Linking Dynamical and Thermal Models of Ultrarelativistic Nuclear Scattering

4 Pages·0.14 MB·English

by  Klaus WERNER

Linking Dynamical and Thermal Models of Ultrarelativistic Nuclear Scattering
xz
Klaus WERNER
Institutfu(cid:127)r TheoretischePhysik,Universita(cid:127)tHeidelberg,Germany
Jo(cid:127)rg AICHELIN
Laboratoirede PhysiqueSubatomiqueet desTechnologiesAssocies,
Universit(cid:19)ede Nantes{ EMN { IN2P3/CNRS,Nantes,France
To analyse ultrarelativistic nuclear interactions, usually either dynamical models like the string model are employed, or a
thermal treatment based on hadrons or quarks is applied. String models encounter problems due to high string densities,
thermal approaches are too simplistic considering only average distributions, ignoring (cid:13)uctuations. We propose a completely
newapproach, providing alink between thetwotreatments, andavoiding theirmain shortcomings: based onthestring model,
connected regions of high energy density are identi(cid:12)ed for single events, such regions referred to as quark matter droplets.
Each individual droplet hadronizes instantaneously according to theavailable n-body phase space. Due tothe huge number of
possible hadron con(cid:12)gurations, special Monte Carlo techniques have been developed tocalculate this disintegration.
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Preprint HD{TVP{94{23
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Studyingnuclear collisionsat ultrarelativistic energies duce particle densities that high that the hadrons are

n
(Ecms=nucleon(cid:29)1 GeV)is motivatedmainlybythe ex- overlapping. Sotheindependentstringmodeliscertainly
a
pectationthatathermalizedsystemofquarksandgluons too simplistic, and also considering secondary interac-
J
(quark{gluonplasma)iscreated[1]. Thereareessentially tionsasbinarycollisionsamonghadronscantheoretically
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twodirectionsformodellingsuchinteractions: dynamical not be justi(cid:12)ed. On the other hand it is also not real-
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and thermalapproaches. The formerones refer to string istic to consider a homogeneous plasma occupying the

models [2{7] or related methods [8], supplemented by whole available volume, what is assumed in thermody-
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semihardinteractions at very highenergies [9{12]. Here, namic models. To illustrate this, we show in (cid:12)g. 1 a
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awellestablishedtreatmentofhadron-hadronscattering, typical event of a string model simulation. We consider
0
1 based on Pomerons and AGK rules [13], is extended to a central S+S collision at 200 GeV (Ecms=A(cid:25)10 GeV),
0 nuclear interactions. Thermal methods [14{19] amount thetransversecoordinatesbeingxandy,thelongitudinal
5 to assuming thermalization after some initial time (cid:28)0, one (= beam direction) being z; it is useful to consider
9 with evolutionand hadronizationbeing mostlybased on space-timerapidity(cid:16) =0:5ln(t+z)=(t(cid:0)z) ratherthanz.
/
h ideal gas assumptions. In the (cid:12)gure, the hatched regions represent high energy
t density ("(cid:21)"c = 1 GeV) inthe x(cid:0)(cid:16) plane (y =0). We
-
l (cid:12)nd a couple of intermediate size regions of high energy
c
density, representing rest masses of few GeV up to few
u
n tens of GeV. This demonstrates that neither the inde-
pendent string modelis correct, since these high density
regions cannot possibly be treated as independent had-
rons nor the thermal approaches, since we do not have
onebighighdensityobjectbutratheracoupleofmedium
sizeobjectsinadditiontoplentyofordinaryhadronsand
resonances in particular in the periphery.
