arXiv:0704.1410v1  [nucl-th]  11 Apr 2007
QCD thermodynamics and confinement from
a dynamical quasiparticle point of view
W. Cassing a,∗
aInstitut für Theoretische Physik, Universität Giessen, Heinrich–Buff–Ring 16,
D–35392 Giessen, Germany
Abstract
In this study it is demonstrated that a simple picture of the QCD gluon liquid
emerges in the dynamical quasiparticle model that specifies the active degrees of
freedom in the time-like sector and yields a potential energy density in the space-like
sector. By using the time-like gluon density (or scalar gluon density) as an indepen-
dent degree of freedom - instead of the temperature T as a Lagrange parameter -
variations of the potential energy density lead to effective mean-fields for time-like
gluons and an effective gluon-gluon interaction strength at low density. The latter
yields a simple dynamical picture for the gluon fusion to color neutral glueballs when
approaching the phase boundary from a temperature higher than Tc and paves the
way for an off-shell transport theoretical description of the parton dynamics.
Key words: Quark gluon plasma, General properties of QCD, Relativistic
heavy-ion collisions
PACS: 12.38.Mh, 12.38.Aw, 25.75.-q
1 Introduction
The formation of a quark-gluon plasma (QGP) and its transition to interacting
hadronic matter – as occurred in the early universe – has motivated a large commu-
nity for several decades (cf. [1] and Refs. therein). Early concepts of the QGP were
guided by the idea of a weakly interacting system of partons (quarks, antiquarks
and gluons) since the entropy s and energy density ǫ were found in lattice QCD to
be close to the Stefan Boltzmann (SB) limit for a relativistic noninteracting system
[2]. However, this notion had to be given up in the last years since experimental
∗ corresponding author
Email address: Wolfgang.Cassing@theo.physik.uni-giessen.de (W. Cassing).
Preprint submitted to Elsevier 24 November 2018
http://arxiv.org/abs/0704.1410v1
observations at the Relativistic Heavy Ion Collider (RHIC) indicated that the new
medium created in ultrarelativistic Au+Au collisions was interacting more strongly
than hadronic matter. Moreover, in line with earlier theoretical studies in Refs. [3–
5] the medium showed phenomena of an almost perfect liquid of partons [6,7] as
extracted from the strong radial expansion and elliptic flow of hadrons as well the
scaling of the elliptic flow with parton number etc. The latter collective observables
have been severely underestimated in conventional string/hadron transport models
[8–10], but hydrodynamical approaches did quite well in describing (at midrapidity)
the collective properties of the medium generated during the early times for low and
moderate transverse momenta [11,12]. Soon the question came up about the con-
stituents of this liquid; it might be some kind of i) ”epoxy” [13], i.e. a system of
resonant or bound gluonic states with large scattering length, ii) a system of chirally
restored mesons, instanton molecules or equivalently giant collective modes [14], iii)
a system of colored bound states of quarks q and gluons g, i.e. gq, qq, gg etc. [15], iv)
some ’string spaghetti’ or ’pasta’ etc. In short, many properties of the new phase are
still under debate and practically no dynamical concepts are available to describe
the freezeout of partons to color neutral hadrons that are subject to experimental
detection.
Lattice QCD (lQCD) calculations provide some guidance to the thermodynamic
properties of the partonic medium close to the transition at a critical temperature
Tc up to a few times Tc, but lQCD calculations for transport coefficients presently
are not accurate enough [16] to allow for firm conclusions. Furthermore, it is not
clear whether the partonic system really reaches thermal and chemical equilibrium
in ultrarelativistic nucleus-nucleus collisions and nonequilibrium models are needed
to trace the entire collision history. The available string/hadron transport models
[17–19] are not accurate enough - as pointed out above - nor do partonic cascade
simulations [20–23] (propagating massless partons) sufficiently describe the reaction
dynamics when employing cross sections from perturbative QCD (pQCD). This also
holds - to some extent - for the Multiphase Transport Model AMPT [24] since it
includes only on-shell massless partons in the partonic phase as in Ref. [21]. The
same problem comes about in the parton cascade model of Xu and Greiner [25] where
additional 2↔ 3 processes like gg ↔ ggg are incorporated. On the other hand it is
well known that strongly interacting quantum systems require descriptions in terms
of propagators D with sizeable selfenergies Π for the relevant degrees of freedom.
Whereas the real part of the selfenergy gives contributions to the energy density, the
imaginary parts of Π provide information about the lifetime and/or reaction rate
of time-like ’particles’ [4]. In principle, off-shell transport equations are available in
the literature [26–28], but have been applied only to dynamical problems where the
width of the quasiparticles stays moderate with respect to the pole mass [29]. On
the other hand, the studies of Peshier [30,31] indicate that the effective degrees of
freedom in a partonic phase should have a width γ in the order of the pole mass M
already slightly above Tc.
The present study addresses essentially three questions: i) Do we understand the
QCD thermodynamics in terms of dynamical quasiparticles down to the phase bound-
ary in a ’top down’ scenario and what are the effective degrees of freedom as well
as energy contributions? ii) Can such a quasiparticle approach help in defining an
off-shell transport model that - at least in thermal equilibrium - reproduces the
thermodynamic results from lQCD? iii) Are there any perspectives in modeling the
transition from partonic to hadronic degrees of freedom in a dynamical way?
The present work is exploratory in the sense that it is restricted to a pure gluonic
system of N2c −1 gluons with two transverse polarisations, i.e. degeneracy dg = 16 for
the gluonic quasiparticles that are treated as relativistic scalar fields. Note, however,
that the qualitative features stay the same when adding light quark degrees of free-
dom [31]; this finding is well in line with the approximate scaling of thermodynamic
quantities from lQCD when dividing by the number of degrees of freedom and scaling
by the individual critical temperature Tc which is a function of the different number
of parton species [32].
The outline of the paper is as follows: After a short recapitulation of the dynamical
quasiparticle model in Section 2 new results on the space-like and time-like parts
of observables are presented that allow for a transparent physical interpretation. In
Section 3 we will examine derivatives of the space-like part of the quasiparticle energy
density with respect to the time-like (or scalar) density which provides information on
gluonic mean fields and their effective interaction strength. The implications of these
findings with respect to an off-shell transport description are pointed out throughout
the study. A summary and extended discussion closes this work in Section 4.
2 Off-shell elements in the DQPM
2.1 Reminder of the DQPM
The Dynamical QuasiParticle Model (DQPM) 1 adopted here goes back to Peshier
[30,31] and starts with the entropy density s in the quasiparticle limit [33],
sdqp = −dg
(2π)3
Im ln(−∆−1) + ImΠRe∆
, (1)
where n(ω/T ) = (exp(ω/T )− 1)−1 denotes the Bose distribution function, ∆ stands
for the scalar quasiparticle propagator and Π for the quasiparticle selfenergy which is
considered here to be a Lorentz scalar. In principle, the latter quantities are Lorentz
tensors and should be evaluated in a nonperturbative framework. However, a more
1 DQPM also stands alternatively for Dynamical-Quasiparticle-Peshier-Model
practical procedure is to use a physically motivated Ansatz with a Lorentzian spectral
function,
ρ(ω) =
(ω −E)2 + γ2
(ω + E)2 + γ2
, (2)
and to fit the few parameters to results from lQCD. With the convention E2(p) =
2+M2−γ2, the parameters M2 and γ are directly related to the real and imaginary
parts of the corresponding (retarded) self-energy, Π = M2 − 2iγω. It should be
stressed that the entropy density functional (1) is not restricted to quasiparticles of
low width γ and thus weakly interacting particles. In fact, in the following it will
be shown that a novel picture of the hot gluon liquid emerges because γ becomes
comparable to the quasiparticle mass already slightly above Tc [30,31].
Following [34] the quasiparticle mass (squared) is written in (momentum-independent)
perturbative form,
M2(T ) =
g2T 2 , (3)
with a running coupling (squared),
g2(T/Tc) =
11Nc ln(λ2(T/Tc − Ts/Tc)2
, (4)
which permits for an enhancement near Tc [34,35]. It will be shown below that an
infrared enhancement of the coupling - as also found in the lQCD calculations in
Ref. [36] for the long range part of the q − q̄ potential - is directly linked to the
gluon fusion/clustering scenario. In order to quantify this statement the coupling
αs(T ) = g
2(T )/(4π) is shown in Fig. 1 as a function of T/Tc in comparison to the long
range part of the strong coupling as extracted from Ref. [36] from the free energy of a
quark-antiquark pair in quenched lQCD. For this comparison the actual parameters
λ = 2.42, Ts/Tc = 0.46 have been adopted as in Ref. [4]. The parametrization (4) is
seen to follow the lQCD results - also indicating a strong enhancement close to Tc -
as a function of temperature reasonably well. One should recall that any extraction
of coupling constants αs(T ) from lQCD is model dependent and deviations from (or
agreement with) lattice ’data’ have to be considered with care. The argument here is
that the specific ’parametric form’ of Eq. (4) is not in conflict with lQCD and that
the coupling αs and consequently the quasiparticle mass M(T ) has the right order
of magnitude.
