Table Of ContentTHEORY OF QUARK-GLUON PLASMA
AND PHASE TRANSITION.
E.V.Komarov , Yu.A.Simonov
∗ †
8
0 State Research Center
0
Institute of Theoretical and Experimental Physics,
2
n Moscow, 117218 Russia
a
J
February 2, 2008
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2
]
h
p
Abstract
-
p
e Nonperturbative picture of strong interacting quark-gluon plasma
h is given based on the systematic Field Correlator Method. Equa-
[
tion of state, phase transition in density-temperature plane is derived
2
and compared to lattice data as well as subsequent thermodynamical
v
quantities of QGP.
1
5
2
2 1 Introduction
.
1
0
8 The perturbative exploring of quark-gluon plasma (QGP) has some difficul-
0 ties in describing the physics of QGP and phase transitions. However, it
:
v was realized 30 years ago that nonperturbative (np) vacuum fields are strong
i
X ([1]) and later it was predicted ([2]) and confirmed on the lattice ([3]) that
r the magnetic part of gluon condensate does not decrease at T > T and even
a c
grows as T4 at large T ([4]).
Therefore it is natural to apply the np approach, the Field Correlator
Method (FCM) ([5]) to the problem of QGP and phase transitions, which
was done in a series of papers ([6]-[10]). As a result one obtains np equation
of state (EoS) of QGP and the full picture of phase transition, including an
unbiased prediction for the critical temperature T (µ) for different number
c
of flavors n .
f
∗e-mail: [email protected]
†e-mail: [email protected]
1
2 Nonperturbative EoS of QGP
We split the gluonic field A into the background field B and the (valence
µ µ
gluon) quantum field a : A = B +a bothsatisfying theperiodic boundary
µ µ µ µ
conditions.
The partition function averaged both in perturbative and np fields is
Z(V,T) = Z(B +a) (1)
B,a
h i
Exploring free energy F(T,µ) = T ln Z(B) that contains perturba-
B
− h i
tive and np interactions of quarks and gluons (which also includes creation
and dissociation of bound states) we follow so-called Single Line Approxima-
tion (SLA). Namely, we assume that quark-gluon system for T > T stays
c
gauge invariant, as it was for T < T , and neglect all perturbative interac-
c
tions in the first approximation. Nevertheless in SLA already exist a strong
interaction of gluons (and quarks) with np vacuum fields. This interaction
consists of colorelectric (CE) and colormagnetic (CM) parts. The CE part
in deconfinement phase creates np self-energy contribution for every quark
and gluon embedded in corresponding Polyakov line. An important point
is that Polyakov line is computed from the gauge invariant qq¯ (gg) Wilson
loop, which for np DE interaction splits into individual quark (gluon) con-
1
tributions. As for CM part - its consideration is beyond the SLA, because
as has been recently shown in paper ([11]) strong CM fields are responsible
for creation of bound states of white combinations of quarks and gluons.
To proceed with FCM we apply the nonabelian Stokes theorem and the
Gaussian approximation to compute the Polyakov line in terms of np field
correlators
1 β
4
L = tr P exp ig B (z)dz =
fund 4
N
c (cid:18) Z0 (cid:19)
1 g2
tr exp dσ (u)dσ (v)D
µν λσ µν,λσ
N − 2
c (cid:18) ZSnZSn (cid:19)
with
2
D g F (u)Φ(u,v)F (v)Φ(v,u) (2)
µν,λσ µν λσ
≡ h i
DE and DE arise from CE field strengths:
1
1 ∂DE ∂DE
D = δ DE +DE +u2 1 +u u 1 (3)
N 0i,0k ik 1 4 ∂u2 i k ∂~u2
c (cid:20) 4 (cid:21)
2
As a result the Polyakov loop can be expressed in terms of ”potentials”
V and V
1 D
V (T)+2V
L = exp 1 D ,L = (L )9/4, (4)
fund adj fund
− 2T
(cid:18) (cid:19)
with V (T) V ( ,T), V V (r ,T) ([6])
1 1 D D ∗
≡ ∞ ≡
r
V (r,T) = ∞dν(1 νT) dξ ξDE( ξ2 +ν2) (5)
1 1
−
0 0
Z Z
p
r
V (r,T) = 2 ∞dν(1 νT) dξ (r ξ)DE( ξ2 +ν2) (6)
D
− −
0 0
Z Z
p
In what follows we use the Polyakov line fit ([9, 10])
T .175Gev
L x = ,T = exp (7)
fund
T −(1.35x 1)2T
(cid:18) c (cid:19) (cid:18) − (cid:19)
The free energy F(T) of quarks and gluons in SLA can be expressed as a
sum over all Matsubara winding numbers n with coefficients Ln and Ln
fund adj
for quarks and gluons respectively. For nonzero chemical potential µ one can
keep L independent of µ, treating np fields as strong and unchanged
fund,adj
by µ in the first approximation.
