The role of the entropy in an expanding hadronic gas

NUCLEAR P H Y S I CS B

Nuclear Physics B 378 (1992) 95—128 North-Holland

The role of the entropy in an expanding hadronic gas H. Bebie

~,

P. Gerber

a,

J.L. Goity

b.c

*

and H. Leutwyler

Institute for Theoretical Physics, Unii’ersity of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland “Paul Scherrer Institut (PSI), CH-5232 Villigen, Switzerland CEBAF, Physics Diiirion, 12000 Jeffersson A,’e., Newport News, VA 23606, USA Received 21 May 1991 (Revised 18 November 1991) Accepted 18 February 1992

The evolution of the hot central region produced in central heavy ion collisions is analyzed. It is shown that the main properties of the final state can be understood as a result of an adiabatic expansion process. The role of the entropy per particle asa global characteristic of the system is emphasized. Our main concern is the evolution of the chemical composition.

1. Introduction In heavy-ion collisions, a hot central region is formed, which is generally assumed to consist of a quark—gluon plasma if the collision energy is sufficiently high [1]. The minimal energy density necessary for the formation of this plasma is estimated to be around 2 GeV/fm3. Such densities of energy are presumably attained in present fixed-target collisions with projectile energies of up to 200 GeV/nucleon [2]. As the hot zone expands and cools down, a phase transition to a hadron gas takes place. Soon after the transition, which is estimated to occur at a critical temperature around 200 McV, the energy density is mostly stored in hadronic resonances (p, w,...) [3—51. As the expansion and cooling proceed, these excited states decay, leading to a final state at freeze-out consisting mainly of pions. At presently available energies, central collisions generate final states containing several hundred pions (e.g.: in 0 + Au (S + S) central collisions at 200 GeV/nucleon 400 (300) pions are observed [6]). Two-pion interferometry indicates that the radius of the “source” as measured with pions close to central rapidity is about 7 fm [61. The processes which take place during the collision are discussed e.g. in refs. [1,7—9],on the basis of the Landau hydrodynamic model [101. In the initial phase the hot spot is expected to look like a pancake whose thickness is swelling, an *

Work supported in part by Schweizerischer Nationalfonds.

0550-3213/92/$05.OO © 1992

—

Elsevier Science Publishers B.V. All rights reserved

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appreciable expansion in the transverse directions only setting in when the pancake starts resembling a sphere. If the energy of the colliding nuclei is sufficiently high, this initial state of the fireball presumably consists of a plasma of quarks and gluons well above the critical point. In this phase, kinetics can be analyzed within perturbation theory which indicates that the relaxation times are short. The behaviour of the plasma in the vicinity of the critical point, however, is not well explored and the evolution of the hot spot from a phase where quarks and gluons are liberated to a phase where they are confined is far from being understood. For an account of the problems connected with hadronization we refer to ref. [11]; this paper also contains references to related work. In the present paper, we are not concerned with the processes which occur in the initial phase of the collision or during hadronization. If the energy of the colliding ions is sufficiently high, these processes lead to the formation of a dense hadronic gas which extends over a sizeable region of space. Our analysis concerns the evolution of such a hadronic fireball and is independent of the processes which give rise to it. The essential ingredient of this analysis is the requirement that the fireball is large compared to the mean free path of the particles, such that the evolution can be traced on the basis of hydrodynamics. One may question whether the configurations produced in heavy-ion collisions at presently available energies meet this requirement. We do not doubt, however, that this will be the case once the collision energy is sufficiently high. The hydrodynamics of a hadronic gas is complicated by the fact that the system consists of many different particle species. It turns out that, unless the temperature is close to the critical point, the relaxation times of the processes which control the chemical equilibrium among the various species are large compared to the collision times. The gas therefore gets out of chemical equilibrium at a rather early stage of the expansion. We will show that this leads to an overpopulation of the pionic components, thus confirming the picture proposed in refs. [8,12—141. Our analysis has the virtue of being model-independent. Several problems, however, call for a more detailed study to arrive at a realistic picture: (i) The hydrodynamic analysis breaks down when freeze-out occurs. Before that happens, the terms associated with viscosity and heat conduction start playing a significant role. To reliably calculate the momentum distributions which emerge after freeze-out, the hydrodynamic evolution must be supplemented with an analysis of the processes which occur at decoupling. We have carried out a numerical simulation of elastic collisions in an expanding spherical gas and have found that the freeze-out process does not strongly affect the transverse momentum distribution, but more work in this direction is needed to arrive at quantitative conclusions. In particular, the modification of the pion spectra generated by the decay of excited states needs to be accounted for [151. (ii) In the following, we focus our attention on the evolution of the mean entropy per particle (for a discussion of entropy production in various stages of the

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evolution see refs. [8,16—19]).In this context, neither the shape of the hot region nor the temperature profile nor the geometry of the mean velocity field play an important role. For other observables, however, such as e.g. the momentum distributions, the profiles of temperature and velocity do matter. Phenomenology indicates that, at the currently available energies, the longitudinal component of the mean velocity dominates over the transverse components when freeze-out occurs. The evolution of the velocity field in the course of the expansion can be studied by solving the hydrodynamic differential equations [7,9]. In particular, the gradual shift from an expansion in the longitudinal direction to a radial flow can be analyzed on this basis provided the mean free path remains small compared to the size of the fireball. (iii) Under the present experimental conditions, the central rapidity interval contains a significant excess of baryons over antibaryons which also manifests itself in the relative abundance of K~/K. To account for this contamination, one may introduce chemical potentials for net haryon number and strangeness [15,20]. In the present paper, we do not address these problems. We analyze the expansion of a hot hadronic fireball with the quantum numbers of the vacuum in the hydrodynamic regime. The initial configuration is assumed to be a state of local thermal and chemical equilibrium. We discuss the range of temperatures and volumes where the hydrodynamic description is internally consistent. Our main concern is is the evolution of the chemical composition. —

2. Equilibrium We analyze the configuration of the gas in terms of distributions in phase space, i.e. we use a classical picture where the positions and the momenta of the particles are specified simultaneously. This is justified provided the typical wavelengths are small compared to the size of the system. For the configurations generated in heavy-ion collisions at 200 GeV/nucleon, the classical analysis does represent a decent approximation, but quantum effects are not entirely negligible. In particular, the classical picture does not provide an adequate framework for particles of low momenta (large wavelengths) wave mechanics may play a significant role, e.g. in the emission of soft photons [21]. Throughout what follows, we however ignore the uncertainty relation and use classical (relativistic) kinematics; quantum effects are accounted for only insofar as we use Bose- or Fermi-statistics to calculate the phase space distributions. In the present paper, we study evolutions which are sufficiently slow for local thermal equilibrium to be maintained, at least approximately. This is the case only if the mean free path A of the particles is smaller than the radius R of the fireball (the probability for a particle emitted from a random point in the interior of a sphere to escape without collision is 0.3 for A -~R,0.5 for A R and 0.7 for —

=

=

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A 2R). One may also compare the collision rate l/T with the expansion rate R/R. Since the mean time ~rbetween two collisions is related to the mean free path by A T~r,where i~is the most probable velocity of the thermal distribution [22], the requirement l/T> R/R amounts to A
=

=

=

=

A~

l2F~n~ T5

(2.1)

iT

As emphasized in ref. [261, the cross section is proportional to E2 only at low energies. Asymptotically, a-.... is expected to roughly become constant, with a value comparable to u~ 40 mb or a-iTp 24 mh. These energies are however only reached at temperatures where the gas does not exclusively consist of pions, but contains a significant number of other particles (K, p, w, N,...). At T 150 MeV, the pionic component accounts for about one half of the density, indicating that the net collision rate for a pion traversing the gas may well be twice as large as the iri~t-collision rate discussed above. Using the formula A 1/a-n and assuming that the cross section between any two hadrons is of order 20 mb, one arrives at A 1.3 fm for T= 160 MeV and at A ~0.6 fm for T= 180 MeV. This indicates that the mean free path continues to rapidly shrink also at higher temperatures, roughly with the power T5. =

=

—

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Our analysis is based on the hypothesis that, when hadronization is over, the fluid can approximately be characterized by a state of local thermal and chemical equilibrium [11,27]. The processes which give rise to this state are then of no relevance, except for the values of temperature and velocity which characterize the local equilibrium configuration. In support of this hypothesis, we remind that the most likely distribution of the energy over the various degrees of freedom is the equilibrium distribution. The estimates given above show that, if equilibrium prevails at a temperature not far below the critical point, then the mean free paths are indeed small compared to the size of the hot spot, indicating that the hypothesis is self-consistent. If local thermal equilibrium prevails, the energy momentum tensor is of the form =