1 2 3 4 5 6 7 8 9 10
Fig. 1. The coupling αs(T ) = g
2(T )/(4π) (solid red line) as a function of T/Tc in comparison
to the long range part of the strong coupling as extracted from Ref. [36] from the free energy
of a quark-antiquark pair in quenched lQCD (for Nτ = 8).
The width γ is adopted in the form γ ∼ g2T ln g−1 [37] or, equivalently, in terms of
M [30], as
γ(T ) =
M2(T )
(M(T )/T )2
, (5)
where c = 14.4 (from [4]) is related to a magnetic cut-off. In case of the pure Yang-
Mills sector of QCD the physical processes contributing to the width γ are both
gg ↔ gg scattering as well as splitting and fusion reactions gg ↔ g or gg ↔ ggg,
ggg ↔ gggg etc. Note that the ratio γ(T )/M(T ) ∼ g ln(c/g2) approaches zero only
asymptotically for T → ∞ such that the width of the quasiparticles is comparable to
the mass for all practical energy scales on earth; the ratio γ(T )/M(T ) drops below
0.5 only for temperatures T > 1.25 · 105 Tc (for the parameters given above).
For the choice (2) for the spectral function the scalar effective propagator reads,
∆dqp(ω,p) =
ω2 − p2 −M2 + 2iγω
, (6)
which can easily be separated into real and imaginary parts. The entropy density (1)
then reads explicitly [31],
sdqp(T ) = dg
(2π)3
− ln(1− e−ωp/T ) +
n(ωp/T )
(2π)3
arctan(
ω2p − ω2
2γω(ω2p − ω2)
(ω2p − ω2)2 + 4γ2ω2
, (7)
using ωp =
p2 +M2. The first line in (7) corresponds to the familiar on-shell quasi-
particle contribution s0 while the second line in (7) corresponds to the contribution
originating from the finite width γ of the quasiparticles and is positive throughout
but subleading (see below).
The pressure P now can be evaluated from
by integration of s over T , where from now on we identify the ’full’ entropy density s
with the quasiparticle entropy density sdqp. Note that for T < Tc the entropy density
drops to zero (with decreasing T ) due to the high quasiparticle mass and the width γ
vanishes as well because the interaction rate in the very dilute quasiparticle system
becomes negligible. Since the pressure for infinitely heavy (noninteracting) particles
also vanishes the integration constant for the pressure P - when integrating (8) -
may safely be assumed to be zero, too.
The energy density ǫ then follows from the thermodynamical relation [34,38]
ǫ = Ts− P (9)
and thus is also fixed by the entropy s(T ) as well as the interaction measure
W (T ) := ǫ(T )− 3P (T ) = Ts− 4P (10)
that vanishes for massless and noninteracting degrees of freedom.
In Ref. [4] a detailed comparison has been presented with the lattice results from
Ref. [39] for the pure gluonic sector to the quasiparticle entropy density (7) for the
parameters given above. The agreement with the lattice data is practically perfect
[4,30]. Needless to point out that also P (T ), ǫ(T ) and W (T ) well match the lattice
QCD results for 1 ≤ T/Tc ≤ 4 [4,31] due to thermodynamical consistency. The same
parameters are also adopted for the following calculations.
2.2 Time-like and space-like quantities
For the further argumentation it is useful to introduce the shorthand notation
P · · · = dg
(2π)3
2ω ρ(ω) Θ(ω)n(ω/T ) Θ(±P 2) · · · (11)
with P 2 = ω2 − p2 denoting the invariant mass squared. The Θ(±P 2) function in
(11) separates time-like quantities from space-like quantities and can be inserted for
any observable of interest.
As the first quantity we consider the entropy density (7). Its time-like contribution
is almost completely dominated by the first line in (7) - that corresponds to the on-
shell quasiparticle contribution s0 - but also includes a small contribution from the
second line in (7) which is positive for T below about 1.5 Tc and becomes negative for
larger temperature. This time-like part s+ is shown in Fig. 2 by the dotted blue line
(multiplied by (Tc/T )
3). The second line in (7) - as mentioned above - corresponds
to the contribution originating from the finite width γ of the quasiparticles and also
has a space-like part s− which is dominant (for the second line in (7)) and displayed
in Fig. 2 by the lower red line (multiplied by (Tc/T )
3). Though s− is subleading in
the total entropy density s = s+ + s− (thick solid green line in Fig. 2) it is essential
for a proper reproduction of s(T ) close to Tc (cf. [31]). Note that the total entropy
density s is not very different from the Stefan Boltzmann entropy density sSB for
T > 2Tc as shown in Fig. 2 by the upper thin line (multiplied by (Tc/T )
2 4 6 8 10
=0.26 GeV
Fig. 2. The time-like contribution to the entropy density s+ (dotted blue line), the space-like
contribution s− (lower red line) and the total entropy density s = s++s− (thick solid green
line) as a function of T/Tc. All quantities have been multiplied by the dimensionless factor
(Tc/T )
3) assuming Tc = 0.26 GeV for the pure gluonic system [40]. The upper solid black
line displays the Stefan Boltzmann limit sSB for reference.
Further quantities of interest are the quasiparticle ’densities’
N±(T ) = T̃r± 1 (12)
that correspond to the time-like (+) and space-like (-) parts of the integrated distri-
bution function. Note that only the integral of N+ over space has a particle number
interpretation. In QED this corresponds to time-like photons (γ∗) which are virtuell
in intermediate processes but can also be seen asymptotically by dileptons (e.g. e+e−
pairs) due to the decay γ∗ → e+e− [17].
A scalar density Ns, which is only defined in the time-like sector, is given by
Ns(T ) = T̃r
and has the virtue of being Lorentz invariant. Moreover, a scalar density can easily
be computed in transport approaches for bosons and fermions [17,41] which is of
relevance for the argumentation in Section 3.
0 2 4 6 8 10
0 2 4 6 8 10
=0.26 GeV
 T/TC
Fig. 3. Upper part: The scalar density Ns (lower orange line), the time-like density N
(blue line), the space-like quantity N− (red line) and the sum N = N+ +N− (thick solid
green line) as a function of T/Tc assuming Tc = 0.26 GeV for the pure gluonic system
[40]. The upper solid black line displays the Stefan Boltzmann limit NSB for reference. All
quantities are multiplied by the dimensionless factor (Tc/T )
3. Lower part: The ratio of the
scalar density Ns to the time-like density N
+ as a function of the scaled temperature T/Tc.
The actual results for the different ’densities’ (multiplied by (Tc/T )
3) are displayed
in the upper part of Fig. 3 where the lower orange line represents the scalar density
Ns, the blue line the time-like density N
+, the red line the space-like quantity N−
and the thick solid green line the sum N = N++N− as a function of T/Tc assuming
(as before) Tc = 0.26 GeV for the pure gluonic system [40]. It is seen that N
substantially smaller than N− in the whole temperature range up to 10 Tc where it
is tacitly assumed that the DQPM also represents lQCD results for T > 4Tc, which
is not proven explicitly, but might be expected due to the proper weak coupling
limit of (3), (5) (cf. Fig. 1). The application of the DQPM to 10 Tc is presented
in Fig. 3 since the initial state at Large Hadron Collider (LHC) energies might be
characterized by a temperature above 4 Tc; note that the properties of the partonic
phase will be explored from the experimental side in the near future at LHC. Quite
remarkably the quantity N follows closely the Stefan Boltzmann limit NSB for a
massless noninteracting system which is given in Fig. 3 by the upper thin solid line
and has the physical interpretation of a gluon density. Though N differs by less than
15% from the Stefan Boltzmann (SB) limit for T > 2TC the physical interpretation
is essentially different! Whereas in the SB limit all gluons move on the light cone
without interactions only a small fraction of gluons can be attributed to quasiparticles
with density N+ within the DQPM that propagate within the lightcone. The space-
like part N− corresponds to ’gluons’ exchanged in t-channel scattering processes
and thus cannot be propagated explicitly in off-shell transport approaches without
violating causality and/or Lorentz invariance.
The scalar density Ns follows smoothly the time-like density N
+ as a function of
temperature which can be explicitly seen in the lower part of Fig. 3 where the ratio
+ is shown versus T/Tc. Consequently, the scalar density Ns uniquely relates
to the time-like density N+ or the temperature T in thermal equilibrium which will
provide some perspectives for a transport theoretical treatment (see Section 3).
The separation of N+ and N− so far has no direct dynamical implications except
for the fact that only the fraction N+ can explicitly be propagated in transport as
argued above. Thus we consider the energy densities,
(T ) = T̃r± ω , (14)
that specify time-like and space-like contributions to the quasiparticle energy density.
It is worth pointing out that the quantity T00 = T
00 + T
00 in case of a conventional
quasi-particle model with vanishing width γ in general is quite different from ǫ in (9)
because the interaction energy density in this case is not included in (14), i.e.
T00 = T
(2π)3
2ω δ(ω2 −M2 − p2) Θ(ω) Θ(±P 2)n(ω/T ) ω (15)
since ω2 − p2 = M2 = P 2 > 0 due to the mass-shell δ-function.
How does the situation look like in case of dynamical quasiparticles of finite width?