The final formulas for pressure of qgp are ([8, 10])
PSLA n µ V1 µ+ V1
p q = f Φ − 2 +Φ 2 (8)
q ≡ T4 π2 ν T ν − T
" ! !#
where ν = m /T and
q
z4 1
∞
Φ (a) = (9)
ν √z2 +ν2(e√z2+ν2 +1)
0
Z
PSLA 8 z3dz
gl ∞
p = = (10)
gl T4 3π2 ez+9/4V1 1
0
Z −
The energy density is ε = T2 ∂ P and the speed of sound in plasma
∂T T V
is c2 = ∂P. In Fig.5 c2 is shown calculated with the use of (8), (10) and
s ∂ε s (cid:0) (cid:1)
compared to lattice data for µ = 0 from ([12]). No lattice calculations has
yet been done for sound speed at nonzero baryon density, though our theory
allows to do that and as is shown in ([13]) the result for µ > 0 does not differ
much from the case µ = 0.
3
Figure 2: Analytic (8), (10)
Figure 1: Fit (7) of and lattice ([12]) curves for
Polyakov line for n = 0 pressure of QGP with n =
f f
and n = 2)(black curves) 0,2+1,3 from ([10]).
f
to the lattice data ([12]).
3 Phase transition
To obtain the curve of phase transition one needs to define pressure P in
I
the confined phase and P in the deconfined phase, taking into account
II
that vacuum energy density contributes to the free energy, and hence to the
pressure:
P = ε +P , P = εdec +(p +p )T4. (11)
I | vac| hardon II | vac| q gl
Having formulas for pressure (which contain parameter of L (x) (7)) we
fund
may write for the phase transition curve T (µ):
c
(11 2n )∆G 1/4
T (µ) = − 3 f 2 (12)
c
32(p +p )
(cid:18) q gl (cid:19)
here ∆ε = ε εdec = (11 2n )/32∆G . In particular, for the ex-
vac | vac − vac| − 3 f 2
pected value of ∆G /G (stand) 0.4 one obtains T = 0.27 GeV (n =
2 2 c f
≈
0), 0.19 GeV (n = 2), 0.17 GeV (n = 3) in good agreement with lattice
f f
data.
4 Summary
The EoS of QGP is written, where the only np input is the Polyakov line.
It should be stressed, that only the modulus of the Polyakov line enters in
4
Tc
0.15
0.1
0.05
0
0 0.1 0.2 0.3 0.4 0.5 0.6 µ
Figure 4: The phase tran-
sition curve T (µ) (in GeV)
c
from (9) as function of
Figure 3: Analytic and lat- quark chemical potential µ
tice ([12]) curves for energy (in GeV) for nf = 2 (up-
density of QGP with nf = per curve) and nf = 3
2+1 andn = 3 from([10]). (lower curve) and ∆G2 =
f
4
0.0034 GeV from ([9]).
Figure 5: Sound speed for µ = 0 and n = 3 (blue dashed curve) compared
f
to lattice data from ([12]).
EoS due to gauge invariance. The phase transition curve T (µ) and speed
c
of sound c2(T) are obtained and agree well with lattice data. An important
s
point of the work is that the only parameter used to receive the final physical
quantities from the initial QCD Lagrangian is the Polyakov line taken from
5
lattice data, and is in agreement with analytic estimate for T = T ([6]).
c
5 Acknowledgments
The financial support of RFFI grant 06-02-17012 is acknowledged.
References
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