Thermalization temperature in Pb+Pb collisions at SpS energy from hadron yields and midrapidity pt distributions of hadrons and direct photons

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by  D. Yu. Peressounko

Thermalization temperature in Pb+Pb collisions at SpS energy from hadron yields
and midrapidity p distributions of hadrons and direct photons
t
∗
D. Yu. Peressounko and Yu. E. Pokrovsky
Russian Research Center ”Kurchatov Institute”, Kurchatov sq. 1, 123182 Moscow, Russia.
SpS(hadronyields,midrapidityp distributionsofdirect
t
In the frame of 2+1 Bjorken hydrodynamics we describe
photons, unflavored and strange hadrons), fix uniquely
simultaneouslyhadronyieldsandmidrapidityptdistributions all model parameters from these data and obtain reli-
of direct photons, unflavored, and strange hadrons measured
able estimate of the thermalization temperature with its
inPb+PbcollisionsatSpS.Wefind,thatareasonablyquanti-
uncertainty concerned with experimental data bars.
tativedescriptionofthesedataispossibleonlyifweintroduce
In this way we find, that if we use a realistic equa-
some radial velocity at the beginning of the one-fluid hydro-
1 dynamicstage. Wefixuniquelyall parametersofthismodel, tion of state (EoS) of hadronic gas incorporating con-
0 and estimate an initial temperature of the just thermalized tributions of all known hadrons and resonances, then to
0 state Tin=200+−4200MeV in thecase of EoS with phasetran- describe midrapidity pt distributions of final hadrons it
2 sitiontoQuark-GluonPlasma,andTin =230+−6300MeV inthe is necessary to assume an essentially nonzero initial ra-
n pure hadronic scenario. We conclude that SpS data consid- dial collective velocity at the beginning of the one-fluid
a ered in our study can not distinguish between a production hydrodynamic stage. This velocity is created during the
J of QGP, and its absence. pre-equilibrium (two-fluid) stage of the collision due to

2
significantradialenergydensitygradientcomingfromthe
A finite volume of rapidly expanding Quark-Gluon
shapes of the colliding nuclei. Nevertheless, in all previ-
3 Plasma (QGP) can be possibly created for a very short
v ous estimations of the thermalization temperature this
time in ultrarelativistic heavyion collisions as an almost
5 initial radial velocity was absolutely neglected, and only
locally equilibrated thermodynamic state of deconfined
2 the simplest case (zero initial velocity) was considered.
0 quarksandgluonswith strongresidualinteraction. Tak-
Evaluating the direct photon yield we assume the fol-
9 ing into account a very short time available for QGP
lowingscenarioofthecollision. Initially,onthepre-equi-
0 formationinthis processanecessaryconditionofitscre-
0 ationsisthataninitial(thermalization)temperatureT librium stage, colliding nuclei penetrate through each
0 in other like two fluids with a friction. On this stage the
have to be visibly higher than the quark-hadron phase
/ ‘prompt’photonsareemittedandradialcollectiveveloc-
h transition temperature, which is estimated from lattice
p ityiscreated. Ifthethermalizationtimeisnotnegligibly
calculations to be T =150 170MeV [1]. Therefore an
- c − small then the both these effects should be taken into
p estimation of a temperature at which the local thermal
account. The next stage is a one-fluid-like expansion of
e equilibrium is just established – thermalization temper-
h the locally thermalized matter with an emission of the
ature – is of great importance.
: ‘thermal’ photons and its decay onto final hadrons.
v There are a lot of attempts to extract this tempera-
Tobeginwith wecalculatecontributionofthe prompt
i ture from direct photon momentum distributions mea-
X photons. We take into account Compton, annihilation
sured in S+Au [2] and Pb+Pb [3] collisions. However,
r andbremsstrahlungprocesses. WeuseGRV-94structure
a to extract the thermalization temperature from experi-
functions [7], two-loop expression for α and K-factor,
mental data reasonably well, it is necessary to separate s
K=2, accounting higher order corrections. Comparing
accuratelythe collectiveandchaoticmotions. Todothis
calculated yield of prompt photons in pp and pA colli-
one needs to know the evolution of the collision start-
sionswithexperimentaldataat√s=19GeV–theclos-
ing fromthe firstinteractionstill the finalparticlestage.
