Qucosa – Hemholtz-Zentrum Dresden-Rossendorf

Extended calculations of the deuteron's static properties, based on the numerical solution of the Bethe-Salpeter equation,.are presented. A formalism is developed, which provides a comparative analysis of the covariant amplitudes in various representations and nonrelativistic wave functions. The magnetic and quadrupole moments of the deuteron are calculated in the Bethe-Salpeter formalism and tlie role of relativistic corrections is discussed

Kämpfer, Burkhard

Khanna, F. C.

Kaptari, L. P.

Bondarenko, S. G.

Umnikov, A. Yu.

Kazakov, K. Yu.

Bethe-Salpeter amplitudes and static properties of the deuteron

Soff, G.

Peshier, André

Pavlenko, O. P.

FORSCHUNGSZENTRUM 
ROSSENDORF e.v. FZR 
September 1995 
Preprint 
A.  Peshier, B. Kämpfer, 
O.P. Pavlenko und G. Soff 
A massive quasi-particle model 
of the SU(3) gluon plasma 
Forschungszentrum Rossendorf e.V. 
Postfach 51 01 19 . D-01314 Dresden 
BundesrepubEk Deutschland 
Telefon (0351) 591 3258 
Telefax (0351) 591 3700 
E-Mail kaempfer@fz-rossendorf.de 
A massive quasi-particle model of the SU(3) gluon plasma 
l Research Center Rossendorf Inc., PF 51 01 19, 01 314 Dresden, Germany 
Institut für Theoretische Physik, Technische Universität Dresden, 
Mommsenstr. 13, 01062 Dresden, Germany 
31nstitute for Theoretical Physics, 252143 Kiev - 143, Ukraine 
Abstract 
Recent SU(3) gauge field lattice data for the equation of state are interpreted 
by a quasi-particle model with effective thermal gluon masses. The model 
is motivated by lowest-order perturbative QCD and describes very well the 
data. The proposed quasi-particle approach can be applied to study color 
excitations in the non-perturbative regime. As an example we estimate the 
temperature dependence of the Debye screening mass and find that it declines 
sharply when approaching the confinement temperature from above, while the 
thermal mass continuously rises. 
PACS number(s): 12.38.Gc, 11.15.Ha 
I. INTRODUCTION 
The recent Progress in calculating properties of quarks and gluons by numerical meth- 
ods on space-time lattices also provides new information on the equation of state. It is 
!generally believed that at sufficiently high temperature the strongly interacting matter ap- 
pears as plasma of quarks and gluons, while at low temperatures the matter constituents 
are represented by hadrons (we consider here charge symmetric matter). Due to subtleties 
in incorporating fermions on a lattice the most accurate information is available for the pure 
+on Plasma. There are extensive studies of the SU(2) [1,2] and SU(3) [3,4] gluon plasma. 
D 
An intriguing question concerns finite-size effects and the extrapolation to the continuum 
Iimit, which has been investigated recently [2,4]. 
Even if precision data for the equation of state are available, one must ask whether they 
a1fo-w for an interpretation in terms of physical quantities. Indeed, there are various attempts 
to find suggestive interpretations of the lattice numerology. In an early attempt the SU(2) 
data [I] are described by a low-momentum cut-off model. In ref. [5] a finite gluon mass 
arid a vacuum pressure are fitted to previous SU(3) data. More accurate SU(3) data [3] are 
described in ref. [6] by a modified cut-off model with perturbative corrections and a bag 
constant, while in refs. [7,8] a thermal mass alone is found to be sufficient for describing the 
data. The latter approach has also proven to be successful for the SU(2) data 191. Despite 
the accuracy of the SU(3) data on a 163 X 4 lattice [3], by now there are data on larger 
lattices available [4]. These new data seem to permit a safe extrapolation to the continuum 
limit and are worth to be interpreted. 