To this aim we consider the integrand in the energy density (14) which reads as (in
spherical momentum coordinates with angular degrees of freedom integrated out)
I(ω, p) =
p2 ω2 ρ(ω, p2)n(ω/T ) . (16)
Here the integration is to be taken over ω and p from 0 to ∞. The integrand I(ω, p) is
shown in Fig. 4 for T = 1.02Tc (l.h.s.) and T = 2Tc (r.h.s.) in terms of contour lines.
For the lower temperature the gluon mass is about 0.91 GeV and the width γ ≈ 0.15
GeV such that the quasiparticle properties are close to a ρ-meson in free space. In
this case the integrand I(ω, p) is essentially located in the time-like sector and the
integral over the space-like sector is subdominant. This situation changes for T = 2Tc
where the mass is about 0.86 GeV while the width increases to γ ≈ 0.56 GeV. As one
observes from the r.h.s. of Fig. 4 the maximum of the integrand is shifted towards
the line ω = p and higher momentum due to the increase in temperature by about
a factor of two; furthermore, the distribution reaches far out in the space-like sector
due to the Bose factor n(ω/T ) which favors small ω. Thus the relative importance of
the time-like (+) part to the space-like (-) part is dominantly controlled by the width
γ - relative to the pole mass - which determines the fraction of T−00 with negative
invariant mass squared (P 2 < 0) relative to the time-like part T+00.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
I(ω,p)
T=1.02 T
time-like
space-like
p [GeV/c]
0 1 2 3 4 5
5 p=ω
I(ω,p)
T=2 T
time-like
space-like
p [GeV/c]
Fig. 4. The integrand I(ω, p) (16) for T = 1.02Tc (l.h.s.) and T = 2Tc (r.h.s.) in terms of
contour lines. The straight (blue) line (ω = p) separates the lime-like from the space-like
sector. Note that for a convergence of the energy density integral the upper limits for ω
and p have to be increased by roughly an order of magnitude compared to the area shown
in the figure.
The explicit results for the quasiparticle energy densities T+00 and T
00 are displayed
in Fig. 5 by the dashed blue and dot-dashed red lines (multiplied by (TC/T )
respectively. As in case of N+ and N− the space-like energy density T−00 is seen
0 2 4 6 8 10
0.9 1.0 1.1 1.2 1.3 1.4 1.5
=0.26 GeV
=0.26 GeV
Fig. 5. Upper part: The time-like energy density T+
(dashed blue line), the space-like energy
density T−
(dot-dashed red line) and the total energy density T00 = T
(thick solid
green line) as a function of T/Tc. The thin black line displays the energy density ǫ(T/Tc)
from (9); it practically coincides with T00 within the linewidth and is hardly visible. All
densities are multiplied by the dimensionless factor (Tc/T )
4 in order to divide out the
leading temperature dependence. Lower part: Same as the upper part in order to enhance
the resolution close to Tc.
to be larger than the time-like part T+00 for all temperatures above 1.05 Tc. Since
the time-like part T+00 corresponds to the independent quasiparticle energy density
within the lightcone, the space-like part T−00 can be interpreted as an interaction
density V if the quasiparticle energy T00 matches the total energy density ǫ(T ) (9)
as determined from the thermodynamical relations (8) and (9). In fact, the DQPM
yields an energy density T00 - adding up the space-like and time-like parts - that
almost coincides with ǫ(T ) from (9) as seen in Fig. 5 where both quantities (multiplied
by (TC/T )
4) are displayed in terms of the thin black and thick solid green lines,
respectively; actually both results practically coincide within the linewidth for T >
2Tc. An explicit representation of their numerical ratio gives unity within 2% for
T > 2Tc; the remaining differences can be attributed to temperature derivatives
∼ d/dT (ln(γ/E)) etc. in order to achieve thermodynamic consistency but this is
not the primary issue here and will be discussed in a forthcoming study [42]. The
deviations are more clearly visible close to Tc (lower part of Fig. 2) where the variation
of the width and mass are most pronounced. However, for all practical purposes one
may consider T00(T ) ≈ ǫ(T ) and separate the kinetic energy density T+00 from the
potential energy density T−00 as a function of T or - in equilibrium - as a function of
the scalar gluon density Ns or N
+, respectively.
3 Dynamics of time-like quasiparticles
Since in transport dynamical approaches there are no thermodynamical Lagrange
parameters like the inverse temperature β = T−1 or the quark chemical potential
µq, which have to be introduced in thermodynamics in order to specify the average
values of conserved quantities (or currents in the relativistic sense), derivatives of
physical quantities with respect to the scalar density ρs = Ns (or time-like gluon
density ρg = N
+) are considered in the following (cf. Ref. [43]). As mentioned above
one may relate derivatives in thermodynamic equilibrium via,
, (17)
if the volume and pressure are kept constant. For example, a numerical evaluation
of dρs/d(T/Tc) gives
d(T/Tc)
− a2 exp(−b(
)) (18)
with b= 5, a1 = 1.5fm
−3 and a2 = 104fm
−3, which follows closely the quadratic
scaling in T/Tc as expected in the Stefan Boltzmann limit. The additional exponential
term in (18) provides a sizeable correction close to Tc. The approximation (18) may
be exploited for convenient conversions between ρs and T/Tc in the pure gluon case
but will not be explicitly used in the following.
The independent quasiparticle energy density TK := T
00 and potential energy density
V := T−00 now may be expressed as functions of ρs (or ρg) instead of the temperature
T . The interaction energy density then might be considered as a scalar energy density
which - as in the nonlinear σ-model for baryonic matter [44] - is a nonlinear function
of the scalar density ρs. As in case of nuclear matter problems the scalar density
ρs does not correspond to a conserved quantity when integrating over space; it only
specifies the interaction density parametrically, i.e. V (ρs). Alternatively one might
separate V into parts with different Lorentz structure, e.g. scalar and vector parts
as in case of nuclear matter problems [44], but this requires additional information
that cannot be deduced from the DQPM alone.
0 2 4 6 8 10
0.9 1.0 1.1 1.2 1.3 1.4 1.5
Fig. 6. Upper part: The quasiparticle energy per degree of freedom TK/N
+ (dashed blue
line) and the space-like potential energy per degree of freedom V/N+ (dot-dashed red line)
as a function of T/Tc. All energies are multiplied by the dimensionless factor (Tc/T ). Lower
part: Same as the upper part in order to enhance the resolution close to Tc.
It is instructive to show the ’quasiparticle’ and potential energy per degree of free-
dom TK/N
+ and V/N+ as a function of e.g. N+, Ns or T/Tc. As one might have
anticipated the kinetic energy per effective degree of freedom is smaller than the
respective potential energy for T/Tc > 1.05 as seen from Fig. 6 where both quan-
tities are displayed as a function of T/Tc in terms of the dashed and dot-dashed
line, respectively. It is seen that the potential energy per degree of freedom steeply
rises in the vicinity of Tc whereas the independent quasiparticle energy rises almost
linearly with T . Consequently rapid changes in the density - as in the expansion
of the fireball in ultrarelativistic nucleus-nucleus collisions - are accompanied by a
dramatic change in the potential energy density and thus to a violent acceleration of
the quasi-particles. It is speculated here that the large collective flow of practically
all hadrons seen at RHIC [6] might be attributed to the early strong partonic forces
expected from the DQPM.
1 10 100
Fig. 7. The mean-field potential U(N+) = U(ρg) as a function of the time-like gluon density
N+ = ρg in comparison to the fit (19) (solid blue line). The densities N
+= 1, 1.4, 5, 10,
50, 100 fm−3 correspond to scaled temperatures of T/Tc ≈ 1.025, 1.045, 1.25, 1.5, 2.58,
3.25, respectively (cf. Fig. 3).
In order to obtain some idea about the mean-field potential Us(ρs) (or U(ρg) in the
rest frame) one can consider the derivative dV/ρs = Us(ρs) or dV/N
+ = U(N+) =
U(ρg). The latter is displayed in Fig. 7 as a function of N
+ = ρg and shows a distinct
minimum at ρg ≈ 1.4 fm−3 which corresponds to a temperature T ≈ 1.045Tc. The
actual numerical results can be fitted by the expression,
U(ρg) =
≈ 39 e−ρg/0.31 + 2.93 ρ0.21g + 0.55 ρ
g [GeV] , (19)
where ρg is given in fm
−3 and the actual numbers in front carry a dimension in order
to match to the proper units of GeV for the mean-field U . By analytical integration
of (19) one obtains a suitable approximation to V (ρg). The approximation (19) works
sufficiently well as can be seen from Fig. 7 - showing a comparison of the numerical
derivative dV/dN+ with the fit (19) in the interval 0.7 fm−3 < N+ ≤ 300 fm−3
- such that one may even proceed with further analytical calculations. Note that
a conversion between the time-like quasiparticle density N+ = ρg and the scalar
density ρs is easily available numerically (cf. lower part of Fig. 3) such that derivatives
with respect to ρs are at hand, too; the latter actually enter the explicit transport
calculations [45] while derivatives with respect to ρg in the rest frame of the system
are more suitable for physical interpretation and will be used below.
