A reanalysis of Finite Temperature SU(N) Gauge Theory
ABSTRACT We revise the $SU(N_c)$, $N_c=3,4,6$, lattice data on pure gauge theories at
finite temperature by means of a quasiparticle approach. In particular we
focus on the relation between the quasiparticle effective mass and the order
of the deconfinement transition, the scaling of the interaction measure with
$N^2_c 1$, the role of gluon condensate, the screening mass.

Article: Thermodynamics of the QuarkGluon Plasma in Terms of Quasiparticles and Polyakov Line Condensates
[Show abstract] [Hide abstract]
ABSTRACT: We study a model of quarkgluon plasma of 2+1 flavors Quantum Chromodynamics in terms of quasiparticles propagating in a condensate of Polyakov loops. The Polyakov loop is coupled to quasiparticles by means of a gaslike effective potential. This study is useful to identify the effective degrees of freedom propagating in the medium above the critical temperature. Our finding is that a dominant part of the phase transition dynamics is accounted for by the Polyakov loop, hence the thermodynamics can be described without the need for rapidly increasing quasiparticle masses as $T \rightarrow T_c$, at variance respect to standard quasiparticle models.Physical Review D 09/2013; · 4.69 Impact Factor  SourceAvailable from: Marco Ruggieri[Show abstract] [Hide abstract]
ABSTRACT: We study the interpretation of Lattice data about the thermodynamics of the deconfinement phase of SU(3) YangMills theory, in terms of gluon quasiparticles propagating in a background of a Polyakov loop. A potential for the Polyakov loop, inspired by the strong coupling expansion of the QCD action, is introduced; the Polyakov loop is coupled to tranverse gluon quasiparticles by means of a gaslike effective potential. This study is useful to identify the effective degrees of freedom propagating in the gluon medium above the critical temperature. A main general finding is that a dominant part of the phase transition dynamics is accounted for by the Polyakov loop dynamics, hence the thermodynamics can be described without the need for diverging or exponentially increasing quasiparticle masses as $T \rightarrow T_c$, at variance respect to standard quasiparticle models.Physical review D: Particles and fields 04/2012; 86(5).  SourceAvailable from: Marco Ruggieri[Show abstract] [Hide abstract]
ABSTRACT: In this talk we review, the quasiparticle description of the hot YangMills theories, in which the quasiparticles propagate in (and interact with) a background field related to Z(N)lines. We compare the present description with a more common one in which the effects of the Z(N)lines are neglected. We show that it is possible to take into account the nonperturbative effects at the confinement transition temperature even without a divergent quasiparticle mass.09/2012;
Page 1
arXiv:1105.5902v1 [hepph] 30 May 2011
A reanalysis of Finite Temperature SU(N) Gauge Theory
P. Castorina1,2, V. Greco1,3, D. Jaccarino1,3,4, D. Zappal` a2
1Dipartimento di Fisica, Universit` a di Catania, Via Santa Sofia 64, I95123 Catania, Italy.
2INFN, Sezione di Catania, I95123 Catania, Via Santa Sofia 64, Italy.
3INFN, Laboratori Nazionali del Sud, I95123 Catania, Italy.
4Scuola Superiore di Catania  Via Valdisavoia 9 I95123 Catania, Italia
(Dated: May 31, 2011)
Abstract
We revise the SU(Nc), Nc= 3,4,6, lattice data on pure gauge theories at finite temperature by means of
a quasiparticle approach. In particular we focus on the relation between the quasiparticle effective mass
and the order of the deconfinement transition, the scaling of the interaction measure with N2
of gluon condensate, the screening mass.
c− 1, the role
PACS numbers:
I. INTRODUCTION
A careful investigation of the quarkgluon
plasma phase needs an understanding of the de
tails of the deconfinement transition which oc
curs above a critical temperature Tcand in this
respect the data provided by lattice simulations
represent the best tool for testing various dy
namical models close to Tc.