Inthispaperweintroduceacompletelynewapproach,
more realistic than the string model and more realistic
than thermal approaches, providing a link between the
two. Based on the string model, we (cid:12)rst determine con-
nected regions ofhighenergy density ("(cid:21)"c,fora given
"c). These regions are referred to as quark matter(QM)
droplets. Forsuchregions,theinitiallyproducedhadrons
serve only as a mean to produce the proper (cid:13)uctuations
FIG. 1. High density regions in the x-(cid:16) plane for a intheenergydensity. Presently,apurelylongitudinalex-
typical event. pansionofthe QMdroplets isassumed. Once the energy
density falls beyond "c, the droplet D decays instantan-
Both methods have serious theoretical drawbacks. eouslyintoan n-hadroncon(cid:12)gurationK =fh1h2:::hng
Even for nuclei as light as sulfur the string models pro- withaprobabilityproportionalto(cid:10),with(cid:10)representing
1
themicrocanonicalpartitionfunctionofann-hadronsys- Here,Cdegaccountsfordegeneracies(giisthedegeneracy
tem. Due to the huge con(cid:12)guration space, sophisticated of particle i), and Cident accounts for the occurence of
methodsofstatisticalphysics[21,22]havetobeemployed identical particles in K (n(cid:11) is the number of particles of
to solve the problemwithout further approximations. species (cid:11)). The last factor
Our hadronizationscenario is referred to as \microca-
Z n
Y
nonical hadron gas (MHG) scenario". It is certainly not 3
(cid:30)= d pi(cid:14)(E(cid:0)(cid:6)"i)(cid:14)((cid:6)~pi)(cid:14)Q;(cid:6)qi (4)
the only one and probably not the most realistic one.
i=1
However, westart withthisscenario for acouple ofreas-
p
ons: thehadrongasscenarioisabenchmark,widelyused istheso-calledphasespaceintegral,with"i = m2i +p2i
in the literature (in a simpli(cid:12)ed fashion though); the being the energy and p~i the 3-momentum of particle i.
MHG scenario can be solved exactly; for massless had- Thevectorqi =(qiu;qid;:::)representsthe(cid:13)avourcontent
rons even an analytical treatment is possible, providing ofhadroni. Theexpressioneq. (4)isvalidforthecentre-
very useful tests for the complicated numerical proced- of-massframe of the droplet D.
ures. After having gained experience in the techniques We have to de(cid:12)ne a set S of hadron spe-
tosolvetheMHGscenario, weplantoinvestigatealtern- cies; we include the pseudoscalar and vector mesons
atives as well. So the purpose of this paper is not so ((cid:25);K;(cid:17);(cid:17)0;(cid:26);K(cid:3);!;(cid:30)) and the lowest spin-1 and spin-
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much to promote this particular scenario, but rather to 3 (cid:3) (cid:3)
baryons (N;(cid:3);(cid:6);(cid:4);(cid:1);(cid:6) ;(cid:4) ;(cid:10)) and the correspond-
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showhowadynamicalandastatisticaltreatmentcan be
ingantibaryons. A con(cid:12)gurationis then anarbitrary set
combined.
fh1;:::;hng with hi 2S.
The (cid:12)rst stage of our approach is the identi(cid:12)cation of
We are interested in droplet masses from few GeV up
high energy density regions, based on the string model, 3
to 10 GeV, corresponding to particle numbers n = jKj
which is already discussed elsewhere [20]. Due to the 3
between 2 and 10 . So we have to deal with a huge
empirically found correlation, y(cid:22) = (cid:16); between the av-
con(cid:12)guration space, which requires to employ Monte
erage rapidity y(cid:22) and space-time rapidity (cid:16), a hypersur-
Carlo techniques, well known in statistical physics. The
face H(cid:28) of constant proper time(cid:28) maybe introduced, in method at hand is to construct a Markov process, spe-
2 2 2
the central region simply de(cid:12)ned by t (cid:0)z = (cid:28) . Ap-
ci(cid:12)ed by an initial con(cid:12)guration K0, and a transition
propriate coordinates on H(cid:28) are the space-time rapidity probability matrix p(Kt ! Kt+1). In generating a se-
(cid:16) = 0:5ln(t+z)=(t(cid:0)z) and the transverse coordinates
quence K0;K1;K2;:::, two fundamental issues have to
x and y. After having used the string model (VENUS
be payed attention at:
5.08) to get complete informationof hadron trajectories
in space and time, we may now, for given (cid:28), determine (cid:15) initialtransient: startingusuallyo(cid:11)equilibrium,it
energy densities on H(cid:28) and thus locate high density re- takesanumberofiterations,Ieq,beforeonereaches
gions on H(cid:28) (with " > "c), as shown in (cid:12)gure 1 for a equilibrium;
typical example.