(

+

P)u”u’

—

Pg°”,

(2.2)

where we have neglected the viscosity of the gas. The four-vector u°(x,t) is related to the velocity v(x, t) of the local comoving frame by u yv, y (1 v2y’~2.The quantity (x, t) denotes the energy density in the comoving frame and P(x, t) is the pressure. Conservation of energy and momentum implies the continuity equations =

=

=

—

(2.3) Electromagnetic or weak interactions are not expected to significantly affect the evolution. We disregard them as well as the contributions to energy density and pressure generated by photons or leptons. The collisions occurring in the gas then conserve the number of quarks (minus antiquarks), separately for each one of the flavours u, d, s,... For simplicity, we assume that the initial configuration of the gas is neutral with respect to these quantum numbers, the number of quarks and antiquarks being the same. The evolution of the gas then of course preserves this property. Note that this assumption in particular requires the number of antibaryons to equal the number of baryons. At the energies explored so far in heavy ion collisions even the central rapidity region is contaminated by remnants of the colliding nuclei, indicating that the above simplification is not fully justified. We do not investigate this complication here, but refer to the literature (see e.g. refs. [11,27]). The contamination should become less significant with increasing collision energies. We divide the evolution into three phases and assume that during the initial, hot phase, which we refer to as phase I, the gas is in local equilibrium, both thermally and chemically. In equilibrium, the state of the system is fully characterized by the local temperature T(x, t) and by the velocity profile v(x, t). In particular, the chemistry of the gas, i.e. the abundance of the various particle species K, p, N, N,... is controlled by the temperature. For the evolution to ~-,

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maintain thermal equilibrium, it suffices that elastic collisions frequently occur. The expansion, however, only preserves full equilibrium if collisions which affect the chemical composition also occur at a sufficient rate which is the case only if the temperature is rather high [12,14]. As the gas cools, inelastic collisions become rare and, since the elastic ones conserve particle number, chemical equilibrium among the particle species is lost. Phase II is the period during which the mean free path for elastic collisions is still smaller than the radius of the expanding fireball (such that the gas remains in local thermal equilibrium, at least approximately) while the mean free path of the inelastic collisions fails to meet this condition (i.e. chemical equilibrium is not maintained). As the gas cools further, even elastic collisions become rare. Phase III is the final period of the evolution which starts when the mean free path becomes comparable to the radius of the fireball. If there are any excited states left over, they now decay, collisions stop and the distribution of the momenta freezes out. In phase I, the local configuration of the gas is determined by T. Given the temperature and velocity profiles at time t, the four differential equations (2.3) therefore specify the cooling rate dT/dt as well as the acceleration dv/dt at that time and hence fully determine the evolution of these quantities from a given initial configuration as long as this evolution maintains full local equilibrium. The functions e(T) and P(T) are not independent of one another. According to thermodynamics, P(T) determines the energy density as e(T) T dP/dT P (note that we are considering thermal as well as chemical equilibrium here, ~x 0). Conservation of energy then implies that, during phase I, the expansion process is adiabatic: denoting the local entropy density in the comoving frame by s dP/dT, eqs. (2.2) and (2.3) entail the conservation law —

=

—

=

=

0~(su~c) =0.

(2.4)

As mentioned above, the expression (2.2) for the energy—momentum tensor is valid only if viscosity can be neglected. In @~,viscosity gives rise to a correction proportional to the gradient of the velocity field [281. Since friction generates entropy, it shows up on the r.h.s. of eq. (2.4) as a positive contribution, proportional to the square of the velocity gradient. In order of magnitude, the correction is determined by the ratio A/R. The role of viscosity in the context of high-energy collisions is discussed in refs. [8,19,26,29].

3. Loss of chemical equilibrium We now turn to the second phase of the evolution where chemical equilibrium is lost. Let us first consider a gas consisting exclusively of pions. If inelastic collisions are ineffective, the total number of pions remains conserved. Denoting

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the pion number density in the comoving frame by niT(x, t), the evolution therefore obeys the conservation law i9~,.(niTu’~) 0. Elastic collisions only equilibrate the velocity distribution the local equilibrium configuration is the most probable distribution at given energy density and number density. According to statistical mechanics this distribution is characterized by two parameters: temperature and chemical potential jx~ ~x,,.(x, t). Consequently, energy density, pressure and number density now are functions of the two variables T and p.~.Conservation of energy and momentum alone does not determine the evolution of the these quantities, but together with the conservation law 3,~(niTu’~) 0, we again have a closed system of differential equations which fully controls the expansion. Excited states play a significant role in the evolution and their decays also affect the momentum distributions at freeze-out [11,15]. At a temperature of 150 MeV e.g., the fraction of the energy stored in these states is larger than the fraction carried by the pionic component of the gas [3,5]. We therefore need to extend the above discussion, accounting for p-mesons and other excited states. In their presence, the evolution does not conserve the number of pions. The conservation law a,~~(niTuIa) 0 is however not abolished it is only modified. The process p 7r77, e.g., conserves neither niT nor n,,, but it conserves the combination + 2n,,. If, apart from elastic collisions, only this process occurs at a significant rate, then the relevant local equilibrium distribution of the two particle species is the most probable one permitted by a given value of + 2n,,. This distribution is still described by two parameters T, ,a, the chemical potentials of the ir and p-components being given by p~,. p., p.,, 2p., respectively. Similarly, the decay w ~r~ir~- conserves niT + 2n~+ 3n 0 and the same is true of the processes w p~ 3p.. There are and pp w~r: these processes are in equilibrium provided p.,,, however also reactions which are not in equilibrium at these values of the chemical potentials, such as irir wir. Quite generally, the occurrence of inelastic reactions implies that chemical equilibrium only occurs if all of the potentials are zero. The mere existence of processes which drive the configuration towards chemical equilibrium does however not imply that this equilibrium is actually maintained: only those reactions are significant for which the reaction rate is comparable to or exceeds the expansion rate. As the gas cools, the excited states decay and the pionic component grows. The collision rates decrease, the inelastic collisions get out of equilibrium earlier than the elastic ones and a chemical potential starts building up. As far as the hydrodynamic evolution goes, the local properties of the system are characterized by the function P(T, p.) which determines the energy density , the number density ~ and the entropy density s according to =

—

=

=

=

—

~

~iT

=

=

—~

~-‘

~

=

~

dP=s dT+ñ dp.,

e=sT+np. —P.

(3.1)

In particular, ñ is the variable conjugate to the chemical potential, ~i l~lP/9p. TWith p.,,. p., p.,, 2p., p.,,, 3p.,... the various components of the gas contribute =

=

=

=

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according to ~i niT + 2n,, + 3n,,, + Hence ~i is the particular combination of particle numbers which remains conserved in the decays of the excited states. At high temperatures, 11 receives a substantial contribution from these states, while at low temperatures where they are rare ~iessentially coincides with the number density of the pions. Inserting the relations (3.1) in the conservation law for energy and momentum, one now obtains =

.....

—

—

3~(su~)=—~8~(ñu~).

(3.2)

This relation shows that entropy is conserved if either inelastic collisions are at work such that the chemical potential vanishes (phase I) or if only elastic collisions and decays occur such that the number ñ is conserved (phase II). Conversely, an adiabatic expansion at non-zero chemical potential necessarily conserves ~i B~(su~) 0, =

3~(niu~)0. =

(3.3)

Note that if p. is positive, an increase in the number of particles [3,,,(iiu”~)>01 requires a decrease in entropy [3,.,,(suI*)< 01 which is absurd: the second law of thermodynamics implies that the number of particles counted by ~icannot grow. [This statement is to be taken with a grain of salt: for the number of particles to grow, the divergence 30(ñut*) need not be positive everywhere. The conclusion strictly holds only if the ratio p./T is constant.] We emphasize that for the relation (3.2) to hold it is important that the number density ñ is the quantity conjugate to the chemical potential. This is the case only if the excited levels acquire a chemical potential proportional to the number of pions liberated in their decays. Otherwise, the right-hand side of eq. (3.2) would contain the number density niT rather than ~i; since the decay of the excited states into pions leads to an increase in their number, 3,.,(niTu°)> 0, the evolution would then be in conflict with the second law. It is clear that the transition from chemical equilibrium where the number

N=fd3xyni

(3.4)

is decreasing to a phase where N remains constant does not take place instantly. During the transition period, where N continues to fall even though a chemical potential is building up, eq. (3.2) requires the entropy to grow. In the following, we will however ignore these transition periods and analyze the expansion in the adiabatic limit where the entropy remains conserved. Note also that we are neglecting heat conduction, which does not show up in the energy—momentum

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tensor ~, but in the currents associated with particle number and entropy [281. In eq. (3.2), the entropy generated by heat conduction manifests itself through a term proportional to the square of the gradient 3,/p./T). The contribution is of the same order of magnitude as the terms generated by viscosity. The above schematic discussion assumes that, during phase II, only a single conserved particle number occurs. The observation of baryons and strange particles in the final state however indicates that, in reality, additional conservation laws play a role. The local configuration of the gas is then characterized by several chemical potentials which start building up when the corresponding particle numbers freeze out. We will discuss this complication in some detail below.

