Dynamical equilibration of strongly interacting "infinite" parton matter within the parton-hadron-string dynamics transport approach

0.78 MB·English

by  V. Ozvenchuk

Dynamical equilibration of strongly interacting “infinite” parton matter within the
Parton-Hadron-String Dynamics (PHSD) transport approach
V. Ozvenchuk∗
Frankfurt Institute for Advanced Studies, 60438 Frankfurt am Main, Germany
O. Linnyk
Institut fu¨r Theoretische Physik, Universita¨t Giessen, 35392 Giessen, Germany
M. I. Gorenstein
Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine, and
Frankfurt Institute for Advanced Studies, 60438 Frankfurt am Main, Germany
3
E. L. Bratkovskaya
1
Institut fu¨r Theoretische Physik, Johann Wolfgang Goethe-Universita¨t, 60438 Frankfurt am Main, Germany, and
0
Frankfurt Institute for Advanced Studies, 60438 Frankfurt am Main, Germany
2

n
W. Cassing
a
Institut fu¨r Theoretische Physik, Universita¨t Giessen, 35392 Giessen, Germany
J
(Dated: January 29, 2013)
8
2 We study the kinetic and chemical equilibration in “infinite” parton matter within the parton-
hadron-string dynamics off-shell transport approach, which is based on a dynamical quasiparticle

] model (DQPM) for partons matched to reproduce lattice QCD results—including the partonic
h
equation of state—in thermodynamic equilibrium. The “infinite” parton matter is simulated by
t
- a system of quarks and gluons within a cubic box with periodic boundary conditions, at various
l
c energy densities, initialized out of kinetic and chemical equilibrium. We investigate the approach
u of the system to equilibrium and the time scales for the equilibration of different observables. We,
n furthermore, study particle distributions in the strongly interacting quark-gluon plasma (sQGP)
[ including partonic spectral functions, momentum distributions, abundances of the different parton

species, and their fluctuations (scaled variance, skewness, and kurtosis) in equilibrium. We also
2
compare the results of the microscopic calculations with the ansatz of the DQPM. It is found that
v
the results of the transport calculations are in equilibrium well matched by the DQPM for quarks
4
and antiquarks, while the gluon spectral function shows a slightly different shape due to the mass
3
dependenceofthegluonwidthgeneratedbytheexplicitinteractionsofpartons. Thetimescalesfor
7
the relaxation of fluctuation observables are found to be shorter than those for the average values.
4
. Furthermore, in the local subsystem, a strong change of the fluctuation observables with the size
3 ofthelocal volumeisobserved. These fluctuationsnolongercorrespond tothoseof thefullsystem
0
andarereducedtoPoissonian distributionswhenthevolumeofthelocalsubsystembecomessmall.
2
1
:
v I. INTRODUCTION It is thus tempting to characterize the experimental ob-
i servables by global thermodynamical quantities such as
X
“temperature,” chemicalpotentials or entropy [1–8]. We
ar Nucleus-nucleus collisions at ultrarelativistic energies notethattheuseofmacroscopicmodelssuchashydrody-
are studied experimentally and theoretically to obtain namics[9–12]employsasabasicassumptiontheconcept
informationaboutthepropertiesofhadronsathighden- oflocalthermalandchemicalequilibriuminthe infinite-
sityand/ortemperatureaswellasaboutthephasetran- volume limit, althoughby introducing different chemical
sition to a new state of matter, the quark-gluon plasma potentials one may treat chemical off-equilibrium also
(QGP). Whereas the early “big bang” of the universe in hydrodynamics. The crucial question, however, of
most likely evolved through steps of kinetic and chem- howandonwhattimescalesthermodynamicequilibrium
ical equilibrium, the laboratory “tiny bangs” proceed can be achieved is presently a matter of debate. Thus
through phase-space configurations that initially are far nonequilibriumapproacheshavebeenusedinthepastto
fromanequilibriumphaseandthenevolvebyfastexpan- addresstheproblemoftimescalesassociatedwithglobal
sion. Ontheotherhand,manyobservablesfromstrongly or local equilibration [13–20]. Another question is the
interacting systems are dominated by many-body phase influence of finite-size effects on fluctuation observables
space such that spectra andabundances look “thermal.” and the possibility of relating experimental observations
in relativistic heavy-ion collisions to the theoretical pre-
dictionsobtainedinthethermodynamiclimit. Therefore,
a thorough microscopic study of the questions of ther-
∗Electronicaddress: ozvenchuk@fias.uni-frankfurt.de malization and equilibration of confined and deconfined
2
matter within a nonequilibrium transport approach, in- correct description of dilepton decays of ρ mesons with
corporating both hadronic and partonic degrees of free- masses close to the two-pion decay threshold.
dom and the dynamic phase transition, appears timely.
The paper is organized as follows. In Sec. II we pro-
vide a brief reminder of the off-shell dynamics and the A. Off-shell transport
ingredients of the transport approach. We then present
in Sec. III the actual results on the chemical equilibra- Let us recall the off-shell transport equations (see [29]
tion of the partonic matter in parton-hadron-string dy- fordetails). Onestartswithafirst-ordergradientexpan-
namics (PHSD). In Sec. IV we investigatethe properties sionoftheWigner-transformedKadanoff-Baymequation
of the partonic matter in chemical and kinetic equilib- andarrivesatthegeneralizedtransportequation[23,27]
rium and compare the particle properties in equilibrium
with the dynamical quasiparticle model (DQPM), which 2pµ∂x iG¯>< ReΣ¯R,iG¯>< iΣ¯><,ReG¯R
µ −{ }−{ }
has been developed to describe the thermodynamics of
{M¯, iG¯><}
lattice QCD. In Sec. V we study (within the dynamical
approach)the parton properties at finite quark chemical | {z =iΣ}¯<iG¯> iΣ¯>iG¯< , (1)
−
potential µ , while in Sec. VI higher moments of parton
q
where the curly brackets denote the relativistic general-
distributions and the equilibrationof fluctuation observ-
ization of the Poisson bracket
ablesaswellasthesizeoffluctuations inequilibriumare
investigated. WethenshowinSec.VIIthetimescalesfor F¯,G¯ =∂pF¯(p,x)∂µG¯(p,x) ∂µF¯(p,x)∂pG¯(p,x) .
therelaxationoffluctuationobservablesincomparisonto { } µ x − x µ
thetime scalesfortheequilibrationofthe averagevalues One additionally obtains a generalized mass-shell equa-
of the observables. Finally, a summary and conclusions tion
are given in Sec. VIII.
