PUBLISHED VERSION
Kalloniatis, Alexander Constantine; Nedelko, Sergei N.
Realization of chiral symmetry in the domain model of QCD Physical Review D, 2004;
69(7):074029
© 2004 American Physical Society
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10th April 2013
Realization of chiral symmetry in the domain model of QCD
Alex C. Kalloniatis
*
Special Research Centre for the Subatomic Structure of Matter, University of Adelaide, South Australia 5005, Australia
Sergei N. Nedelko
†
Institute of Theoretical Physics III of Erlangen-Nuremberg University, Erlangen, Germany
and Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia
共Received 27 November 2003; published 28 April 2004兲
The domain model for the QCD vacuum has previously been developed and shown to exhibit confinement
of quarks, a strong correlation of the local chirality of quark modes, and a duality of the background domain-
like gluon field. Quark fluctuations satisfy a chirality violating boundary condition parametrized by a random
chiral angle
␣
j
on the jth domain. The free energy of an ensemble of N→⬁ domains depends on
兵
␣
j
,j
⫽ 1,...,N
其
through the logarithm of the quark determinant. Its parity odd part is given by the axial anomaly.
The anomaly contribution to the free energy suppresses continuous axial U(1) degeneracy in the ground state,
leaving only a residual axial Z(2) symmetry. This discrete symmetry and the flavor SU(N
f
)
L
⫻ SU(N
f
)
R
chiral
symmetry in turn are spontaneously broken with a quark condensate arising due to the asymmetry of the
spectrum of the Dirac operator. In order to illustrate the splitting between the
⬘
from octet pseudoscalar
mesons realized in the domain model, we estimate the masses of the light pseudoscalar and vector mesons.
DOI: 10.1103/PhysRevD.69.074029 PACS number共s兲: 12.38.Aw, 12.38.Lg, 14.65.Bt, 14.70.Dj
I. INTRODUCTION
A mechanism that simultaneously provides for confine-
ment of color, spontaneously broken chiral symmetry, and a
resolution of the U
A
(1) problem remains one of the open
problems in QCD today. Partial solutions 关1兴 based on spe-
cific semiclassical or topologically stable configurations can
go some way to manifest this triplet of phenomena, but
founder either on generating all three or in allowing for an
effective model of the vacuum from which hadron spectros-
copy can be derived. In any case, one expects that topologi-
cal objects of various dimensions—pointlike, stringlike, and
sheetlike—should contribute 关2兴, complete with significant
quantum fluctuations, in a way that would be difficult to
describe via an interacting microscopic model. In this paper,
we continue the exploration of the ‘‘domain model’’ for the
vacuum, originally proposed in 关3兴, as a scenario for simul-
taneous appearance of all three phenomena: confinement,
spontaneous chiral symmetry breaking via the appearance of
a quark condensate, and a continuous SU(N
f
)
L
⫻ SU(N
f
)
R
degeneracy of the vacuum for N
f
massless quarks, but with-
out a U
A
(1) continuous degeneracy of ground states that
would be indicative of an unwanted Goldstone boson. The
purpose of studying a model of this type is to identify the
typical features of the relevant nonperturbative gluonic con-
figurations. Such configurations would provide for as many
gross features of nonperturbative QCD as possible. But the
model should preserve simultaneously the well-studied short
distance regime and should be expressed in terms of quark-
gluon degrees of freedom as well as in terms of colorless
hadron bound states.
The model under consideration provides for confinement
of both static 共area law兲 and dynamical 共propagators are en-
tire functions of momentum兲 quarks 关3兴. It also displays spe-
cific chiral properties of quark eigenmodes; namely, as will
be discussed in more detail below, the spectrum of the Dirac
operator is asymmetric with respect to →⫺ and zero
quark modes are absent, but the local chirality of all nonzero
modes at the center of domains is correlated with the duality
of the background field 关4兴. This has been observed on the
lattice 关5兴 and is usually considered as an indication of spon-
taneous breakdown of flavor chiral symmetry. The purpose
of this article is to study the details of chiral symmetry real-
ization in the domain model. The nonzero quark condensate
and axial anomaly are generated as a result of spectral asym-
metry and the definite mean chirality of the eigenmodes. We
compute the quark condensate, study the degeneracies of the
minima of the free energy of the domain ensemble with re-
spect to chiral transformations, and estimate the spectrum of
pseudoscalar mesons.