Some information on the properties of the effective gluon-gluon interaction vgg may
be extracted from the second derivative of V with respect to ρg, i.e.
vgg(ρg) :=
≈ −125.8 e−ρg/0.31 + 0.615/ρ0.79g + 0.2/ρ
g [GeVfm
3], (20)
where the numbers in front have again a dimension to match the units of GeV
fm3. The effective gluon-gluon interaction vgg (20) is strongly attractive at low den-
sity 0.003 fm−3 < ρg and changes sign at ρg ≈ 1.4 fm−3 to become repulsive at
higher densities. Note that the change of quasiparticle momenta (apart from col-
lisions) will be essentially driven by the (negative) space-derivatives −∇U(x) =
−dU(ρg)/dρg ∇ρg(x) (or alternatively by −dUs(ρs)/dρs ∇ρs(x)). This implies that
the gluonic quasiparticles (at low gluon density) will bind with decreasing density,
i.e. form ’glueballs’ dynamically close to the phase boundary and repell each other
for ρg ≥ 1.4 fm−3. Note that color neutrality is imposed by color-current conser-
vation and only acts as a boundary condition for the quantum numbers of the
bound/resonant states in color space.
This situation is somehow reminiscent of the nuclear matter problem [44] where a
change in sign of the 2nd derivative of the potential energy density of nuclear matter
at low density indicates the onset of clustering of nucleons, i.e. to deuterons, tritons,
α-particles etc., which form the states of the many-body system at low nucleon
densities (and not a low density nucleon gas). This is easy to follow up for the
simplified nonrelativistic energy density functional ǫN for nuclear matter,
ǫN ≈ Aρ
ρ2N +
N , (21)
where the first term gives the kinetic energy density and the second and third term
correspond to attractive and repulsive interaction densities. For A ≈ 0.073GeV fm2,
B ≈ −1.3 GeV fm3 and C ≈ 1.78 GeV fm4 a suitable energy density for nuclear
matter is achieved; it gives a minimum in the energy per nucleon E/A = ǫN/ρN ≈
−0.016 GeV for nuclear saturation density ρ0N ≈ 0.168 fm−3. The mean-field potential
UN = BρN+Cρ
N has a minimum close to ρ
N such that the effective nucleon-nucleon
interaction strength vNN = B + 4/3Cρ
N changes from attraction to repulsion at
this density. Note that in the gluonic case the minimum in the mean-field potential
U (19) occurs at roughly 8 times ρ0N and the strength of the gluonic interaction is
higher by more than 2 orders of magnitude!
The confining nature of the effective gluon-gluon interaction vgg (20) becomes ap-
parent in the limit ρg → 0, where the huge negative exponential term dominates for
ρg > 0.003 fm
−3; for even smaller densities the singular repulsive terms take over.
Note, however, that the functional extrapolation of the fit (19) to vanishing gluon
density ρg has to be considered with care and it should only be concluded that the
interaction strength becomes ’very large’. On the other hand the limit ρg → 0 is only
academical because the condensation/fusion dynamically occurs for ρg ≈ 1 fm−3.
A straight forward way to model the gluon condensation or clustering to confined
glueballs dynamically (close to the phase transition) is to adopt a screened Coulomb-
like potential vc(r,Λ) with the strength
d3r vc(r,Λ) fixed by vgg(ρg) from (20) and
the screening length Λ from lQCD studies. For the ’dilute gluon regime’ (ρg < 1.4
fm−3), where two-body interactions should dominate, one may solve a Schrödinger
(or Klein-Gordon) equation for the bound and/or resonant states. This task is not
addressed further in the present study since for the actual applications (as in the
Parton-Hadron-String-Dynamics (PHSD) approach [45]) dynamical quark and anti-
quarks have to be included. The latter degrees of freedom do not change the general
picture very much for higher temperatures T > 2Tc but the actual numbers are dif-
ferent close to Tc since the quarks and antiquarks here dominate over the gluons due
to their lower mass. The reader is referred to an upcoming study in Ref. [42].
Some comments on expanding gluonic systems in equilibrium appear in place, i.e.
for processes where the total volume Ṽ and pressure P play an additional role. For
orientation we show the entropy per time-like particle s/N+ in Fig. 8 as a function of
N+ (upper) and T/Tc (lower part) which drops close to the phase boundary since the
quasiparticles become weakly interacting (cf. Fig. 6). Note that this is essentially due
to the low density and not due to the interaction strength (20); a decrease of the width
γ (as encoded in (5)) implies a decrease in the interaction rate! An expansion process
with conserved total entropy S = sṼ leads to a change in the total gluon number
N+Ṽ since s/N+ changes with density (or temperature) (Fig. 8). The same holds
for an expansion process with constant total energy ǫṼ since also ǫ/N+ is varying
with density (or temperature). Other scenarios involving e.g. S = P/T also involve
a change of the gluon number N+Ṽ during the cooling process such that reactions
like gg ↔ g, ggg ↔ gg etc. are necessary ingredients of any transport theoretical
approximation. We do not further investigate different expansion scenarios here since
the reactions g ↔ qq̄, i.e. the gluon splitting to a quark and antiquark as well as
the backward fusion process, are found to play a dominant role in the vicinity of the
phase transition as well as for higher temperatures [42,45].
4 Conclusions and discussion
The present study has provided a novel interpretation of the dynamical quasiparticle
model (DQPM) by separating time-like and space-like quantities for ’particle densi-
ties’, energy densities, entropy densities ect. that also paves the way for an off-shell
transport approach [45]. The entropy density s in (7) is found to be dominated by the
on-shell quasiparticle contribution (first line in (7)) (cf. [31]) while the space-like part
of the off-shell contribution (second line in (7)) gives only a small (but important)
enhancement (cf. Fig. 2). However, in case of the ’gluon density’ N = N+ +N− and
the gluon energy density T00 = T
00 the situation is opposite: here the space-like
1 2 3 4 5 6 7 8 9 10
1 10 100 1000
 T/TC
Fig. 8. The entropy per degree of freedom s/N+ as a function of N+ (upper part) or T/Tc
(lower part).
parts (N−, T−00) dominate over the time-like parts (N
+, T+00) except close to Tc where
the independent quasiparticle limit is approximately regained. The latter limit is a
direct consequence of the infrared enhancement of the coupling (4) close to Tc (in
line with the lQCD studies in Ref. [36] ) and a decrease of the width γ (5) when
approaching Tc from above.
Since only the time-like part N+ can be propagated within the lightcone the space-
like part N− has to be attributed to t-channel exchange gluons in scattering processes
that contribute also to the space-like energy density T−00. The latter quantity may
be regarded as potential energy density V . This, in fact, is legitimate since the
quasiparticle energy density T00 very well matches the energy density (9) obtained
from the thermodynamical relations. Only small deviations close to Tc indicate that
the DQPM in its straightforward application is not thermodynamically consistent.
However, by accounting for ’rearrangement terms’ in the energy density - as known
from the nuclear many-body problem [46] - full thermodynamical consistency may
be regained [42].
It is instructive to compare the present DQPM to other recent models. In the
PNJL 2 model [47] the gluonic pressure is build up by a constant effective potential
U(Φ,Φ∗;T ) which controls the thermodynamics of the Polyakov loop Φ. It is ex-
panded in powers of ΦΦ∗ with temperature dependent coefficients in order to match
lQCD thermodynamics. Thus in the PNJL there are no time-like gluons; the effec-
tive potential U(Φ,Φ∗;T ) stands for a static gluonic pressure that couples to the
quark/antiquark degrees of freedom. The latter are treated in mean-field approxi-
mation, i.e. without dynamical width, whereas the DQPM incorporates a sizeable
width γ.
Another approach to model lQCD thermodynamics has been suggested in Ref. [43]
and is based on an effective Lagrangian which is nonlinear in the effective quark and
gluon fields. In this way the authors avoid a parametrization of the interaction density
in terms of Lagrange parameters (T, µ) and achieve thermodynamical consistency.
The latter approach is closer in spirit to the actual interpretation of the DQPM and
may be well suited for an on-shell transport theoretical formulation. The on-shell
restriction here comes about since effective Lagrangian approaches should only be
evaluated in the mean-field limit which implies vanishing scattering width for the
quasiparticles. This is sufficient to describe systems is thermodynamical equilibrium,
where forward and backward interaction rates are the same, but might not provide
the proper dynamics out-of-equilibrium.
Some note of caution with respect to the present DQPM appears appropriate: the pa-
rameters in the effective coupling (4) and the width (5) have been fixed in the DQPM
by the entropy density (7) to lQCD results assuming the form (2) for the spectral
function ρ(ω). Alternative assumptions for ρ(ω) will lead to slightly different results
for the time-like density, energy densities etc. but not to a qualitatively different
picture. Independent quantities from lQCD should allow to put further constraints
on the more precise form of ρ(ω) such as calculations for transport coefficients [16];
unfortunately such lQCD studies are only at the beginning. A more important issue
is presently to extend the DQPM to incorporate dynamical quark and antiquark
degrees of freedom (as in [31]) in order to catch the physics of gluon splitting and
quark-antiquark fusion (g ↔ q + q̄, g + g ↔ q + q̄ + g) reactions [42,45].
Coming back to the questions raised in the Introduction concerning i) the appropri-
ate description of QCD thermodynamics within the DQPM and ii) the possibility to
develop a consistent off-shell partonic transport approach as well as iii) the perspec-
tives for a dynamical description of the transition from partonic to hadronic degrees
of freedom, we are now in the position to state: most likely ’Yes’.
