ORIGINAL RESEARCH ARTICLE
(QCD)
TIS VERY GOOD FOR QUARK – GLUON - PLASMA
*Dr. Salman Al- Chatouri
Associate Prof. in the Department of Physics, Faculty of Science in Tishreen University, Latakia – Syria
ARTICLE INFO ABSTRACT
We have defined the non-abelian pure gauge theory SU (3) on a torus. Fourier modes are discrete throughout this definition. For enough small size, we have treated the non-glue ball modes as a perturbation of zero modes. Infra-red singularity is not appearing throughout the discrete momentums. The temperature depending contributions of the effective potential of the non-abelian glueball gauge fields are continuously calculated by us, for the first time on an asymmetric torus 3
L , till the fourth grade of gauge fields. So, L is the length of the torus in space direction and is the length in time direction (the inverse of temperature). The Phase transition is indicated by the coefficient
2
instead of the coupling constant g. The critical temperature is10
12 6827150752.
5
K
Copyright © 2019,Dr. Salman Al- Chatouri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
INTRODUCTION
The thermodynamic properties of systems of the quantum fields theories are great interest. These systems are described in case of equilibrium by the formalism of the imaginary time which Matsubara introduced. There are active algorithms for the numerical investigation equilibrium problems. The formalism of the real time or Minkowski’s space which was introduced to the perturbation theory by relationship with statistic of the non-equilibrium by Schwinger and others is a useful formalism for the calculation of correlation functions depended on time. Treatment of non-equilibrium problems is very important [1-37]. From the point of view of elementary particles physics, these problems must be exposed, a description of the heating of the early universe (according to an available expositing phase) or a description of hadronic material under extreme conditions for studying the experimental results for a short transition to a quark-Gluon-plasma-phase. The third problem is called the anomaly Baryon number violation processes in the stander model. One, principally, can try to treat such problems by the analytical continuation to imaginary time, but in the practice the return of the analytical continuation to real time in many cases, is rarely able to practice in a special case that can not be done when approximations come for use in Euclidean formalism (that can rarely be avoided). The aim of this work is to develop suitable algorithm to describe non-equilibrium processes. The physical background was built by the heating of the early universe, and by a description collision of heavy ions at high energies. The algorithm that is to be developed is based on a combination of the background fields method and one-loop-approximation. This method has been developed for the pure gauge theory (without fermions) with the gauge group theory SU (3). As the effective potential can be calculated in Mincowski’s space or in Euclidean-space, the Euclidean formalism was chosen because of plainness. This means that we have calculated the effective potential at finite temperature on the asymmetric torus 3
L , Meanwhile (L) is the length of the torus in
all the three-space direction and
is the length in the time direction.*Corresponding author: Dr. Salman Al- Chatouri,
Associate Prof. in the Department of Physics, Faculty of Science in Tishreen University, Latakia – Syria.
ISSN: 2230-9926
International Journal of Development Research
Vol. 09, Issue, 07, pp. 28598-28618, July, 2019
Article History:
Received 27th April, 2019 Received in revised form 17th May, 2019
Accepted 11th June, 2019 Published online 28th July, 2019
Key Words:
Real time in Non – Equilibrium
Phase Transition to Quark – Gluon – Plasma Non – Equilibrium in the Quantum field Theory.
Citation: Dr. Salman Al- Chatouri, 2019. “(QCD)t is very good for quark – gluon - plasma”, International Journal of Development Research, 09, (07), 28598-28618.
The gauge theory is considered on the torus in 1979 by the scientist G.T. Hooft, after that, Lusher [22-23], Van Baal [24-28], J.Kripfganz and C.Michael[29-30], have worked in this field. All these works deal with the glueball spectrum in a small or medium size. Fremionic contributions were considered by J.Kripfganz, C.Michael and Van Baal. The pure gauge theory on the asymmetric torus:
L
3
was studied and discussed the finite temperature by Al- Chatouri, S.[17] . We followed [17] and [28] when we have calculated the effective potential. That means we have used the one-loop-approximation.RESEAARCH METHODOLOGY
Calculation of temperature contributions for the effective potential.