(cid:15) autocorrelationinequilibrium: eveninequilibrium,
High density regions are considered as QM droplets,
subsequent con(cid:12)gurations, Ka and Ka+i, are cor-
presently it is assumed that they expand purely longit-
related for some range Iauto of i.
udinally. Whenever other clusters or hadrons cross their
way, the two objects fuse to form a new, more energetic In general, both Ieq and Iauto should be as smallas pos-
cluster. Due to the expansion, the energy density of a sible.
cluster willatsomestagedropbelow"c,whichcauses an Weare goingtoproceed asfollows: foragivendroplet
instantaneous decay. D with mass E, volumeV, and (cid:13)avour Q, we start from
We employ the \microcanonical hadron gas (MHG) some initial con(cid:12)guration K0, and generate a sequence
scenario" for the hadronization: the probability of a K0;K1;:::;KIeq,withIeq beingsu(cid:14)cientlylargetohave
droplet D | charcterized by the invariant mass E, the reached equilibrium. If we repeat this procedure many
volumeV, and the (cid:13)avour Q=(Qu;Qd;:::) |to decay (1) (2)
times,gettingcon(cid:12)gurationsKIeq;KIeq;:::;thesecon(cid:12)g-
intoahadroncon(cid:12)gurationK =fh1;:::;hngofhadrons
urationsaredistributedas(cid:10)(K). Soforourproblem,we
hi is given as
have only to deal with the initialtransient, not with the
prob(D !K)(cid:24)(cid:10)(K) ; (1) autocorrelation in equilibrium. We have to (cid:12)nd a trans-
ition probability p such that it leads to an equilibrium
with (cid:10)(K) being the microcanonical partition function distribution(cid:10)(K), withthe initialtransient Ieq being as
of an ideal, relativistic gas of the n hadrons hi, smallas possible.
Su(cid:14)cient for the appropriate convergence to (cid:10)(K) is
(cid:10)(K)=CvolCdegCident(cid:30) ; (2) the detailed balance condition,
with
(cid:10)(Ka)p(Ka !Kb)=(cid:10)(Kb)p(Kb !Ka) ; (5)
Vn Yn Y 1
Cvol = (2(cid:25))3n ; Cdeg =i=1gi ; Cident =(cid:11)2S n(cid:11)! : (3) amnudsteregxoisdticsiotmy,ewrhiwchithmtehaensprtohbaatbfiolirtyatnoygKeta;inKbr tshteepres
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from Ka to Kb being nonzero. Henceforth, we use the (cid:12)xed lattice size. We therefore introduce \zeros", i.e. we
abbreviations supplement the sequences fh1;:::;hng by adding L(cid:0)n
zeros, as fh1;:::;hn;0;:::;0g, to obtain sequences of
(cid:10)a :=(cid:10)(Ka); pab :=p(Ka !Kb): (6) (cid:12)xed length L. The zeros may be inserted at any place,
not necessarily at the end. Therefore the weight of a
FollowingMetropolis [21],we make the ansatz microcon(cid:12)guration Kaj with zeros relative to the one
without, K~aj, is one divided by the number of possib-
pab =wabuab ; (7)
ilities to insert L(cid:0)n zeros, so fromeq. (11) we get
with a so-called proposal matrix w and an acceptance ( )
matrixu. Detailed balance now reads 1 Y n!(L(cid:0)n)!
(cid:10)(Kaj)= n(cid:11)! (cid:10)(Ka) : (12)
n! L!
uab (cid:10)b wba (cid:11)2S
= ; (8)
uba (cid:10)awab We now have the analogywith a spin system: we have a
one-dimensional lattice of (cid:12)xed size L, with each lattice
which is obviouslyful(cid:12)lled for
site containing either a hadron or a zero. Henceforth,
(cid:18) (cid:19)
(cid:10)b wba we use for microcon(cid:12)gurations with zeros the notation
uab =F (cid:10)awab ; (9) Kaj =fh1;:::;hLg with hi being a hadron or zero.