QCD simulations, interesting indications on the
gluonic sector can be drawn from lattice stud
ies on the high temperature transition of pure
nonabelian gauge theories SU(Nc), where all
difficulties related to the presence of fermions
and to the details of the chiral simmetry break
ing are absent. Recent lattice simulations on
SU(Nc) gauge theories at finite temperature T
and for large number of colors Nc [1, 2] are
now available and they show a peculiar scaling
of the interaction measure, ∆ = (ǫ − 3p)/T4(
ǫ =energy density and p =pressure), with the
number of gluons N2
c−1. Moreover, in the range
1.1 Tc< T < 4 Tc, the interaction measure has
a O(1/T2) behavior which implies a O(T2) con
tribution in ǫ − 3p [3].
The previous features have interesting con
sequences on the number of effective degrees of
freedom and on the role of the gluon conden
sate above the critical temperature.
served approximate scaling of ∆ with N2
Nc≥ 3, suggests: 1) a quasiparticle behavior of
Besides the full
The ob
c−1 for
the effective degrees of freedom, with the typi
cal degeneracy, N2
c− 1, of the gluons and with
an effective, temperature dependent, mass that
turns out to be divergent or, at least, very large
at Tc[4–7]; 2) the presence of the same degener
acy factor in the gluon condensate contribution,
if any, to ∆; 3) the same proportionality to the
number of gluons N2
c−1 of the dynamical mech
anism which produces the O(T2) contribution.
However, a more accurate analysis shows that
the scaling of interaction measure and of the
other thermodynamic observables with N2
is not exact [2].More precisely one can con
sider two different ranges of temperature. Near
the transition temperature , i.e. for Tc< T <
(1.05 −1.1)Tc, the scaling with N2
violated as shown in Fig.1, whereas above the ∆
peak temperature the scaling is almost exact ,
see Fig.2.
c− 1
c−1 is clearly
In this
SU(3,4,6) lattice data in [2, 8], by means of a
quasiparticle approach in order to discuss the
origin of the deviations from the exact scaling
in relation to the critical behavior. The general
formalism is discussed Sec. II; the comparison
with lattice data is presented in Sec. III; the
role of the gluon condensate is addressed in Sec.
IV and in Sec. V we draw our conclusions.
brief report we reconsiderthe
1
Page 2
11.02 1.04
T/Tc
1.061.08
0
0.1
0.2
0.3
0.4
∆/(Νc
2−1)
SU(3)
SU(4)
SU(6)
FIG. 1: ∆/(N2− 1) close to Tc.
1.5
2
2.5
T/Tc
3
3.5
4
0
0.1
0.2
0.3
0.4
∆ / (Νc21)
SU(3)
SU(4)
SU(6)
FIG. 2: ∆/(N2− 1) versus T/Tcabove the ∆ peak.
II. GENERAL FORMALISM
The partition function for the very simple
case of free quasiparticles in a volume V , at
temperature T and with temperature dependent
mass m(T) is
lnZ(T,V ) = 2V (N2
(2π)3ln
c− 1) ×
?k2+ m2(T)/T
?
d3k
?
fT(k) exp
??
??
(1)
where fT(k) is the distribution
fT(k) =
?
exp
??
?k2+ m2(T)/T
?
− 1
?−1
(2)
and all thermodynamical quantities are obtained
by deriving Eq. (1). For our purpose, the energy
density ǫ, the pressure p, the entropy density s
are respectively, ǫ = (T2/V ) ∂ lnZ/∂T , p =
T ∂ lnZ/∂V and s = (ǫ+p)/T. The interaction
measure ∆ is directly obtained from ǫ and p as
∆ = (ǫ − 3p)/T4.
Obviously the temperature dependence of the
mass must be taken into account when differen
tiating with respect to T. Note also that the
additional effect of a temperature independent
bag pressure (gluon condensate) B corresponds
to the changes p → p − B, and ǫ → ǫ + B, with
no change in the entropy density s.