The aim of our note is to present an interpretation of the recent SU(3) data [4] in terms 
of an ideal gas model of quasi-particles with thermal masses m(T) .  This model can be 
applied for studying various physical quantities (such as Debye or screening mass of heavy 
potential, transport coefficients, dilepton and Photon rates) at physically relevant, 
low temperatures near the confinement temperature Tc, where perturbative &CD can not 
be utilized directly. The particular point we adopt is that the high temperature limit of our 
thermal mass follows essentially from perturbative QCD. Such a functional dependence of 
m(T) turns out to reproduce quite well the newest SU(3) lattice data. We extend our model 
here to estimate the Debye mass in SU(3) gauge theory near T,. 
11. IDEAL QUASI-PARTICLE GAS MODEL 
Our goal is a quasi-particle model for the equation of state of a gluon plasma which is 
compatible with both the continuum extrapolation of lattice data, currently available up to 
4T„ and the perturbative region, i.e., QCD at T 4 co. We utilize the dispersion relation 
(U and k: are the quasi-particle energy and momentum). The effective Hamiltonian for such a 
t system of quasi-particle excitations may be written as [I01 Heff = ~t~ CI ~ ( k ,  T) q i  +
E&!'), with af and a as usuaI creation and destruction operators for bosons, and E. denoting 
the ground state energy; d is the degree of degeneracy of energy eigen states. With the 
distribution function f (k) = [exp{ dk2  + m2(T)} - I]-' the entropy density takes then the 
ideal gas form 
while the primary thermodynamical potential pressure p and the energy density e read 
e(T) = lrn dk f (k) k 2 J k ' + m ; i T j  + B(T): 27r2 o (4) 
These relations are thermodynamically self consistent, i.e., they fulfill e + p = sT and 
Eo T s = ap/aT. The function B(T) = l i m ~ , ~  4 (with V as volume of the system) is a 
necessary quantity when allowing for a temperature dependent quasi-particle energy w(k, T) 
[10]. B(T) is not a second independent function, but related to the thermal mass, due to 
self consistency, via 
The integration constant B. resembles somewhat the bag constant. Note that the previous 
approaches [8,9] used p(T) with B(T) 0, and that consequently neither the entropy density 
nor the energy density take the structure of an ideal gas. Eqs. (3, 4) are used in ref. 151. 
To determine the functional dependence of m(T) on the temperature let us first con- 
sider the perturbative regime. The thermodynamical properties of the gluon plasma depend 
predominantly on the transverse part of the gluon self-energy [11,12]. In the weak coupling 
regime the transversal gluon self-energy in a gluon plasma with N, colors results in a disper- 
sion relation which can be approximated [11,12] by w2 = cuk2 + ßd, with a = 1 (8) and P 
= % (1) at large (small) momenta and gauge invariant plasma frequency W: = % g 2 ~ 2  (here 
g2 denotes the perturbative QCD coupling constant). Numerically, the large momentum ap- 
proximation to the fuH transverse one-loop dispersion relation [U] holds at k/T > 2 J%; 
longitudinal excitations are there overdamped. Otherwise, the large momentum region dom- 
inates the statistical integrals in eqs. (2 - 4), e-g., more than 96.5% of the contribution to 
the energy density come from k/T 2 1. Therefore, eq. (1) represents an excellent approx- 
imation of QCD properties, relevant for evaluating eqs. (2 - 4), and m2(T) = f with 
ß = f is supported within this approximation. We have here included the factor $ nhich, 
according to Goloviznin, Satz and Shuryak [9], accounts for ~artitfoning the self-energy in 
12 lowest order between two interaction partners. Hence, nz2(T) = l?;g2(T)T2, with = N;, 
emerges approximately from perturbative QCD. 
Let us now compare the obtained pressure potential ( 3 )  at high temperature with the 
corresponding pressure obtained within first-order QCD. The high-ternperature expansion 
(i-e., m / T < 1) of eq. (3) reads 
with psB = % T ~ .  From QCD it is known [l l]  that the perturbative contribution to the 
pressure is 
Comparing the leading terms in eqs. (6,7) one reveals that, despite of massive quasi-particles, 
one needs to include only the two transverse degrees of freedom, i.e., d = 2(N: - 1). The 
next-to-leading order terms in the parenthesis confirm our above ansatz for m2(T). In this 
way the perturbative part of QCD pressure (7) is related to the quasi-particle excitations 
with energy w ( k ,  T), while the ground state contribution E0 is related to B(T) and can be 
estimated in such an expansion from eq. (5) as B(T) = - p ~ ~ $  This interpretation 
Points to the non-trivial rde  of the ground state contribution even at high temperatures. 