The investigating about the quark – gluon – plasma phase and determination of the critical temperature
T
cr. The research method and its materials:We have mentioned in the introduction that we took the developed numerical algorithm in the Dissertation [17] and the references [22-29] which is based on a combination of the background fields method and one-loop-approximation for the pure gauge theory with the group SU (3). We will follow the reference [17] in all steps.
The gauge theory:
Introduction
In this term, we will discuss the moving of the pure QCD. When the perturbation theory is employed on the QCD theory, it is necessary to use the infra-red cut-off. It's a very kind way which one considers the theory on a torus with d dimensions and puts extreme periodic conditions. These extreme conditions are not allowed to destroy the invariant of the gauge. The gauge potential is periodic till the gauge transformations. We will use the non-local gauge invariant which is introduced in [28]. The modes are divided into glueball and non-glueball. The integration of the non-glueball modes was done by the one-loop-approximation.
The one-loop –approximation
We will only derive from this passage the effective potential at a finite temperature.
The Division into glueball modes and non-glueball modes We introduce the projector P:
3 31
TA
L
PA
, (2.2.1.1)and function of gauge invariant
:
0 1
])
,
[
)(
1
(
P
A
i
PA
A
L
PA
,
(2.2.1.2)with the definition:
PA
Q
P
A
B
,
(
1
)
(2.2.1.3)
is equivalent to :
,
0
,
0
Q
i
B
Q
B
.
(2.2.1.4)
exp
i
is:
].
,
[
]}
),
)
(
(
[
)
(
)
(
){
1
(
1
0
1
0
P
D
PA
D
A
i
P
D
A
A
L
P
iL
P
A
(2.2.1.5)
D
(A) is the covariant derivative in this relation.When we divide
into
and
1
P
we will find:
P
A
P
L
i
P
L
A
A
P
A
D
P
D
P
,
,
1
,
,
)
(
)
(
)
1
(
0
0
(2.2.1.6)
The operator M is:
D
(
PA
)
D
(
A
)
A
,
P
A
,
M
. (2.2.1.7)It can express Faddeev-popov’s determinant:
(
A
)
(
D
(
))
1 (2.2.1.8)
(
A
)
D
D
d
d
2 0
1
exp(
g
(
Tr
)
(
0
P
A
0,
))
.
L
i
Tr
(2.2.1.9)
and
are the space sections of the ghost-fields,the sign / on D means that
0
. While
and
are constant to the space, it can be explicitly integrated. These integrations about
and
deliver a constant. This identity (2.2.1.8) can be generalized:.
1
]
)
(
1
exp[
)
(
)
(
2
2
0
0
E
Tr
g
E
D
E
D
A
(2.2.1.10)
Meanwhile,
is known throughoutX
0
and / means that
0
. When we put this in the sum of the states, we conclude that:
) ( ))
( 1 exp(
) ( 2 ) ( )) ( ( 2 1 ( 1 exp
2 2
2 2
2 0
E E
Tr g E D
Tr E Tr A F Tr g D
D DA Z
o
(2.2.1.11)
After doing the integrations about
we conclude the expression of Z:
.
))
(
2
)
(
)
(
(
2
1
(
1
exp
2 22 0
Tr
F
A
Tr
Tr
g
D
D
DA
Z
(2.2.1.12)0
B
. (2.2.1.13)This leads to:
D
(
B
)
Q
. (2.2.1.14)We put this in (2.2.1.12):
22 0
2
1
1
exp
Tr
F
B
Q
Tr
D
B
Q
g
D
D
Q
D
DB
Z
2
,
,
2
Tr
D
B
D
B
Q
Tr
Q
p
Q
(2.2.1.15).one can simply derive effective Lagrange function for B.