Since from now on we only consider microcon(cid:12)gura-
with some function F ful(cid:12)lling F(z)=F(z(cid:0)1) = z: Fol- tions with zeros (Kaj) rather than con(cid:12)gurations (Ka),
lowingMetropolis [21],we take wearegoingtowriteKa insteadofKaj,keepinginmind
thatarepresents adoubleindex,andsay\con(cid:12)guration"
F(z)=min(z;1) : (10) rather than \microcon(cid:12)guration with zeros". The ad-
vantageis that we can use the aboveformulasspecifying
Thepowerofthemethodisduetothefactthatanarbit- the Metropolis algorithmwithout changes.
rary w maybe chosen, in connection with u being given We are now in a position to de(cid:12)ne a symmetric pro-
a a
byeq. (9). Soour taskistwofold: we havetodevelopan posal matrixw(Ka !Kb), with Ka =fh1;:::;hLg and
e(cid:14)cient algorithm to calculate, for given K, the weight Kb =fhb1;:::;hbLg, as
(cid:10)(K), andwe have to(cid:12)nd an appropriateproposal mat- 8 9
(cid:12)rirxstwtawskhicchanleabdesstoolvfeads,tacodnevtearigleedncdee(sscmripatllioIneqw).illTbhee 2 >< YL >= a a b b
published soon [23]. In the following we discuss about w(Ka !Kb)= L(L(cid:0)1)>: k=1 (cid:14)hakhbk>;v(hihj !hihj) ;
k6=i;j
constructing an appropriate matrixw.
Mostnatural,thoughnotnecessary,istoconsidersym- (13)
metricproposalmatrices,wab =wba;whichsimpli(cid:12)esthe
acceptance matrix to uab = F((cid:10)b=(cid:10)a): This is usually with
referred to as Metropolis algorithm. Whereas for spin (cid:26)
a a (cid:0)1 b b a a
systems, it is obvious how to de(cid:12)ne a symmetric matrix v(haihaj !hbihbj)= jP(hihj)j if hihj 2P(hihj) ;
w, this is not so clear in our case. We may take spin 0 else
systems as guidance. A con(cid:12)guration K is per def. a set
(14)
of hadrons fh1;:::;hng with the ordering not being rel-
evant,so f(cid:25)0;(cid:25)0;pgisthe sameas fp;(cid:25)0;(cid:25)0g. We intro- whereP(haihaj)isthesetofallpairs(hihj)withthesame
duce \microcon(cid:12)gurations"to be sequences fh1;:::;hng total (cid:13)avour as the pair (haihaj). The symbol jPj refers
of hadrons, where the ordering does matter. So for a to the number of pairs of P. The term fg in eq. (13)
givencon(cid:12)gurationKa =fh1;:::;hngthere existseveral makes sure that up to one pair all hadrons in Ka and
microcon(cid:12)gurations K~aj = fh(cid:25)j(1);:::;h(cid:25)j(n)g, with (cid:25)j Kb are the same, the term 2=L(L (cid:0) 1) is the probab-
representing a permutation. The weight of a microcon- ility to randomly choose some pair of lattice indices i
(cid:12)guration is and j. So our proposal matrix amounts to randomly
( ) choosing a pair in Ka, and replacing this pair by some
1 Y pairwiththesame(cid:13)avour,withallpossiblereplacements
(cid:10)(K~aj)= n(cid:11)! (cid:10)(Ka) ; (11)
n! having the same weight. The proposal matrix is obvi-
(cid:11)2S
ously symmetric,since v is symmetric (the symmetry of
v is crucial!). We have now fully de(cid:12)ned an algorithm,
with n(cid:11) being the number of hadrons of type (cid:11). Taking
0 0 which due to general theorems will converge, but how
for example K = fp;(cid:25) ;(cid:25) g, there are three microcon-
(cid:12)gurations fp;(cid:25)0;(cid:25)0g; f(cid:25)0;p;(cid:25)0g and f(cid:25)0;(cid:25)0;pg, with fast, i.e., how large is Ieq? Considering particle ratios,
weight (cid:10)(K)=3. like n(cid:25)0=n(cid:25)+, we (cid:12)nd immediately that we have a very
So far we deal with sequences fh1;:::;hng of arbit- slow convergence, so Ieq is too large for the method to
be of practical importance. This is obvious, since the
rary length n, to be compared with spin systems with
3
scribe ultrarelativistic nuclear collisions, by linking dy-
namicaland thermal approaches, and to show that such
an approach is technically feasible.