This particular aspect is discussed in ref. [ll] and exemplified in ref. [14]. 
Finally we specify the coupling constant in accordance with perturbative &CD as 
with Ts/Tc as phenomenological regularization as in ref. [8] and limT„ G2(T) + g2(T); 
Tc/X represents the usual regularization scale parameter A. In the following we utilize in 
eqs. (2, 3, 4) the thermal mass 
111. ANALYSIS OF LATTICE DATA 
We apply our model now to the SU(3) lattice data [4]. In fig. 1 we demonstrate that 
our model, defined by eqs. (1, 2 - 5, 8, 9), describes very well the continuum-extrapolated 
data. As fit parameters we obtain X = 1.66, Ts/Tc = -0.53. B. = 0.25T: turns out as an 
optimum choice for the present data. As Seen in fig. 1 the function B(T), which becomes 
small at T > 1-5Tc, changes its sign at 2Tc (a similar observation is made in ref. [10] for 
the older data [3]). This might be considered as a hint on a complicated non-perturbative 
vacuum structure. 
Fig. 2 displays the interaction measure (e - 3p) T-4, which is a sensitive quantity. One 
observes that for T > 1.2Tc the 323 X 6 and 323 X 8 lattice data are nicely reproduced. In 
the region Tc - 1.2TC the scaled energy density is a rapidly varying function. It might turn 
out that our quasi-particle model does not Cover perfectly the very details of forthcoming 
high-precision lattice data in this region. However, it seems that the gross features of the 
equation of state in the physically relevant region are fairly well described. This gives some 
confidence in our quasi-particle interpretation. 
As matter of fact we mention three obvious aspects of our phenomenological approach. 
(i) The flexibility, introduced by the definition of G(T) in eq. (8), allows to some extent for 
the description of the data. (ii) Otherwise, this flexibility is insufficient to describe the data 
on the basis of a different model, wherein eqs. (6) and (7) would be completely identified. 
(iii) Higher order agreement of the corresponding parts of eqs. (6,7) is not achievable due 
to a mismatch .of numerical prefactors, therefore, eq. (3) can not be considered as resumed 
expression. 
IV. SCREENING MASS 
Our model can be applied to study various collective properties of a colored quark- 
gluon system. Since even at comparatively low temperatures the gas of quasi-particles 
still remains weakly interacting, we have a chance to treat this gas in a perturbative way 
down to Tc. Here we estimate as an example the temperature dependence of the Debye 
screening mass. The Debye mass reflects the property of a plasma medium to screen the 
static chromoelectric interactions. Following the standard definition [I 1,151 the Debye mass 
mD for an electromagnetic plasma is given by the small momentum limit of the static 
longitudinal photon self-energy function IIoo(w, k), 
rn; = limIIoo(w = 0, k). 
k - 4  (10) 
It is connected with the longitudinal part of the plasma dielectric tensor eL(u, k) via k2 + 
IIoo(O, k) = k2eL(0, k) [16]. At leading order in a, the above definition is valid also for 
the &CD plasrna [17]. In our model a, is the coupling constant of the color interaction 
between quasi-particles. The chromoelectrical tensor eL(w, k) can be calculated in lowest 
order in a, within the kinetic theory of collective color excitations [18] with the corresponding 
corrections related to the non-zero effective mass m(T) of our quasi-particles. (The analogous 
approach has been employed for calculations within the cut-off model [19].) 
