; Veneziano Ghost Theory of QCD; Local Quark Vacuum Condensate; Nonlocal Quark Condensate; Quantum Chromodynamics-QCD"> ; Veneziano Ghost Theory of QCD; Local Quark Vacuum Condensate; Nonlocal Quark Condensate; Quantum Chromodynamics-QCD"/>
Journal of Modern Physics, 2012, 3, 1172-1177
http://dx.doi.org/10.4236/jmp.2012.329151 Published Online September 2012 (http://www.SciRP.org/journal/jmp)
Prediction of Cosmological Constant Λ in Veneziano
Ghost Theory of QCD*
Lijuan Zhou1, Weixing Ma2, Leonard S. Kisslinger3
1Department of Information and Computing Science, Guangxi University of Technology, Liuzhou, China
2Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, China
3Department of Physics, Carnegie-Mellon University, Pittsburgh, USA
Email: Leonard Kisslinger kissling@andrew.cmu.edu
Received June 15, 2012; revised July 14, 2012; accepted July 24, 2012
ABSTRACT
Based on the Veneziano ghost theory of QCD, we estimate the cosmological constant Λ, which is related to the vacuum
energy density,
, by =8πG
. In the recent Veneziano ghost theory
is given by the absolute value of the
product of the local quark condensate and quark current mass: 2f
NH
m
=<
0::|0>|
q
cm qq
. By solving Dyson-
Schwinger Equations for a dressed quark propagator, we found the local quark condensate

3
:0 235MeVqq
q
m
0: ,
the generally accepted value. The quark current mass is 4.0 Mev. This gives the same result for
as found
by previous authors, which is somewhat larger than the observed value. However, when we make use of the nonlocal
quark condensate,


0:0 :0=0::0qxqgxqq , with g(x) estimated from our previous work, we find Λ is in a
good agreement with the observations.
Keywords: Cosmological Constant Λ; Veneziano Ghost Theory of QCD; Local Quark Vacuum Condensate; Nonlocal
Quark Condensate; Quantum Chromodynamics-QCD
1. The Cosmological Constant Λ and the
QCD Veneziano Ghost Theory
The starting point of most cosmological study is Albert
Einstein’s Equations, which is a set of ten equations in
Einstein’s theory of general relativity. The original Ein-
stein field equations can be written as the form [1]
1=8π
2
RRg GT
 
==1c
39 2
= 6.7087(10)10GeVG
(1)
in units of , where G is the gravitational
constant (

,=0, ,3R


, sometime called
Newton’s constant),

is the Ricci
tensor, R is the trace of Ricci tensor (it is like the radius
of curvature of space-time), x
T
is the energy-momentum tensor, which describes
the distribution of matter and energy. Equation (1) des-
cribes a non-static universe. However, Einstein believed,
at that time, that our universe should be static. In order to
get a static universe, in 1917 Einstein introduced a new
term,
g
, in Equation (1) to balance the attractive
force of gravity, giving his modified equation
g

represents the me-
tric tensor, which is a function of position x in spacetime.
1=8π.
2
RRgg GT
 
 (2)
The
in Equation (2) is the so-called cosmological
constant, which is a dimensional parameter with units of

2
length
. Indeed, Equation (2) allows a static universe
[2], called Einstein’s universe, which is one of the so-
lution [3] of Friedmann’s simplified form of Einstein’s
equation with a
term. However, almost one hundred
years ago the observations of redshifts of galaxies led to
Hubbles Law [4] and the interpretation that the universe
is expanding. This led Einstein to declare his static cos-
mological model, and especially the introduction of the
term to his original field equation theory, his “biggest
blunder”.
Note that the term
*This work was supported in part by National Natural Science Founda-
tion of China (10647002), Guangxi Science Foundation for Young Re-
searchers under contract No. 0991009, and Guangxi Education Depar-
tment with grant No.200807MS112, Department of Science and Tech-
nology of Guangxi under funds No. 2011GXNSFA018140, Department
of Guangxi Education for the Excellent Scholars of Higher Education,
2011-54, Doctoral Science Foundation of Guangxi University of Tech-
nology, 11Z16, and in part by the Pittsburgh Foundation.
g
in Equation (2) corresponds
C
opyright © 2012 SciRes. JMP
L. J. ZHOU ET AL. 1173
T
to adding a vacuum term to

