Proceedings of International Conference on Physics in ... - KEK

confinement-deconfinement transition. It is convenient to
introduce the following quantities [8];
⟨
β(g)
M0(T ) =
2g Ga µνG aµν
⟩
, (3)
T
(
uαuβ − 1
4 gαβ
) ⟨
M2(T ) = −ST G a αµG aµ
⟩
β . (4)
T
These denote the decomposition <strong>of</strong> the gluonic operator
into the trace anomaly part and traceless and symmetric
one, i.e., the scalar gluon condensate and the twist-2 operator
which stand for the leading contribution to the OPE.
Note that the twist-2 operator must be taken into account
for consistency in the case <strong>of</strong> medium in which Lorentz
invariance is absent. The M0 and M2 in the pure gauge
theory are related to the energy density ε(T ) and pressure
p(T ) which were obtained in lattice calculations as
M0 = ε − 3p and M2 = ϵ + p. Then taking the one-loop
expression <strong>of</strong> the beta function leads to the electric condensate
⟨
αs
π ∆E2⟩ =
T
2
11 − 2
3 Nf
M0(T ) + 3
4
α eff
s
π M2(T ). (5)
The equation <strong>of</strong> state is adopted from Ref. [9]. As for α eff
s ,
we use the effective coupling constant αqq(T ) which was
measured by heavy quark free energy [10]. Putting Nf =
0 into Eq. (5), we plot the temperature dependence <strong>of</strong> the
electric condensate near Tc = 264 MeV in Fig. 1. One
sees a rapid rise in the vicinity <strong>of</strong> the transition temperature
which reflects the first order phase transition <strong>of</strong> pure SU(3)
theory. Indeed this behavior could be related with that <strong>of</strong>
the space-time Wilson loop which shows change from the
area law to the perimeter law across Tc [11] through OPE
for the Wilson loop calculated by Shifman [12].
Putting Eq. (5) into Eq. (2), we obtained the mass shift
<strong>of</strong> J/ψ shown in Fig. 2. The downward mass shift occurs
abruptly in the vicinity <strong>of</strong> Tc, signaling the phase transition.
It reaches 40 MeV at Tc and 100 MeV at 1.05Tc.
We note, however, that applicability <strong>of</strong> this method for the
charmonium is questionable beyond this temperature [8].
RESONANCE GAS MODEL
Now we turn to the realistic case including light quarks.
It is known that QCD with physical quark mass exhibits
a crossover transition such that the change <strong>of</strong> the thermodynamic
quantity becomes smoother in the vicinity <strong>of</strong> Tc.
From an experimental point <strong>of</strong> view, the system produced
in heavy ion collisions will be QGP, which undergoes a
transition into a hadronic gas. The temperature and chemical
potential at the hadronization can be extracted from statistical
model analysis <strong>of</strong> particle ratios [13]. We consider
the electric condensate at these points which are just below
Tc.
There are two difficulties to extract the condensate from
the lattice data as done in the case <strong>of</strong> the pure gauge theory.
One is the so-called sign problem. Namely, lattice QCD
(α s /π)E 2 [GeV 4 ]
0.004
0.002
0
-0.002
-0.004
(αs /π)E
0.8 0.9 1
T/Tc 1.1 1.2
2
Figure 1: Electric condensates <strong>of</strong> pure gluonic case near
Tc. The value at low temperature limit is obtained from
that <strong>of</strong> the gluon condensate in vacuum, ⟨(αs/π)G 2 ⟩ =
(0.35GeV) 4 .
∆m [MeV]
0
-50
-100
-150
-200
-250
J/ψ
0.9 0.95 1 1.05
T/Tc 1.1 1.15 1.2
Figure 2: Mass shift <strong>of</strong> J/ψ from the second order Stark
effect with the electric condensate shown in Fig. 1.
cannot perform a simulation at finite chemical potential.
The other is that one has to separate gluonic contribution to
the thermodynamic quantities from those including quarks.
Therefore, we use a resonance gas model based on the linear
density approximation which has been used to estimate
the gluon condensate in the nuclear matter. We define M0
and M2 for a resonance gas, introduced in previous section,
as
M had
0 (T, µ) = ∑
ρi(T, µ)m
i=hadrons
0 i (6)
M had
2 (T, µ) = ∑
ρi(T, µ)miA
i=hadrons
i G. (7)
One sees this expression is linear in the number density <strong>of</strong>
i−th hadrons, ρi. If we put ρi as normal nuclear density
ρ0 = 0.17 fm −3 and pick up only nucleon contribution
in the summation, this formula reduces to the gluon condensates
in the nuclear matter [14]. In Eq. (6), the chiral
limit is taken for the mass <strong>of</strong> hadrons m 0 i to isolate
the gluonic contribution to the trace anomaly. The second
moment <strong>of</strong> the gluon distribution function A i G plays
a similar role that picks up the gluonic part <strong>of</strong> the twist-2
term. Here we take the all resonances given in the Particle
Data Group [15] into account. Masses in the chiral
limit, however, are not known for most hadrons. We use