For the gluon plasma the chromoelectrical tensor is 
where k' = (W, k) is the wave four-vector , and f (P) denotes the above distribution function 
of quasi-particles with four momentum pp = (E, P )  and the dispersion relation (1). The 
factor y = 2 accounts for the spin degrees of freedom of the quasi-particle~ with respect to 
the asymptotic limit above and to ref. [5]. Solving eq. (11) in the limit (10) yields 
J, (T) = T-3 ICo dp p2 f (p) exp 
0 
In the limit m(T)/T -+ 0 one recovers the well known perturbative QCD limit mr> = 
L 3 Nc g2 T2. The perturbative QCD coupling constant can be expressed by 
where the quantity &Pis determined by averaging over the squared quasi-particle momenta 
[ l l ] ,  i.e., 
tVe choose the scale parameter A in accordance with the high-temperature limit of eqs. (8, 
13). Since in this lirnit 1\/i N 3.7 T, we find A/TC = 3.7 X - l  . For the temperature dependence 
of m(T), extracted above from the lattice data, the coupling constant (13) remains as small as 
0.68 at  Tc. So based on the perturbative ansatz for rn; we get in our quasi-particle model the 
non-perturbative behavior of the Debye mass displayed in fig. 3. As Seen in fig. 3 the thermal 
mass increases when approaching Tc from above, while the screening mass rng drops; above 
1.3Tc mD/T stays roughly constant. The obtained behavior of the Debye screening mass 
is confirmed by lattice calculations [20]. Such a sharp dropping of rno near T, might have 
quite interesting consequences for several deconfinement probes in ultrarelativistic heavy-ion 
collisions, e.g., the mono-jets considered in ref. [21]. At the Same time we stress that the 
direct comparison of the screening mass in our model with the lattice data requires a more 
impraved analysis of the static quak  potential due to contributions from the vacuum state 
to the free energy, where the potential is extracted from. This quantitative comparison is 
envisaged, as also an extension to SU(2) [22]. 
V. SUMMARY 
In Summary we present an interpretation of new SU(3) gluon lattice data within a model 
of an ideal gas of quasi-particles with effective thermal masses, which is motivated by per- 
turbative &CD. Such a functional dependence of the effective mass is found to reproduce 
rather perfectly the recent SU(3) lattice data of therm~d~namical  parameters. We utilize 
our model to deduce the behavior of the Debye Screening mass near the confinement tem- 
perature and find a sharply dropping Debye mass, when approaching close to Tc from above, 
while the thermal mass continuously rises. 
Acknowledgments: Discussions with J. Engels, M.I. Gorenstein, U. Heinz, B. Müller, 
K. Redlich, D.H. Rischke, H. Satz, M. Thoma, G.M. Zinovjev and the participants of the 
QCD workshop 1995 in ECT" Trento are gratefully acknowledged. F. Karsch is especially 
thanked for informing us on his lattice results prior to publication. O.P.P. is grateful for 
the warm hospitality of the Research Center Rossendorf. The work is supported in part by 
BMFT under 06 DR 666, and grant No. U4D000 by International Scientific Foundation. 
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A. Peshier, B. Kämpfer, O.P. Pavlenko, G. Soff, in Hot Hadronic Matter: Theory und 
Experiment, (Eds.) H.H. Gutbrod, J. Rafelski, NATO ASI B 346 (1995), p. 139, Plenum 
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Quark-gluon plasma 11, (Ed.) R. Hwa, World Scientific, 1993 
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L.P. Pitayevskii, Physical kinetics, Pergamon New York (1981) 
1161 V.P. Silin, Sov. Phys. J.E.T.P. 11 (1960) 1136 
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[18] H.Th. Elze, U. Heinz, Phys. Rep. 183 (1989) 81 
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FIG. 1. Comparison of our model (thin lines) with continuum-extrapolated lattice data (sym- 
bols, from [4]) of scaled energy density, pressure and entropy density. The dash-dotted curve 
depicts the function B(T)/T4.  
FIG. 2. The interaction measure as function of temperature (heavy full line: our model; sym- 
bols: lattice data [4]. 
FIG. 3. The thermal mass (dashed line) and the estimated Debye screening mass (heavy fuil 
line) in our model. 


Massive quasiparticle model of the SU(3) gluon plasma

FORSCHUNGSZENTRUM 
ROSSENDORF e.v. 
February 1995 
Preprint 
B. Kämpfer, O.P. Puvlenko, A. Peshier und G, Soff 
Parton kinetics for strangeness and 
charm production 
Forschungszentrum Rossendorf e.V. 