x
Heisenberg fellow
z
Internet: [email protected]
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[2] K. Werner, Physics Reports 232 (1993) 87{299
[3] A. Capella, U. Sukhatme, Chung-I Tan and J. Tran
FIG.2. Multiplicity versus the number ofiterations. Thanh Van, Physics Reports 236 (1994) 225
[4] A.Kaidalov, Nucl. Phys. A525 (1991)39c
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0 0
(cid:25) ;(cid:26) or also zeros are much more frequently proposed [6] V. D. Toneev, A.S. Amelin and K.K. Gudima preprint
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(cid:12)ning w such that two pairs are exchanged rather than [7] B. Andersen, G. Gustafson and B. Nielsson-Almqvist,
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rary pair, the second one by some pair to guarantee (cid:13)a- [8] H.Sorge, H. Sto(cid:127)cker and W. Greiner, Nucl. Phys. A498,
vour conservation. In addition it is necessary to weight 567c (1989)
the \zeros" di(cid:11)erently than the hadrons. This improved [9] F. E. Paige, Lecture at \Theoretical Advanced Summer
method violates the symmetryof w, however, the asym- Institute", Boulder, CO,USA, 1989
metrywab=wbacanbecalculatedandproperlytakeninto [10] T.Sjo(cid:127)strand, M. van Zijl, Phys. Rev. D36 (1987) 2019
[11] X. N. Wang and M. Gyulassi, LBL 31036 (1991), LBL
account. Further details of the \asymmetric algorithm"
31159 (1991)
willbe published elsewhere [23].
[12] K. Geiger and B. Mu(cid:127)ller, Nucl. Phys. B369 (1992) 600
The asymmetric method converges quite fast. As a
[13] V. A. Abramovski(cid:21)i, V. N. Gribov, O. V. Kancheli, Sov.
check, we consider massless hadrons, where analytical
J.Nucl. Phys. 18 (1974) 308
results exist. In (cid:12)g. 2 we plot the iterated total mul-
[14] P.Koch,B.Mu(cid:127)ller, andJ.Rafelski, Physics Reports142
tiplicity N for a droplet with E = 10 GeV and V = 10
fm3, compared to the average multiplicity N(cid:22) from the (1986) 167
[15] U. Heinz, Kang. S. Lee, E. Schnedermann, in \Quark
analytical calculation (dashed line). The \equilibration
Gluon Plasma",ed.R.Hwa,WorldScienti(cid:12)c, 1990,page
time"Ieq is roughly given as 471
[16] J.Zima(cid:19)nyi,P.L(cid:19)evai,B.Luc(cid:19)acs,andA.Ra(cid:19)cz,in\Particle
Ieq=N(cid:22) (cid:25)10 ; (15)
ProductioninHighlyExcitedMatter",ed.H.H.Gutbrod
and J.Rafelski, Plenum Press, 1993, page 243
with the initialcon(cid:12)guration being two (cid:25)0's.
[17] K.Redlich, J.Cleymans, H.Satz,andE.Suhonen, in [1]
A major advantage of our method is the fact that it
[18] H.W. Barz, B.L. Friman, J. Knoll, and H. Schulz, Nucl.
is parameterfree. The (cid:12)naldistribution of particles from
Phys. A484 (1988) 661
the decay of a QM droplet is given by the phase space
[19] L.P. Cernai, J.I. Kapusta, G. Kluge, E.E. Zabrodin, Z.
only. The necessary technical parameters Ieq and L are
Phys. C58 (1993) 453
presently systematicallystudied. Foralltypes ofclusters
[20] Klaus Werner, Phys. Rev. Lett.73 (1994) 1594
(di(cid:11)erent E;V; (cid:13)avour) they have to be chosen as small
[21] N.Metropolis,A.W.Rosenbluth,M.N.Rosenbluth, A.H.
as possible but large enough to not a(cid:11)ect the results.
Teller, and E. Teller, J. Chem.Phys. 21 (1953) 1087
Preliminary calculations have shown that the method is [22] Zhang X.Z., D.D.E. Gross, Xu S.H., and Zheng Y.M.,
su(cid:14)ciently fast (1 min per S+S event on a DEC Alpha) Nucl. Phys. A461 (1987) 668
tobeofusefortheinvestigationofultrarelativisticheavy [23] Klaus Werner and Jo(cid:127)rg Aichelin, in progress
ion collisions. Another systematic study aims at a com-
parisonofourmethodwithacanonicalhadrongas,which
may be useful to optimize L and the starting con(cid:12)gura-
tion.
Beingparameterfree,andwiththehadrongasscenario
representing a benchmark, it will be very interesting to
simulate nuclear collisions and compare with data. The
scope of our method,however, is muchlarger. Our main
intention is to introduce a completely new way to de-
4