[p2 m2 ReΣ¯R]iG¯><
− −
M¯
II. THE PARTON-HADRON-STRING
DYNAMICS TRANSPORT APPROACH | =i Σ¯{>z< ReG¯R}+ 14{iΣ¯>,iG¯<}− 14{iΣ¯<,iG¯>} (2)
with the mass function M¯ = p2 m2 ReΣ¯R. In Eqs.
In this work we study the kinetic and chemical equili- (1) and (2) the Green’s functions−G¯><−stand for the ex-
bration in “infinite” parton matter within the Parton-
pectation values of the quantum fields (denoted here by
Hadron-String Dynamics (PHSD) transport approach
Φ),
[21, 22], which is based on generalized transport equa-
tions on the basis of the off-shell Kadanoff-Baym equa- i G¯<(x,y)=η Φ†(y)Φ(x) ,
tions [23, 24] for Green’s functions in phase-space rep- h i (3)
resentation (in the first order gradient expansion, be- i G¯>(x,y)= Φ(x)Φ†(y) ,
h i
yond the quasiparticle approximation). The approach
withη =1forbosonsandη = 1forfermions,whilethe
consistently describes the full evolution of a relativistic −
self-energiesΣ(x,y)aregivenbythefunctionalderivative
heavy-ion collision from the initial hard scatterings and
of with respect to the full propagator G¯:
string formation through the dynamical deconfinement F
phase transition to the strongly interacting quark-gluon δ
plasma (sQGP) as well as hadronization and the subse- Σ=2iδFG¯ . (4)
quent interactions in the expanding hadronic phase. In
the hadronic sector PHSD is equivalent to the Hadron- In (4) the functional is the sum of all closed two-
F
String-Dynamics(HSD)transportapproach[25,26]that particle-irreducible(2PI)diagramsbuilt up by full prop-
has been used for the description of pA and AA colli- agators G¯.
sionsfromGSI heavyionsynchrotron(SIS) to Relativis- The retarded and advanced Green’s functions G¯R and
tic Heavy Ion Collider (RHIC) energies in the past. G¯A are given as
In particular, PHSD incorporates off-shell dynamics
for partons and hadrons. In the off-shell transport G¯R(x1,x2)=Θ(t1 t2) G¯>(x1,x2) G¯<(x1,x2) , (5)
− −
description, the hadron and parton spectral functions G¯A(x1,x2)= Θ(t2 t1(cid:2)) G¯>(x1,x2) G¯<(x1,x2(cid:3)) .(6)
change dynamically during the propagation through the − − −
medium and—in case ofhadrons—evolvetowardthe on- These Green’s functions co(cid:2)ntain exclusively spectral(cid:3)but
shellspectralfunctioninvacuumifthesystemexpandsin no statisticalinformation of the system. Their time evo-
the course of the heavy-ion collisions. As demonstrated lution is determined by Dyson-Schwinger equations (cf.
in[27,28]theoff-shelldynamicsisimportantforhadronic Ref. [27]). Retarded and advanced self-energies Σ¯R and
resonances with a rather long lifetime in vacuum but Σ¯A are defined in analogy to Eqs. (5) and (6).
strongly decreasing lifetime in the nuclear medium (es- In the transport equation (1) one recognizes on the
pecially ω and φ mesons) and also proves vital for the left-hand side the drift term pµ∂x iG¯><, as well as the
µ
3
Vlasovtermwiththerealpartoftheretardedself-energy algebraicsolutionsandcontainno derivativeterms,they
ReΣ¯R. Onthe other handthe right-handside represents are valid up to the first order in the gradients.
the collision term with its typical “gain and loss” struc- Inaddition,subtractionoftherealpartsandaddingup
ture. Thus interaction between the degrees of freedom the imaginaryparts leadto the time evolutionequations
is thus incorporated into the mean fields and collisions
as in the Vlasov-Boltzmann “standard” transport ap- pµ∂x A¯= 1 ReΣ¯R,A¯ + 1 Γ¯,ReG¯R , (16)
µ 2 2
pisroaancahd[d3i0t]i.onIanlctoernmtrast,iΣi¯n>t<h,eRoeffG¯-Rshe,llwthriacnhspisodrtentohteerde pµ∂xµReG¯R =(cid:8)12 ReΣ¯R,(cid:9)ReG¯R (cid:8)− 18 Γ¯,(cid:9)A¯ . (17)
−{ }
as the back-flow term and is responsible for the proper (cid:8) (cid:9) (cid:8) (cid:9)
When inserting (14) and (15) we find that these first-
off-shell propagation. It vanishes in the on-shell quasi-
particle limit. Note, however, that the self-energies Σ¯ ordertime-evolutionequationsaresolvedbythealgebraic
expressions. Accordingly, the time evolution of the sys-
fully determine the dynamics of the Green’s functions
tem is fully defined by ReΣ¯R and the width Γ¯ in (1).
for given initial conditions.
We recall that the off-shell transport equation (1) can
We, further on, represent Eqs. (1) and (2) in terms of
be solved explicitly by employing a generalized test-
realquantities by the decomposition of the retardedand
particle ansatz for the real quantity iG¯<(x,p). For the
advanced Green’s functions and self-energies as
explicit equations of motion for these test particles we
G¯R/A =ReG¯R iImG¯R =ReG¯R iA¯/2 , (7) refer the reader to Ref. [29].