The factor 2 in front of the color multiplicity
factor N2
c−1 in Eq. (1), corresponds to the num
ber of polarization degrees of freedom. In gen
eral, the representations of the Poincar´ e group of
massive and massless particles in this case would
suggest that our massive physical constituents
carry three, rather than two, spin degrees of free
dom. However, this is valid for free particles and,
in fact, the comparison of all predicted thermo
dynamic quantities with the observed high tem
perature lattice results clearly shows much bet
ter agreement when only two polarization states
are considered.
In particular all lattice QCD results, as shown
also in Fig.5, hint at an asymptotic limit of the
ǫ/T4consistent with 2(N2
dom. In other words, a simple shift to massive
gluons with three spin degrees of freedom can
not satisfactorily explain the effects of the inter
action apparently still present in the gluon gas
above the critical temperature.
It is well known that gauge symmetry for
bids a mass term in the lagrangian for the el
ementary gluons and, in order to preserve the
symmetry, one can expect to observe the gener
ation of mass through a dynamical mechanism,
such as the Schwinger mechanism [9] in which
the mass comes from the appearance of a pole in
the selfenergy. In fact this effect has been ex
plicitly pointed out and it has been argued that
the longitudinal polarization component could
be canceled by the scalar massless pole [10–12].
On the other hand, in the modified Hard
Thermal Loop perturbation theory approach,
where each order already includes some aspects
of gluon dressing and which leads to a rather
rapid convergence of the expansion, the con
tribution of longitudinal gluons vanishes in the
limit g → 0, and, in particular, one also obtains
c− 1) degrees of free
2
Page 3
the right number of degrees of freedom for the
StefanBoltzmann form [13].
Moreover, from a comparison of the lattice
glueball spectrum with the predictions of con
stituent models it has recently been argued [14]
that massive gluons should in fact be trans
versely polarized, since two massless gluons can
not combine to form a longitudinally polarized
massive gluon [15]. According to these indica
tions we limit ourselves to consider just to polar
ization degrees of freedom for the effective quasi
particle in Eq. (1).
Let us now turn to the most important in
gredient in our approach, that is the effective
temperature dependent mass m(T) which con
tains the nonperturbative dynamics. Previous
analyses [4–6] show that m(T) strongly increases
near Tcand a qualitative explanation of it has
been outlined in [16]. To illustrate this aspect
one describes the mass of the quasigluon in the
strongly coupled region as the energy contained
in a region of volume Vcor whose characteristic
size is given by the correlation range ξ, so that
in three spatial dimensions one gets (η is the
anomalous dimension and t is the reduced tem
perature t = T/Tc):
m(t) ≃ ǫ(t)Vcor= ǫ(t)
In the case of a second order phase transi
tion, the correlation length shows the power law
divergence ξ(t) = (t − 1)−νat t = 1, which in
dicates that the associated fluctuations have an
infinite range at criticality, and the correspond
ing component of the energy density vanishes as
ǫ(t) ≃ (t − 1)1−αwhere α is the specific heat
critical exponent. In this case Eq. (3) predicts
a power law divergence of the mass m(1). Then,
by focussing on the 3D Ising model universal
ity class, which includes the critical behavior of
the SU(2) gauge theory deconfinement transi
tion, one finds the value of the critical exponent
for our mass: m(t) ≃ (t−1)−0.41In addition the
SU(2) critical behavior suggests that very close
to Tc, ∆ should have the form [17]
?
dr r2exp[−r/ξ(t)]
r1−η
(3)
∆ = Aτ(1 − τ)0.89+ Bτ
with τ = t−1and A, B constants and the expo
nent is given by 0.89 = 1 − α.
(4)
In such transitions m(t) close to t = 1 has
a power law divergent behavior which has to be
considered as an approximation for T near Tc.
By recalling that for t >> 1, the mass is ex
pected to grow linearly with the temperature,
which is the only dimensionful scale available, a
suitable ansatz for m(t) is
m(t) =
a
(t − 1)c+ bt,(5)
where a,b,c are constant parameters.