4. Equation of state Our main assumption is that during phases I and II, the gas remains in local thermal equilibrium, the state being characterized by a maximum of the entropy consistent with the conservation laws of energy, momentum and of the relevant particle numbers. Neglecting dissipative processes, the corresponding continuity equations then imply that entropy is conserved the expansion is an adiabatic process. At low temperatures and low densities, the gas consists of a collection of mesons and baryons in the ground state, the occurrence of excited states being suppressed by the factor exp(—~E/T), where ~E is the excitation energy. In chemical equilibrium and in the range T ~ 100 MeV, the gas almost exclusively contains pions with a mean energy of order E ~ 300 MeV. Chiral symmetry strongly suppresses the interaction among pions of low energy. An evaluation of energy density and pressure within chiral perturbation theory shows that this interaction barely affects the equation of state even at T 150 MeV (see fig. 6 in ref. [5]).At this temperature, excited states however already occur in abundance: although the probability for a given level of mass M. to be populated is small, of order exp(—M,/T), the level density rapidly rises with the mass [3]. While the number of pions roughly grows in proportion to the third power of the temperature, the number of excited states grows much more rapidly. In chemical equilibrium a particle picked out at random is more likely in an excited state than in the ground state, if the temperature exceeds 150 MeV. Moreover, excited states carry more energy the energy density stored in the excited states reaches the energy density of the pionic component already at T 125 MeV. At this temperature, the gas is still rather dilute, the mean distance between two particles being of order —

—

*

In the presence of dissipative contributions, the form of the energy—momentum tensor depends on the choice of the mean velocity. The above statements holds if u° is defined by the elgenvalue equation (9,,,~u’=

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TABLE 1 Population of levels in chemical equilibrium

T[MeV] n

[fm3l dlfm] iT

K N N~

N* N~’

100 0.036 3.0 82.0 10.0 1.7 0.54 0.17 3.9 1.3 0.20 0.054

120 0.084 2.3 67.0 14.0 2.5 1.5 0.69 9.1 3.6 0.97 0.37

140 0.19 1.7 52.0 15.0 2.9 2.7 1.7 15.0 6.8 2.7 1.3

160 0.40 1.4 38.0 14.0 2.8 3.6 2.8 20.0 10.0 5.4 2.9

180 0.83 1.1 27.0 12.0 2.5 4.2 3.9 24.0 13.0 8.4 5.0

The first three rows indicate the temperature, the total number of particles per fm3 and their mean distance d = (nY 1/3 The next five entries give the fractions of particle number (percent of n) occurring in the form of stable particles (N counts nucleons and antinucleons while N~refers to the stable strange baryons A, ~, ~, 12 and their antiparticles). The last four rows show the population of the corresponding excited states (ir* and K* indicate mesons with S = 0 and S = ±1, respectively, and N*, N~*represent the baryonic excitations in an analogous manner).

2.2 fm. In view of the fact that chiral symmetry suppresses the interaction not only among the pions but also between the pions and the excited states, we approximate the gas by a collection of free particles, distributed according to

d~

=

d3xd3p E—p.. (2~)~~i{ex~( T

-t

(4.1)

) ~

where i ir, K, p, N,... labels the particle species and g. is the corresponding statistical weight (g~ 3, g~ 4, g,, 9,...). The chemical potential of the level is denoted by p., and E (M,2 +p2)t~~2 is the energy of the particle in the local comoving frame. The sign occurring in the thermal factor depends on whether the particle is a boson or a fermion. The pressure generated by the distribution (4.1) is given by =

=

=

=

=

P=T)J ~g~f

d3p

(

—

3ln 1+exp

____

T

(4.2)

We take the levels from the tables provided by the particle data group [301, restricting ourselves to the mass range 0
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18

0

50

100

150

200

250

T [MeV] Fig. 1. Hadronic phase in thermal and chemical equilibrium. The figure shows the dimensionless ratios 4, s/T3 and /P as a function of the temperature ( and s represent energy and entropy per unit /T volume and P is the pressure). The horizontal line indicates the value e/T4 15.6 which corresponds to a free gas of gluons and quarks (3 flavours).

Above this point, the interaction among the constituents of the gas rapidly grows with T and inelastic collisions become increasingly important. Around Tc 190 MeV a phase transition is expected to occur and a description in terms of colour-neutral hadronic states then presumably ceases to make sense. For a ball with a radius of 10 fm or less, we expect chemical equilibrium to be lost at a temperature somewhere in the region 150 MeV < T < I~. Unfortunately, this occurs outside the range where the formula (4.2) can be taken at face value. Nevertheless, this representation does provide for a reasonable extrapolation towards T~for the following reason. Consider chemical equilibrium, p., 0. Fig. 1 shows that for the equation of state (4.2) the ratio b c/P approximately becomes temperature independent above T= 150 MeV, with b 6. Thermodynamics then implies that the equation of state is characterized by a power law, =

=

P=aT~1,

=ahT’~’,

(4.3)

s=a(b+1)Th.

Indeed, in the interval 150 MeV < T < 200 MeV these formulae provide an adequate approximation for eq. (4.2) with b 6.3 and a ~ T 1 176 MeV [if baryons and strange particles are omitted, one instead obtains b 4.6, T1 288 =

=

=

=

=

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MeV and the free massless Bose gas corresponds to b 3, a ir2f/90 where f is the number of degrees of freedom]. Accordingly the ratio E/T4 rapidly grows with T, approximately in proportion to T3. This behaviour is to be expected, since the value c/T4 1 characteristic of a gas of pions is small compared to the value l9ir2/l2 15.6 which pertains to a plasma of free quarks and gluons. For the equation of state (4.2), the plasma value is reached at T 225 MeV, not far from the critical point (the energy densities of the hadronic and quark—gluon phases need not match at the critical point the phase transition may liberate latent heat). The above equation of state also implies a behaviour of the mean distance between the particles which is intuitively reasonable. Calculating the number of particles per unit volume, n~ 01,with eq.calculation (4.1) one shows arrivesthat at the t~3 given in table 1 (third row). The thevalues mean for d (n~0~Y distance reaches the value d 1 fm at T 190 MeV, indicating that our equation of state packs the hadrons very densely there, which again suggests that colour cannot remain confined much beyond this point. =

=

=

=

—

=

=

=

5. Chemical potentials Next, we consider the chemical potentials of the various levels, which start building up when chemical equilibrium is lost. We assume that the population of the excited states remains in equilibrium with the particles formed in their decay and set the chemical potential of the mother equal to the sum of the chemical potentials of her daughters; if several decay channels are open, we weigh the various final state configurations with the corresponding branching ratio. For the states p(77O), ~(1231) and a 2p.,,., ~ p.~+ p.~and 2(1320) e.g., this leads to p.,, p.,,, 2.8p.,,. + O.lp.~+ °I5p.~respectively. More generally, the above condition for partial chemical equilibrium determines the chemical potentials of the excited states in terms of the potentials associated with the stable particles a{ir, K, x~,N, A, E, Q} which occur as end products of the decay chain: =

=

=

=

~,

p.~=~

(5.1)

The coefficient d~,e.g., is the mean number of pions emerging in the decay of the level i. Note that we restrict ourselves to configurations for which the number of particles and antiparticles is the same, such that, e.g. p.~ In the SU(3) limit, the situation then simplifies considerably: the chemical potentials of the stable mesons take a common value, p..., while the stable baryons sit at /.tN (note that the 12 is now unstable, decaying into EK). In this limit, the equation of state therefore only involves two independent chemical potentials, P P(T, p.,,., p.N). In reality, mainly as a consequence of the level splittings =

=

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11)7

generated by m,, m,,, the various members of a multiplet however develop somewhat different chemical potentials in the course of the expansion. We assume that the effect is proportional to the number of strange quarks or antiquarks contained in the particle and set —

p.1.