± ∓
A¯= 2ImG¯R/A, (8)
∓
B. Explicit equations for fermions
Σ¯R/A =ReΣ¯R iImΣ¯R =ReΣ¯R iΓ¯/2 , (9)
± ∓
Γ¯ = 2ImΣ¯R/A . (10) In case of fermions—such as baryons or quarks—
∓ the self-energy ReΣ¯R is separated into different Lorentz
We note that in Wigner space the real parts of the re- structures of scalar and vector type:
tarded and advanced Green’s functions and self-energies
are equal, while the imaginary parts have opposite sign ReΣ¯R/mh =UhS(x,p)+γµUhµ(x,p) (18)
and are proportional to the spectral function A¯ and to
the width Γ¯, respectively. for each fermion species h. The mass function for
fermions is then
With the redefinitions (7)–(10) one obtains two alge-
braic relations for the spectral function A¯ and the real
partofthe retardedGreen’sfunction, ReG¯R, in termsof Mh(p,x)=Π20−Π2−m∗h2 , (19)
the widthΓ¯ andthe realpartofthe retardedself-energy,
with the effective mass and four-momentum given by
ReΣ¯R, as [27, 29]
m∗(x,p)=m +US(x,p) , (20)
p20−p2−m2−ReΣ¯R ReG¯R =1 + 14 Γ¯A¯ , (11) Πµh(x,p)=pµh Uµh(x,p) , (21)
(cid:2) p20−p2−m2−Re(cid:3)Σ¯R A¯=Γ¯ ReG¯R. (12) where m stands for the bare−(vahcuum) mass. After in-
h
Notethat(cid:2)alltermswithfirst-ord(cid:3)ergradientshavedisap- serting (19) into the generalized transport equation (1),
peared in (11) and (12). A first consequence of (12) is a thecovariantoff-shelltransporttheoryemerges. Itisfor-
direct relation between the real and the imaginary parts mally written as a coupled set of transportequations for
of the retarded, advanced Green’s function, which reads the phase-space distributions N (x,p) [x = (t,r), p =
h
(for Γ¯ =0) (ω,p)]offermionhwithaspectralfunctionA (x,p)[us-
h
6 ing iG¯<(x,p)=N (x,p)A (x,p)], i.e.,
p2 p2 m2 ReΣ¯R h h h
ReG¯R = 0− − − A¯ . (13)
Γ¯ Πµ−Πν∂µpUhν −m∗h∂µpUhS ∂xµNh(x,p)Ah(x,p)
Inserting(13)in(11)weendupwiththefollowingresult (cid:0) + Πν∂µxUhν +m∗h∂µxUhS(cid:1) ∂pµNh(x,p)Ah(x,p)
forthespectralfunctionandtherealpartoftheretarded (cid:0)iΣ¯<,ReG¯R (cid:1)
Green’s function: −{ }
=(2π)4 tr tr tr [T†T] δ4(Π+Π Π Π )
A¯= p2 p2 m2 Γ¯ReΣ¯R 2+Γ¯2/4 , (14) h2Xh3h4 2 3 4 12→34 2− 3− 4
0− − − × Ah(x,p)Ah2(x,p2)Ah3(x,p3)Ah4(x,p4)
ReG¯R =(cid:2) p20−(cid:2)pp202−−pm2−2−mR2e−Σ¯(cid:3)RRe2Σ¯+R(cid:3)Γ¯2/4 . (15) ×−N(cid:2)Nhh(x3(,xp,)pN3h)2N(xh4,(px2,)pf¯4h)3f(¯xh(,xp,3p)f)¯fh¯h4(2x(x,p,p4)2) (22)
The spectral fun(cid:2)ction (14) shows a typ(cid:3)ical Breit-Wigner with (cid:3)
shape with energy- and momentum-dependent self-
energy terms. Although the above equations are purely f¯(x,p)=1 N (x,p)
h h
−
4
and which are taken into account in PHSD by means of the
generalized off-shell transport equations (cf. Sec. II.A)
d4p
tr = n . thatgo beyondthe mean-fieldor Boltzmannapproxima-
n (2π)4 tions.
Z
In the scope of the DQPM the running coupling con-
Here ∂x (∂ , ) and ∂p (∂ , ) (µ = 0,1,2,3).
µ ≡ t ∇r µ ≡ ω ∇p stant (squared) for T >Tc is approximated by
Thefactor T†T standsforthein-mediumtransitionma-
| |
trix element (squared) for the binary reaction 1+2 48π2
→ g2(T/T )= , (24)
3+4,whichhastobeknownalsooffthe massshell. The c (11N 2N )ln[λ2(T/T T /T )2]
c f c s c
back-flow term in (22) is given by − −
where the parameters λ = 2.42 and T /T = 0.56 have
iΣ¯<,ReG¯R s c
−{ } been extracted from a fit to the lattice QCD equation
∂µ Mh(x,p) ∂x[N (x,p)Γ (x,p)] of state as described in Refs. [36, 40]. In (24), Nc = 3
≈ p(cid:18)Mh(x,p)2+Γh(x,p)2/4(cid:19) µ h h standsforthenumberofcolors,Tc isthecriticaltemper-
ature(=158MeV), while N (=3)denotes the number of
M (x,p) f
−∂µx M (x,p)2h+Γ (x,p)2/4 ∂pµ[Nh(x,p)Γh(x,p)]. flavors.
(cid:18) h h (cid:19) Intheasymptotichigh-momentum(high-temperature)
(23) regime, the functional form of the parton quasiparticle
mass is chosen to coincide with that of the perturbative
As pointed out before this expression stands for the off-
thermal mass, i.e., for gluons
shell evolution, which vanishes in the on-shell limit or
when the spectral function A (x,p) does not change its
h g2 1 N µ2
shape during the propagation through the medium, i.e., M2(T)= N + N T2+ c q , (25)
g 6 c 2 f 2 π2
for rΓ(x,p)=0 and pΓ(x,p)=0. We recall that the (cid:18) (cid:19) q !
∇ ∇ X
transportequation(22)hasbeenthebasisfortheoff-shell
HSD transport approach for the baryon and antibaryon and for quarks (antiquarks)
dynamics.