2 Polyakov-loop-extended Nambu Jona-Lasinio
The author acknowledges valuable discussions with E. L. Bratkovskaya and A. Peshier.
Furthermore he likes to thank S. Leupold for a critical reading of the manuscript and
constructive suggestions.
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  In this study it is demonstrated that a simple picture of the
  QCD gluon liquid emerges in the dynamical quasiparticle model that specifies
the active degrees of freedom in the time-like sector and yields a potential
energy density in the space-like sector. By using the time-like gluon density
(or scalar gluon density) as an independent degree of freedom - instead of the
temperature $T$ as a Lagrange parameter - variations of the potential energy
density lead to effective mean-fields for time-like gluons and an effective
gluon-gluon interaction strength at low density. The latter yields a simple
dynamical picture for the gluon fusion to color neutral glueballs when
approaching the phase boundary from a temperature higher than $T_c$ and paves
the way for an off-shell transport theoretical description of the parton
dynamics.

Introduction
The formation of a quark-gluon plasma (QGP) and its transition to interacting
hadronic matter – as occurred in the early universe – has motivated a large commu-
nity for several decades (cf. [1] and Refs. therein). Early concepts of the QGP were
guided by the idea of a weakly interacting system of partons (quarks, antiquarks
and gluons) since the entropy s and energy density ǫ were found in lattice QCD to
be close to the Stefan Boltzmann (SB) limit for a relativistic noninteracting system
[2]. However, this notion had to be given up in the last years since experimental
∗ corresponding author
Email address: Wolfgang.Cassing@theo.physik.uni-giessen.de (W. Cassing).
Preprint submitted to Elsevier 24 November 2018
http://arxiv.org/abs/0704.1410v1
observations at the Relativistic Heavy Ion Collider (RHIC) indicated that the new
medium created in ultrarelativistic Au+Au collisions was interacting more strongly
than hadronic matter. Moreover, in line with earlier theoretical studies in Refs. [3–
5] the medium showed phenomena of an almost perfect liquid of partons [6,7] as
extracted from the strong radial expansion and elliptic flow of hadrons as well the
scaling of the elliptic flow with parton number etc. The latter collective observables
have been severely underestimated in conventional string/hadron transport models
[8–10], but hydrodynamical approaches did quite well in describing (at midrapidity)
the collective properties of the medium generated during the early times for low and
moderate transverse momenta [11,12]. Soon the question came up about the con-
stituents of this liquid; it might be some kind of i) ”epoxy” [13], i.e. a system of
resonant or bound gluonic states with large scattering length, ii) a system of chirally
restored mesons, instanton molecules or equivalently giant collective modes [14], iii)
a system of colored bound states of quarks q and gluons g, i.e. gq, qq, gg etc. [15], iv)
some ’string spaghetti’ or ’pasta’ etc. In short, many properties of the new phase are
still under debate and practically no dynamical concepts are available to describe
the freezeout of partons to color neutral hadrons that are subject to experimental
detection.
Lattice QCD (lQCD) calculations provide some guidance to the thermodynamic
properties of the partonic medium close to the transition at a critical temperature
Tc up to a few times Tc, but lQCD calculations for transport coefficients presently
are not accurate enough [16] to allow for firm conclusions. Furthermore, it is not
clear whether the partonic system really reaches thermal and chemical equilibrium
in ultrarelativistic nucleus-nucleus collisions and nonequilibrium models are needed
to trace the entire collision history. The available string/hadron transport models
[17–19] are not accurate enough - as pointed out above - nor do partonic cascade
simulations [20–23] (propagating massless partons) sufficiently describe the reaction
dynamics when employing cross sections from perturbative QCD (pQCD). This also
holds - to some extent - for the Multiphase Transport Model AMPT [24] since it
includes only on-shell massless partons in the partonic phase as in Ref. [21]. The
same problem comes about in the parton cascade model of Xu and Greiner [25] where
additional 2↔ 3 processes like gg ↔ ggg are incorporated. On the other hand it is
well known that strongly interacting quantum systems require descriptions in terms
of propagators D with sizeable selfenergies Π for the relevant degrees of freedom.
Whereas the real part of the selfenergy gives contributions to the energy density, the
imaginary parts of Π provide information about the lifetime and/or reaction rate
of time-like ’particles’ [4]. In principle, off-shell transport equations are available in
the literature [26–28], but have been applied only to dynamical problems where the
width of the quasiparticles stays moderate with respect to the pole mass [29]. On
the other hand, the studies of Peshier [30,31] indicate that the effective degrees of
freedom in a partonic phase should have a width γ in the order of the pole mass M
already slightly above Tc.
The present study addresses essentially three questions: i) Do we understand the
QCD thermodynamics in terms of dynamical quasiparticles down to the phase bound-
ary in a ’top down’ scenario and what are the effective degrees of freedom as well
as energy contributions? ii) Can such a quasiparticle approach help in defining an
off-shell transport model that - at least in thermal equilibrium - reproduces the
thermodynamic results from lQCD? iii) Are there any perspectives in modeling the
transition from partonic to hadronic degrees of freedom in a dynamical way?
The present work is exploratory in the sense that it is restricted to a pure gluonic
system of N2c −1 gluons with two transverse polarisations, i.e. degeneracy dg = 16 for
the gluonic quasiparticles that are treated as relativistic scalar fields. Note, however,
that the qualitative features stay the same when adding light quark degrees of free-
dom [31]; this finding is well in line with the approximate scaling of thermodynamic
quantities from lQCD when dividing by the number of degrees of freedom and scaling
by the individual critical temperature Tc which is a function of the different number
of parton species [32].
The outline of the paper is as follows: After a short recapitulation of the dynamical
quasiparticle model in Section 2 new results on the space-like and time-like parts
of observables are presented that allow for a transparent physical interpretation. In
Section 3 we will examine derivatives of the space-like part of the quasiparticle energy
density with respect to the time-like (or scalar) density which provides information on
gluonic mean fields and their effective interaction strength. The implications of these
findings with respect to an off-shell transport description are pointed out throughout
the study. A summary and extended discussion closes this work in Section 4.
2 Off-shell elements in the DQPM
2.1 Reminder of the DQPM
The Dynamical QuasiParticle Model (DQPM) 1 adopted here goes back to Peshier
[30,31] and starts with the entropy density s in the quasiparticle limit [33],
sdqp = −dg
(2π)3
Im ln(−∆−1) + ImΠRe∆
, (1)
where n(ω/T ) = (exp(ω/T )− 1)−1 denotes the Bose distribution function, ∆ stands
for the scalar quasiparticle propagator and Π for the quasiparticle selfenergy which is
considered here to be a Lorentz scalar. In principle, the latter quantities are Lorentz
tensors and should be evaluated in a nonperturbative framework. However, a more
1 DQPM also stands alternatively for Dynamical-Quasiparticle-Peshier-Model
practical procedure is to use a physically motivated Ansatz with a Lorentzian spectral
function,
ρ(ω) =
(ω −E)2 + γ2
(ω + E)2 + γ2
, (2)
and to fit the few parameters to results from lQCD. With the convention E2(p) =
2+M2−γ2, the parameters M2 and γ are directly related to the real and imaginary
parts of the corresponding (retarded) self-energy, Π = M2 − 2iγω. It should be
stressed that the entropy density functional (1) is not restricted to quasiparticles of
low width γ and thus weakly interacting particles. In fact, in the following it will
be shown that a novel picture of the hot gluon liquid emerges because γ becomes
comparable to the quasiparticle mass already slightly above Tc [30,31].
Following [34] the quasiparticle mass (squared) is written in (momentum-independent)
perturbative form,
M2(T ) =
g2T 2 , (3)
with a running coupling (squared),
g2(T/Tc) =
11Nc ln(λ2(T/Tc − Ts/Tc)2
, (4)
which permits for an enhancement near Tc [34,35]. It will be shown below that an
infrared enhancement of the coupling - as also found in the lQCD calculations in
Ref. [36] for the long range part of the q − q̄ potential - is directly linked to the
gluon fusion/clustering scenario. In order to quantify this statement the coupling
αs(T ) = g
2(T )/(4π) is shown in Fig. 1 as a function of T/Tc in comparison to the long
range part of the strong coupling as extracted from Ref. [36] from the free energy of a
quark-antiquark pair in quenched lQCD. For this comparison the actual parameters
λ = 2.42, Ts/Tc = 0.46 have been adopted as in Ref. [4]. The parametrization (4) is
seen to follow the lQCD results - also indicating a strong enhancement close to Tc -
as a function of temperature reasonably well. One should recall that any extraction
of coupling constants αs(T ) from lQCD is model dependent and deviations from (or
agreement with) lattice ’data’ have to be considered with care. The argument here is
that the specific ’parametric form’ of Eq. (4) is not in conflict with lQCD and that
the coupling αs and consequently the quasiparticle mass M(T ) has the right order
of magnitude.
1 2 3 4 5 6 7 8 9 10
Fig. 1. The coupling αs(T ) = g
2(T )/(4π) (solid red line) as a function of T/Tc in comparison
to the long range part of the strong coupling as extracted from Ref. [36] from the free energy
of a quark-antiquark pair in quenched lQCD (for Nτ = 8).