—

p.N

=

p.~ p.N p.~ p.~ p.J( 3(p.K p.r). —

=

—

=

—

(5.2) p.f2 p.~= In this approximation, reactions such as 7T. KN are in equilibrium, because they conserve the number of strange quarks or antiquarks. For the s~we use (4p.~ p.~)/3,the value suggested by ideal mixing where the probability for this particle to contain an s~pair is 2/3. [Actually, the s~does not play a significant role in our analysis, because of its low statistical weight changing the value of p.,., to p.,,. or to /i~ leaves our results practically untouched]. With the above assignments, the chemical potentials of the various levels can now be expressed in terms of p.,, ~ and p.~, —*

=

—

—

p.

d~p.,,.+dJKp.~+d,Np.~.

(5.3)

The coefficients are determined by the conditions of partial chemical equilibrium specified above d~’=d”—~-d,°—dt —d~—2d~—2df1, d,~=d~+~d~+d;’ +d~+2d~+3d[2, (5.4) As compared to the SU(3) limit, we thus need an additional chemical potential, p.~,to characterize the state of the gas. The distinction between p.~and p.,~allows us to separately count strange and non-strange particles this is important particularly at low T where the mass difference between ~r and K strongly affects their abundance. We denote the number densities conjugate to p.,,., p.~ and p.~by —

fliT’

~K

and

~N’

respectively, Th,.=8P/0p.~= Erfl flK~’~/dp.K=

~

fl~=aP/ap.~=Ed~n~,

(5.5)

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where n, is the occupation number of the level i per unit volume. Since the coefficient d,~” is equal to one for baryonic levels and equal to zero for mesonic ones, the number density ~N simply counts the number of baryons and antibaryons per unit volume. Similarly, the quantity ~K enumerates the strange quarks and antiquarks in the following sense: let the excitations decay according to the branching ratios given in the particle data tables and count the strange valence quarks contained in the stable particles which emerge in these decays. At low temperatures where strange baryons are rare, 11K essentially counts the kaons occurring after disintegration of the excited states. Finally, the expressions (5.4) for the decay coefficients show that, disregarding the 12, the sum ñ,,. + ~1K is the number of stable mesons (ir, K, i~) per unit volume, including those “hidden” in excited states. If strange baryons are rare, ii,,. counts the pions emerging if all resonances have decayed.

6. Main features of the evolution We now discuss the gross features of the expansion process on the basis of the global conservation laws for energy, entropy and particle number, defering a detailed analysis of the local conservation laws to a separate paper [31]. Consider first phase I where chemical equilibrium insures that the chemical potentials vanish. The expansion process conserves the total energy E and the total entropy S of the system, given by E

=

fd3x

y2(e

+

r’2P),

S

=

fd3x ys.

(6.1)

The entropy density decreases with the distance from the center while the velocity increases with distance. The total entropy is given by Vys, where V is the volume of the hot region while y and s are suitable mean values of the distribution. Conservation of entropy thus implies that Vys a VyT” remains constant in the course of the expansion. Similarly, energy conservation requires the product Vy2(b + v2)T”~ to remain invariant. Assuming that the velocities at which the gas expands during phase I are small such that y I + ~L’2, this gives T

h/(b+2)

T h(b±I)/3(h±2) RT~Ro(-~) ,

(6.2)

where T 0 and R0 are the initial values of temperature and radius when y 1. The radius approximately growsdensity in proportion to T oft.82 while y is proportional 8 athus R°4.A drop in energy by a factor corresponds to a drop to in T° =

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109

temperature by 10%, an increase in radius by 20% and a rise in the y-factor of the expansion velocity by 7%. As the evolution enters phase II, one or several of the particle numbers discussed above freeze-out and chemical potentials start building up. To simplify the discussion, let us first ignore the complications due to the occurrence of baryons and strange particles disregard these components of the gas altogether. In terms of the equation of state set up in sect. 5, this limiting case occurs if the chemical potentials p.~and p.~are given large, negative values. In theoretical terms, the simplified situation we are considering here corresponds to the limit where the number of colours and the mass of the strange quark are sent to infinity, N~—s m5 In this limit, baryons and strange particles acquire infinite mass. —

~,

—

~.

The local configuration of the gas is then described by temperature, velocity and a single chemical potential, p.,,.. The corresponding particle number is given by

(6.3) where ~iiT + 2n,, + is defined in eq. (5.5). While N.. decreases during the initial phase where the number of particles is determined by chemical equilibrium, its value freezes out when the evolution enters phase II. Estimating the integral (6.3) in the manner discussed above, we conclude that the ratio s/niT of entropy density to number density approximately remains constant from then on. The behaviour of this quantity as a function of temperature and chemical potential is shown in fig. 2a which is based on the equation of state discussed in sect. 4, except that we are now omitting the contributions of baryons and strange particles. (Note that the trajectories in the plane of temperature and chemical potential are determined by the value of s/niT. In a full hydrodynamic description, temperature and chemical potential are not uniform; in this context, the numbers displayed in our figures merely represent typical values). The entropy per pion decreases if the chemical potential is increased at fixed T. At low temperatures, the gas predominantly consists of pions and the ratio s/nL,. therefore agrees with the entropy per particle in a gas of free bosons of mass MiT (dashed lines). At high temperatures and densities, a substantial fraction of the entropy is stored in the excited states, but these states also manifest themselves in the number density ~L,..The figure shows that for T> 100 MeV, p.~< 100 MeV, the contribution of the excited states to the ratio s/ni,,. is smaller than the contribution of the pionic component. Suppose now that chemical equilibrium is lost at T 180 MeV (point A). The entropy per particle at this temperature and for p.,,. 0 is s/n,,. 3.5. If the reactions would maintain chemical equilibrium, the evolution would follow the curve p.,,. 0 and the entropy per particle would grow, reaching the value s/n,,. 4.5 if the temperature falls to 100 MeV (point B) not because the entropy grows, but because the number N,, N,,. + 2N~+ decreases. For s/n,, to remain =

...

=

=

=

—

—

=

...

110

H. Bebie et al.

/

Expanding hadronic gas

constant, the chemical potential must therefore rise in such a way that the evolution follows the horizontal line shown in the figure: it rises to 40 MeV by the time the gas has cooled to T 150 MeV and to p.,,. 86 MeV when the temperature reaches 100 MeV (point C). This shows how the conservation of particle number drives the system out of chemical equilibrium, leading to an overpopulation of the pionic states when the excitations have disappeared. =

=

0 o

50

100 T

150

200

[MeVI

0.00 0

50

100 T

150

200

[MeVI

Fig. 2. (a) Entropy per particle as a function of temperature. The number density fi,, = n,,. +2n~+3n,, + ... also counts the pions “hidden” in the excited states. For comparison, the dashed lines indicate the entropy per particle for the pionic component of the gas and the arrow points to the value s/n 3.6 characteristic of the free massless Bose gas. (b) Decay of the excited states for an evolution where chemical equilibrium is lost at T = 180 MeV. The ratio e,,/e is the fraction of the energy density stored in the pionic component and ~iT/~ is the corresponding fraction of the entropy density. (c) Evolution in the (T, j.c)-diagram. The curves shown correspond to constant entropy per pion. If chemical equilibrium is lost at T = 180 MeV, the evolution follows the trajectory with s/b, = 3.5. Note that (a), (b) and (c) refer to a simplified model where baryons and strange particles are omitted.

H. Behie ci a!.

/

Expanding hadronic gas

160—’-

Ill —

Bose

140

Id

condensation

20L~

7.3

0

50

100 T

150

200

[MeV[

Fig. 2. Continued.