Postfach 51 01 19 . D-01314 Dresden 
Bundesrepublik Deutschland 
Telefon (0351) 591 3258 
Telefax (0351) 591 3700 
E-Mail kaempfer@gamma.fz-roscendorf.de 
Parton kinetics for strangeness and 
charrn production' 
"Forschungszenh-um Rossendorfe.V., PF 510119, 01314 Dresden, Germony 
*Innstitut für Theoretische Physik, Technische Uniuersität Desden,  
Monrrnsenstr. 13, 01062 Dresden, Germony 
'Institute for Theonoticnl Physics, Kieu, Ukraine 
Strangeness 2nd charm degrees of freedom are considered on the 
partonic level. First we present a model of the fuiiy equilibrated qua& 
gluon placma with thermal parton maxses and find a. considerable con- 
tribution of strangeness and also some charm. Then the kinetics of 
chemical equdibration orocesses of in an earlv hot aiue svstem is mod- 
- .  
elled, i.e., the cooking of light and strange quarlic in an expanding 
giuon gas is foiiowed. The degree of equilibrvtion depends on uncer- 
tainly known ratec, however, we find that such penetrating probes as 
hard phatons are less sensitive to details of the equilibration. Finally, 
we estimate the very early charm production in not yet thermalized 
parton matter and find indicationc for a clear dominance of the glue 
fusion mocessec. i.e.. charm orabes Wrelv the earls due  distribution. 
INTRODUCTION 
In ultrarelativistic heavy-on collisions (say at RHIC or LHC energies) the 
suhnuclear degrees of freedoni, i.e., quarks and gluons or partons, are elvpected 
to dominate the early dynamics. After loosing the coherence of partons, which 
are initially described by the nuclear structure functioiis, the very early stage 
of the collision dynamics is characterized by entropy production and secondary 
partiele production. Due to the larger gluon-gluon cross sections t~he gluons 
are estimated to exceed the quark and anti-quark nuinbers in the charge- 
symmetric midrapidity region (1). Therefore, the thermalization processes in 
the hot glne, with a few quarks and anti-quarks immersed, sho~ild proceed on 
rather short time scales. Later on also the quark componetit can thermalize. 
Hut even if thermalization is achieved to some degree, chemical 'quilibration 
processes usuaily need longer time scales. Kinetic models, such a$ tiie parton 
caccade model (2) or the HIJING code (3)! predict irideed approximate early 
thermalization but chemicaloff-equilihrium. Supposed the chemical eqiiilihra- 
tion is fast enough, the parton system evolves towards a quark-gluon plasma 
'Supported by BMFT grants 06DR107 and 06DR666 I@). 
Invited talk at "Strangeness'95", Tuscon, Arizonn, Jariuary 4 - 6, 10% 
(QGP), which hadronizes afterwards. 
In the present paper we focus on three of the mentioned Stages of parton 
matter evolution. We consider, going backwards in time, 
(i) the QGP mithin a model with thermal parton masses and analyze the 
röle of Strange and charm quarks; this stage relies on the assnmption of 
thermal and chemical equilibrium, 
(ii) the chemical equilibration processes in an initially undersaturated gluon 
gas and follow in particular the cooking of Strangeness in the hot glue; 
for such considerations tbe thermal equilibrium is supposed, 
(iii) the thermalization process and address the question whether early 
charm production can probe this stage. 
The aim of such investigations is to understand on a qualitative level relevant 
subprocesses in tlie rather involved evolution of partonic degrees of freedom. 
We also calculate penetrating probes (dileptons in items (i, iii), photons in 
item (ii)) in order to elucidate how these ones are affected by peculiarities of 
the parton evolution. 