est available c.m. energy to Pb+Pb at SpS (√s = 17.3
Unfortunately, the most part of the estimations of the
GeV) we find, that both experimental data and theoret-
thermalization temperature [4,5] are based on hydrody-
ical predictions have approximately the same slope, but
namic description with too simplified initial conditions
differenceinnormalizationreachesfactor 7. Nowadays
(zero initial radial collective velocity), and other model ≈
thisdifferenceisattributedtotheintrinsictransversemo-
parameters fixed partially by some experimental data,
mentumofcollidingpartons[8]. Inthisletterweaccount
and partially – ad hoc from more or less reasonablecon-
these corrections by introducing a proper factor, assum-
siderations. Only in the recent paper [6] a first attempt
ingthatitisindependenton√swhilemovingfrompAto
toestimatetheinfluenceofnonzeroinitialradialvelocity
Pb+Pb energy. To go from pA to AA collisions we need
was undertaken.
thenumberofnucleon-nucleoncollisionsoccurredduring
In contrast to this, in the present letter we use initial
the interpenetrationofthe collidingnuclei. In ourcalcu-
conditions for hydrodynamic model in the most general
lations we use Glauber approximation,what gives in the
form, describe simultaneously a large amount of avail-
case of Pb+Pb collision N 3 A. Comparison with
ableexperimentaldata,measuredinPb+Pbcollisionsat Gl ≈ ·
1
WA98 data [3] shows, that prompt photons contribute in hadronic gas, the chemical freeze-out takes place be-
approximately 30% of total direct photon yield. fore the thermal one. Following paper [13] we use the
Toevaluateyieldofthermalphotonsweintegrateaver- following values of the temperature of chemical freeze-
ageemissionrateofdirectphotonsfromunitvolumeper out, and baryon, strange and I3 chemical potentials –
unittimeoveraspace-timeevolutionofhotmattercalcu- T = 168MeV, µ = 266MeV, µ = 71MeV and
ch b s
lated in a one-fluid hydrodynamics. In the present anal- µ3 = 5MeV. As a result we well describe yields of
−
ysis we use recently calculated emission rates account- π, K, η, p, Λ, Ξ and Ω hadrons measured by NA44 [14],
ing next to leading effects in QGP [9] and dominating NA49 [15], WA97 [16] and WA98 [17] collaborations in
contribution in hadronic gas – reaction πρ a1 πγ Pb+Pb collisions at SpS energy.
→ →
[10]. Fordescriptionoftheevolutionweuse2+1Bjorken Adescriptionofourresultswestartfrompurehadronic
hydrodynamics, which works sufficiently well at midra- scenario. Despitethebeliefthathadronicgascannotex-
pidity. It has several parameters which should be fixed ist at temperatures higher than 200 MeV (because of
∼
to define the evolution: a) initial conditions: initial tem- the essential overlapping of hadrons in it) we adopt its
perature T , initial time τ , initial energy density and existence and consider the consequences of this assump-
in in
velocity distributions; b) EoS of hot matter: transition tionfromthepointofviewofdescriptionofdirectphoton
temperatureT (if EoSincludes phasetransition),chem- production. To begin with, we assume zero initial radial
c
ical freeze-out temperature T and thermal freeze-out velocity and try to describe momentum distributions of
ch
temperature T . final hadrons. We choose some initial temperature and
f
For the shape of energy density distribution we use calculate initial time to describe multiplicity of final π0.
Woods-Saxon distribution, integrated along beam axis. After that we perform the hydrodynamic calculations
As for initial distribution of radial velocity, it deserve with various values of chemical and thermal freeze-out
separate discussion. As we find, to describe momentum temperatures and obtain χ2 of the fit of the calculated
π
distributionsoffinalhadronsusingrealisticEoSitisnec- to the experimental π0 distributions as a function of T
ch
essary to introduce some radial velocity already at the andT . We find, that it is impossible to describe midra-
f
beginning of the one-fluid hydrodynamics. This velocity pidity p distributions of π0 within any reasonable set
t
is created during interpenetration of colliding nuclei due of model parameters: because of the much steeper slope
to significant radial energy density gradient, originated of the calculated p distribution of π0 with respect to
t
from spherical shapes of colliding nuclei. Two-fluid hy- the experimental one, we obtain χ2 102/point. We
π ∼
drodynamics or kinetic models might predict value and findthe same situationforallinitial temperaturesin the
radial dependence of this collective velocity but, in this range 160MeV <T <300MeV. A reasonable way to
in
letter we use ‘phemenologic’ approach and use a simple improve this situation is to introduce some initial radial
lineardistributionv(r)=(r/R) Θ(R r) Vmax charac- velocity to the one-fluid hydrodynamic stage.