=.ac g
,
Tv

(3)
Therefore, the cosmological constant is related to
the vacuum energy density,
by [3]
=8πG.
(4)
The vacuum energy density, called dark energy den-
sity, and a model with representing dark energy were
reintroduced about three decades ago. See Ref. [5] for a
review of the physics and cosmology of , with refe-
rences to the many models that have been published. To
explain our uniform and flat universe via inflation a cos-
mological constant was added to the Friedmann equation
[6]. From studies of radiation from the early universe, the
Cosmic Microwave Background Radiation (CMBR), by
a number of projects, including WMAP [7], the inflation
scenerio was verified, and it was shown that about 73%
of the total energy in the universe is dark energy. As
clearly shown by Friedmann’s equation with a cosmolo-
gical constant, dark energy corresponds to negative pre-
ssure, or anti-gravity. This was confirmed by studies of
distant type 1a supernovae [8,9], which showed an acce-
leration of the expansion of the universe, and was con-
sistent with dark energy being 73% of the energy in the
universe. Also, dark energy causes distant galaxies to
accelerate away from us, in contrast to the tendency of
ordinary forms of energy to slow down the recession of
distant objects. See Ref. [5] for other of the many refe-
rences to CMBR, supernovae, galaxy and other studies of
dark energy.
The existence of a non-zero vacuum energy would, in
principle, have an effect on gravitational physics on all
scales. The value of in our present universe is not
well known, and it is an empirical issue which will ulti-
mately be settled by observation. A precise determination
of this number () or
will be one of the primary
goals of observational cosmology in the near future. Re-
cently the possiblity of determining the cosmological
constant by observations has been discussed [10].
A major outstanding problem is that most quantum
field theories predict a huge cosmological constant
from the energy of the quantum vacuum. This conclusion
also follows from dimensional analysis and effective
field theory down to the Planck scale, by which we
would expect a cosmological constant of the order of
4
p
l
M
(
p
l
M
is the Planck mass with 12
==MG
19
1.22 10 Gepl
. The Planck energy is thought to be the
energy where conventional physical theories break down
and a new theory of quantum gravity is required ). We
know that the measured value is on the order of
,or , or
V
47
10 35 2
s

10 4
GeV 3
cm
energy expected from zero -point fluctuations and scalar
potential, 3
110
=210ergcm,
theory
and the observed
value, 3
10
=210ergcm
observe
120.
observe
11
10
, a discrepancy of a
factor of 10 This is the largest discrepancy—the
worst theoretical prediction in the history of physics. At
the same time, some supersymmetric theories require a
cosmological constant that is exactly zero. Therefore, we
face a big difficulty in understanding the observational
. This problem has been referred to as the long-
standing cosmological constant problem.
Vacuum energy is predicted to be created in cosmolo-
gical phase transitions. In the standard model of particle
physics with the temperature (T) of the universe as a
function of time (t), there are two important phase tran-
sitions. At t
seconds, with T 140 GeV the
universe undergoes the electroweak phase transition
(EWPT), with the vacuum expectation value of the Higgs
0: :0
Higgs
5
10
field, , going from zero to a finite value
corresponding to a Higgs mass 140 GeV. At
t
seconds, with T 150 MeV, the universe undergoes
the QCD phase transition (QCDPT), when a universe
consisting of a dense quark-gluon plasma becomes our
current universe with hadrons. The latent heat for this
phase transition is the quark condensate, 0: :0qq ,
also a vacuum energy, which is an essential part of the
present work.
First we review the work of F. R. Urban, A. R. Zhit-
nitsky [11,12], which is based on the QCD Veneziano
ghost theory [13-16] In this model the cosmological va-
cuum energy density
can be expressed in terms of
QCD parameters for light flavors as follows
[10,11]
=2
f
N

29 g
10 , or about
in reduced planck units (
120
10
p
l
M
). That is, there is a
large difference between the magnitude of the vacuum
2
=0:00:0,
f
q
HN
cmqq
m
=ccc
(5)
where mq is the current quark mass and .QCD grav
.
The first factor D is a dimensionless coefficient with
value of QCD
c [10,11], which is entirely of QCD
origin and is related to the definition of QCD on a
specific finite compact manifold such as a torus,
QC
c
1
2fq
QCD
Nmqq
cLm
L
m
with being the size of the
manifold and
the mass of
meson. A precise
computation of QCD
c has been calculated in a
conventional lattice QCD approach by studying corre-
ctions of order
s
1 to the vacuum energy [10,11]. Note
that QCD depends on the manifold where the theory is
defined. The second factor
c
g
rav has a purely gravita-
tional origin and is defined as the relation between the
size L of the manifold we live in, and the Hubble
constant H,
c
1
.0
=grav
LcH
. One can define this size of
the manifold as 1
0
17LH
42
0=2.1 10
where
h
GeV =0.71h0
and (
H
, Hubble
H
Copyright © 2012 SciRes. JMP
L. J. ZHOU ET AL.
1174
constant today). Therefore, one can explicitly obtain an
estimate for the linear length of the torus, and then
obtain the value of
L
.
g
rav .grav
In Section 2 we briefly review our previous calculation
of the quark condensate [17] using Dyson-Schwinger
equations (DSEs) [18,19], and discuss the quark current
mass q, which are needed to calculate
c c
with .
= 0.0588
m
, as shown
in Equation (5). Since our values for the local quark
condensate