The model is defined by a partition function describing an
ensemble of hyperspherical domains, each characterized by a
background covariantly constant self-dual or anti-self-dual
gluon field of random orientation. Summing over all orien-
tations and both self-dual and anti-self-dual fields guarantees
Lorentz and CP invariance. Quarks are confined as demon-
strated in the original work 关3兴. On the boundaries of each
hypersphere, fermion fluctuations satisfy a chirality violating
boundary condition
i
”
共
x
兲
e
i
␣␥
5
共
x
兲
⫽
共
x
兲
共1兲
which is 2
periodic in the chiral angle
␣
. Here
is a unit
radial vector at the boundary. Integrating over all such chiral
angles guarantees chiral invariance of the ensemble. As a
consequence of Eq. 共1兲, the spectrum of eigenvalues of the
Dirac operator in a single domain is asymmetric under
→⫺ . Such asymmetries have been studied in other con-
*
Electronic address: akalloni@physics.adelaide.edu.au
†
Electronic address: nedelko@thsun1.jinr.ru
PHYSICAL REVIEW D 69, 074029 共2004兲
0556-2821/2004/69共7兲/074029共23兲/$22.50 ©2004 The American Physical Society69 074029-1
texts, for example, by 关6兴. In the case of the domain model,
the above boundary conditions are combined with the 共anti-兲
self-dual gluon field which leads to a strong correlation be-
tween the local chirality of quark modes at the centers of
domains with the duality of the background gluon field 关4兴.
In this paper, we study how these aspects contribute to quark
condensate formation and the pattern of chiral symmetry
breaking.
The vacua of the quantum problem associated with an
ensemble of domains are the minima of the free energy de-
termined from the partition function. The problem of the
quark contributions to the free energy requires calculation of
the determinant of the Dirac operator in the presence of
chirality violating boundary conditions. For a choice of
boundary condition with
␣
→⫺ i
⫺
/2 this problem was
tackled in 关7兴 without taking into account the spectral asym-
metry, where the parity odd part of the logarithm of the de-
terminant was identified as ln det(iD” )⬃2q
with q the to-
pological charge 共not necessarily integer兲 of the underlying
gluon field, namely, the axial anomaly.
Our first goal in this paper is to address the analogous
problem for the specific gluon field relevant to the domain
model, taking into account the asymmetry of the spectrum.
For the parity odd part we obtain
ln det
共
iD”
兲
⬃2iq
共
␣
mod
兲
. 共2兲
This result is consistent with 关7兴 up to a contribution coming
from the asymmetry spectral function. However, we obtain
an additional parity even part which also turns out to be
␣
dependent. We consider this to be more an artifact of the
incompleteness of our calculation than an established prop-
erty of the determinant.
In the partition function all possible sets of chiral angles
兵
␣
1
,...,
␣
N
其
are summed, ensuring the chiral invariance of
the ensemble. Summation over all degrees of freedom in
addition to chiral angles defines the free energy as a function
of these chiral angles. In the limit N→⬁ the minima of the
free energy density in
兵
␣
1
,...,
␣
N
其
determine the preferred
chiral angles. More specifically, when self-dual and anti-self-
dual configurations are summed, the anomaly Eq. 共2兲 leads
to a contribution to the free energy of the form
⫺ ln cos
关
2q arctan(tan
␣
)
兴
which vanishes when
␣
⫽ 0,
.