The width γ is adopted in the form γ ∼ g2T ln g−1 [37] or, equivalently, in terms of
M [30], as
γ(T ) =
M2(T )
(M(T )/T )2
, (5)
where c = 14.4 (from [4]) is related to a magnetic cut-off. In case of the pure Yang-
Mills sector of QCD the physical processes contributing to the width γ are both
gg ↔ gg scattering as well as splitting and fusion reactions gg ↔ g or gg ↔ ggg,
ggg ↔ gggg etc. Note that the ratio γ(T )/M(T ) ∼ g ln(c/g2) approaches zero only
asymptotically for T → ∞ such that the width of the quasiparticles is comparable to
the mass for all practical energy scales on earth; the ratio γ(T )/M(T ) drops below
0.5 only for temperatures T > 1.25 · 105 Tc (for the parameters given above).
For the choice (2) for the spectral function the scalar effective propagator reads,
∆dqp(ω,p) =
ω2 − p2 −M2 + 2iγω
, (6)
which can easily be separated into real and imaginary parts. The entropy density (1)
then reads explicitly [31],
sdqp(T ) = dg
(2π)3
− ln(1− e−ωp/T ) +
n(ωp/T )
(2π)3
arctan(
ω2p − ω2
2γω(ω2p − ω2)
(ω2p − ω2)2 + 4γ2ω2
, (7)
using ωp =
p2 +M2. The first line in (7) corresponds to the familiar on-shell quasi-
particle contribution s0 while the second line in (7) corresponds to the contribution
originating from the finite width γ of the quasiparticles and is positive throughout
but subleading (see below).
The pressure P now can be evaluated from
by integration of s over T , where from now on we identify the ’full’ entropy density s
with the quasiparticle entropy density sdqp. Note that for T < Tc the entropy density
drops to zero (with decreasing T ) due to the high quasiparticle mass and the width γ
vanishes as well because the interaction rate in the very dilute quasiparticle system
becomes negligible. Since the pressure for infinitely heavy (noninteracting) particles
also vanishes the integration constant for the pressure P - when integrating (8) -
may safely be assumed to be zero, too.
The energy density ǫ then follows from the thermodynamical relation [34,38]
ǫ = Ts− P (9)
and thus is also fixed by the entropy s(T ) as well as the interaction measure
W (T ) := ǫ(T )− 3P (T ) = Ts− 4P (10)
that vanishes for massless and noninteracting degrees of freedom.
In Ref. [4] a detailed comparison has been presented with the lattice results from
Ref. [39] for the pure gluonic sector to the quasiparticle entropy density (7) for the
parameters given above. The agreement with the lattice data is practically perfect
[4,30]. Needless to point out that also P (T ), ǫ(T ) and W (T ) well match the lattice
QCD results for 1 ≤ T/Tc ≤ 4 [4,31] due to thermodynamical consistency. The same
parameters are also adopted for the following calculations.
2.2 Time-like and space-like quantities
For the further argumentation it is useful to introduce the shorthand notation
P · · · = dg
(2π)3
2ω ρ(ω) Θ(ω)n(ω/T ) Θ(±P 2) · · · (11)
with P 2 = ω2 − p2 denoting the invariant mass squared. The Θ(±P 2) function in
(11) separates time-like quantities from space-like quantities and can be inserted for
any observable of interest.
As the first quantity we consider the entropy density (7). Its time-like contribution
is almost completely dominated by the first line in (7) - that corresponds to the on-
shell quasiparticle contribution s0 - but also includes a small contribution from the
second line in (7) which is positive for T below about 1.5 Tc and becomes negative for
larger temperature. This time-like part s+ is shown in Fig. 2 by the dotted blue line
(multiplied by (Tc/T )
3). The second line in (7) - as mentioned above - corresponds
to the contribution originating from the finite width γ of the quasiparticles and also
has a space-like part s− which is dominant (for the second line in (7)) and displayed
in Fig. 2 by the lower red line (multiplied by (Tc/T )
3). Though s− is subleading in
the total entropy density s = s+ + s− (thick solid green line in Fig. 2) it is essential
for a proper reproduction of s(T ) close to Tc (cf. [31]). Note that the total entropy
density s is not very different from the Stefan Boltzmann entropy density sSB for
T > 2Tc as shown in Fig. 2 by the upper thin line (multiplied by (Tc/T )
2 4 6 8 10
=0.26 GeV
Fig. 2. The time-like contribution to the entropy density s+ (dotted blue line), the space-like
contribution s− (lower red line) and the total entropy density s = s++s− (thick solid green
line) as a function of T/Tc. All quantities have been multiplied by the dimensionless factor
(Tc/T )
3) assuming Tc = 0.26 GeV for the pure gluonic system [40]. The upper solid black
line displays the Stefan Boltzmann limit sSB for reference.
Further quantities of interest are the quasiparticle ’densities’
N±(T ) = T̃r± 1 (12)
that correspond to the time-like (+) and space-like (-) parts of the integrated distri-
bution function. Note that only the integral of N+ over space has a particle number
interpretation. In QED this corresponds to time-like photons (γ∗) which are virtuell
in intermediate processes but can also be seen asymptotically by dileptons (e.g. e+e−
pairs) due to the decay γ∗ → e+e− [17].
A scalar density Ns, which is only defined in the time-like sector, is given by
Ns(T ) = T̃r
and has the virtue of being Lorentz invariant. Moreover, a scalar density can easily
be computed in transport approaches for bosons and fermions [17,41] which is of
relevance for the argumentation in Section 3.
0 2 4 6 8 10
0 2 4 6 8 10
=0.26 GeV
 T/TC
Fig. 3. Upper part: The scalar density Ns (lower orange line), the time-like density N
(blue line), the space-like quantity N− (red line) and the sum N = N+ +N− (thick solid
green line) as a function of T/Tc assuming Tc = 0.26 GeV for the pure gluonic system
[40]. The upper solid black line displays the Stefan Boltzmann limit NSB for reference. All
quantities are multiplied by the dimensionless factor (Tc/T )
3. Lower part: The ratio of the
scalar density Ns to the time-like density N
+ as a function of the scaled temperature T/Tc.
The actual results for the different ’densities’ (multiplied by (Tc/T )
3) are displayed
in the upper part of Fig. 3 where the lower orange line represents the scalar density
Ns, the blue line the time-like density N
+, the red line the space-like quantity N−
and the thick solid green line the sum N = N++N− as a function of T/Tc assuming
(as before) Tc = 0.26 GeV for the pure gluonic system [40]. It is seen that N
substantially smaller than N− in the whole temperature range up to 10 Tc where it
is tacitly assumed that the DQPM also represents lQCD results for T > 4Tc, which
is not proven explicitly, but might be expected due to the proper weak coupling
limit of (3), (5) (cf. Fig. 1). The application of the DQPM to 10 Tc is presented
in Fig. 3 since the initial state at Large Hadron Collider (LHC) energies might be
characterized by a temperature above 4 Tc; note that the properties of the partonic
phase will be explored from the experimental side in the near future at LHC. Quite
remarkably the quantity N follows closely the Stefan Boltzmann limit NSB for a
massless noninteracting system which is given in Fig. 3 by the upper thin solid line
and has the physical interpretation of a gluon density. Though N differs by less than
15% from the Stefan Boltzmann (SB) limit for T > 2TC the physical interpretation
is essentially different! Whereas in the SB limit all gluons move on the light cone
without interactions only a small fraction of gluons can be attributed to quasiparticles
with density N+ within the DQPM that propagate within the lightcone. The space-
like part N− corresponds to ’gluons’ exchanged in t-channel scattering processes
and thus cannot be propagated explicitly in off-shell transport approaches without
violating causality and/or Lorentz invariance.
The scalar density Ns follows smoothly the time-like density N
+ as a function of
temperature which can be explicitly seen in the lower part of Fig. 3 where the ratio
+ is shown versus T/Tc. Consequently, the scalar density Ns uniquely relates
to the time-like density N+ or the temperature T in thermal equilibrium which will
provide some perspectives for a transport theoretical treatment (see Section 3).
The separation of N+ and N− so far has no direct dynamical implications except
for the fact that only the fraction N+ can explicitly be propagated in transport as
argued above. Thus we consider the energy densities,
(T ) = T̃r± ω , (14)
that specify time-like and space-like contributions to the quasiparticle energy density.
It is worth pointing out that the quantity T00 = T
00 + T
00 in case of a conventional
quasi-particle model with vanishing width γ in general is quite different from ǫ in (9)
because the interaction energy density in this case is not included in (14), i.e.
T00 = T
(2π)3
2ω δ(ω2 −M2 − p2) Θ(ω) Θ(±P 2)n(ω/T ) ω (15)
since ω2 − p2 = M2 = P 2 > 0 due to the mass-shell δ-function.
How does the situation look like in case of dynamical quasiparticles of finite width?
To this aim we consider the integrand in the energy density (14) which reads as (in
spherical momentum coordinates with angular degrees of freedom integrated out)
I(ω, p) =
p2 ω2 ρ(ω, p2)n(ω/T ) . (16)
Here the integration is to be taken over ω and p from 0 to ∞. The integrand I(ω, p) is
shown in Fig. 4 for T = 1.02Tc (l.h.s.) and T = 2Tc (r.h.s.) in terms of contour lines.