DB
exp(
d
L
eff
(
B
))
DB
exp(
S
eff
)
Z
. (2.2.1.16)This means:
(
)
1
exp(
1
2 3(
,
,
,
))
0
D
D
g
d
d
xL
B
Q
Q
D
og
B
L
d
S
eff eff (2.2.1.17)Meanwhile,
L
B
,
Q
,
,
will take the following form:
.
]
,
[
]
,
[
2
)
(
)
(
2
)
)
(
(
))
(
(
2
1
(
)
,
,
,
(
2 2
Q
P
Q
Q
B
D
B
D
Q
B
D
Q
B
F
Tr
Q
B
L
(2.2.1.18)
When we develop
[
F
B
Q
]
2 till the second grade of Q, we get:
(
(
))
(
)
((
)
))
.
2
1
(
)
)
(
(
2
1
2 2 2
Tr
F
B
Q
Tr
F
ijB
Tr
Q
W
Q
Tr
D
Q
(2.2.1.19)So, it is:
Q
D
B
Q
i
F
Q
W
2
(
)
2
,
.When we put this in
L
B
,
Q
,
,
and take terms till the second grade ofQ
,
and
we get:)
)
(
(
2
)
(
))
(
2
1
(
)
,
,
,
(
B
Q
Tr
F
2
B
Tr
Q
W
Q
Tr
D
2
B
L
(2.2.1.20)From (2.2.1.17) and (2.2.1. 20), we get:
. ))] ) ( ( 2 ) (
)) ( 2 1 ( ( 1
exp[ 1
) (
0
2 2
3 2
0
0
3
T
eff d d xTr F B Tr Q W Q Tr D B
g D
D Q D og B
L
d (2.2.1.21)
0 2
1 2
1
]
))
(
t
(de
))
(
(
t
de
[
1
B
W
B
D
og
V
d
eff . (2.2.1.22)The index (1) is to one-loop-approximation, so
D
B
is inverse ghost-propagator and:)
(
2
)
(
)
(
B
D
2
B
iadF
B
W
ij
, (2.2.1.23)the propagation of the inverse vector propagator.
adF
ij
B
isF
ij
B
in the adjoint representation which is known in the appendix C.2
2
2
)
(
2
iadB
i
i
adB
i
D
(2.2.1.24)So,
adB
i
is the vector potentialB
i
in the adjoint representation.In the momentum representation , it confirms :
2 2
2
)
(
2
adB
iK
iadB
iK
D
. (2.2.1.25)The equation (2.2.1. 22) is written as :
1
de
t
(
)
2
1
))
(
(
t
de
1
2
0
1
og
D
B
og
W
B
V
d
eff
(2.2.1.26)Development with the grades of B
In order to calculate both the determinants , we have to use the following identity:
.
)
)
((
)
1
(
log
)
1
log(
log
)
log(
)
det(
log
1
1
1
n
n n
CA
Tr
n
A
Tr
CA
Tr
A
Tr
C
A
Tr
C
A
(2.2.2.1)
In order to calculate
log
det
W
B
2
1
, (2.2.2.1) is written as :.
)
)
((
8
1
)
)
((
6
1
)
)
((
4
1
)
(
2
1
log
2
1
)
(
`
det
log
2
1
4 1 3
1
2 1 1
A
C
Tr
CA
Tr
CA
Tr
CA
Tr
A
Tr
C
A
(2.2.2.2) Meanwhile, it is:
2
A
(2.2.2.3))
(
2
)
)
(
2
(
iadB
adB
2
iadF
B
C
i
i
i
ij
. (2.2.2.4)This means that we are calculating the determinant till the forth grade of
B
i
a
. We introduce Fourier transformations:
0
0
2
2
0
1
1
1
exp
)
2
(
1
)
,
(
k k
d
k
k
x
x
ik
x
x
A
A
2
2
.