In order to specify the dynamics of partons one has N2 1 µ2
M2 (T)= c − g2 T2+ q , (26)
to evaluate/specify the related self-energies for quarks q(q¯) 8N π2
c !
and antiquarks as well as gluons that enter the spectral
functions (14) andretardedGreen’s functions (15). This butwiththecouplinggivenin(24). Theeffectivequarks,
taskhas beencarriedoutwithina dynamicalquasiparti- antiquarks,and gluons in the DQPM have finite widths,
cle model. which for µ =0 are approximated by
q
1 g2T 2c
C. The dynamical quasiparticle model Γg(T) = 3Nc 8π ln g2 +1 , (27)
(cid:18) (cid:19)
1N2 1g2T 2c
The basis of the partonic phase description is the dy- Γ (T) = c − ln +1 , (28)
q(q¯) 3 2N 8π g2
namical quasiparticle model [31, 32], which has been c (cid:18) (cid:19)
matched to reproduce lattice QCD results (lQCD)—
where c=14.4 (from Refs. [34]) is related to a magnetic
including the partonic equation of state—in thermody-
cutoff. Note that for µ =0 the DQPM gives
namic equilibrium. The DQPM allows for a simple q
and transparent interpretation of thermodynamic quan- 2 4
titiesaswellascorrelators—measuredonthelattice—by Mq(q¯) = 3Mg, Γq(q¯) = 9Γg . (29)
meansofeffectivestronglyinteractingpartonicquasipar-
ticles with broadspectral functions. The essential quan- Fromthe expressions(24)–(29), one cansee that athigh
titiesintheDQPMare“resummed”single-particleprop- temperature, T , the masses and the interaction
agators G¯ , G¯ , and G¯ . We stress that a nonvanishing strengthofthequ→asip∞articlesintheDQPMareapproach-
q q¯ g
widthΓ¯ inthepartonicspectralfunctionisthe maindif- ingtheone-loopperturbativeQCDresults. However,the
ferencebetweenthe DQPMandconventionalquasiparti- one-loop functional form is not the relevant description
cle models [33]. Its influence onthe collisiondynamics is attemperaturesclosetoT orevenbelow. Thetransition
c
essentially seen in the correlation functions; e.g., in the region(approximately0.9T <T <1.1T )isdominated
c c
stationarylimit,thecorrelationinvolvingtheoff-diagonal by nonperturbative phenomena. Therefore, we imple-
elements of the energy-momentum tensor Tkl define the ment in PHSD for the transitional values of T/T (the
c
shearviscosityηofthemedium[34]. Hereasizablewidth region0.9T <T <1.1T ) functional forms for M and
c c q,g
ismandatorytoobtainasmallratiooftheshearviscosity Γ , which are growing softer with decreasing T/T as
q,g c
toentropydensity,η/s[35],whichresultsinaroughlyhy- compared to the perturbative logarithmic divergence in
drodynamical evolution of the partonic system in PHSD (24). TheactualvaluesofM andΓ havebeenshown
q,g q,g
[22]. The finite width leads to two-particle correlations, as functions of temperature as well as the scalar parton
5
density in Ref. [40]. A comparison to the lQCD interac- partial widths of the microscopic scattering and decay
tionmeasure[39]hasbeenpresentedalsoinRef.[40](cf. channels have to be known, while the DQPM provides
Figs. 1, 2, and 4). only the total widths of the dynamical quasiparticles
We note in passing that the smooth parametrizations that have been fixed by lattice QCD calculations as de-
for M and Γ at T close to T nicely reproduce the scribed in Sec. II.C and in more detail in Refs. [31, 32].
q,g q,g c
recent lQCD calculations from the Wuppertal-Budapest Furthermore, the explicit shape of the partonic spectral
group [39]. As a consequence, we obtain not only a functions—takenasLorentziansintheDQPM(30)—will
quantitativelygooddescriptionofthephasetransitionre- dependonthe decompositionofthe interactionintopar-
gionbut alsoa smooth“interpolation”fromthe hadron- ticular channels within the coupled-channel dynamics of
dominated systems to those with dominant partonic de- PHSD.
grees of freedom. Inordertofixthepartialcrosssectionsfortheinterac-
With the partonmassesandwidths fixedby (24)–(29) tionsbetweenthedynamicalquarksandgluons(asfunc-
thespectralfunctions canbewritten[inalternativeform tionsofenergydensity ε)weperformPHSDcalculations
to (14)] as inacubicfiniteboxwithperiodicboundaryconditions—
simulating “infinite” hadronic or partonic matter. In
Γ 1 1 this particular case the derivatives of the retarded self-
A¯ = ρ (ω,p)= j
j j Ej (ω−Ej)2+Γ2j − (ω+Ej)2+Γ2j! ewneeregssieesntwiaitllhyrdeesaplewctitthotshpeapcearvtoannisdhyninam(2ic2s)dsuuechtoththaet
4ωΓj collision terms in (22).
= , (30)
(ω2 p2 M2)2+4Γ2ω2 The following (quasi)elastic interactions among
− − j j
quarks, antiquarks, and gluons (q,q¯,g) are implemented
separately for quarks, antiquarks, and gluons (j = in PHSD:
q,q¯,g), with the notation E2(p2) = p2+M2 Γ2. We
j j − j
may identify (cf. Sec. II.A) q(m1)+q(m2) q(m3)+q(m4), (33)
→
q+q¯ q+q¯, (34)
ReΣ¯R =M2, Γ¯ =2ωΓ . (31) →
j j j j q¯+q¯ q¯+q¯, (35)
→
Thespectralfunction(30)isantisymmetricinωandnor- g+q g+q, (36)
→
malized as g+q¯ g+q¯, (37)
→
∞ ∞ g+g g+g. (38)
dω dω →
ωρ (ω,p)= 2ωρ (ω,p)=1 . (32)
j j
2π 2π The (quasi)elastic processes (33)–(38) play a crucialrole
Z Z
−∞ 0 for the thermalization in PHSD due to the possibility to
changethemassesofinteractingpartonsinthefinalstate
The parameters Γ and M from the DQPM have been
j j as shown in Eq. (33).