For the lower temperature the gluon mass is about 0.91 GeV and the width γ ≈ 0.15
GeV such that the quasiparticle properties are close to a ρ-meson in free space. In
this case the integrand I(ω, p) is essentially located in the time-like sector and the
integral over the space-like sector is subdominant. This situation changes for T = 2Tc
where the mass is about 0.86 GeV while the width increases to γ ≈ 0.56 GeV. As one
observes from the r.h.s. of Fig. 4 the maximum of the integrand is shifted towards
the line ω = p and higher momentum due to the increase in temperature by about
a factor of two; furthermore, the distribution reaches far out in the space-like sector
due to the Bose factor n(ω/T ) which favors small ω. Thus the relative importance of
the time-like (+) part to the space-like (-) part is dominantly controlled by the width
γ - relative to the pole mass - which determines the fraction of T−00 with negative
invariant mass squared (P 2 < 0) relative to the time-like part T+00.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
I(ω,p)
T=1.02 T
time-like
space-like
p [GeV/c]
0 1 2 3 4 5
5 p=ω
I(ω,p)
T=2 T
time-like
space-like
p [GeV/c]
Fig. 4. The integrand I(ω, p) (16) for T = 1.02Tc (l.h.s.) and T = 2Tc (r.h.s.) in terms of
contour lines. The straight (blue) line (ω = p) separates the lime-like from the space-like
sector. Note that for a convergence of the energy density integral the upper limits for ω
and p have to be increased by roughly an order of magnitude compared to the area shown
in the figure.
The explicit results for the quasiparticle energy densities T+00 and T
00 are displayed
in Fig. 5 by the dashed blue and dot-dashed red lines (multiplied by (TC/T )
respectively. As in case of N+ and N− the space-like energy density T−00 is seen
0 2 4 6 8 10
0.9 1.0 1.1 1.2 1.3 1.4 1.5
=0.26 GeV
=0.26 GeV
Fig. 5. Upper part: The time-like energy density T+
(dashed blue line), the space-like energy
density T−
(dot-dashed red line) and the total energy density T00 = T
(thick solid
green line) as a function of T/Tc. The thin black line displays the energy density ǫ(T/Tc)
from (9); it practically coincides with T00 within the linewidth and is hardly visible. All
densities are multiplied by the dimensionless factor (Tc/T )
4 in order to divide out the
leading temperature dependence. Lower part: Same as the upper part in order to enhance
the resolution close to Tc.
to be larger than the time-like part T+00 for all temperatures above 1.05 Tc. Since
the time-like part T+00 corresponds to the independent quasiparticle energy density
within the lightcone, the space-like part T−00 can be interpreted as an interaction
density V if the quasiparticle energy T00 matches the total energy density ǫ(T ) (9)
as determined from the thermodynamical relations (8) and (9). In fact, the DQPM
yields an energy density T00 - adding up the space-like and time-like parts - that
almost coincides with ǫ(T ) from (9) as seen in Fig. 5 where both quantities (multiplied
by (TC/T )
4) are displayed in terms of the thin black and thick solid green lines,
respectively; actually both results practically coincide within the linewidth for T >
2Tc. An explicit representation of their numerical ratio gives unity within 2% for
T > 2Tc; the remaining differences can be attributed to temperature derivatives
∼ d/dT (ln(γ/E)) etc. in order to achieve thermodynamic consistency but this is
not the primary issue here and will be discussed in a forthcoming study [42]. The
deviations are more clearly visible close to Tc (lower part of Fig. 2) where the variation
of the width and mass are most pronounced. However, for all practical purposes one
may consider T00(T ) ≈ ǫ(T ) and separate the kinetic energy density T+00 from the
potential energy density T−00 as a function of T or - in equilibrium - as a function of
the scalar gluon density Ns or N
+, respectively.
3 Dynamics of time-like quasiparticles
Since in transport dynamical approaches there are no thermodynamical Lagrange
parameters like the inverse temperature β = T−1 or the quark chemical potential
µq, which have to be introduced in thermodynamics in order to specify the average
values of conserved quantities (or currents in the relativistic sense), derivatives of
physical quantities with respect to the scalar density ρs = Ns (or time-like gluon
density ρg = N
+) are considered in the following (cf. Ref. [43]). As mentioned above
one may relate derivatives in thermodynamic equilibrium via,
, (17)
if the volume and pressure are kept constant. For example, a numerical evaluation
of dρs/d(T/Tc) gives
d(T/Tc)
− a2 exp(−b(
)) (18)
with b= 5, a1 = 1.5fm
−3 and a2 = 104fm
−3, which follows closely the quadratic
scaling in T/Tc as expected in the Stefan Boltzmann limit. The additional exponential
term in (18) provides a sizeable correction close to Tc. The approximation (18) may
be exploited for convenient conversions between ρs and T/Tc in the pure gluon case
but will not be explicitly used in the following.
The independent quasiparticle energy density TK := T
00 and potential energy density
V := T−00 now may be expressed as functions of ρs (or ρg) instead of the temperature
T . The interaction energy density then might be considered as a scalar energy density
which - as in the nonlinear σ-model for baryonic matter [44] - is a nonlinear function
of the scalar density ρs. As in case of nuclear matter problems the scalar density
ρs does not correspond to a conserved quantity when integrating over space; it only
specifies the interaction density parametrically, i.e. V (ρs). Alternatively one might
separate V into parts with different Lorentz structure, e.g. scalar and vector parts
as in case of nuclear matter problems [44], but this requires additional information
that cannot be deduced from the DQPM alone.
0 2 4 6 8 10
0.9 1.0 1.1 1.2 1.3 1.4 1.5
Fig. 6. Upper part: The quasiparticle energy per degree of freedom TK/N
+ (dashed blue
line) and the space-like potential energy per degree of freedom V/N+ (dot-dashed red line)
as a function of T/Tc. All energies are multiplied by the dimensionless factor (Tc/T ). Lower
part: Same as the upper part in order to enhance the resolution close to Tc.
It is instructive to show the ’quasiparticle’ and potential energy per degree of free-
dom TK/N
+ and V/N+ as a function of e.g. N+, Ns or T/Tc. As one might have
anticipated the kinetic energy per effective degree of freedom is smaller than the
respective potential energy for T/Tc > 1.05 as seen from Fig. 6 where both quan-
tities are displayed as a function of T/Tc in terms of the dashed and dot-dashed
line, respectively. It is seen that the potential energy per degree of freedom steeply
rises in the vicinity of Tc whereas the independent quasiparticle energy rises almost
linearly with T . Consequently rapid changes in the density - as in the expansion
of the fireball in ultrarelativistic nucleus-nucleus collisions - are accompanied by a
dramatic change in the potential energy density and thus to a violent acceleration of
the quasi-particles. It is speculated here that the large collective flow of practically
all hadrons seen at RHIC [6] might be attributed to the early strong partonic forces
expected from the DQPM.
1 10 100
Fig. 7. The mean-field potential U(N+) = U(ρg) as a function of the time-like gluon density
N+ = ρg in comparison to the fit (19) (solid blue line). The densities N
+= 1, 1.4, 5, 10,
50, 100 fm−3 correspond to scaled temperatures of T/Tc ≈ 1.025, 1.045, 1.25, 1.5, 2.58,
3.25, respectively (cf. Fig. 3).
In order to obtain some idea about the mean-field potential Us(ρs) (or U(ρg) in the
rest frame) one can consider the derivative dV/ρs = Us(ρs) or dV/N
+ = U(N+) =
U(ρg). The latter is displayed in Fig. 7 as a function of N
+ = ρg and shows a distinct
minimum at ρg ≈ 1.4 fm−3 which corresponds to a temperature T ≈ 1.045Tc. The
actual numerical results can be fitted by the expression,
U(ρg) =
≈ 39 e−ρg/0.31 + 2.93 ρ0.21g + 0.55 ρ
g [GeV] , (19)
where ρg is given in fm
−3 and the actual numbers in front carry a dimension in order
to match to the proper units of GeV for the mean-field U . By analytical integration
of (19) one obtains a suitable approximation to V (ρg). The approximation (19) works
sufficiently well as can be seen from Fig. 7 - showing a comparison of the numerical
derivative dV/dN+ with the fit (19) in the interval 0.7 fm−3 < N+ ≤ 300 fm−3
- such that one may even proceed with further analytical calculations. Note that
a conversion between the time-like quasiparticle density N+ = ρg and the scalar
density ρs is easily available numerically (cf. lower part of Fig. 3) such that derivatives
with respect to ρs are at hand, too; the latter actually enter the explicit transport
calculations [45] while derivatives with respect to ρg in the rest frame of the system
are more suitable for physical interpretation and will be used below.