For the gauge groups SU(Nc), with Nc =
3,4,6 here considered, a first order phase transi
tion and consequently a finite correlation length,
is expected at Tcand power law behavior at criti
cality is modified by the finite scale ξ. However,
in the case of weak first order transitions one
should expect a behavior of the thermodynam
ical quantities at Tc not totally different from
that observed in second order transitions, and
therefore a finite but large correlation length and
a corresponding large m(1).
In particular, as discussed in [18], the thermo
dynamical quantities approach Tc (from larger
values of the temperature t > 1) as in a sec
ond order phase transition with the critical tem
perature shifted to a lower value: 1 → δ with
0 ∼ (1 − δ) << 1. According to this sugges
tion the ansatz (5) for weakly first order phase
transition must be changed into
m(t) =
a
(t − δ)c+ bt (6)
which yields a large but finite value of the effec
tive mass and nonvanishing interaction measure
at the transition point t = 1.
The quasiparticle mass m(t) should not to be
confused with the screening mass mD(T). The
relation between m(t) and mD(T), has been clar
ified in [4] where it is shown that
mD=g2Nc
π2T
?∞
0
dk k2f2
T(k)exp
?
?k2+ m(T)2
T
(7)
and the leading order QCD coupling g2is evalu
ated at the average, M2, over the squared quasi
particle momenta, i.e.
M2(T) =4
3
?dk k4fT(k)
?dk k2fT(k)
(8)
3
Page 4
Nc
3
4
6
cm(Tc)/Tc
6.33 (6.64)
6.02 (7.68)
4.33 (5.91)
δχ2
dof
0.5 (0.46)
0.5 (0.35)
0.5 (0.33)
0.940 (0.952)
0.944 (0.982)
0.888 (0.969)
2.6 (2.0)
7.7 (0.8)
6.5 (5.8)
TABLE I: Mass, shift δ, and χ2/dof from the fit to
SU(3,4,6) lattice data with a,b,δ as free parameters
and c = 0.5 fixed. In brackets, the same quantities
from the fit with a,b,c,δ as free parameters.
To illustrate this point in the next Section we
display m(t) and mD(T) which show totally dif
ferent behaviors when approaching Tc.
III. ANALYSIS OF THE LATTICE DATA
Now we check the simple model outlined in
Sect. II against the lattice data for SU(3,4,6)
[2, 8] . These theories undergo a weakly first
order transition and we shall resort to the large
but finite mass in Eq. (6) for computing the
various thermodynamical quantities.
In addition, as discussed above, the singular
behavior of the 3D Ising model corresponds to
the value or the critical exponent in Eq. (5) c =
0.41, which is close to the mean field exponent
c = 0.5. Therefore, it seems reasonable to verify
whether the mean field behavior produces a good
fit to the data.
In Table 1 we collect the values of the mass
m(1), of the shift δ and of the χ2per degree
of freedom (dof), obtained by fitting the data
with a,b,δ free parameters and c = 0.5 fixed.
As an example the SU(3) values of the other
parameters a and b, turn out to be a/Tc= 1.42
and b/Tc = 0.53. The critical mass decreases
when Nc is increased and the values of the χ2
indicate a reasonably good agreement with the
data.
However no clear dependence on Nccan be
traced in the values of m(t) and this is expected
because the dependence of the interaction mea
sure ∆, shown in Figs. 1, 2, on m(t) is highly
nonlinear. On the other hand the different be
havior of ∆ in Fig. 1, i.e. near the critical tem
perature, is essentially due to the dependence of
m(Tc) on Ncgiven in Table I.
It is then remarkable that the simple
1
1.5
2
2.5
3
3.5
T/Tc
0
2.5
5
7.5
10
12.5
∆
SU(3)
SU(4)
SU(6)
FIG. 3: The interaction measure as obtained from a
fit to the SU(3,4,6) lattice data with c = 0.5 fixed.