THERMAL PARTON MASSES IN A STRANGE QGP 
Recently the observation has been made (4,5) that the pure SU(3) gauge 
theory lattice data (6,7) can rather perfectly be fitted by a non-interacting 
quasi-particle model with effective thermal gluon mass 
1 1 6 2  
m"T) = - g g " ( ~ )  T" ,'(T) = 
P# (16 - $Ni) log (F)%' (1) 
The running coupling g is phenomenologicaliy regularized by the sh'ift param- 
eter T, = 0.023TC. fg = 3.3 is astonishing near the high-temperature QCD 
prediction 3/(1 + LN ) (8,Q). Only tbe physical transversal gluon modes 
6. (dg = 16) are to be included in the primary thermodynamical potential 
The energy density and entropy are self consistently calculated from the pres- 
Sure p(T) and lmk not as the usual ideal gas formulae. Using the somewhat 
involved dispersion relation for the transversal in-medium glnon modes in one- 
loop order (8,9), instead of the free one Eq. (2), one does not find a noticeabie 
change of the thermodynamical quantities. 
Lacking precise lattice data with fermions we extend our thermal mass 
model to the QGP as folloivs: the Fermi statistics is properly accounted for; 
tlie known flavor and color degeneracies are included; the thermal masses are 
FIG. 1. The partial pressures (left upper panel, in units of T*&), the partial 
eneigy densities (right upper pmel, in units of T'&), the partial particie dencities 
(left lower panel, in nnits of the Stephan Boltzmann values), and the total energy 
density and pressure (right lawer pantl, fu!.l/d~otted lines - 213 flavar plasma with 
the same p(T,)). 
8 
4 
4 
as in high-temperatnre QCD, i.e., TC) = 6; finite quark rest masses m, give rise 
t o  expressions like rn = J v m  for the effective mass. The resulting 
equation of state is displayed in Fig. 1. One observes that the Strange quarks 
behave in this model very much as the u,d quarks. This is to be contrasted 
with the bag model parametrization, whexe Strangeness is suppressed, and 
the u,d quarks are taken as in the Stephan Boltzmann limit, and e/T4 is an 
overali decreasing function. The local maximum of our quark energy density 
resembles recent two-flavor lattice resnlts (10). 
When including also charm (bat leaving Arj = 3 in the above quoted equa- 
tions) we find at  least a 10% contribution t o  the thermodynamical qtiantit,ies 
shown in Fig. 1. Therefore, if our model equation of state is correct and the 
numerical values of the decisive values of rgSp are near the high-tcniperature 
one-loop QGD calculations also charm contribiites to a fully equilibrated QGP 
and should be included properly in precision lattice calculations. 
T i m e  are distinct observable effects of finite thermal masses. For example, 
the & scaling property of the dilepton spectrum is violated (11). :\ detailed 
analysis (5) shows the appearance of a threshold effect in the transverse-rnass 
spectrum at transwerse dilepton mornentum qi = \/M: t m2(T) - i G L .  In 
this region the dileptoii production is strongly suppressed (5). Also the cooling 
beliavior of tiie QGP is changed; tliis st.ems from the fact that the latent 
heat in tlle present model is much smaller than in the bag modei. Aiiotlier 
consequence is the higher temperature a t  given ent,ropy density. This i ~ i  turu 
affects estimates of rates at  given particle rapidity densitiea. 
.................. 
... 
30 
....... 
..... 
,/' 20 ' 
6 
- -u,d . -.__........_........------Y 
S 
- - 
........... L.. ......... 
.............................. 
8 
4 
CHEMICAL EQUILIBRATION PROCESSES IN AN EXPANDING 
QGP 
Qualitative estimates and parton cascade simulations indicate substantial 
quark undersaturation (1-3,12,13) at early times in the Course of an ultra- 
relativistic heavy-iou collision; in the Same moment the gluon distribution 
looks isotropicalfy aud can be considered as thennalized. To follow the chem- 
ical equilibration processes in the hot glue oue can formulate rate eqnations 
(14). Bere we utilize rate equations similar to those nsed in Refs. (12,15). 