defined above in the Eqs. (25)–(29). Note that the
The flavor exchange of partons is possible only within
DQPM assumes Γ = const(ω); we will discuss the con-
j the inelastic interactions in PHSD, which are:
sequences of this approximation in Sec. IV. Also, the
decomposition of the total width Γ into the collisional
j g q+q¯, (39)
width (due to elastic andinelastic collisions)andthe de- ↔
g g+g, (40)
cay width is not addressed in the DQPM. Therefore, we
↔
dedicate the next section to this question and to a brief g+g q+q¯. (41)
↔
description of the microscopicimplementation of the de-
confined phase of QCD within the PHSD. The inelastic interactions (39)–(41) are the basic pro-
cesses for the chemical equilibration in PHSD; however,
the inelastic processes [(40 and (41)] are strongly sup-
D. Reaction rates and effective cross sections pressed (<1%) kinematically in PHSD due to the large
masses of gluons.
Werecallthatforbinarychannelswehaveexplicitfor-
In this section we present the effective cross sections
mulasforthepartialwidths,e.g. [fromthecollisionterm
for each of the various partonic channels as a function
in (22)],
of energy density ε; these cross sections determine the
partial widths of the dynamical quasiparticles as well as
Γelastic(p )= tr tr tr T†T 2
the various interactionrates. This analysisis important, 1 2 3 4| |1+2→3+4
because, although the DQPM provides the basis of the 2X,3,4
description of the strongly interacting quark-gluon sys- ×Ah2(p2)Ah3(p3)Ah4(p4)Nh2(p2)f¯h3(p3)f¯h4(p4)
tem in PHSD in equilibrium, the dynamical transport (2π)4δ4(P +P P P ) , (42)
1 2 3 4
approach (i.e., PHSD) goes beyond the DQPM in sim- × − −
ulating hadronic and partonic systems also out of equi- where h is an index, which can be equal to “q ”, “q¯” or
i i i
librium. For the microscopic transport calculations, the “g ,” where i = 1,2,3. Since we study partons at high
i
6

temperaturethefermionblockingtermscanbeneglected,
i.e., approximated by f¯=1, and one ends up with 0.6
Γelastic(p )= tr T†T 2
1 2 | |1+2→3+4 PHSD
2X,3,4 fit
A (p )N (p )R (p +p ;M ,M ) , (43)
× h2 2 h2 2 2 1 2 3 4
wherethefour-momentaofparticle4arefixedbyenergy-
q
momentumconservationandR standsforthetwo-body q0.5
2
phase-spaceintegral(cf. [41]). Werecallthatthesquared
matrix element times the two-body phase-space integral
defines abinarycrosssectionσ times akinematic factor,
i.e.,
T†T 2 R (p +p ) =4E E v σ, (44)
| |1+2→3+4 2 3 4 1 2 rel 0.4
X3,4 0 2 4 6 8 10
3
with the relativistic relative velocity for initial invariant [GeV/fm ]
energy squared, s, given by FIG.1: (Color online) Theenergy density dependenceofthe
coefficient αqq extracted from the PHSD simulations in the
v = (s M2 M2)2 4M2M2/(2E E ) . (45) box (bluedots) and corresponding fit (red line).
rel − 1 − 2 − 1 2 1 2
q
In (44) stands for a summation over discrete final
3,4 while the factor 2/4 corresponds to the ratio of final to
channels.
P initialspins(assumingtwotransversedegreesoffreedom
If the cross section σ is essentially independent of the
forthegluoninlinewiththeDQPM).Notethatformula
momenta, which should hold for low-energy binary scat-
(48) providesan off-shell cross section which depends on
tering, we may write (43) as
the four-momenta of the incoming quark and antiquark
Γelastic(p )= v σ N˜ , (46) as well as on the spectral properties of the gluon. We
1 12 2
h i recall that in the actual simulation the quark and anti-
which corresponds to the Boltzmann limit relating the quark masses are distributed according to the spectral
collision rate to the average velocity between the collid- function (30) and their three-momenta vary in a broad
ing partners (in the center-of-mass frame) and the cross range roughly in line with thermal Boltzmann distribu-
section for scattering as well as the density N˜ (summed tions.
2
over the discrete quantum numbers of particle 2). We We point out that the iteration of the coupled equa-
employtheserelationsindeterminingtheeffectiveelastic tions has been performed with the additional boundary
cross sections between partons in the PHSD. Note that conditions
the totalnumber ofcollisionsbetweenparticlesoftype 1 4
and 2 are obtained from (46) (in our case) by multipli- σgq(qg) = 9σgg(ε), σqq =αqq(ε)σgg(ε) (50)
cation with the volume V and the particle density N˜ ,
1 as suggested by lattice QCD, which roughly follows a
i.e.,
scaling with the color Casimir operators. This is also
dNcoll reflected in the DQPM ansatz (29). We mention that
12 =V v12σ N˜1N˜2 . (47) this scalingmightbe violatedandrequireafurtherinde-
dt h i
pendent parameter,which, however,presently cannotbe
Boththenumberofcollisionsbetweentheindividualpar- fixed appropriately by lQCD calculations. The function
ticlespeciesaswellastheirdensitiesareeasilyaccessible α (ε) has to be determined by the iteration procedure
qq
in the transport approach. until self-consistency has been reached for each value of
The cross section for gluon formation from flavor- energy density ε. Note that for µ = 0 we have identical
q
neutral q +q¯ interactions in the color octet channel is phase-space distributions for quarks and antiquarks and
calculated by the resonant cross section at invariant en- also identical interaction rates, which simplifies substan-
ergy squared, s=(pq+pq¯)2, tially the iteration process. Additionally, we assume for
the present study that the elastic scattering process is
2 4πsΓ2(ε) 1
g isotropic.