Some information on the properties of the effective gluon-gluon interaction vgg may
be extracted from the second derivative of V with respect to ρg, i.e.
vgg(ρg) :=
≈ −125.8 e−ρg/0.31 + 0.615/ρ0.79g + 0.2/ρ
g [GeVfm
3], (20)
where the numbers in front have again a dimension to match the units of GeV
fm3. The effective gluon-gluon interaction vgg (20) is strongly attractive at low den-
sity 0.003 fm−3 < ρg and changes sign at ρg ≈ 1.4 fm−3 to become repulsive at
higher densities. Note that the change of quasiparticle momenta (apart from col-
lisions) will be essentially driven by the (negative) space-derivatives −∇U(x) =
−dU(ρg)/dρg ∇ρg(x) (or alternatively by −dUs(ρs)/dρs ∇ρs(x)). This implies that
the gluonic quasiparticles (at low gluon density) will bind with decreasing density,
i.e. form ’glueballs’ dynamically close to the phase boundary and repell each other
for ρg ≥ 1.4 fm−3. Note that color neutrality is imposed by color-current conser-
vation and only acts as a boundary condition for the quantum numbers of the
bound/resonant states in color space.
This situation is somehow reminiscent of the nuclear matter problem [44] where a
change in sign of the 2nd derivative of the potential energy density of nuclear matter
at low density indicates the onset of clustering of nucleons, i.e. to deuterons, tritons,
α-particles etc., which form the states of the many-body system at low nucleon
densities (and not a low density nucleon gas). This is easy to follow up for the
simplified nonrelativistic energy density functional ǫN for nuclear matter,
ǫN ≈ Aρ
ρ2N +
N , (21)
where the first term gives the kinetic energy density and the second and third term
correspond to attractive and repulsive interaction densities. For A ≈ 0.073GeV fm2,
B ≈ −1.3 GeV fm3 and C ≈ 1.78 GeV fm4 a suitable energy density for nuclear
matter is achieved; it gives a minimum in the energy per nucleon E/A = ǫN/ρN ≈
−0.016 GeV for nuclear saturation density ρ0N ≈ 0.168 fm−3. The mean-field potential
UN = BρN+Cρ
N has a minimum close to ρ
N such that the effective nucleon-nucleon
interaction strength vNN = B + 4/3Cρ
N changes from attraction to repulsion at
this density. Note that in the gluonic case the minimum in the mean-field potential
U (19) occurs at roughly 8 times ρ0N and the strength of the gluonic interaction is
higher by more than 2 orders of magnitude!
The confining nature of the effective gluon-gluon interaction vgg (20) becomes ap-
parent in the limit ρg → 0, where the huge negative exponential term dominates for
ρg > 0.003 fm
−3; for even smaller densities the singular repulsive terms take over.
Note, however, that the functional extrapolation of the fit (19) to vanishing gluon
density ρg has to be considered with care and it should only be concluded that the
interaction strength becomes ’very large’. On the other hand the limit ρg → 0 is only
academical because the condensation/fusion dynamically occurs for ρg ≈ 1 fm−3.
A straight forward way to model the gluon condensation or clustering to confined
glueballs dynamically (close to the phase transition) is to adopt a screened Coulomb-
like potential vc(r,Λ) with the strength
d3r vc(r,Λ) fixed by vgg(ρg) from (20) and
the screening length Λ from lQCD studies. For the ’dilute gluon regime’ (ρg < 1.4
fm−3), where two-body interactions should dominate, one may solve a Schrödinger
(or Klein-Gordon) equation for the bound and/or resonant states. This task is not
addressed further in the present study since for the actual applications (as in the
Parton-Hadron-String-Dynamics (PHSD) approach [45]) dynamical quark and anti-
quarks have to be included. The latter degrees of freedom do not change the general
picture very much for higher temperatures T > 2Tc but the actual numbers are dif-
ferent close to Tc since the quarks and antiquarks here dominate over the gluons due
to their lower mass. The reader is referred to an upcoming study in Ref. [42].
Some comments on expanding gluonic systems in equilibrium appear in place, i.e.
for processes where the total volume Ṽ and pressure P play an additional role. For
orientation we show the entropy per time-like particle s/N+ in Fig. 8 as a function of
N+ (upper) and T/Tc (lower part) which drops close to the phase boundary since the
quasiparticles become weakly interacting (cf. Fig. 6). Note that this is essentially due
to the low density and not due to the interaction strength (20); a decrease of the width
γ (as encoded in (5)) implies a decrease in the interaction rate! An expansion process
with conserved total entropy S = sṼ leads to a change in the total gluon number
N+Ṽ since s/N+ changes with density (or temperature) (Fig. 8). The same holds
for an expansion process with constant total energy ǫṼ since also ǫ/N+ is varying
with density (or temperature). Other scenarios involving e.g. S = P/T also involve
a change of the gluon number N+Ṽ during the cooling process such that reactions
like gg ↔ g, ggg ↔ gg etc. are necessary ingredients of any transport theoretical
approximation. We do not further investigate different expansion scenarios here since
the reactions g ↔ qq̄, i.e. the gluon splitting to a quark and antiquark as well as
the backward fusion process, are found to play a dominant role in the vicinity of the
phase transition as well as for higher temperatures [42,45].
4 Conclusions and discussion
The present study has provided a novel interpretation of the dynamical quasiparticle
model (DQPM) by separating time-like and space-like quantities for ’particle densi-
ties’, energy densities, entropy densities ect. that also paves the way for an off-shell
transport approach [45]. The entropy density s in (7) is found to be dominated by the
on-shell quasiparticle contribution (first line in (7)) (cf. [31]) while the space-like part
of the off-shell contribution (second line in (7)) gives only a small (but important)
enhancement (cf. Fig. 2). However, in case of the ’gluon density’ N = N+ +N− and
the gluon energy density T00 = T
00 the situation is opposite: here the space-like
1 2 3 4 5 6 7 8 9 10
1 10 100 1000
 T/TC
Fig. 8. The entropy per degree of freedom s/N+ as a function of N+ (upper part) or T/Tc
(lower part).
parts (N−, T−00) dominate over the time-like parts (N
+, T+00) except close to Tc where
the independent quasiparticle limit is approximately regained. The latter limit is a
direct consequence of the infrared enhancement of the coupling (4) close to Tc (in
line with the lQCD studies in Ref. [36] ) and a decrease of the width γ (5) when
approaching Tc from above.
Since only the time-like part N+ can be propagated within the lightcone the space-
like part N− has to be attributed to t-channel exchange gluons in scattering processes
that contribute also to the space-like energy density T−00. The latter quantity may
be regarded as potential energy density V . This, in fact, is legitimate since the
quasiparticle energy density T00 very well matches the energy density (9) obtained
from the thermodynamical relations. Only small deviations close to Tc indicate that
the DQPM in its straightforward application is not thermodynamically consistent.
However, by accounting for ’rearrangement terms’ in the energy density - as known
from the nuclear many-body problem [46] - full thermodynamical consistency may
be regained [42].
It is instructive to compare the present DQPM to other recent models. In the
PNJL 2 model [47] the gluonic pressure is build up by a constant effective potential
U(Φ,Φ∗;T ) which controls the thermodynamics of the Polyakov loop Φ. It is ex-
panded in powers of ΦΦ∗ with temperature dependent coefficients in order to match
lQCD thermodynamics. Thus in the PNJL there are no time-like gluons; the effec-
tive potential U(Φ,Φ∗;T ) stands for a static gluonic pressure that couples to the
quark/antiquark degrees of freedom. The latter are treated in mean-field approxi-
mation, i.e. without dynamical width, whereas the DQPM incorporates a sizeable
width γ.
Another approach to model lQCD thermodynamics has been suggested in Ref. [43]
and is based on an effective Lagrangian which is nonlinear in the effective quark and
gluon fields. In this way the authors avoid a parametrization of the interaction density
in terms of Lagrange parameters (T, µ) and achieve thermodynamical consistency.
The latter approach is closer in spirit to the actual interpretation of the DQPM and
may be well suited for an on-shell transport theoretical formulation. The on-shell
restriction here comes about since effective Lagrangian approaches should only be
evaluated in the mean-field limit which implies vanishing scattering width for the
quasiparticles. This is sufficient to describe systems is thermodynamical equilibrium,
where forward and backward interaction rates are the same, but might not provide
the proper dynamics out-of-equilibrium.
Some note of caution with respect to the present DQPM appears appropriate: the pa-
rameters in the effective coupling (4) and the width (5) have been fixed in the DQPM
by the entropy density (7) to lQCD results assuming the form (2) for the spectral
function ρ(ω). Alternative assumptions for ρ(ω) will lead to slightly different results
for the time-like density, energy densities etc. but not to a qualitatively different
picture. Independent quantities from lQCD should allow to put further constraints
on the more precise form of ρ(ω) such as calculations for transport coefficients [16];
unfortunately such lQCD studies are only at the beginning. A more important issue
is presently to extend the DQPM to incorporate dynamical quark and antiquark
degrees of freedom (as in [31]) in order to catch the physics of gluon splitting and
quark-antiquark fusion (g ↔ q + q̄, g + g ↔ q + q̄ + g) reactions [42,45].
Coming back to the questions raised in the Introduction concerning i) the appropri-
ate description of QCD thermodynamics within the DQPM and ii) the possibility to
develop a consistent off-shell partonic transport approach as well as iii) the perspec-
tives for a dynamical description of the transition from partonic to hadronic degrees
of freedom, we are now in the position to state: most likely ’Yes’.
2 Polyakov-loop-extended Nambu Jona-Lasinio
The author acknowledges valuable discussions with E. L. Bratkovskaya and A. Peshier.
Furthermore he likes to thank S. Leupold for a critical reading of the manuscript and
constructive suggestions.
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