exp
2
1
)
,
(
0
2 2
2 0 0 1
1
k
ij i
i i k
d
adB
k
adB
iadF
B
k
k
x
x
ik
x
x
CA
(2.2.2.5) Now, we calculate the trace on space – time:
2 2 2
0 0
1 2
1
2
0 2 2
0 1
1
1
2
1
4
1
1
2
2
1
2
1
0 0
k
k
k
k
d
x
d
CA
Tr
adB
Tr
k
k
x
d
d
CA
Tr
j i k
k d d
i k
k d d
2 22 2
0
1
4
1
i j
i
Tr
adB
k
k
d
adB
adB
Tr
22 2 2
0
2
1
B
adF
Tr
k
k
adB
j ij
0 0
3 2 2
0 2
1 3
1
2
1
2
1
6
1
k k d d
k
k
k
d
d
x
d
CA
Tr
2 2
j i
adB
adB
Tr
0 0
4 2 0
1 4
1
1
2
2
1
8
1
k k
k j i d
d
k
k
k
k
k
k
d
x
d
CA
Tr
adB
adB
adB
adB
Tr
i j k . (*)We use the same identity to calculate the other determinants
D
A
C
Tr
A
C
Tr
A
log
de
t
2log
de
t
log
log
nn
n
A
C
Tr
n
1
1
1
1 1 22
1
.
4
1
3
1
1 3 1 4
A
C
Tr
A
C
Tr
(2.2.2.6)It is by this:
2
A
22
iadB
iadB
iC
. exp
2 1
, 2
2 2
0 0
1 1
0
j i
i k
k
d adB k adB
k k
x x ik x
x A
C
While calculating the
trace on space. time, we get the following equations:
22 2
0 0 1
2 1
1
1
0
1 j
k k d
adB
Tr
k
k
x
d
A
C
Tr
d
2 2 2
0 0
1 1
2
1
2
2
1
2
1
0
k
k
k
K
x
d
A
C
Tr
i jk k d
d
2
2
2 2 2
0
1
2
1
j i
j
i
Tr
adB
adB
k
k
adB
adB
Tr
.
4
2
1
3
1
2 2
3 2 2
0 2
0 1
1 3
1
0
j i
k k d d
adB
adB
Tr
k
k
k
d
x
d
A
C
Tr
.
4
2
1
4
1
4 2 2 0
1 0
1 1
4 1
0
Tr
adB
adB
adB
adB
k
k
k
k
k
k
x
d
A
C
Tr
i j k i j kk k d
d
(**)
We put (*) in (2.2.2.5) and (**) in (2.2.2.6) , then we get the following equations :
B
A
C
W
log
det
2
1
t
de
log
2
1
2 2
0 0 1
1
1
2
1
2
1
0
k
k
x
d
d
k k d
d
0 1
1 2
0
2
1
k k d d
i
d
x
adB
Tr
2 2 2
0 2
2 2
0
4
1
1
k
k
k
k
d
adB
adB
Tr
k
k
k
k
d
i jj i
j i
22 2
0 2
2
1
k
k
adB
adB
Tr
i j
0 0
1 1
2
1
k k
d d
ij
B
d
x
adF
Tr
2 2
3 2 2
0 2
.
1
2
j
i
adB
adB
Tr
k
k
k
d
d
0
4 2 2
0 0
1
1
2
1
2
1
k
k j i
k d
d
k
k
k
k
k
k
d
x
d
adB
adB
adB
adB
o
B
6
Tr
i
j
k
. (2.2.2.7)
D
B
A
C
log
de
t
2
log
de
t
0
0
2
2
0
1
1
1
2
1
k
k
d
d
k
k
x
d
).