σ (s,ε,M ,M )= ,
qq¯→g q q¯ 4 s−Mg2(ε) 2+sΓ2g(ε)Pr2el(48) tioTnhoefnthuemcerroicssalsercetsiuolntssoafntdhweisdetlhf-scocnasnisbteenptadraemteermtriiznead-
with (cid:2) (cid:3) in the following form (with the cross sections given in
square femtometers):
s (M +M2 s (M M )2
Pr2el = − q q¯ 4s − q− q¯ , (49) σgg(ε) 7.6e−ε/0.8+106.2e−ε/0.2+1.7e−ε/3.7+0.3, (51)
(cid:2) (cid:3)(cid:2) (cid:3) ≈
7
E. Dynamical hadronization
30
elastic gg In the present manuscript we essentially consider
25 systems in the partonic phase where the dynamical
qq hadronization plays no substantial role. However, we
describe here in short the implementation of the tran-
20

] qg sition from the partonic to hadronic degrees of freedom
b
m (hadronization)andviceversa(deconfinement)inPHSD.
[ 15 Hadronization is described in PHSD by covariant tran-
sition rates for the fusion of quark-antiquark pairs to
10 mesonicresonancesorthreequarks(antiquarks)tobary-
onicstates[22],e.g.,forq+q¯fusiontoamesonmoffour-
5 momentum p=(ω,p) at space-time point x=(t,x):
dN (x,p) x +x
0 d4mxd4p =TrqTrq¯δ4(p−pq−pq¯)δ4 q 2 q¯ −x
0 5 10 15 20 (cid:18) (cid:19)
3
p p
[GeV/fm ] ω ρ (p )ω ρ (p )v 2W x x , q− q¯
q q q q¯ q¯ q¯ qq¯ m q q¯
× | | − 2
FIG. 2: (Color online) The gluon-gluon (solid blue line), (cid:18) (cid:19)
quark(antiquark)-quark(antiquark) (dashed red line), and Nq(xq,pq)Nq¯(xq¯,pq¯)δ(flavor,color) . (54)
×
quark(antiquark)-gluon or gluon-quark(antiquark) (dash-
In(54)we haveintroducedthe shorthandoperatornota-
dotted green line) elastic cross sections as functions of the
tion
energy density.
d4p
Tr ...= d4x j ..., (55)
j j (2π)4
where ε is given in units of GeV/fm3. The solution of Xj Z Z
the coupled equations then give the coefficient where denotes a summation over discrete quantum
j
numbers (spin, flavor, and color); N (x,p) is the phase-
αqq(ε) 0.3 e−ε/2.6+0.4 . (52) space dPensity of parton j at space-tjime position x and
≈
four-momentum p. In (54) δ(flavor, color) stands sym-
This fit is shown in comparison to the numerical results bolicallyfor the conservationofflavorquantumnumbers
of the iteration in Fig. 1. Accordingly, the expressions aswellascolorneutralityoftheformedhadronm,which
for the partonic elastic scatterings may be parametrized canbeviewedasacolordipoleor“pre-hadron.” Further-
as more,v (ρ )isthe effectivequark-antiquarkinteraction
qq¯ p
from the DQPM (displayed in Fig. 10 of Ref. [31]) as a
4
σgq(qg) = 9σgg(ε), σqq ≈(0.3e−ε/2.6+0.4)σgg(ε). (53) functionof the localparton(q+q¯+g)density ρp (oren-
ergy density). Furthermore, W (x,p) is the dimension-
m
less phase-space distribution of the formed pre-hadron,
In Fig.2 we displaythe resulting gluon-gluon(solidblue
i.e.,
line), quark-quark (dashed red line), and quark-gluon
(dash-dotted green line) elastic cross sections as func- ξ2 (M M )2
W (ξ,p )=exp exp 2b2 p2 q− q¯ ,
tions of the energy density. Note that these cross sec- m ξ 2b2 ξ − 4
tions are moderate at high energy density and typically (cid:18) (cid:19) (cid:20) (cid:18) ((cid:19)5(cid:21)6)
intheorderof2–3mbbutbecomelargeclosetothecrit-
ical energy density. This behavior basically reflects the
with ξ = x x = x x and p = (p p )/2 =
1 2 q q¯ ξ 1 2
infraredenhancementofthe strongcoupling(24)around − − −
(p p )/2. The width parameter b has been fixed by
q q¯
T andimpliesthatpartons“seeeachother”atdistances −
c r2 = b = 0.66 fm (in the rest frame), which corre-
ofabout1fm(andevenmore)inthevicinityofthephase h i
sponds to an average rms radius of mesons. We note
transition. The physics interpretation is that color sin- p
that the expression (56) corresponds to the limit of in-
glet qq¯pairs form “rotating strings” whereas qq or (q¯q¯)
dependent harmonic oscillator states and that the final
pairs form resonant (and colored) diquark (antidiquark)
hadron-formation rates are approximately independent
states that may fuse with another quark (or antiquark)
ofthe parameterb within reasonablevariations. By con-
to form baryonic resonances.
struction the quantity (56) is Lorentz invariant; in the
Although the cross sections (53) have been extracted
limit of instantaneous “hadron formation,” i.e., ξ0 = 0,
forµ =0inthermalequilibriumwemayadoptthesame
q it provides a Gaussian dropping in the relative distance
cross sections also out of equilibrium and for µ = 0 in
q 6 squared, (r1 r2)2. The four-momentum dependence
the PHSD transport approach. This appears legitimate −
reads explicitly (except for a factor 1/2)
forphase-spaceconfigurationsslightlyoutofequilibrium
as well as for moderate µ . (E E )2 (p p )2 (M M )2 0 (57)
q 1 2 1 2 1 2
− − − − − ≤
8
ter, since we are primarily interested in the time scales
300 3 forkineticandchemicalequilibrationinthesQGPwithin
= 0.35 GeV/fm
thePHSD.Thuswewillstudythesystemsatenergyden-
sitieshigherthanthecriticalonefortheremainderofthis
work.