The minima are degenerate with respect to discrete Z
2
chiral
transformations. Each of these minima are characterized by a
quark condensate of opposite sign, which arises due to the
spectral asymmetry. An infinitesimally small quark mass re-
moves the degeneracy between the two discrete minima, and
a nonzero quark condensate is generated with the value
具
¯
共
x
兲
共
x
兲
典
⫽⫺
共
237.8 MeV
兲
3
with no additional modifications of the two model param-
eters after fixing in the gluonic sector of the theory. This
gives a model with the chiral Z
2
discrete subgroup of U
A
(1)
being spontaneously broken, and not the continuous U
A
(1)
itself. In the absence of the mass term the ensemble average
of
¯
correctly vanishes. A similar argument based on mini-
mization of the free energy and thereby a relaxation of the
effective
parameter of QCD to zero is discussed in detail in
关8兴 in the context of the strong CP problem.
Moreover, the form of Eq. 共2兲 means that the free energy
does not depend on flavor nonsinglet chiral angles when
more than one massless quark flavors are introduced. This
allows for the correct degeneracy of vacua with respect to
continuous SU(N
f
)
L
⫻ SU(N
f
)
R
chiral transformations. This
vacuum structure implies the existence of Goldstone bosons
in the flavor nonsinglet pseudoscalar channel but not in the
singlet channel. To unveil more explicitly the singlet-octet
splitting, we analyze the structure of pseudoscalar correlation
functions in the context of the domain model and estimate
the masses of light pseudoscalar and vector mesons. The
qualitative conclusion of this analysis is that the area law
共confinement of static quarks兲 and the singlet-octet splitting
in the model have the same origin: the finite range correla-
tions of the background gluon field.
In the next section we briefly review the model, followed
by a summary of the properties of the spectrum of the Dirac
operator in the domainlike gluon field. We then discuss in
detail the calculation of the logarithm of the quark determi-
nant for one massless quark flavor, including the role of
spectral asymmetry in domains in giving the anomaly for the
parity odd part. This is followed by an analysis of the sym-
metries of the ground state of the domain ensemble and the
computation of the condensate. In Sec. V we generalize the
result to N
f
massless flavors in order to verify the spontane-
ous breaking of SU(N
f
)
L
⫻ SU(N
f
)
R
in the ensemble. The
last section is devoted to calculation of meson masses. De-
tails of calculations are relegated to the Appendix.
II. THE DOMAIN MODEL
For motivation and a detailed description of the model we
refer the reader to 关3兴. The essential definition of the model is
given in terms of the following partition function for N
→⬁ domains of radius R:
Z⫽ N lim
V,N→⬁
兿
i⫽1
N
冕
⌺
d
i
冕
F
i
D
(i)
D
¯
(i)
⫻
冕
F
Q
i
DQ
i
␦
关
D
共
B
˘
(i)
兲
Q
(i)
兴
⌬
FP
关
B
˘
(i)
,Q
(i)
兴
⫻ e
⫺ S
V
i
QCD
[Q
(i)
⫹ B
˘
(i)
,
(i)
,
¯
(i)
]
共3兲
where the functional spaces of integration F
Q
i
and F
i
are
specified by the boundary conditions (x⫺ z
i
)
2
⫽ R
2
n
˘
i
Q
(i)
共
x
兲
⫽ 0, 共4兲
i
”
i
共
x
兲
e
i
␣
i
␥
5
(i)
共
x
兲
⫽
(i)
共
x
兲
, 共5兲
¯
(i)
e
i
␣
i
␥
5
i
”
i
共
x
兲
⫽⫺
¯
(i)
共
x
兲
. 共6兲
Here n
˘
i
⫽ n
i
a
t
a
with the generators t
a
of SU
c
(3) in the adjoint
representation and the
␣
i
are chiral angles associated with
the boundary condition Eq. 共5兲 with different values ran-
domly assigned to domains. We shall discuss this constraint
A. C. KALLONIATIS AND S. N. NEDELKO PHYSICAL REVIEW D 69, 074029 共2004兲
074029-2
in detail in later sections. The thermodynamic limit assumes
V,N→⬁ but with the density
v
⫺ 1
⫽ N/V taken fixed and
finite. The partition function is formulated in a background
field gauge with respect to the domain mean field, which is
approximated inside and on the boundaries of the domains
by a covariantly constant 共anti-兲self-dual gluon field with the
field-strength tensor of the form
F
a
共
x
兲
⫽
兺
j⫽1
N
n
(j)a
B
(j)
„1⫺
共
x⫺ z
j
兲
2
/R
2
…,
B
(j)
B
(j)
⫽ B
2
␦
.
