and gluons 173 hot hadronic matter 225 1.4. Phase boundary of quark-gluon plasma 176 6. Strangeness abundance in hadronic gas in chemical 1.5. Evolution of quark-gluon plasma 178 equilibrium 227 2. Strange quark mass and hadronic spectrum 181 6.1. Grand canonical treatment 227 2.1. Phenomenological mass formula 181 6.2. State of chemical equilibrium in hadron—nucleus reac- 2.2. Fitting procedures 182 tions 232 2.3. Strange quark energy and mass 183 6.3. Relative chemical equilibrium 236 3. Heavy quark production in quark-gluon plasma 186 6.4. Measurement of the baryochemical potential 238 3.1. Quark pair production 186 7. Strangeness evolution in an expanding fireball 240 3.2. Results for static quark-gluon plasma 192 7.1. Fireball scenario 240 3.3. Influence of volume expansion 195 7.2. Strange particle abundances: Baryon-rich plasma 245 3.4. influence of Pauli blocking of final states 198 7.3. Numerical results 247 3.5. Pre-equilibrium production of JId, particles 201 8. Highlights 250 4. Conversion from the quark-gluon phase to the hadronic Appendix A: Master equations for strangeness evolution in phase (Hadronization of quark-gluon plasma) 203 hadronic gas 252 4.1. Hadronization scenarios 203 Appendix B: Parameterization of strangeness production 4.2. Fragmentation of quarks and gluons 204 cross sections in hadronic gas 255 4.3. Combinatoric break-up model 208 References 259 4.4. Hadrons from quark-gluon plasma 210 Notes added in proof 261 5. Strange hadrons in hadronic gas — Evolution and formation 217 Abstract: Abundances of strange antibaryons formed in nuclear collisions at above 10 GeVIA are considered as a most accessible diagnostic tool for the study of the possible formation and physical properties of the quark-gluon plasma phase of hadronic matter. In this report we describe the current status and develop a dynamical approach in order to describe strange particle formation in nuclear collisions at high energy. Single orders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 142, No. 4 (1986) 167—262. Copies of this issue may be obtained ai the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 64.00, postage included. 0370-15731861$23.60 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) P. Koch et a!., Strangeness in relativistic heavy ion collisions 169 1. Introduction 1.1. Overview Nearly all matter around us is built of up (u) and down (d) quarks. However, as soon as adequate excitation energy becomes available in hadronic interactions, it becomes apparent that further quark flavours exist and are easily accessible. The lightest of the heavy flavours of matter was called ‘strange’ as it appeared so to its discoverers, and today this name is associated with the strange ~-antiquark. s-quarks are identical in their properties (charge q = — ~, baryon number b = + ~, spin ~, colour ~)to the d-quarks but different in mass. In the laboratory experiments excitation of strange matter is understood in terms of matter— antimatter formation as it is the s—~quark pair which is created in reaction processes by strong or electromagnetic interactions. The formation mechanism of strange quarks is highly correlated to the reaction mechanism governing the hadronic collisions — most of this review will deal with our current understanding of the strangeness generation in nuclear collisions and how occurrence of various strange and multiply strange hadrons can tell us the story about the formation of the new phase of matter, the quark-gluon plasma [RH8O,Ra81, Ra82, RM82, Ra83, Ra841. Alternate approaches to diagnose quark gluon plasma, e.g. by dilepton spectra [DG81, Ch82, SC8S, HK85j are also of considerable interest, but it has not been established beyond reasonable doubt that in this case the usual backgrounds will permit measurement. We will not discuss further other interactions and properties of strange quarks such as the Cabibbo mixing of the down and strange quarks with respect to the weak interactions. Note, however, that this would become relevant if the difference in mass of d and s quarks would be generated, e.g. by the properties of the vacuum state of strong interactions. Should this speculation be true, it is quite imaginable that in nuclear collisions various fundamental properties of strange quarks could be altered. This remark is a particular case of the observation that the physical properties of matter and even the appearance of laws of physics may undergo a drastic change within the environment of a quark-gluon plasma. This we take as primary motivation for a detailed and exhaustive study of the properties of the ground state of strong interactions and its melted form, the quark-gluon plasma. We will describe how certain strange particle abundances created in nuclear collisions can be used as a tool to study the properties of this, still hypothetical form of matter. Our view of the relativistic nuclear collision reaction [DR82J is in agreement with the recently observed large nuclear stopping power [BG841which implies that a domain in space arises in a centre of momentum frame of two nuclei in which an important part of the longitudinal energy is transferred to transverse degrees of freedom, as also recently confirmed experimentally [KS85]. Such regions of space we call ‘fireball’. Our reaction picture and the mentioned experimental results contradict earlier theoretical proposals [AKM8O]based on p—p collision phenomenology and assuming essential nuclear transparency. More recent theoretical studies [CK84,DGS8S] find high p—A stopping power and therefore suggest that nuclear collisions at energies under consideration will be truly inelastic. The physical variables characterising the fireball domain are: energy (density), baryon number (density), entropy (density), and volume rather than the usual individual degrees of freedom of particles. The methods required in the theoretical study of these quantities involve straightforward generalisations of the usual statistical thermodynamic theory. However, the fireballs are rather isolated and not in temperature or particle ‘baths’. Further, the size of the fireball, and in particular, its baryon number may be relatively small. Hence, one may justly question the validity of the statistical approach. 170 P. Koch et al., Strangeness in relativistic heavy ion collisions The concept of ‘temperature’ is almost certainly useful in this context, due to the large number of particles contained in a hadronic fireball. It is less obvious whether particle conservation laws may be considered in the framework of the grand canonical ensemble through the introduction of various associated chemical potentials p.r. Exact ‘counting’ of particle members in the ‘canonical’ ensemble in comparison with the statistical average in the ‘grand canonical’ approach leads to correction factors of the order 1 / N, where N is the relevant number of particles [HR841.Here, N may be the baryon number b, strangeness S = s — = 0 or even be a synonym for required colour conservation. Correction factors for exact strangeness and colour conservation have been studied [RD82, EGR83, EGR84, DGER85] and it has been found that for fireball volumes corresponding to a size of 2 fm in radius or more, these corrections are not significant. We will be content hence with the grand canonical approach and use the baryochemical potential ~.th next to temperature T, as the physically observable statistical variables throughout this work. We further note that all particle quantum numbers that are invariant under the strong interaction are conserved on the timescale of interest to us (1024 s ~ t ~ 1021 s). Such quantum numbers are, aside from baryon number b, electric charge Q, and all flavour quantum numbers, in particular, strangeness S. The third component of isospin is hence also conserved due to the Gell-Mann—Nishijima relation, but the conservation of isospin may be violated to some extent due to electromagnetic interactions (we note that the isospin-violating reactions, e.g. the decay of ~°—~ A~yhave a characteristic time constant of —20 e.g. 10 s). A prerequisite in our work is rather precise knowledge of certain empirical properties of strange quarks in a large quark bag. We will discuss at length in the next section how the study of the spectrum of strange particles helps in understanding the mass of confined strange quarks [AR84, AR85]. While the physical origin of the strange quark mass m~remains a puzzle, its precise value is of considerable importance to us because 2m~is the threshold for the formation of a strange quark pair. This threshold is near to the hadronic ‘temperature’ T —~ 160 MeV observed in the slopes of the particle spectra [Ha65, HR681 and which is a measure of the mean energy available per particle in hadronic reactions.