arXiv:1105.5902v1 [hep-ph] 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, I-95123 Catania, Italy.
2INFN, Sezione di Catania, I-95123 Catania, Via Santa Sofia 64, Italy.
3INFN, Laboratori Nazionali del Sud, I-95123 Catania, Italy.
4Scuola Superiore di Catania - Via Valdisavoia 9 I-95123 Catania, Italia
(Dated: May 31, 2011)
We revise the SU(Nc), Nc= 3,4,6, lattice data on pure gauge theories at finite temperature by means of
a quasi-particle approach. In particular we focus on the relation between the quasi-particle 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
A careful investigation of the quark-gluon
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
non-abelian 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 .
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 quasi-particle behavior of
Besides the full
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 . 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 ,
c−1 is clearly
SU(3,4,6) lattice data in [2, 8], by means of a
quasi-particle 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
FIG. 1: ∆/(N2− 1) close to Tc.
∆ / (Νc2-1)
FIG. 2: ∆/(N2− 1) versus T/Tcabove the ∆ peak.
II. GENERAL FORMALISM
The partition function for the very simple
case of free quasi-particles in a volume V , at
temperature T and with temperature dependent
mass m(T) is
lnZ(T,V ) = 2V (N2
c− 1) ×
where fT(k) is the distribution
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
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
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  in which
the mass comes from the appearance of a pole in
the self-energy. 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-
the right number of degrees of freedom for the
Stefan-Boltzmann form .
Moreover, from a comparison of the lattice
glueball spectrum with the predictions of con-
stituent models it has recently been argued 
that massive gluons should in fact be trans-
versely polarized, since two massless gluons can-
not combine to form a longitudinally polarized
massive gluon . 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 non-perturbative dynamics. Previous
analyses [4–6] show that m(T) strongly increases
near Tcand a qualitative explanation of it has
been outlined in . To illustrate this aspect
one describes the mass of the quasi-gluon 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 
∆ = Aτ(1 − τ)0.89+ Bτ
with τ = t−1and A, B constants and the expo-
nent is given by 0.89 = 1 − α.
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
(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 , 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
(t − δ)c+ bt (6)
which yields a large but finite value of the effec-
tive mass and non-vanishing interaction measure
at the transition point t = 1.
The quasi-particle 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  where it is shown that
and the leading order QCD coupling g2is evalu-
ated at the average, M2, over the squared quasi-
particle momenta, i.e.
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
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
non-linear. 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
FIG. 3: The interaction measure as obtained from a
fit to the SU(3,4,6) lattice data with c = 0.5 fixed.
FIG. 4: The same fit as in Fig.3 but optimizing also
on the parameter c.
p / T4
FIG. 5: Fit to the pressure of SU(3,4,6) optimizing
also on the parameter c.
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
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,
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
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) corrections in
IV. THE ROLE OF THE GLUON CONDEN-
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
m / T
SU(3) - Debye Mass
SU(4) - Debye Mass
SU(6) - Debye Mass
FIG. 6: The screening mass MDcompared with m(t)
as obtained from the fit to the lattice data.
FIG. 7: The speed of sound as obtained from the fit
to the lattice data.
unquenched QCD  . It turns out that:
1) for T < Tcthe gluon condensate is almost
2) for T > Tc the chromo-electric 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 energy-momentum tensor
by the general expression:
µ= 4B =β(g)
where B is the bag pressure, β(g) is the QCD
β-function, a = 1,2,..,(N2
Above Tcits contribution to interaction mea-
sure is about a half of the zero temperature
value ( because the electric-part melts) and,
as discussed in ref., 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 
B(T) = B0+ B1T2
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
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
c− 1) near Tc
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  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 quasi-particle 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 quasi-particle approach means that the
relevant dynamics is contained in the two-point
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)  in the so-called
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
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 
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 .
The meaning of the effective gluon mass as
discussed in Sect. II, which is similar to a ”col-
ored glue-lump”, is probably related with the dy-
namical mechanism of the transition, bubble nu-
cleation for example, and a deeper understand-
ing in this direction would be extremely appeal- Download full-text
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
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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