As the temperature falls, the fraction of energy, entropy and particle number contained in the excited states decreases. This is illustrated in fig. 2b where we plot the fractions stored in the pionic component of the gas as a function of temperature, again assuming that chemical equilibrium is lost at T= 180 MeV and disregarding the occurrence of baryons or strange particles. It is striking that the three ratios c,,./, s,,/s and n,,./n1,,. are almost the same. The physical significance of this observation is the following. If the energy stored in the excited states is distributed over the pions which emerge in their decays, each one on average receives the energy (E c,,)/(ii,, n,,). In view of e,,/e n,,/n,,., this value roughly agrees with E,,/n,,, i.e. with the mean energy of a pion in thermal equilibrium at the temperature of interest. This property is analogous to the rule of thumb proposed in ref. [32], according to which the number of pions which on average emerge in the annihilation of two particles at a given temperature is such that each one roughly obtains the energy which corresponds to thermal equilibrium at this temperature (see sect. 8). Fig. 2b shows that, in the decays of the excited states, this property not only applies to the energy but also to the entropy. The evolution depends on the temperature at which chemical equilibrium is lost. This is illustrated in fig. 2c where we plot curves of constant entropy per particle in the (T, p.)-diagram. The example discussed in detail above corresponds to the curve with s/a,, 3.5 where chemical equilibrium (p.,, 0) occurs for T>~180 MeV. If the decoupling temperature if higher, the chemical potential is given more time to build up and the expansion leads to a more pronounced overpopulation of the pionic component at thermal freeze-out. At low temperatures, the chemical potential approaches the Bose condensation limit p. M,,. otherwise the entropy per particle would tend to infinity there [see eq. (9.1)]. Actually, this aspect of the figure is academic, because the mean free path grows —

—

=

=

=

—

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/ Expanding hadronic gas

rapidly as the temperature falls and thermal equilibrium is lost much before T 0 is reached. In principle, Bose condensation may set in already at a finite temperature. In our model, this happens if the initial temperature is taken sufficiently high. The figure shows, however, that this possibility is also academic because our equation of state certainly fails at these temperatures: excited states of high mass then play an important role, while in our approach, levels with M.> 2 GeV are simply disregarded. To study evolutions where chemical equilibrium is lost immediately after the temperature falls below Te, one might invoke the Hagedorn model [3], supplemented with an analysis of the decay characteristics of the high-mass excitations. We did not attempt to carry out such an analysis and accordingly have to restrict ourselves to evolutions where the temperature at which chemical equilibrium is lost does not exceed 180 MeV (compare sect. 4). Note that if the initial energy density rises, equilibrium is maintained down to lower temperatures. Hence, once the ions collide at sufficiently high energy, chemical equilibrium should prevail until the temperature falls to values where our neglect of high mass excitations is justified. Thermal freeze-out occurs when the mean free path of the pions becomes comparable to the radius of the hot region such that collisions become rare. At the same time, the excited states disappear, because their abundance is controlled by the equilibrium between their decays and the reactions which produce them. Consider, e.g. the p-mesons which at low temperatures are mainly formed in the reaction irir p ir~-.Denote the collision rate of this reaction by y,,. In equilibrium, the number of p-mesons produced per unit time and unit volume, is equal to the number of decays, T,n,,. Hence the contribution to the irir collision rate due to resonance scattering is determined by the width T, 150 MeV and by the relative abundance p/ir =

—~

—~

—

=

2[’~n,,/n,,..

(6.4)

The direct evaluation of the collision rate to first order in the pion density indeed confirms this relation [24]. This shows that if the collisions become rare, the abundance of excited states is not governed by the equilibrium density but by their lifetime the p-mesons disappear at thermal freeze-out within 1/T, 1.3 fm. Conversely, detailed balancing implies that thermal decoupling does not occur at a temperature where excited states are abundant. For the values of p.,, which result if the pions decouple chemically at T 180 MeV, the pion density runs through the values n,, 0.19, 0.10 and 0.05 fm3 as the temperature reaches 140, 120 and 100 MeV, respectively. According to eq. (2.1), the corresponding mean free path rapidly rises, from A,,.,, 1.7 fm at T= 140 MeV to 4 fm at 120 MeV and to 10 fm at 100 MeV. Fig. 2b shows that in this temperature interval, the fraction of the energy or of the entropy stored in the excited states falls from one half to one quarter. [Note that the above values for —

=

=

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Expanding hadronic gas

113

the mean free path neglect collisions with excited states. To account for these, one may replace the excited states by the pions into which they decay, using the density n, instead of n,.. Fig. 2b indicates that this reduces the mean free path at T 140 MeV by about a factor of two.] We conclude that in the simplified model under consideration here, a ball of the size of a few Fermi is expected to freeze out somewhere around T 120 MeV. In the above discussion, neither the expansion velocity nor the volume of the hot region occurred, because these quantities drop out when the ratio S/N,, is evaluated within the approximation used here. In particular, we expect the evolution in the (T, p.)-diagram not to be sensitive to the expansion velocities occurring during phase II. For the same reason, it should not make an essential difference whether the expansion velocities in the directions parallel and orthogonal to the beam axis are the same or not (at the collision energies explored so far, the momentum distribution in the central rapidity region is found to be broader along the beam direction than transverse to it). A rough idea of the velocities built up during the expansion is obtained by considering the ratio E/S which is given by (6.5) E/S y(e + i2P)/s const., =

=

=

where c, P, s and y, are mean values. Since the ratio c/s falls, roughly in proportion to temperature, the cooling from T 180 MeV to T 100 MeV must lead to relativistic expansion velocities. For a radial expansion, the issue is discussed in detail in ref. [31]. The above simplified discussion serves to demonstrate the main point of the present paper: if the expansion is too fast for chemical equilibrium to be maintained, then the entropy per pion observed in the final state must be low. This conclusion is in accord with the analysis of the observed transverse momentum distributions given in ref. [141. As pointed out there, the requirement that the mean free path at freeze-out is of the order of the radius of the ball can be met only if the pions are out of chemical equilibrium, with p.,, 120 or 130 MeV. The chemical potentials found in ref. [13] are also in this range. Fig. 2a shows that these values of the chemical potential correspond to an entropy per pion of s/n,, 3, considerably less than the number s/n,, 5 which one finds if the pions are assumed to remain in chemical equilibrium until thermal freeze-out occurs (the observed PT distribution then requires T’~76 MeV, with a mean free path of order 75 fm this assumption does not lead to a coherent picture [14]). =

=

=

—

7. Baryons and strange particles In sect. 6 we discussed the limiting case N~ m~ ~ where baryons and strange particles do not occur. Let us now see what happens if these particles are taken in account. —* ~,

—~

114

H. Bcbie ci a!.

1.00

‘~

‘.

——‘

/

Expanding hadronic gas

,-~-..-.-

-S

5

‘‘i’’’

n

~ 0.80

-

0.60

-

0

K V

~

~i_:—

50

........

100

150

200

T 1MeV] 0K’ nN count the number of pions, Fig. 3. Hadronic gas in chemical equilibrium. The quantities n,. kaons and nucleons per unit volume. The density . = n, + 2n 5 + ... includes the pions emerging from the decay of the excited states; hK counts the number of strange quarks and antiquarks in the final state and ~N enumerates the baryons and antibaryons (compare table 1).

Table 1 shows how the number of particles is distributed over the various components of the gas if chemical equilibrium prevails, p., = 0, and fig. 3 gives the ratios of the corresponding number densities n,,, nK, ~N as a function of temperature. As long as the expansion proceeds sufficiently slowly, chemical equilibrium is maintained such that the relative abundances follow the curves shown in this figure. Accordingly, the particle numbers N,=fd3xyn,,,

a=ir,K,N

(7.1)

fall with temperature until the point is reached where chemical equilibrium is lost. The temperature at which this occurs depends on the rate of those reactions which violate the conservation of N,,., NK and NN. Note that, by construction, the decay of the excited states into stable particles conserves all three of these numbers. The processes which maintain N,, in equilibrium were discussed in sect. 3. In the case of NK, chemical equilibrium is controlled by reactions such as KK ~ 1TIT7T, or K.~~-s ~rN while in the case of NN, processes like NN irir, KK, ~ are relevant. Equilibrium is lost when the mean free path of these reactions becomes comparable to the size of the hot region. ~

~-*

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Expanding hadronic gas

115

In the case of p~5 annihilation reactions, there is a wealth of experimental information regarding the cross section as well as the final-state distribution [33]. At low momenta, the annihilation cross section rises, roughly according to the empirical formula a- a + b/PIab, with a 17 mb, b 39 mb GeV. For the temperatures of interest here, the protons can approximately be treated as non-relativistic particles, such that PI,Ih p1 —p2 , where Pt and p2 are the momenta of =

the collision partners. Averaging the product adistribution, the proton annihilation rate becomes 1 —=

r

4ai/— V ‘zrm

+—.

m

V1

—

I n over the Maxwell

n,

where n is the number of antiprotons per unit volume and m is the proton mass. The corresponding mean free path is defined in terms of the most probable velocity I ~2T/m as =

AN

IT

l/a-effn.