The change of densities of different particle species oheys the master type 
equations 
where 
is the comovmg derivative of particle density which includes boost invariant 
longitudinal and transversal expansion (7 is the transverse Lorentz factor, 
cf. Ref. (15); ' = d l d r ,  and r is the proper time). The thermally velocity- 
averaged cross sections ü account for inelastic glue production, glue flows 
into u,d channel, glue flows into s channel, n,d production from glue, u,d 
flow into s channel, glue flovvs into s channel, n,d transmutates into s and 
C? - 
the corresponding hack reactions. The remaining quantities b G 3, b G 
*' i iZ $, c G are so defined that in eqnilibrium the reactions cease. A tilde 
indicates the corresponding satnration density, e.g., Ag = na/fiy is the gluon 
fugacity. These coupled three equations are snpplemented by one equation for 
t:itc:rg coriscriari«ri. nnd tivo cquatioiis for transvcrse expaiisioii (ior dcrails CF 
Ilef. (1.5)). \Vc so1i.r rhis systerri i r i i h  rlie folloiririgsiiii~~lifications: neglecting 
the q@ - sS channel (14), and using ugg„, = (this neglects the 
mais differente of u,d and s quarks, see previous section, as also done in 
fi„ energy density, and pressure). These simplifications seem to he harmless 
compared to uncertainties in the thermally averaged cross sections and higher 
order processes (16). We use here the cross sections of Ref. (12) in one set 
of calculations (set I), and in a second Set we multipty these cross sections 
by a factor 10 (Set 11). The initial conditions are also matter of debate (cf. 
refs. (1,12,13,15,16)for widely differing estimates). 
In Fig. 2 two representative examples are displayed. In hoth ones the u,d 
quarks are initially undersaturated, while the gluons are either undersaturated 
too (12,13,16) or rveu oversaturated (17). We assume that there is initially no 
r [fni/ej T [fm/c] 
FIG. 2. Tune evolution of the temperature and the various fugacities for tivo 
different initid conditions (heavyjthin curves - cet 1/11 calculations). The initid 
trsnsverse expansioii velocity is zero, and the transverse radius ic 5 fm. Transverse 
expitnsion is dealt with by exploiting the global relativistic hydradynamics Ref. (15). 
strangeness. One ohserves that in the set I calculation the System remains far 
from chemical equilibrium. If gluons are initially rare the giue production is 
not efficient enough to overcompensate the gluon loss in the gg - q f ,  sS reac- 
tions. The Strange channel, which is not included in the previous calculations 
(12,15), mainly reduces the gluon fugacity, while for the temperature and light 
quark fugacities it is not so important. Our set I1 calculation displays fast 
approach to equilihrium even in the Strangeness channel. 
We mention that we do not include the g - q f ,  s8 reaction channel. Wheri 
utilizing the thermal masses of the previous sections, tliis channel is closed. 
Giving the particles, however, a width then via the tails of tlie distributions 
such reactions can proceed, c t  Ref. (18). Our thermal masses have some sup- 
port from the lattice calculations; the estimates of tlie pdrticle decay widths 
rely completely on the extrapolation from the high-temperature evaluat,ions. 
While the drastically different flavor evolution scenarios I and I1 demon- 
strate the need to constrain the reaction Cross sections, as also thc iiiitial 
conditions, one may investigate whether penetrating prohes are sensitive to 
such differences. In Ref. (15) we have calculated the photori spectra for I;: $ 
1 GeV. We found that the slopes are rather insensitive to the different flavor 
evolutions. The decisive quantities, ivliich nearly cornpletely deterinine the 
photon spectra, are the poorly knovrn initial temperature arid Fugacities. 
One might also include the charm evolution in the above master equat,ions. 
Recent estimates (13), however, indicate that the pre-equilibriurn contribution 
is rnuch larger than tlie thermal rate. 
The proper inctusion of in-medium effects in the kinetic treatrnerit of fliwnr 
evolution needs obviously more investigations too. 