· − ·
terized by one new parameter – Vmax – the velocity on
the surfaceofthejustthermalizedsystem. We find,that
our results are not sensitive to reasonable variations of
the shape of this distribution.
Concerning equation of state we consider two differ-
ent cases: EoS with phase transition to QGP, and pure
hadronic EoS. In the case of phase transition for QGP
phase we use EoS of ideal gas of massless quarks and
gluons with degeneracy g = 41.5, what corresponds to
QGPconsistingof2.5masslessquarkflavorsandgluons.
To calculate EoS of hadronic phase both in the case of
phasetransitionandpure hadronicscenariowetake into
account contributions of all hadrons listed in the Parti-
cle Data Book [11]. Inclusion of heavy hadrons leads to
significant reduction of speed of sound in the hadronic
phase: v2 0.15 in our case should be compared with
v2 0.3s3≈for massless hadronic gas or v2 0.25 for
s ≈ s ≈
pure pion gas. The decreasing of speed of sound results
in decreasingofaccelerationinhadronic gasandstepper
slopes of p distributions of final hadrons [12].
t
To model freeze-out of hadronic gas we use two differ- FIG. 1. Levels of constant χ2 of the fit of the ratio of
π
ent parameters: chemical (Tch) and thermal (Tf) freeze- calculated tomeasured pt distributionsofπ0 asafunctionof
out temperatures. It is known, that due to large dif- maximal initial velocity (Vmax) and thermal freeze-out tem-
ference in cross-sectionsof elastic and inelastic reactions peratures (Tf) for purehadronic gas EoS, Tin =230 MeV.
2
On the next step, we repeat calculations with nonzero We find, an agreement with experimental data at ini-
+60
initial radial velocity. Again we choose initial tempera- tial temperature T = 230 MeV: T = 230MeV
in −30 in
ture,fixinitialtime fromthemultiplicity considerations, corresponds to χ2 = 3.7/point, χ2 = 0.67/point,
π γ
fix temperature of chemical freeze-out Tch = 168MeV while curves, calculated for Tin = 200MeV (χ2π =
to describe hadron yields and plot levels of constant χ2 4.9/point, χ2 = 0.67/point), and T = 300MeV (χ2 =
π γ in π
χin2π(tThef,TVfm−axV)mhaaxscaooshrdaipneatoefsa–vsaeellefiyg.w1i.thTshteeefpunwctailolsn, 3er.5ro/rpso.inTth,eχ2γre=aso2n.3o/fpotihnist)waeraekstsielnlswitiitvhiitnyetxoptehreimineinttiaall
while any point on its bottom (thick line), corresponds temperature is following. As far as we introduce the ini-
to a good description of midrapidity pt distributions of tial radial velocity, all time stages contribute with more
π0 with χ2 1/point. To fix a point on this line we use or less the same effective slopes. But, the difference be-
π ∼
spectra of heavier hadrons (K, p, Λ, Ξ). It was shown tween initial time for T =300 MeV – τ =0.12 fm/c,
in in
[18]thatspectraofthe heavierhadronscanbe described andforT =230MeV–τ =0.6fm/cisnotlarge,and
in in
wellifweassumefreeze-outtemperatureTf =120MeV. thereforecontributionofthis stageisnotveryimportant
The description of the hadron pt distributions at with respect to the subsequent evolution.
midrapidity obtained in this way is shown on the fig. 2.
One can see, that shapes of all available experimental
spectra: π0 [17], K and Λ [19], protons [20], and Ξ [21]
are described very well, although we overestimate total
yields of p and Λ in this rough model.
FIG. 3. Yield of direct photons in Pb+Pb collisions at
SpSenergywithpurehadronicgasEoSfordifferentTin: 200
(solid line), 230 (dashed line) and 300 MeV (dotted line).
Dots – WA98 data.