1
1.5
2
2.5
3
3.5
T/Tc
0
2.5
5
7.5
10
12.5
∆
SU(3)
SU(4)
SU(6)
FIG. 4: The same fit as in Fig.3 but optimizing also
on the parameter c.
1
1.5
2
2.5
3
3.5
4
T/Tc
0
2
4
6
8
p / T4
SU(6)
SU(4)
SU(6)
SB Limits
FIG. 5: Fit to the pressure of SU(3,4,6) optimizing
also on the parameter c.
4
Page 5
parametrization of m(t) in Eq.(5,6) produces the
correct behavior of the interaction measure ∆
which is plotted in Fig. 3.
An improvement on these results is obviously
obtained by releasing the constraint on c and
leaving it as another free parameter of the fit.
Results are again reported in Table 1 (in brack
ets) and the interaction measure ∆ and the pres
sure p are plotted in Figs. 4, 5. In this case the
χ2/dof shows an excellent agreement with the
data.
We note that even if we have not forced any
specific dependence of the mass on the coupling
g and on Ncthe results of the fit shown in Fig.6
manifests an independence of m(T) on the the
SU(Nc) gauge group for temperature above the
peak in the interaction measure. This could be
expected if one considers the parametric depen
dence of the mass in a perturbative approach,
m2
the coupling, g2≃ 1/Nc.
Moreover, the comparison between the gluon
effective mass m(t) and the Debye screeening
mass mD(T) according to Eqs.
played in Fig. 6. Finally in Fig. 7 c2
reported.
As mentioned in the Introduction, lattice re
sults very close to Tcseem to scale approximately
with ≃ Nc(N2
Accordingly also the behavior of m(t) very close
to Tcbreaks its independence on Nc, even if not
clearly visible from Fig. 6.
This signals that the t’Hooft condition of
QCD at large Nc, i.e. g2≃ 1/Nc, is violated
at the transition region. The effective mass at
Tcdepends on Ncin a way which is not consis
tent with the standard O(1/N2
QCD at large NC. This interesting aspect will
be reconsidered in the next Section.
g≃ g2NcT2, along with the t’Hooft scaling of
(7,8), is dis
s= ∂p/∂ǫ is
c− 1) rather than with (N2
c− 1).
c) corrections in
IV. THE ROLE OF THE GLUON CONDEN
SATE
A fundamental ingredient of the non per
turbative QCD regime is the gluon condensate
which has been evaluated by lattice simulations
at zero and finite temperature, in quenched and
1
1.5
2
2.5
T/Tc
3
3.5
4
0
1
2
3
4
5
m / T
SU(3)  Debye Mass
Gluon
SU(4)  Debye Mass
Gluon
SU(6)  Debye Mass
Gluon
FIG. 6: The screening mass MDcompared with m(t)
as obtained from the fit to the lattice data.
012345
T/Tc
0
0.1
0.2
0.3
cs
2
SU(3)
SU(4)
SU(6)
FIG. 7: The speed of sound as obtained from the fit
to the lattice data.
unquenched QCD [19] . It turns out that:
1) for T < Tcthe gluon condensate is almost
T independent;
2) for T > Tc the chromoelectric part of
the gluon condensate quickly decreases to zero
whereas the magnetic one is constant. This cor
responds to the deconfinement transition.
In QCD the gluon condensate is related to the
trace anomaly of the energymomentum tensor
by the general expression:
θµ
µ= 4B =β(g)
g
< Ga
µνGµνa> (9)
where B is the bag pressure, β(g) is the QCD
βfunction, a = 1,2,..,(N2
Above Tcits contribution to interaction mea
sure is about a half of the zero temperature
c− 1).
5
Page 6
value ( because the electricpart melts) and,
as discussed in ref.[16], a temperature inde
pendent gluon condensate/bag pressure is not
able to fit lattice data and in particular the be
havior ∆ ∗ T2≃ const. observed in the range
1.1Tc< T < 4Tc.