Here z
j
are the positions of the centers of domains in Eu-
clidean space.
The measure of integration over parameters characterizing
domains is
冕
⌺
d
i
...⫽
1
48
2
冕
V
d
4
z
i
V
冕
0
2
d
␣
i
冕
0
2
d
i
冕
0
d
i
sin
i
⫻
冕
0
2
d
i
兺
l⫽ 0,1,2
3,4,5
␦
冉
i
⫺
共
2l⫹ 1
兲
6
冊
⫻
冕
0
d
i
兺
k⫽ 0,1
␦
共
i
⫺
k
兲
..., 共7兲
where (
i
,
i
) are the spherical angles of the chromomag-
netic field,
i
is the angle between chromoelectric and chro-
momagnetic fields, and
i
is an angle parametrizing the color
orientation.
This partition function describes a statistical system of the
domainlike structures of density
v
⫺ 1
where the volume of a
domain is
v
⫽
2
R
4
/2. Each domain is characterized by a set
of internal parameters and whose internal dynamics are rep-
resented by fluctuation fields. Most of the symmetries of the
QCD Lagrangian are respected, since the statistical ensemble
is invariant under space-time and color gauge transforma-
tions. For the same reason, if the quarks are massless then
the chiral invariance is respected. The model involves only
two free parameters: the mean field strength B and the mean
domain radius R. These dimensionful parameters break the
scale invariance present originally in the QCD Lagrangian.
In principle, they should be related to the trace anomaly of
the energy-momentum tensor 关9,10兴 and, eventually, to the
fundamental scale ⌳
QCD
. Knowledge of the full quantum
effective action of QCD would be required for establishing a
relation of this kind.
A straightforward application of Eq. 共3兲 to the vacuum
expectation value of a product of n field strength tensors,
each of the form
B
a
共
x
兲
⫽
兺
j
N
n
(j)a
B
(j)
„1⫺
共
x⫺ z
j
兲
2
/R
2
…,
gives for the connected n-point correlation function
具
B
1
1
a
1
共
x
1
兲
•••B
n
n
a
n
共
x
n
兲
典
⫽ lim
V,N→⬁
兺
j
N
冕
V
dz
j
V
冕
d
j
n
(j)a
1
•••n
(j)a
n
B
1
1
(j)
•••
B
n
n
(j)
„1⫺
共
x
1
⫺ z
j
兲
2
/R
2
…•••
„1⫺
共
x
n
⫺ z
j
兲
2
/R
2
…
⫽ B
n
t
1
1
,...,
n
n
a
1
...a
n
⌶
n
共
x
1
,...,x
n
兲
,
where the tensor t is given by the integral
t
1
1
,...,
n
n
a
1
...a
n
⫽
冕
d
j
n
(j)a
1
•••n
(j)a
n
B
1
1
(j)
•••B
n
n
(j)
,
and can be calculated explicitly using the measure Eq. 共7兲.
This tensor vanishes for odd n. In particular, the integral over
spatial directions is defined by the generating formula
1
4
冕
0
2
d
j
冕
0
d
j
sin
j
e
B
(j)
J
⫽
sin
冑
2B
2
关
J
J
⫾ J
˜
J
兴
冑
2B
2
关
J
J
⫾ J
˜
J
兴
.