0:00 :0qq and the current quark
mass are approximately the same as in Ref. [10,11] we
find the same value for
no
as in that work, with a
factor 6 discrepancy when compared to the observed
vacuum energy density. In Section 3 we use a nonlocal
quark condensate, based on earlier research, and find
good agreement between and .
Finally, we give our Summary and concluding remarks in
Section 4.
nlocaltheoryobserved

2. Local Quark Condensate, Current Quark
Mass, ρΛ
In this section we review our previous work on the quark
condensate, the current quark mass, and the resulting va-
lue for the cosmological constant/vacuum energy density.
2.1. The Local Quark Condensate
The quark propagator is defined by

()=0
ab a
q
Sx Tq0 0,
b
xq



x

bx
(6)
where (q) is a quark field with color a (b),
and T is the time-ordering operator. The nonperturbative
part of the quark propagator is given by
a
q
 
1
=0
12
Sx
: 0:00:0:0.
NP
q
qx
qxqx q



(7)
For short distances, the Taylor expansion of the scalar
part,

0:qx

NP
q
Sx0 :0q, of can be written as
( see, e.g., Refs.[17,20] )
 
 
2
0:0 :0=0:
0: 00
4s
qxq q
xqigG


00:0
0 :0.
q
q
(8)
In Equation (8) the vacuum expectation values in the
expansion are the local quark condensate, the quark-
gluon mixed condensate, and so forth.
The Dyson-Schwinger Equations [18,19] were used to
derive the local quark condensate in Ref. [17]. See this
reference for details and a discussion of approximations.
Note that as shown in Equation (8), the quark-gluon
mixed condensate provides the small-x dependence of the
nonlocal

0:0 :0qxq quark condensate. How-
ever, for the present work this small-x expansion is not
useful, and we shall use a known expression for the
nonlocality, described below. Therefore we only give the
results for the local quark condensate. Also note that the
vacuum condensates can act as a medium [21,22], which
influences the properties of particles propagating through
it.
Using the solutions of DSEs with three different sets
of the quark-quark interaction parameters (see Ref.[17])
leads to our theoretical predictions for the local quark
vacuum condensate listed in Table 1.
Set 1 results are consistent with many other calcula-
tions, such as QCD sum rules [23-25], Lattice QCD
[26-28] and Instanton model predictions [29-31]. These
numerical results will be used to calculate
in the
Subsection 2.3 below.
2.2. The Current Mass of Light Quarks
As we have seen from Equation (5) to predict
we
need to know the basic quark current mass q. Since
one cannot produce a beam of quarks, it is difficult to
determine the quark masses. Using various models the
effective quark masses have been estimated, but we need
the current quark masses of the light u and d quark.
Estimates of these masses and references can be foud in
the Particle Data Physics booklet [32]. They are
m
1.7<< 3.3 MeV
4.1 <<5.8MeVe.
u
d
m
m
4.0MeV
q
m
(9)
From this we estimate that the current quark mass is
(10)
2.3. Cosmological Constant Λ with
::0000qq
=8πG
and mq
From Equation (2),
, the vacuum energy
density, while
, is given in Equation (5) as

2
=0:00:0
f
q
HN
cmqq
m
(11)

Since our values for mq and 0:00 :0qq are
the standard ones, we find the same value for
as in
Table 1. Predictions of local quark condensate in QCD
vacuum, ::00
f
qq
μ
with f standing for quark flavor and
μ denotes renormalization point, μ2 = 10 GeV2.
Set no. of quark
interactions ,
0::0ud
qq for u and d quarks
Set 1
 
33
0.0130GeV235MeV
Set 2
 
33
0.0078GeV198 MeV
 
33
0.0027GeV139 MeV
Set 3
Copyright © 2012 SciRes. JMP
L. J. ZHOU ET AL. 1175
Ref.[11]