(
)
(
)
(
4
)
2
(
1
)
)
(
)
((
)
(
4
)
2
(
1
)
(
)
((
)
(
2
1
))
)(
((
)
(
2
)
2
(
1
6
0 02 2 4
1 1
1 2
2
0 02 2 3
2 1
1 2
2
0 02 2 2 02 2 2
1 1 2
0 0
0
B
o
adB
adB
adB
adB
Tr
k
k
k
k
k
k
x
d
adB
adB
Tr
k
k
d
k
x
d
adB
adB
Tr
k
k
adB
adB
Tr
k
k
k
k
x
d
adB
Tr
k
j i k k
k j i d
d j
i k k
d
d j
i
K K
j i
J i d
d i
(2.2.2.8)
when we put (2.2.2.7) and (2.2.2.8) in (2.2.1.27) , then we get – for the effective potential – the following expression
0
0 0
0
2 2
2 0 2
0 1 2
0 2
2 2 0
2
0
2 2 0 1
) 1 (
1 8
) 2 ( ) ) ((
) (
) 1 ( 1 2
1 )
2 (
1
k k
ij d
i
k k
k k
d d eff
B adF Tr k k g
adB Tr
k k
k d
k k d
x d v
.
)
(
1
2
)
((
)
(
1
2
1
4
1
4 2 2
0 0
2 2
0 2 2 3
0 2
0 2 2
0
0
0 0
adB
adB
adB
adB
Tr
k
k
K
K
K
K
d
adB
adB
Tr
k
k
k
d
d
k
k
d
K j
i K
j i K K
j i
K K K K
(2.2.2.9)
The case of the vanish temperature
The sum
0
K
is considered integration on
K
0
. After doing the integration onK
0 , the effective potential ofone-loop-approximate will take the following expression:
(2.2.3.1)
1
4
1
^ ^ ^ ^ 4
^ ^ ^ ^ 3
2 ^ ^ 2
2 ^
^ 1 1
d
i
c
i
b
i
a
i
S
d
j
c
j
b
i
a
i
S
c
j
b
i
f
L
g
a
i
a
i
V
B
B
B
B
B
B
B
B
B
B
B
B
abcd abcd
abc eff
Meanwhile, the coefficients are
0 2 1
1
4
1
3
2
1
K d
d
K
d
d
x
d
(2.2.3.2)
3 0 2
2
1
8
6
17
2
1
K
d
d
d
x
d
K d
d
(2.2.3.3)
222 1 4
7 0
3
6
15
16
1
2
1
k
dk
K
d
K
d
d
x
d
K d
d
(2.2.3.4)
.
3
16
1
5
0
7 2 2 2 1 4 1
4
K
k
k
k
k
d
(2.2.3.5)
This conclusion accords to the reference [28].
The case of the non- vanish temperature From (2.2.2.9) results :
d i c i b i a i S d j c j b i a i S
c j b i f L
g a i a i V
B
B
B
B
B
B
B
B
B
B
B
B
abcd abcd
abc eff
^ ^ ^ ^
4 ^ ^ ^ ^
3
2 ^ ^
2 2 ^
^
1 1
1 4 1
So, the coefficients:
1 .