The translation-invariant function
⌶
n
共
x
1
,...,x
n
兲
⫽
1
v
冕
d
4
z
„1⫺
共
x
1
⫺ z
兲
2
/R
2
…•••
„1⫺
共
x
n
⫺ z
兲
2
/R
2
… 共8兲
can be seen as the volume of the region of overlap of n
hyperspheres of radius R and centers (x
1
,...,x
n
), normal-
ized to the volume of a single hypersphere
v
⫽
2
R
4
/2,
⌶
n
⫽ 1 for x
1
⫽ •••⫽ x
n
.
It is obvious from this geometrical interpretation that ⌶
n
is a
continuous function and vanishes if the distance between any
two points
兩
x
i
⫺ x
j
兩
⭓2R; correlations in the background field
have finite range 2R. The Fourier transform of ⌶
n
is then an
entire analytical function, and thus the correlations do not
have a particle interpretation. It should be stressed that the
statistical ensemble of background fields is not Gaussian
since all connected correlators are independent of each other
and cannot be reduced to the two-point correlations.
Within this framework the gluon condensate to lowest or-
der in fluctuations is 4B
2
, the absolute value of the topologi-
cal charge per domain reads q⫽ B
2
R
4
/16, and the topologi-
cal susceptibility turns out to be
⫽ B
4
R
4
/128
2
. An area
law is obtained for static quarks. Computation of the Wilson
loop for a circular contour of a large radius LⰇ R gives a
string tension
⫽ Bf(
BR
2
) where f is given for color
SU(2) and SU(3) in 关3兴. The area law emerges due to the
finite range of background field correlators Eq. 共48兲. On the
other hand, the model cannot account for such a subtle fea-
ture as Casimir scaling: the adjoint Wilson loop naturally
REALIZATION OF CHIRAL SYMMETRY IN THE... PHYSICAL REVIEW D 69, 074029 共2004兲
074029-3
shows perimeter law, but trivially because of the Abelian
character of the domain mean field.
Estimations of the values of these quantities are known
from lattice calculation or phenomenological approaches and
can be used to fit B and R. As described in 关3兴 these param-
eters are fixed to be
冑
B⫽ 947 MeV, R⫽ (760 MeV)
⫺ 1
⫽ 0.26 fm with the average absolute value of topological
charge per domain turning out to be q⬇0.15 and the density
of domains
v
⫺ 1
⫽ 42 fm
⫺ 4
. The topological susceptibility is
then
⫽ (197 MeV)
4
, comparable to the Witten-Veneziano
value 关11兴. This fixing of the parameters of the model re-
mains unchanged in this investigation of the quark sector.
The quark condensate at the origin of a domain where angu-
lar dependence drops out was estimated in paper 关3兴 with the
result of ⫺ (228 MeV)
3
.
III. DIRAC OPERATOR AND SPECTRUM
The eigenvalue problem
D”
共
x
兲
⫽
共
x
兲
,
i
”
共
x
兲
e
i
␣␥
5
共
x
兲
⫽
共
x
兲
, x
2
⫽ R
2
was studied in 关4兴. The dirac matrices are in an anti-
Hermitian representation. For
␣
assumed to be real a bior-
thogonal basis has to be constructed. Solutions can be la-
beled via the Casimirs and eigenvalues
K
1
2
⫽ K
2
2
→
k
2
冉
k
2
⫹ 1
冊
, k⫽ 0,1,...,⬁,
K
1,2
z
→m
1,2
,
m
1,2
⫽⫺k/2, ⫺ k/2⫹ 1,...,k/2⫺ 1, k/2,
corresponding to the angular momentum operators
K
1,2
⫽
1
2
共
L⫾ M
兲
with L the usual three-dimensional angular momentum op-
erator and M the Euclidean version of the boost operator.