Numerically, this gives a-,.,.,.= 90 mb and 100 mb for T= 180 MeV and 140 MeV, respectively, indicating that the cross section is not sensitive to temperature. In contrast, the number density very rapidly drops as T falls, making it increasingly difficult for the protons to find annihilation partners. The equilibrium values for the number of antibaryons per unit volume, nB X (baryon fraction) >< n, can be taken from table 1. Note that for T ~ 150 MeV, the majority of the baryons is in an excited state. Assuming that the p~cross section is typical for baryon annihilations, the corresponding mean free path becomes AN 1.2 fm at T= 180 MeV, rising to AN 3.5 fm at T falls to 160 MeV and to AN 13 fm for T= 140 MeV. This indicates that baryons decouple chemically at an early stage; for a region of the size of a few Fermi, we expect decoupling to occur around T” 160 MeV. For the reactions which equilibrate the number of strange particles such as KK annihilations, direct experimental information is not available. Presumably, the cross sections are of a similar order of magnitude, perhaps somewhat smaller. Note that the corresponding mean free path AK rises less rapidly as the temperature falls than it is the case with AN (in equilibrium at T 180 MeV, the density of strange particles exceeds the baryon density by nK/nN 2.2; this factor increases to 4.0 if the temperature falls to 140 MeV). Since the energy stored initially in the central region of a heavy ion collision is expected to rise with the collision energy, this region should grow to larger sizes and stay in equilibrium down to lower temperatures the number of baryons, NN, should then freeze out earlier than the number of strange quarks, NK. The rates of the reactions which equilibrate N,, are difficult to estimate. The decays of the excited states conserve this number only in the statistical mean. For highly excited states, individual decay channels violate the conservation law, thus =

=

=

—

116

H. Bcbic ci a!.

/

Expanding hadronic gas

affecting the chemical composition. States for which the decays lead to significant fluctuations in N,, however only occur abundantly at high temperatures. We therefore expect N,, to remain in chemical equilibrium only in the very early phase of the evolution. In the simplified model discussed before, chemical equilibrium is lost in a single step, N,, freezing out as the evolution enters phase II. In the presence of baryons and strange particles, however, several conservation laws occur and chemical equilibrium is lost in a sequence of steps. If the energy at which the ions collide is sufficiently high, we expect the pion number N,, to get out of equilibrium first, followed by NN and finally by NK. The evolution in phase II then runs through the following stages: (a) In this phase, the pions are chemically decoupled while the abundance of baryons and strange particles is still controlled by chemical equilibrium. (j3) Both pions and baryons are out of equilibrium while NK continues to fall. (y) All three particle numbers N,,, NN and NK are conserved. We briefly analyze the equation of state for these three phases and then discuss the consequences for the evolution. As we will see, the behaviour of the gas does not strongly depend on the uncertainties associated with the equation of state or with the values of the various decoupling temperatures. The main features which characterize the evolution in the simplified model where baryons and strange particles are ignored manifest themselves also in this more realistic framework. In particular, the expansion necessarily leads to an overpopulation of the pionic component, i.e. to a low value for the entropy per pion. The evolution of the strange population is somewhat more sensitive to details of the microscopic picture which are not under good control. We will discuss this aspect of the problem in sect. 10. —

—

8. Partial chemical equilibrium We now turn to the equation of state which characterizes the various phases of partial equilibrium specified in the preceeding section. (ci) If N,, is out of equilibrium while NN and NK are not, the composition of the gas is characterized by a single chemical potential, p.,,. One might be inclined to characterize this situation by p.~ 0. This is not the proper equilibrium condition, however: reactions like NN 2ir and NN 4ir are not in equilibrium 2p.,,, respectively. The value of the for p.,~, 0, but require ‘~‘~ p.,, and p.,.., equilibrium potential is determined by the mean number of pions produced per disappearing baryon. Denoting this number by aN, the baryons are in chemical equilibrium if p.,~ ct~p.,,. Analogously, denoting the mean number of pions generated in the annihilation of a pair of strange quarks by 2ciK, the strange particles are in equilibrium provided p.~ =

=

*-*

=

~-*

=

=

=

=

H. Bebie ct a!.

/

Expanding hadronic gas

117

The annihilation reactions conserve the number N N,, + aKNK + aNN,..,. Entropy conservation therefore implies that the ratio S/N remains constant. Approximating this ratio again by the ratio of the corresponding densities, we conclude that in phase (a) the evolution is characterized by =

s/(n,,

+ aKnK + aNn,..,) = const.,

p.K=a~p.,,,

p.~=a~p.,,.

(a)

The complication generated by the presence of baryons and strange particles manifests itself in the occurrence of the two chemical parameters ciK and aN which may depend on T. We will discuss their values below. (/3) If both pions and baryons are locked in while the abundance of strange particles is controlled by chemical equilibrium, we need two independent chemical potentials p.,,., p.,.., to describe the configuration of the gas. In this phase, the evolution is determined by the mean number of pions (a~) and baryons (a~) generated per strange quark which disappears. The combinations N,, + a~NKand NN + aKNK then remain conserved and the chemical potential of the kaons is now given by p.~ a~p.,,+ ~ Actually, at the temperatures of interest here, we expect reactions like KK NN, which produce baryon pairs, rarely to occur. We therefore neglect the quantity a~ and identify a~with the coefficient ak introduced above. In this approximation, phase (/3) is characterized by the conservation of N,, + aKNK and NN, such that =

—~

s/(n,,

+ aKnK)

=

const.,

S/nN

=

const., (/3)

~

(y) Finally, if all three particle numbers are conserved, the evolution approximately conserves the ratios

s/n,,

=

const.,

s/nN

=

const.,

s/nK const.. =

(y)

From a theoretical point of view, phase (y) is the simplest one of the three, because it does not involve any new parameters, the evolution of the three chemical potentials p.,,, p.~,and p.~needed to describe the configuration of the gas being fully determined by the above three conservation laws. If the equilibrium conditions specified for (/3) apply, the equation of state involves one chemical parameter, aK, while in phase (a) two such parameters occur. Estimates of these

quantities can be found in ref. [32], where it is argued that the mean number of pions produced in an NN annihilation is such that the energy which each one receives roughly agrees with the mean energy of a pion at the temperature of interest, i.e. aN (EN)/
118

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Expanding hadronic gas

T < 180 MeV, NN annihilations produce about five pions, such that aN 2.5, while KK annihilations should generate about three pions, i.e. aK 1.5. Indeed, the estimate ciN 2.5 is confirmed by the data on pji annihilations [33]. For the collision energies which correspond to the above temperature range, these data show that on average, the annihilation indeed produces about five pions (this configuration also represents the most probable final state). Particles other than pions rarely occur. For the temperatures of interest here, the chemical parameter aN is therefore not subject to significant uncertainties. In the following, we use the value aN 2.5. In the case of aK, where direct experimental information is not available, we invoke the following bounds. Kinematics sets the lower limit ciK ~ 1 (annihilations generate at least two particles). On the other hand, if aK is too large, the expansion would not lead to a decay, but to a growth of the strange population. The upper limit is reached when NK remains constant. If the baryons are decoupled, this corresponds to phase y where, starting from equilibrium at T= 180 MeV, the value of a~=p.~/p.,,rises from aK= 1.57 (T= 180 MeV) to 1.64 (T= 140 MeV). Essentially the same upper limit is obtained if the baryons remain in equilibrium. This shows that the estimate ciK 1.5 mentioned above is close to the upper limit. <

=

=

=

=

=

9. Evolution of the pionic component We now wish to demonstrate that the details of the chemical evolution during phase II, discussed in sect. 2, do not strongly affect the behaviour of the pionic component. For this purpose, we fix the temperature at which chemical equilibrium is lost at T 180 MeV and trace the chemical potential p.,, as the temperature falls for three extreme situations, assuming that the chemistry obeys one of the three equilibrium conditions throughout the expansion. The result is shown in fig. 4. Curve a indicates the evolution of p.,, which obtains if only the pion number freezes out while baryons and strange particles remain in equilibrium with ciK 1.5, aN 2.5. The trajectory labelled [3depicts the behaviour if one assumes that pions and baryons simultaneously decouple while the population of the strange levels is controlled by chemical equilibrium with ciK 1.5. Finally, (y) corresponds to an evolution where all three of the particle numbers N,,, NK and NN lock in at T 180 MeV. The figure shows that in all three cases, the evolution leads to a strong overpopulation of the pionic component which is even more pronounced than for the simplified model discussed before (the corresponding trajectory is indicated by the dotted line, taken from fig. 2c). It is not difficult to understand why, in phase (y), the baryons and strange particles accelerate the build-up of a pionic chemical potential. In this phase, the additional degrees of freedom play the role of mere spectators, the conservation =

=

=

=

=

H. Behie ci al.

/

Expanding hadronic gas

119

160

Bose condensatIon

140

0

50

100

150

200

T Fig. 4. Evolution of p~,. resulting if N gets out of equilibrium at T = 181) MeV. (a) Baryons and strange particles remain in equilibrium; (p) only strange particles in equilibrium; (y) all three particle number’, N,, NN. NK conserved. The full lines correspond to a~= 1.5. the dashed ones to a~= I; the 0N = 2.5. For comparison, the dotted curve shows the chemical parameter of the haryons is fixed at evolution in the simplified model, taken from fig. 2c.

laws NN const., NK const. preventing baryons and strange particles from leaving the scenery. The acceleration originates in the fact that heavy particles carry more entropy than light ones. If the temperature is small compared to m p., the entropy per particle is given by the non-relativistic formula =

=

—

5 —=

ii

ifl~p. T

5 (9.1)

+—,

2

and is therefore large. As the temperature falls, the entropy per particle increases more rapidly for heavy particles than for light ones. The conservation of NN and NK therefore implies that at low temperatures, a more substantial fraction of the entropy resides in haryons and strange particles than at the temperature where equilibrium is lost while the heavy spectators watch the scenery, they consume entropy. Hence a smaller fraction of the entropy is left for the pions, or, equivalently, their chemical potential is pushed to higher values. The full lines (a), (/3) shown in fig. 4 are based on the values ciK 1.5, ciN 2.5 of the chemical parameters (recall that 2aK and 2a~ represent the mean number of pions produced in KK and NN annihilations. respectively). As discussed in sect. —

=

=

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Expanding hadronic gas

8, phase (y) corresponds to the upper limit of these parameters. The dashed lines indicate the trajectories which result if ciN is fixed at the experimental value 2.5 while ciK is set equal to the lower limit, ciK 1. The comparison of the various trajectories shows that the behaviour of p.,, is barely affected by the uncertainties associated with the chemistry of the gas. We conclude that the main features of the evolution discussed in sect. 6 persist if baryons and strange particles are taken into account. As before, the pion abundance mainly depends on the temperature at which they get out of chemical equilibrium and increases if this temperature is taken higher. =

10. Composition at thermal freeze-out The abundances of the various particle species are among the key observables and have been studied both experimentally [34] and theoretically [11,27,351. We now discuss the implications of our analysis for the chemical composition of the final state. In our framework, this composition is determined by the chemical potentials of IT, K and N (recall that, throughout the present paper, the number of baryons and antibaryons is taken to be the same). Fig. 5a shows the behaviour of p.~and p.~,again fixing the temperature at which the pions get out of chemical equilibrium at T 180 MeV. Since the parameters which determine the baryon abundance (decoupling temperature and value of aN) are under reasonably good control, the evolution of this component of the gas is determined to within rather small uncertainties. In fact, the value of the temperature at which the baryons decouple does not strongly affect the evolution of p.,,., and the chemical parameters associated with the strange particles play an even less important role. The shaded region indicates the sensitivity of the result to these details, assuming only that chemical decoupling occurs above T 140 MeV and allowing ciK to vary between the lower and upper limits given in sect. 9. The evolution of the strange component of the gas is more sensitive to the uncertainties in the equation of state. The earlier the strange particles get out of equilibrium, the higher is their abundance at low T and the steeper is therefore the growth of p.~. The upper limit indicated by a corresponds to an evolution where NK and N1.., lock in simultaneously with N,, at 180 MeV. The lower limit =

=

indicated by b shows the behaviour which results in the opposite extreme, setting the parameter aK equal to its lower limit ciK 1 and assuming chemical freeze-out of strange particles and baryons to occur only at 140 MeV. We discuss the physical significance of these curves in terms of the corresponding distribution of stable particles emerging at thermal freeze-out. Suppose the mean free path of the elastic collisions becomes comparable to the radius of the hot region when the temperature reaches the value TT. The collisions then stop =

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121

and the excited states decay. To include the stable particles which emerge from these decays, we evaluate the sums h,.,= Ed,~n~a-=(7T,K,’17,N,...}.

(10.1)

K/IT~hK/ñ,,

(10.2)

The ratios N/IT~hN/h,,,

are plotted as a function of the thermal freeze-out temperature in fig. 5b. The dash-dotted lines show the relative abundances in chemical equilibrium. If chemical decoupling of the baryons and strange particles occurs simultaneously with the pions at T= 180 MeV, the ratios nN/n,, and nK/n,, lock in there, at the values 0.21 and 0.46, respectively. The ratios N/IT and K/IT are smaller by about a factor of two, because here, strange baryons only contribute if their decays liberate nucleons, kaons or pions [compare eqs. (5.4) and (5.5)]. In the temperature range of interest, i.e. from chemical to thermal freeze-out, these ratios decrease only little. At T 100 MeV, the relative abundance of the nucleons is in the range 0.09 g~N/IT ~ 0.11, while for the kaons, we find 0.18 K/IT ~ 0.23. As a side remark, we note that if the evolution is traced down to T 0, the gas finally settles in the state of minimal energy consistent with the conservation of N,,, NK and N1~.,. Since chemical freeze-out leads to N,, > NK> NN and since it costs more energy to store a strange quark in a meson than in a baryon (MK M,, > M, MN), all of the baryons wind up as A particles. If thermal equilibrium were maintained down to very low temperatures, the final state would therefore exclusively contain IT, K, A and A. As noted above, the presence of baryons and strange particles accelerates the overpopulation of the pionic component, leading to higher densities and thus delaying thermal freeze-out. Table 2 contains numerical values of the densities at which the various stable particle species occur at thermal freeze-out, for the two cases a, b which correspond to the upper and lower limits of the strange particle abundance, as well as for the case of chemical equilibrium c. The table shows that, at a given temperature, the density is not sensitive to the strange particle abundance and exceeds the equilibrium density by a factor between four and ten. The corresponding mean free path is determined by eq. (2.1). Numerically we obtain A,,,, 2, 3 and 8 fm for T 120, 100 and 80 MeV, respectively. For a ball the size of a few Fermi, these values indicate that thermal freeze-out occurs around T~ 100 MeV. Table 2 shows that the evolution leads to significant departures from chemical equilibrium, not only for the pions, but also for the baryons and strange particles. The origin of the phenomenon is evident: at higher temperatures, a considerable fraction of the gas consists of baryons and strange particles if the expansion =

~

=

—

—

=

—

122

H. Bebic ci a!.

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/

Expanding hadronic gas

‘

(a) 500 400 300 a

200 b 100

0

‘

0

50 T

I

I

100

150

200

[Mcvi

0.40

(b)

0.30

0

50

100

150

200

T [Mcvi 5K Fig. 5. Abundance of baryons and strange particles. In (a) the trajectories of the chemical potentials / and ~ and in (h) the corresponding relative abundances of pions, kaons and nucleons occurring in the final state at thermal freeze-out are shown. The shaded regions indicate the sensitivity to the temperature at which chemical freeze-out takes place and to the chemical parameter 0K~ In the ratio K/~, the upper limit, indicated by a, results if NE and N..~freeze out simultaneously with N,,., while the lower limit, indicated by b. corresponds to an evolution where NK and NN remain in equilibrium down to T = [40 MeV with ~ = I. The dash-dotted lines indicate the relative abundances in chemical equilibrium.

occurs too rapidly for the relevant annihilation reactions to be effective, these particles are still present when the gas has cooled down. In fact, the baryonic levels are far out of equilibrium at low T. According to fig. 5a their chemical potential

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TAH,.E 2 Distribution of the stable particles emerging at thermal freeze-out

a T(MeV) ii

lfm3]

,i*/h K 0 N A

80 0.13 0.27 70.0 15.0 3,4 6.4

2.4 1.8 1.7 0.31

100 0.27 (1.44 68.1) 16.0 3.3 7.2 2.2 1.8 1.2 0.18

b 120 1)47 0.56 67.0 17.0 3.1 7.6 2.0 1.9 1.0 (1.14

80 (1.11 (1.25 74.0 13.0 2.8 (~.l 1.8 1.4 1.0 0.14

100 0.24 0.42 73.0 14.0 2.6 6.8 1.6 1.3 0.69 0.080

c 120 (1.42 0.54 73.1) 14.1) 2.5 7.1 1.4 1.3 (1.55 0.057

80 0.014 0.026 94.0 5.6

1)78 (1.11 0.008 (1.009 0.002 (LO

100 0.038 0.11 86.0 11.1) 1.7 0.70 0.084 0.095 (1.024 0.001

120 0.098 1)27 79.0 16.0 2.5 2.1 0.34 0.40 0.12 0.006

The first three columns correspond to evolution (a) where chemical freeze-out of pions. baryons and strange particles is assumed to occur at T = 180 MeV. The columns labelled (h) represent a scenario where only the pions decouple chemically at T = 181) MeV while baryons and strange particles stay in equilibrium down to T = 140 MeV with 0N = 2.5. 0K = I. The last three columns, labelled by (c). indicate the corresponding distribution in chemical equilibrium (compare table 1). The first two rows give the thermal freeze-out temperature and the number si of stable particles per fm3. including those which are generated in the decay of the excited states. The share ,~‘/h contributed by these decays is shown in the third row. The remaining entries indicate the distribution over the various stable particle species (percent of ii).