4
3
10 eV,

4
3
10 eV
3.6
theory
(12)
while the value observed [33] is
2.3
observed
. (13)
Although the theoretical and observed values are
similar, they still differ by
6.0
observed

theory

3. Cosmological Constant Λ with Nonlocal
Quark Condensate
As mentioned above, the expression
 
 
2
0:0 :0=0:
0: 00
4s
qxq q
xqigG


00 :0
0 :0
q
q
does not work except for very small x. Therefore we shall
use the nonlocal quark condensate derived from the
quark distribution function (see Refs.[34,35]). Using the
form in Ref.[35],



2
0:0 :0=0:qxqgx q00 :0,q (14)
with


2
22
1
=
18
gx
x
2
.
(15)
The value of
estimated in Ref.[36] is
. Using
22
GeV0.8
1QCD as the length scale, or
2
.2GeV
0
2=1x, one obtains

2
11
1==.
6.25
2.25
QCD
g (16)
From this we obtain
 
1
0:0 :0=0:
6.25
qxq q00:0 ,q (17)
and


4
4
eV
Vobserved
quantity, wh was introduced by A. Einstein who mo-
3
3
13.6 10
6
=2.310 e
nonlocal theory
(18)
Therefore, using the modification of the quark conden-
sate via the nonlocal condensate, one obtains excellent
agreement between the theoretical and observed cosmo-
logical constants.
4. Summary and Concluding Remarks
The cosmological constant is an important physical
dified the field equations of his general theory of rela-
tivity to obtain a stationary universe. The constant has
recently been used to explain the observed accelerated
expansion of the universe, but its observational value is
about 120 orders of magnitude smaller than the one
theoretically computed in the framework of the currently
accepted quantum field theories. Namely, quantum field
theory predicted that vacuum energy density,
ich
, is of
the order of 4
p
l
M
, with 19
=1.22 10 GeV
pl
M,hich is
about 120 order of mag observed
value of

4
3
=2.310 eV
observed
. This difference is
w
snitude larger than the
ed cosmological constan
eory of QCD, using a
lo
the so callt problem, the worst
problem of fine-tuning in physics.
Based on the Veneziano ghost th
cal quark condensate, we obtained the same result for
as in Refs[11,12], about a factor of 6 larger than
observed . However,

0:00 :0qq is just an appro-
n to ximatio
0:qx the nonlocal quark
0 :0q. Using
condensate
 
0:0 :0qxq
the theoretical and observed values of
=0:00 :0gxq q
we find that
are approximately equal.
The cosmological constant is a potentially impor-
ta
might doubt the correctness of the Veneziano
Q
nt contributor to the dynamical history of the universe.
Unlike ordinary matter, which can clump together or dis-
perse as it evolves, the vacuum energy is a property of
spacetime itself, and is expectd to be the same every-
where. If the cosmological costant is the valid model of
dark energy, a sufficiently large cosmological constant
will cause galaxies and supernovae to accelerate away
from us, as has been observed, in contrast to the tendency
of ordinary forms of energy to slow down the recession
of distant objects. The value of in our present uni-
verse is not well known. A precise determination of this
constant will be one of the primary goals of both theore-
tical cosmology and observational cosmology in the near
future.
One
CD ghost theory that we used in this work, since it is an
analogue of two-dimensional theory based on the Schw-
inger model [18,19], replacing the vector gauge field by
two scalar fields. These scalar fields have positive and
negative norms and cancel with each other, leaving no
trace in the physical subspace. They have small contribu-
tion to the vacuum energy in the curved space. It is
known that the QCD ghost must be an intrinsically vector
field in order for the
1U problem to be consistently
resolved within the fraork of QCD. It seems to be
necessary to examine if the Veneziano mechanism works
in terms of the vector ghost fields instead of the scalar
fields used here. However, Ohta and others in Refs.
[36-38] have discussed the same problem in more rea-
listic four dimensional models, and show that the QCD
mew
Copyright © 2012 SciRes. JMP
L. J. ZHOU ET AL.
1176
ghost produces vacuum energy density
proportional
to the Hubble parameter which has aoximately the
right magnitude

4
3
310eV
.
ppr
considerable evence that the universe
be
There is now id
gan as fireball in the cosmological vacuum, the so-
called “Big Bang”, with extremely high temperature and
high energy density. One knows that the quark conden-
sate is vastly changed by the QCD phase transition, and
this implies that there is a tempreature (T) dependence of

0:0 :0qxq and .
is probably dependent
and moentm p of virtual particles
which produce vacuum condensateas mentioned above.
We can predict the dependence on temperature T
and momentum p by solving the temperature depen-
dent Dyson-Schwger Equations. In this case,
on temperat T mu
s,
in
ure
is a
function of T and p. Such a new study could shw the
behavior of the uring the evolution of the universe.
This work is und its way and should be complete soon.
o
er
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