The solutions for the self-dual background field are then
km
1
⫺
⫽ i
”
km
1
⫺
⫹
km
1
⫺
, 共9兲
where
and
must both have negative chirality in the self-
dual field and
is related to the polarization of the field
defined via the projector
O
⫽ N
⫹
⌺
⫹ N
⫺
⌺
⫺
共10兲
with
N
⫾
⫽
1
2
共
1⫾ n
ˆ
/
兩
n
ˆ
兩
兲
, ⌺
⫾
⫽
1
2
共
1⫾ ⌺B/B
兲
being respectively separate projectors for color and spin po-
larizations. Significantly, the negative chirality for
and
is
the only choice for which the boundary condition Eq. 共5兲 can
be implemented for the self-dual background. The explicit
form of the spinors
and
can be found in 关4兴, where it is
demonstrated that the eigenspinor Eq. 共9兲 has definite chiral-
ity at the center of domains correlated with the duality of the
gluon field. The boundary condition reduces to
⫽⫺e
⫿ i
␣
,
¯
⫽
¯
e
⫿ i
␣
, x
2
⫽ R
2
, 共11兲
where upper 共lower兲 signs correspond to
(
) with chirality
⫿ 1, which, using the solutions, amounts to equations for the
two possible polarizations, for ⌳
k
⫺⫹
:
e
⫺ i
␣
M
共
k⫹ 2⫺ ⌳
2
,k⫹ 2,z
0
兲
⫺
冑
z
0
i⌳
冋
M
共
k⫹ 2⫺ ⌳
2
,k⫹ 2,z
0
兲
⫺
k⫹ 2⫺ ⌳
2
k⫹ 2
M
共
k⫹ 3⫺ ⌳
2
,k⫹ 3,z
0
兲
册
⫽ 0, 共12兲
and for ⌳
k
⫺⫺
:
e
⫺ i
␣
M
共
⫺ ⌳
2
,k⫹ 2,z
0
兲
⫹
i⌳
冑
z
0
k⫹ 2
M
共
1⫺ ⌳
2
,k⫹ 3,z
0
兲
⫽ 0,
共13兲
where z
0
⫽ B
ˆ
R
2
/2 and ⌳⫽ /
冑
2B
ˆ
. For the present work
Eqs. 共12兲, 共13兲 are the starting point, from which we see by
inspection that a discrete spectrum of complex eigenvalues
emerges for which there is no symmetry of the form
→⫺ . For given chirality and polarization and angular mo-
mentum k, an infinite set of discrete ⌳ are obtained labeled
by a ‘‘principal quantum number’’ n.
IV. QUARK DETERMINANT AND FREE ENERGY FOR A
SINGLE DOMAIN
A. Massless case
We consider the one-loop contribution of the quarks to the
free energy density F(B,R
兩
␣
) of a single 共anti-兲self-dual do-
main of volume
v
⫽
2
R
4
/2,
exp
兵
⫺
v
F
共
B,R
兩
␣
兲
其
⫽ det
␣
冉
iD”
i
”
冊
⫽
兿
,k,n,m
1
冉
kn
共
B
兲
kn
共
0
兲
冊
⫽ exp
兵
⫺
⬘
共
s
兲
其
s⫽ 0
.
共14兲
The normalization is chosen such that lim
B→0
F(B,R
兩
␣
)
⫽ 0. The free energy is then F⫽
v
⫺ 1
⬘
(0).
In the zeta-regularized determinant an arbitrary scale
appears, and it is convenient to work with scaled variables

⫽
冑
2B
ˆ
/
,
⫽
R,
⫽
共
B
兲
/
,
0
⫽
共
0
兲
/
,
and where the dimensionless quantity z⫽ BR
2
/2⫽

2
2
/4 ap-
pears prominently. Moreover, it is convenient to analytically
A. C. KALLONIATIS AND S. N. NEDELKO PHYSICAL REVIEW D 69, 074029 共2004兲
074029-4