rises to l.tN 380 MeV when T reaches 100 MeV. Since the nucleons can be treated non-relativistically there, this value of the chemical potential implies that their density exceeds the equilibrium density by the factor exp(p.~/T) 45. The magnitude of this factor merely illustrates the fact that the equilibrium density at thermal freeze-out is not relevant what counts instead is the equilibrium density at the temperature where the baryons decouple chemically. Finally, let us compare our results with the properties of the final state observed in heavy ion collisions at the presently available energies. As mentioned above, the transverse momentum distribution indicates that thermal freeze-out occurs around T 100 MeV with a very low value of the entropy per pion, s,,/n,, 3. A value as low as this is reached only if the pions get out of equilibrium very early in the evolution. We have set the temperature at which this occurs equal to 180 MeV (this is at the upper end of the temperature interval where the approximations underlying our equation of state make sense). With this input, the evolution indeed leads to freeze-out somewhere around T 100 MeV, at a value of p.,, between 100 and 120 MeV; this corresponds to an entropy per pion in the range 3.25 s,,/n,, 2.90. Regarding the chemical composition observed at thermal freeze-out, the situation is less rosy. At the collision energies available now, the rapidity distribution of the baryons does not allow one to clearly isolate the central region the —

—

—

124

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contamination from nucleons which were present in the projectile and in the target even before the collision occurred appears to also affect the central rapidity interval. In the case of the kaons, the rapidity distribution does show a central peak. The data of the NA35 group [34] indicate that in the central region, the ratio K~/(negativelycharged particles) is 10.7/105. Table 2 shows that, in our idealized picture where the net baryon number is taken to vanish, the negatively charged particles mainly consist of IT, K and such that for our final state configuration the abundance of negatively charged particles is approximately given by ~IT + ~K + ~N. Using N/IT 0.1, the experimental number quoted above therefore implies K/IT 0.16. A glance at fig. 5b shows that this value is on the low side, the lower limit b corresponding to K/IT 0.18. This may indicate that the contamination referred to above also affects the K/IT ratio. At any rate, the problems associated with the fact that the initial configuration is characterized by a nonzero baryon number should become less significant at larger collision energies. It is difficult to imagine that strange particles continue annihilating below 140 MeV at thermal freeze-out, the K/IT ratio must therefore be substantially higher than in chemical equilibrium (for T= 100 MeV, the equilibrium value is ñK/h,, 0.13). We can also compare our predictions for K/IT and N/IT with recent results obtained in pj5 collisions at 1800 GeV at the Tevatron [36]. The highest multiplicities are here somewhat smaller than in the heavy ion data of NA35 we have considered. They correspond to about 120 charged particles. It is, therefore, not clear that our general framework can be applied to these events. Nevertheless, it is amusing to see that for the highest multiplicity events the observed ratios K/IT 0.145 ±0.02 and N/IT 0.086 ±0.013 are not far from our predictions. ~,

—

—

=

—

=

=

=

=

11. Summary and conclusions (i) In the present paper we have analyzed the evolution of the hadronic plasma which is generated in the central rapidity region of a heavy ion collision. The main hypothesis underlying this analysis is that the evolution passes through an equilibrium phase, during which the mean free path of the particles is small compared to the size of the hot region. In this phase, the local configuration of the gas is fully determined by the densities of energy and momentum, or, equivalently by temperature and expansion velocity. As long as equilibrium prevails, the rate at which temperature and velocity are changing is determined by the conservation of energy and momentum, which, furthermore, implies that the expansion is an adiabatic process. (ii) We argue that chiral symmetry suppresses the interaction at low temperature and describe the gas as a collection of free particles, using the mass spectrum given in the particle data tables. This limits the temperatures accessible to our analysis to T ~ 180 MeV. (Beyond this point, highly excited states become increas-

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125

ingly important and an extrapolation which corrects for their absence in the data tables is required [3]; also our neglect of the interaction energy is then not justified). Once the energy density of the initial configuration is sufficiently high, this limitation will not be of serious concern, because equilibrium will then be maintained down to lower temperatures. To compare our analysis with data obtained at the presently available collision energies we do however need to consider temperatures which are at the upper end of the range where our equation of state is adequate. (iii) As the hot region expands, the temperature falls and the mean free path A grows. For elastic ITIT-collisions, A is proportional to T~’ while for inelastic collisions, A grows with the power T9 [14]. Since elastic collisions only equilibrate energy and momentum and do not affect the chemical composition of the gas, chemical equilibrium is lost at an earlier stage than thermal equilibrium. (iv) We point out that the momentum distribution of the particles observed at thermal freeze-out indicates a remarkably low value of the entropy per pion and we show that this property of the final state is indeed to be expected if the expansion conserves entropy, but occurs too quickly for chemical equilibrium to be maintained. The mechanism at work here is readily understood: at high temperatures, most of the entropy is stored in excited states; since their number is considerably smaller than the number of pions which emerge from their decay, each pion only gets a small share. (v) The quantitative analysis of an expansion out of chemical equilibrium requires an understanding of the processes which determine the evolution of the chemical composition. We discuss this problem in some detail. In particular, we analyze the role played by baryons and strange particles. The cross sections which determine the baryon annihilation rate indicate that, if the size of the hot region is of the order of a few Fermi, these particles decouple chemically at T 160 MeV. The reactions which control the abundance of strange quarks are less well explored. We give upper and lower limits and discuss the consequences for the ratio K/IT. (vi) At 200 GeV/nucleon our analysis suggest that the evolution of the central region can be characterized as follows. The expansion starts in a phase of approximate local equilibrium at a temperature above or only slightly below the critical temperature. The pions get out of chemical equilibrium at a very early stage (T 180 MeV) and a corresponding chemical potential p.,, starts building up. Soon thereafter, baryons and strange particles also decouple and two further chemical potentials p.,.,,, p.~develop. After chemical freeze-out, the kinetics of the gas is dominated by decays and elastic collisions (including resonance formation, such as ITIT p, ITK ~ K* or ITN ~). At a temperature of the order of 100 MeV, even elastic collisions become rare, the excited states left over decay and the momentum distribution freezes out. The value which results for the entropy per pion in the final state is in the range 2.9 ~ s,,/n,, s~ 3.25. Note that the number of ~

~-÷

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pions per unit volume is substantially larger than if they were in chemical equilibrium at the same temperature: at T 100 MeV, the equilibrium density in the comoving frame is n,, 0.22T3, less than a quarter of the value n,, 0.94T3 which corresponds to p.,, 120 MeV. The chemical potential built up in the course of the expansion therefore also reduces the mean free path and hence delays freeze-out. (vii) The implications of this picture for collisions occurring at higher energies are the following. Let us first disregard the chemical potentials. Thermal freeze-out takes place when the mean free path of elastic collisions reaches values comparable to the radius R of the system. The mean free path approximately grows in proportion to T5 while the density falls with T3. The temperature and the size of the system at freeze-out therefore approximately scale with the number of pions produced in the collision according to Ta N;~2, R aN,~2. An increase in N,, by a factor of 10 corresponds to a drop in the freeze-out temperature by about 20% and an increase in size by a factor of two or three. The temperature at which the system gets out of chemical equilibrium also falls, but we expect this change to be small, because the density of the gas and hence also the rate of the collisions which control the chemistry are very sensitive to temperature. If the temperature at which the pions get out of equilibrium drops by 10 MeV, the value of p.,, reached by the time the temperature falls to 100 MeV diminishes by about 10 MeV. We therefore expect the entropy per pion to remain low. The consequences of chemical disequilibrium which we are discussing in the present paper should persist to very high collision energies. (viii) The pressure generates considerable expansion velocities. A crude estimate based on the ratio E/S indicates that the mean y-factor reaches values between 1.2 and 1.3 when freeze-out occurs [cf. eq. (6.5)]. This result agrees with the phenomenological values extracted from the transverse momentum distribution in [14]. For a ball to contain the observed number of pions, its radius must be of order 6 fm which is also consistent with observation (see ref. [6]). Note however that these crude estimates must be taken with a grain of salt, because here the profiles of velocity and temperature play a more important role than in the ratios S/N which determine the evolution of the chemical potentials. For an analysis of the evolution of these profiles and for a discussion of the physics of thermal freeze-out we refer to a subsequent paper [31], which also contains our conclusions concerning the transverse momentum distribution. =

=

=

=

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