To explain this behavior one has to include
in B a term proportional to T2, as already sug
gested in [3]
B(T) = B0+ B1T2
(10)
Then, it is probably possible to fit the lattice
data because B1T2gives the correct O(1/T2) be
havior above the peak and at the same time B0
can be adjusted to optimize the curve below the
peak.
But, as ∆ scales with ≃ Nc(N2
and with (N2
c− 1) for larger temperature, the
scaling behavior of B0 and B1 with Nc should
be different. In particular the scaling near the
critical point can be understood as a violation of
the t’Hooft limit condition g2≃ 1/Ncbecause
B0≃β(g)
g
c− 1) near Tc
< Ga
µνGµνa>≃ g2γNc(N2
c−1) (11)
where the constant γ contains numerical factors
and an average, Ncindependent, condensate per
gluon. Therefore, ∆ ≃ Nc(N2
plies g2?= O(1/Nc) as seen before for the mass
term. However it must be remarked that there is
no indication in lattice computation [19] of the
T2behavior in Eq. (10) up to T ≃ 1.5Tc.
The points discussed above indicate that it
is very unlikely that the condensate alone could
explain the behavior of the interaction measure
∆. On the contrary, it is reasonable to treat
the condensate as a small additional piece which
has to be added to the quasiparticle contribu
tion to ∆, discussed in the previous Sections. In
fact lattice data allow the insertion of a physi
cally acceptable constant B0, with a (minor) role
only in the region near Tc, without qualitatively
changing the fit to the data shown in Section III.
c− 1) near Tcim
V. COMMENTS AND CONCLUSIONS
The previous results show that a quasi
particle approach, where the effective mass is
related with the features of the deconfinement
transition, gives a very good desciption of the in
teraction measure and of the thermodynamical
quantities for weak first order phase transition.
In our opinion this is not so surprising. In
deed, a quasiparticle approach means that the
relevant dynamics is contained in the twopoint
function and fluctuactions have a minor role. A
general framework to describe this behavior in
quantum field theory is the effective potential for
composite operators (CJT) [20] in the socalled
Gaussian approximation. It has been extensively
applied at finite temperature for scalar, fermion
and gauge theories and naturally leads to a first
order phase transition to restore the symmetries,
although this conclusion has to be confirmed by
other more reliable methods( ǫ expansion, lattice
simulations,etc).
We find that the scaling of the interaction
measure with (N2
c−1) is observed for T > 1.1Tc
and clearly violated near Tc. Accordingly, above
the peak temperature of the interaction mea
sure, the mass behavior is independent on Nc
in agreement with a perturbative parametric de
pendence on g2Ncand the 1/√Nct’Hooft scal
ing of the coupling which, on the other hand, is
broken very close to Tc. Moreover by combining
this aspect with the almost constant behavior of
∆∗T2in the range 1.1Tc< T < 4Tc, it turns out
difficult, in our opinion, to describe the interac
tion measure by a temperature dependent bag
pressure and/or gluon condensate above Tc. An
interesting point is to find the connection with
the confined phase below Tc which can be de
scribed by a glueball gas plus bag pressure [21]
It would be also useful to analyze the SU(2)
case because there is a second order phase tran
sition , with a corresponding divergent effective
mass at the critical point, and a small number
of colors. However the SU(2) data are quite old
and ,unfortunately, there does not seem to ex
ist any lattice study allowing an extrapolation
to the continuum, thus eliminating finite lattice
size effects [17].
The meaning of the effective gluon mass as
discussed in Sect. II, which is similar to a ”col
ored gluelump”, is probably related with the dy
namical mechanism of the transition, bubble nu
cleation for example, and a deeper understand
6
Page 7
ing in this direction would be extremely appeal
ing.
Following the same strategy here applied, the
next step will be the analysis of full QCD to
study the role of fermions in modifing the effec
tive gluon properties related to the deconfined
phase crossover.
Acknowledgements The authors thank M.P.
Lombardo and H. Satz, for useful comments.
This work has been partially supported by the
FIRB Research Grant RBFR0814TT provided
by the MIUR.
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