e
nc200
a
d III. CHEMICAL AND THERMAL
n partons
u EQUILIBRATION
b hadrons
a
100 Beforeweproceedtotheactualresultsonthechemical
equilibration and kinetic thermalization of the partonic
matter in PHSD, let us note that the PHSD transport
approach has been tested in comparison to various data
fromrelativisticheavy-ioncollisionsandhasledtoa fair
0
description of particle production [21], elliptic flow [37],
0 5 10 15 20
and dilepton production [38] both at Super Proton Syn-
time [fm/c]
chrotron (SPS) and top RHIC energies. In particular,
FIG. 3: (Color online) PHSD calculations for the system
the comparison of PHSD calculations to the data of the
GineitVia/lfimze3d. byThqeuanrkusmabnedrsgolufopnasrtaotnsµq(s=olid0 arendd lεin=e) 0a.n3d5 NA60, PHENIX, and STAR Collaborations in Ref. [38]
hasshownthatthepartonicdileptonproductionchannels
hadrons (dashed blueline) are shown as functions of time.
should be visible in the intermediate-mass region (from
1 to 3 GeV). The partonic contribution to the dilepton
radiationappearstobeexponentialinmassfrom1to2.5
and leads to a negative argument of the second expo-
GeV so that an interpretation in terms of thermal radi-
nential in (56) favoring the fusion of partons with low
ation from the sQGP might appear appropriate. How-
relative momenta p p =p p .
q− q¯ 1− 2 ever, such an interpretation is subject to the question
Note that due to the off-shell nature of both partons
of whether or not the PHSD dynamics shows that ki-
andhadrons,the hadronizationprocessobeys allconser-
netic equilibriumis achievedonthe partoniclevelwithin
vationlaws(i.e.,four-momentumconservationandflavor
thecharacteristiclifetimeofthepartonicsysteminthese
currentconservation)in eachevent, the detailed balance
collisions. We will address this question in the present
relations, and the increase in the total entropy S in case
section.
ofarapidlyexpandingsystem. Thephysicsbehind(54)is
As mentioned above, we simulate the “infinite” mat-
thatthe inversereaction,i.e.,the dissolutionofhadronic
terwithinacubicboxwithperiodicboundaryconditions
states to quark-antiquark pairs (in case of mesons), at
at various values for the quark density (or chemical po-
low energy density is inhibited by the huge masses of
tential) and energy density. The size of the box is fixed
thepartonicquasiparticlesaccordingtotheDQPM.Vice
to 93 fm3 for most of the following calculations. How-
versa, the resonant q-q¯pairs have a large phase space to
ever,wewillstudyalsolargerboxsizesinordertodeter-
decay to several 0− octet mesons. We recall that the
mine whetherthe thermodynamiclimit isapproximately
transition matrix element becomes huge below the criti-
reached, in particular when addressing the fluctuation
cal energy density [32]. For further details on the PHSD
measures. The initialization is done by populating the
off-shell transport approach and hadronization we refer
box with light (u and d) and strange (s) quarks, an-
the reader to Refs. [21, 22, 28, 29, 40].
tiquarks, and gluons. The system is initialized out of
If the system is initialized by an ensemble of partons,
equilibriumandapproacheskineticandchemicalequilib-
but the energy density in the system is below the crit-
riumduringitsevolutionbyPHSD.Wearenotinterested
ical energy density (ε 0.5 GeV/fm3), the evolution
c ≈ here in very far nonequilibrium configurations, such as,
proceeds throughthe dynamical phase transition (as de-
for example, the result of the initial hard scatterings in
scribed in Sec. IIE) and ends up in an ensemble of
a heavy-ion collision. Instead, we study here configura-
hadrons. InFig.3weshowtheresultsofthePHSDcalcu-
tions which are reasonably close to equilibrium, because
lationsforthe systeminitializedbyquarksandgluonsat
in this case the approach to equilibrium will have uni-
µ = 0 and ε= 0.35 GeV/fm3. The numbers of partons
q versalcharacteristicsthat will not depend on the precise
(solidredline)andhadrons(dashedblueline)areshown
choice of the initial state. We will see in the end of the
asfunctionsoftime. Weobservethatthetransitionfrom
present section (see Fig. 7 and its description) that this
partonictohadronicdegreesoffreedomiscompleteafter
is indeed the case for our choice ofinitializations. Let us
about 9 fm/c. A small nonvanishing fraction of partons
describe our initial state in detail.
remains due to local fluctuations of energy density from
cell to cell. The equilibration of hadron-dominated mat- 1. The initial space coordinates for the quarks, anti-
terisaninterestingtopic. However,weconcentrateinthe quarks, and gluons are chosen at random within
presentworkonthepropertiesofparton-dominatedmat- the finite box.
9

103 (a) = 1.1 GeV/fm3 104 (b) = 4.72 GeV/fm3
elastic collisions
3 elastic collisions
V] 2 V]10
Ge10 Ge
dt [ q + q g t [102
g q + q d
/
N /
N
d 1
10 d
1 q + q g
10
g q + q
0 0
10 10
1 10 100 1 10 100
time [fm/c] time [fm/c]
FIG.4: (Coloronline)Thereactionratesforelasticpartonscattering(dashedgreenlines),gluonsplitting(solidbluelines),and
flavor-neutral qq¯fusion (short-dashed red lines) as functions of time for systems at different energy densities initially slightly
out of equilibrium. (a) ε = 1.1 GeV/fm3; (b) ε = 4.72 GeV/fm3.
2. The spectral properties (pole masses and the 4. The initial momentum distributions and abun-
widths) of the quarks, antiquarks, and gluons are dances of partons are given by the thermal distri-
initially taken in the simple Lorentzian form (23) butions
with two parameters for each parton type (M, Γ).
Note that in the DQPM model one also assumes f(ω,p)=Cip2ωρi(ω,p)nF(B)(ω/Tinit), (58)
Lorentzianshapesforthepartonspectralfunctions;
however, we choose to start the system evolution where ρi(ω,p) are the spectral functions (with i=
notfromtheDQPMequilibriumspectralfunctions. q,q¯,g)andnB(F)(ω/Tinit)aretheBose(Fermi)dis-
Forthispurposewedeliberatelyemployanaverage tributions with a “temperature” parameter Tinit,
value for the pole mass parameter in the spectral which should not be misidentified with the final
function of the strange quark at initialization (i.e., temperature T, which will be characteristicfor the
wechooseM =M =M ). Theotherparameters energy distributions of the particles after the ther-
s u d
(M , M , M , Γ ) are initially as in the DQPM. malization. The latter, “true” temperature T is
u d g i
The spectral functions of the partons then evolve well defined for the final, thermalized state, and in
dynamically in time and in the final state may de- Sec. IV it will be extracted from the final particle
viate noticeably from the initial ones. We will see spectra by fitting their high-energy tails. We will
in the results of Section IV that indeed in the fi- usethis extractedfinaltemperatureT tostudythe
nalthermalizedstate the dynamicalgluonspectral equation of state of the partonic matter in PHSD
functions deviate from the Lorentzian input and in Sec. IV. On the other hand, the value of the
thus are not described by the DQPM ansatz. On “temperature”parameterTinitoftheinitialenergy-
the other hand, the pole massof the strange quark momentum distributions and the numbers of par-
dynamically reaches the correct value in equilib- tons (determined by the coefficients Ci) just define
rium. We stress here the importance of using off- the total energy of the system (and in equilibrium
shell transport for our studies. Only in case of the the quark chemical potentials).
generalizedtransportpropagationcanwestudythe
5. The dynamical quarks, antiquarks, and gluons
evolution of the spectral functions!
within the PHSD interact also via the mean fields.
3. Weexpectthatinthechemicallyequilibratedstate Notethatthepotentialenergyofthisinteractionis
the ratio of strange quarks to the number of light taken into account at initialization, so that it con-
(u or d) quarks is governed by the ratio of their tributes to the total energy density. The strength
masses (their flavor decomposition). We start our ofthequarkandgluonpotentialenergyinPHSDis
simulation from a flavor ratio, which is far from givenbythespacelikepartofthe00componentsof
equilibrium; i.e, in the initial state the ratio of the the energymomentumtensor T00 as inthe DQPM
number of s quarks to the number of u quarks and (see Ref. [40]).
to the number of d quarks as 1:3:3 is takenas such
that the strangeness is clearly undersaturated ini- In the course of the subsequent transport evolution
tially. of the system by PHSD, the numbers of gluons, quarks,
10

400 (a) = 1.1 GeV/fm3 1200 = 4.72 GeV/fm3
u + u (b) u + u
d + d d + d
300
900
ce s + s e s + s
n c
n
a a
nd200 nd 600
bu bu gluons
a a
100 300
gluons
0 0
0 30 60 90 120 150 0 30 60 90 120 150
time [fm/c] time [fm/c]
FIG.5: (Coloronline)Abundancesoftheu(solidredlines),d(short-dashedblacklines),ands(dash-dottedbluelines)quarks
+antiquarksandgluons(dashedgreenlines)asfunctionsoftimeforsystemsatdifferentenergydensities. (a)ε=1.1GeV/fm3;
(b) ε = 4.72 GeV/fm3.
andantiquarkschangedynamicallythroughinelasticand
1 3
elastic collisions to equilibrium values. We observe in 10 = 1.1 GeV/fm
Fig. 4 that after about 20 fm/c (for ε = 1.1 GeV/fm3)
or 3 fm/c (for ε = 4.72 GeV/fm3) the reactions rates
s + s g
are practically constant and obey a detailed balance for
g s + s
gluonsplittingandqq¯fusion. InFig.4thereactionrates ]
V
for elastic parton scattering (dashed green lines), gluon
e
splitting (solid blue lines), and flavor-neutral qq¯ fusion G
0
(short-dashedredlines)arepresentedasfunctionsoftime [ 10
t
d
at energy densities of 1.1 and 4.72 GeV/fm3. We find /
N
that the rate of inelastic collisions relative to the elastic
d
rate is larger at higher energy density; this is due to a
larger gluon fraction with increasing energy density (or
temperature) since gluons are more suppressed at low
temperatureduetotheirlargermassdifferencerelativeto -1
10
thequarks. Nevertheless,itisworthmentioningthatthe
0 30 60 90 120 150
elastic scattering between partons dominates in PHSD.
time [fm/c]
A signfor chemicalequilibrationis the stabilizationof FIG.6: (Color online) Thereactions ratesfor gluon splitting
the numbers of partons of the different species in time topairsofstrangequarksandantiquarks(solidblueline)and
for t . In Fig. 5 we show the particle abundances of flavor-neutralss¯fusion(short-dashedredline)asfunctionsof
→∞ time for a system at an energy density of 1.1 GeV/fm3.
u, d, and s quarks+antiquarks (solid red, short-dashed
black, and dash-dotted blue lines, respectively) and glu-
ons (dashed green lines) for systems at energy densities
of1.1and4.72GeV/fm3,whichareabovethecriticalen- Thesefindings appearto be incontradiction;however,
ergydensity(asinthe previousfigure). We noteinpass- the time scalesfromthe box calculationscannotdirectly
ingthatenergyconservationwithinPHSDholdswithan be applied to nucleus-nucleus collisions since the initial
accuracybetter than10−3 in these cases,whichis a nec- conditionsareverydifferent. The initialstate inthe box
essaryrequirementforourstudy. Theslowincreaseofthe is chosen close to thermal parton equilibrium. This sup-
totalnumberofstrangequarksandantiquarksduringthe presses the production of strange quark-antiquark pairs
time evolution reflects long equilibration times through due to kinematics or available energy. The strangeness
inelasticprocessesinvolvingstrangepartons. Thesetime productionin A+A collisionsoccursmainly inthe early
scales are significantly larger than typical reaction times stage of A+A reactions where the system is rather far
of nucleus-nucleus collisions at SPS or RHIC energies. awayfromlocalthermalequilibriumandkinematical(en-
Note, however,that the rapidity and transverse momen- ergy) constraints are subleading, i.e., particle collisions
tum spectra of strange hadrons are well described by with large center-of-mass energies take place. These en-
PHSD from lower SPS to top RHIC energies [21, 40]. ergies are much larger than those in local thermal equi-