We confirm the result found in [5] – ‘reach’ hadron
FIG. 2. Comparison of experimental pt distributions
gas EoS allows to describe the experimental p distribu-
of several hadrons with hydrodynamic predictions for pure t
hadronic gas EoS with Tin = 230MeV and initial radial ve- tion of direct photons [3] within pure hadronic scenario.
locity Vmax =0.26. Our best fit value of Tin =230−+3600MeV is smaller, than
the initial temperature estimated in [5] T 260MeV.
in
≈
To fix finally all model parameters we estimate T There are two reasons of this discrepancy: first is, the
in
using p distribution of direct photons. We depict on initial radialvelocity which we included to describe final
t
the fig. 3 the sum of the p distribution of prompt and hadron spectra. In contrast to our approach, authors of
t
thermalphotons,calculatedatthreeinitialtemperatures. [5]madenotattempttodescribehadronspectra,anddid
For each initial temperature we repeat calculations, de- notconsideranonzeroinitialradialvelocity. Thesecond
scribedaboveandobtainthefollowingsetsofparameters: reason is different normalization to final hadron multi-
plicity. The authors of [5] used the well known Bjorken
T , MeV τ , fm/c Vmax, c Curve on fig. 3
in in formula, relating initial time and temperature with final
200 1.4 0.36 solid
multiplicity(formula(5)in[5]). But,thisformulaisvalid
230 0.6 0.29 dashed
onlyfor masslessfinalparticlesandbeing appliedto real
300 0.12 0.23 dotted
hadronic gas leads to an underestimation of the initial
3
time and, as a consequence,ofthe thermalphotonyield. of hadrons and direct photons, measured in Pb+Pb col-
NowletusconsiderEoSincorporatingfirstorderphase lisionatSpSenergy. Wefind,thatcalculationswithzero
transition. In this case there is an additional parameter initial velocity, and EoS of hadronic gas accounting con-
T whichvalueisestimatedfromlatticeQCDsimulations tributionofallknownhadronsresultintoosoftp distri-
c t
to be 150 170 MeV [1]. On the other hand T T . butionsoffinalhadrons,andareasonablewaytodescribe
c ch
− ≥
Therefore we use T = T = 168 MeV to consider an experimental slopes is to introduce nonzero radialveloc-
c ch
extreme case with the most intensive QGP production. ityatthebeginningoftheone-fluidhydrodynamicstage.
As in the case of pure hadronic gas, first we try to de- Thisinitialradialvelocity,whichmightbeproduceddur-
scribeexperimentalp distributionofπ0 withzeroinitial ingthepre-equilibriumstagebecauseofthestrongradial
t
radial velocity, and find almost the same situation: for energy density gradientoriginatedfromspherical shapes
anyreasonablesetofmodelparametersthe calculatedp of colliding nuclei, is not small ( 0.3 c near the sur-
t
distributionofπ0 goesmuchsteeper,andresultinunrea- face). Therefore the pre-equilibriu≈m stage is expected to
sonably large χ2 102/point. So, we find again, that it be long enough, and, in particular the ‘prompt photons’
π ∼
isnecessarytointroduceinitialradialvelocitytodescribe have to be taken into account.
momentumdistributions offinalhadrons. Repeating the An introduction of the initial radial velocity to this
same procedure as for pure hadronic gas EoS we obtain model allows to describe simultaneously direct photon
the following sets of model parameters: and hadron p distributions for the both EoS includ-
t
ing phase transition and without it. We estimate an
Tin, MeV τin, fm/c Vmax, c Curve on fig. 4 initial temperature of the just thermalized state Tin =
180 2.0 0.40 solid 200+40MeV in the case of EoSwith phase transitionto
−20
200 1.5 0.38 dashed +60
QGP, and T = 230 MeV in the pure hadronic sce-
in −30
230 1.0 0.29 dotted
nario. Because the lower bounds of the estimated ther-
250 0.7 0.26 dash-dotted
malization temperature are very close to the expected
transition temperature we conclude that the SpS data
We find the best agreementwith experimentaldata at
T = 200+40 MeV (χ2 = 3.3/point, χ2 = 0.8/point), considered in this letter can not distinguish between a
in −20 π γ productionof QGP,and its absence: The initial temper-
what means very short QGP phase and much more pro-
atureincentralPb+PbcollisionsatSpSistoosmall,and
longed mixed phase.
experimentalerrorsforthedirectphotonp distributions
t
are too large.
This work was supported by the INTAS under Con-
tract INTAS-97-0158,and by grant RFBR 00-15-96590.
∗
Presentaddress: SUBATECH,EcoledeMinesdeNantes,
4, rueAlfred Kastler, BP 20722-44307, Nantes,France.
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