6 
PARTON KINETICS FOR CHARM PRODUCTION 
Hard probes come from early Stages of the parton evolution (19,20). There- 
fore, one has to look at  heavy particles to learn about early parton distribution 
evolution. Here we are going to present an analytical model for the early off- 
equilibrium parton kinetics and estimate the eharm production. Previoiis 
estimates (13,21) rely on parametrizations of HIJING resuits. \?Je adopt the 
Boltzmann equation in relaxation time approximationfor the one-dimensio~ial 
boost invariant evolution of the parton distrihution function f 
where T and 5 are proper time and rapidity, and T„, denotes the relaxation 
time. The fiducial distribution faq is the usual Jüttner function needed for lin- 
earizing the collision operator. The four-velocity flow pattern U, is prescribed 
by the symmetry of the System. p-s the parton four momentum. The re- 
maining temperature parameter in feq is deterrnined by an integral equation 
(cf. Ref. (19) for technical details) and depends on the initial distribution 
fo = f(m). The exact solution 
might be used, e.g., for the charmed pair production rate 
dNec 
-d42 - J~MZ-- 'P 'P' f(p1) f (pa) u(M2) 6(lpi + pzI2 - M') (9) ( 2 ~ ) s  ( 2 ~ ) 3  
with standard cross sections (14) U = u g s - ~ ~  + unr-s. 
As an aside we mention that such an approach has been exercised for dilep- 
tons with U = uq„+,-. Also in this case there a e  three terms in the rate 
and we found (22) that, uuless the relaxation time is estremely short, the term 
X f$ don~inates, i.e., hard prohes measure indeed the initial distributions. 
An analog calculation for charm is not yet performed. Instead we utilize a 
simplified 2-component model, where the gluons are thermalized, fg = fiq, 
rvhile the quarks are not therrnalized, fq = fi. In case of f = feq one recovers 
the thermal rate (cf. Refs. p2,23) for further explanations) 
where the " X" means g. To compare this thermal rate with a non-thermal one, 
we use the evolution equation Eq. (8) hut neglect tlie fe1 component (see the 
3 7 0 
hl [GeY 
FIG. 3. The charm pair spectrum (in units oi r (u ,Rnor~)~;  TO 7 hc/p;;  R is the 
transverse area (22)) as iunction of invariant mass X. 
above remark on dileptons). Tbe quarks obey initially a mini-jet produced 
parton distribution 
~O(PL,Y,TO) = NQ(PL -v~ )~ (Y)P ; ' ,  1 %  7 P; = 2GeV. (12) 
The corresponding rate Eq. (9) for quarks ("X" stands for q or #) 
can be simplified somewhat, cf. Ref. (22). To compare the charm yields from 
thermalized gluons and not yet thermalized quarks we normalize the corre- 
sponding distribution functions feP und fo on tbe same energy densities and 
particle densities, i.e., eeq = ea and n„ = no. This relates the parameters 
in both distributions, in particular it causes a ratber high initial temperature 
To of the hot glue (22). Interestingly, the spectra for gg i cC and q p  - cc 
look very similar, see fig. 3. However, due to tbe low quark fngacities (here 
we assume X q  - $Ag in line usith Refs. (1,12,13,16)) tbe rates behave as 
where X = 1 - (whicli meaus in full chemicai equilibrinm I„ = 0.4). 
"s+~+o  
The present fugacities imply a strong qnark nndersaturatiori, i.e., X - 0.8, 
&ich results in a suppressioii factor & for light quark-created Charm. We 
mention that rougbly the Same results are obtained if qnarks would be ther- 
malized too, or if the gluons also obey the mini-jet initial distribution). That 
means, charm measures the glnon initial distribution. 
While the model needs iniprovemeutsfor haudling thermalizatiou and chem- 
ical equilibration together, it might be sufficient for qualitative estimates aud 
for comparison with initial glnon fusion (13,21). 
8 
SUMMARY 
Our results may be  summarized as follorvs: 
(i) If the thermal mass model is correct then s quarks behave nearly as 
u,d quarks; there is no need of a bag constant; the latent heat is much 
smaller than in the bag model. 
(ii) The cbemical Bavor evolution depends sensitively on the cross sections 
for gluon multiplication gg - ggg, quarkization gg - qrj arid gg - si, 
aud quark transmutation qrj - ss; cither Strangeness remains, as u,d 
quarks, undersaturated, or s gets similar to u,d saturated. Penetratiug 
probes, e.g., photons do not depend on these differences. 
(iii) If quarks are really strongly suppressed in the early Stages then charm 
probes the initial (pre-equilibrium) gluon distribution fy (from other 
studies ones knows that photons probe f, f„ and dileptons fp). 
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