3 1 2 1 3 2 1 0 0 0 2 2 2 0 2 0 2 2 0 1 1 K K K K d d k k k d k k d x d
(2.2.4.2)
0 0 0 3 2 2 0 2 2 2 0 0 1 2 2 1 8 1 2 23 2 1 K K K K d d k k k d d k k d x d (2.2.4.3)
2 3
2 0 2 0 2 2 2 0 0 1 3 0 0 1 3 1 8 1 3 2 1 k k k d d k k d x d K K K K d d (2.2.4.4)
. 3 1 2 1 4 2 2 0 2 2 2 1 4 1 0 1 4 0
k k k k k d x d K K dd
(2.2.4.5)
One can calculate these coefficients by the helping of the heat kernel. Up from now, we will omit
d
d
x
because thisintegration delivers only the constant
L
3 . The definition of the kernelsg
1 andg
2 , which appear in the calculation is that one can find in the appendix A. We will divide the coefficients into: related to heat parts and others are not so. By this, we can write)
1
(
eff
v
as:
eff
eff
T
eff
V
V
V
1
0
So,
1
0
eff
V
is the unrelated to heat part andV
eff
T
1
is the one which is related to heat.From (2.2.4.2) , (B.7) and (B. 8) we result to :
1 2 0 2 1 0 2 1 12
1
1
3
1
1
2
1
3
d d dd
dt
tg
g
g
L
d
g
g
dt
L
d
. (2.2.4.6)Then, we put (A. 12) in ( 2.2.4.6) :
At the end,
1becomes into two parts: one is related to heat
1
T
0
and other which is not related to heat
1:
T
O
1 1 1
. (2.2.4.8)So, it is:
22 2 2 1 3 3 2 2 1 3 2 2 2 0 3 2 2 1 3 1 3 1 2 3 8 3 3 g g t dt L V g t dt L h h t dt t L
V t t
t
1
6 .3 4 3 3 2 2 2 2 1 3 3 2 2 1 3 2 2 2 3 0 2 0 2 1 3 1 g g h t dt L g h t dt L h h h t dt h t dt L o t t t t t
(2.2.4.9)From (2.2.4.3) , (B.7) and (B.8) results :
3 .1 4 1 2 2 23 0 1 2 2 1 2 0 2 1
2
d d d
d d L dt t g g g
d d g g t dt L d
(2.2.4.10)
We put, after that, (A.12) in (2.2.4.10) and find:
. 4 exp 2 1 4 1 4 exp 1 2 23 4 1 4 1 2 1 2 23 1 2 2 1 2 0 2 2 3 0 2 1 2 0 2 0 2 1 1 2 2 2 3 0 2 2 1 0 2 0 0
d n d d n d d d d d g g n t t dt L d d d g n t t dt L d g g t dt L d d d g t dt L d (2.2.4.11) This means:
0
2 2
2
T
, (2.2.4.12)that:
0 1 2 2 2 3 0 2 2 1 24
1
4
1
2
1
2
23
d d d dg
g
t
dt
L
d
d
d
g
t
dt
L
d
(2.2.4.13) and:
.
4
exp
2
1
4
1
4
exp
1
2
23
0
0 1 1 2 2 2 0 2 2 3 0 2 1 2 0 2 1 2 0 0
n d d d n dg
g
n
t
t
dt
L
d
d
d
g
n
t
t
dt
L
d
T
(2.2.4.14)
4 log 16 11 log 16 11 48 1 2 1 2 13 4 4 3 13 1 4 1 3 4 2 11 2 2 2 2 2 2 2 3 0 3 2 2 1 2 2 2 2 2 2 3 0 3 2 1 2 2 2
t g g t dt L g t dt L h h t dt t L h t dt d t t d t d d t (2.2.4.15)The related to heat part
2
T
0
reads:
. 2 1 4 1 1 2 23 4 hh 4 2 d -1 4 1 4 2 2 23 0 1 2 2 2 3 2 2 1 0 0 1 2 2 2 1 2 1 d 2 2 1 2 2 1 0 2 1 0 2 2 1 2 1 2
d t d t d d t t d d d d d t t d d d d g g h t dt L d d d g h t dt L d h h h t dt d t dt d d h t dt L h h t dt d T (2.2.4.16)When we put (B.7), (B.8) and (B. 9) in (2.2.4.4) we get the following expression of
3:
4
.
9
1
3
1
3
1
2
1
8
1
3
0 1 2 2 1 3 0 1 2 2 1 2 2 0 1 3
d d d d d dg
g
g
t
dt
L
d
g
g
g
t
dt
dL
d
d
g
g
t
dt
L
d
(2.2.4.17)We put, after that (A. 12) in (2.2.4.17).
3 , at that time, is divided into two parts:
0
3 3
3
T
(2.2.4.18)By this, it is: