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229
Progress of Theoretical Physics, Vol. 81, No.1, January 1989
An Infrared Effective Theory of Quark
Confinement Based on Monopole Condensation
Shinji MAEDAN and Tsuneo SUZUKI*
Department of Physics, Kanazawa University, Kanazawa 920
and
*Departamento de Ffsica, Ponti/fcia Universidade Cat6lica
C. p. 38071, 22453, Rio de Janeiro, RJ
(Received August 3, 1988)
We propose an infrared effective theory of quark confinement developing the 't Hooft's idea of
abelian projection. QeD is regarded as a V(l) x V(l) abelian gauge theory with magnetic
monopoles. The effective theory contains dual abelian vector fields and complex scalar monopole
fields in addition to quarks and gluons. A static potential between quarks or between gluons is
mainly composed of the linear and the Yukawa interaction. A relation VQQ = VQQ/2 is derived with
regard to the static potentials of quark-quark and quark-antiquark. Our model predicts the existence of neutral axial-vector bosons and heavy scalar bosons both of which are not composed of
quarks.
§ 1.
Introduction
Recent computer simulationsO show that quantum chromodynamks (QeD) is the
true theory of strong interaction. But they do not explain why and how quarks and
gluons are confined in QeD_ It is very important to understand the mechanism of
confinement so that we may predict analytically hadron physics such as the lowenergy chiral Lagrangian from QeD.
It is necessary to find dynamical quantities which play a dominant role in the
confinement problem and also to construct an infrared effective theory like the
Ginzburg-Landau theory of superconductivity in terms of such quantities. In 1981,
't Hoofe proposed an interesting idea of how to isolate the relevant dynamical vari'
ables at the hadron mass scale in QeD. Fixing the non-abelian part of the gauge
redundancy reduces the SU(3) gauge symmetry to that of the maximal abelian torus
group U(l) X U(l). This is called abelian projection. Quarks and off-diagonal
gluons have various combinations of these U(l) x U(l) charges. One also gets
point-like singularities which are magnetic monopoles with respect to the U(l) XU(l).
QeD is reduced to the abelian U(l) x U(l) theory with the magnetic monopoles. If
the monopoles make Bose condensation, both quarks and off-diagonal gluons which
are charged are expected to be confined.
An abelian theory with charges and magnetic charges has been studied by many
authors. 3 )-5l Especially Zwanzig~r's formalism 4l is interesting. When· one introduces another abelian vector potentials, the theory is written in a local form and also
is symmetric with respect to electric and magnetic parts. Moreover the monopole
interactions take a simple form.
Since the monopoles are abelian, one may perform the summation over the
230
S. Maedan and T. Suzuki
monopole trajectories. 6)-8) The monopole contribution is reexpressed in terms of
complex scalar fields with magnetic charges. We call them monopole fields. Selfinteractions among the monopole fields are expected to arise naturally.6)-8) It is our
basic assumption of this paper that there arise such self-interactions which cause
monopole condensation. Then our infrared effective Lagrangian is very similar to
the Ginzburg-Landau theory. Charged particles are confined.
A short report of this approach was published by one of the authors (T. S.) in the
simpler case of SU(2).9) To extend it to more realistic SU(3) and to give the details
of the approach are the purpose of this paper.
The layout of the paper is as follows: In § 2, we review shortly the idea of
abelian projection. Then we rewrite QCD with monopoles using the Zwanziger's
formalism. The summation over the monopole trajectories is tried in § 3. In § 4, we
derive the static potential between quarks or between gluons. In § 5 comparison with
experimental data is done and we determine the value of parameters of our model.
The final section is devoted to concluding remarks.
§ 2.
Abelian projection and theory of electric and magnetic charges
Let us introduce a (composite) operator X (not specified here) which transforms
non-trivially under SU(3). Then the non-abelian part of gauges is fixed if gauge
transformations are restricted such that X becomes diagonal everywhere. 2) QCD is
reduced to a U(l) x U(l) theory with magnetic monopoles.
First let us write the QCD Lagrangian:
(2'1)
where .£~.F. is the above non-abelian gauge-fixing term not specified in this paper and
GI-'J)=a~J)-aJ)AI-'-ie[AI-',AJ)]
,
The SU(3) generators T a are redefined as follows:
iJ=(Hl, H2)=(T3, Ts),
E±l =
lz (Td iT2) ,
E±2= lz(T4+iT5) '
E±3= lz(T6±iT7).
Then root vectors € a are defined through
An Infrared Effective Theory of Quark Confimement
Table I.
231
U(l) x U(l) charges of off·diagonal gluons and quarks.
C p'
C/
C/
A p3 charge
1
-Z-
1
-Z-
-Z-
Ap· charge
0
-/3
-2
-/3
2
- 2-/3
1
rf2
rf'
1
1
1
Z1
- 2-/3
rf3
0
1
-/3
[11, Ea]= €aEa,
[11, E-a]= - €aE- a , (a=l, 2, 3)
(2·2)
and are written explicitly as €l =(1,0), €2=( -1/2, - -/3/2) and €3=( -1/2,-/3/2).
Also we rewrite the gauge field Ap in terms of new variables,
(2·3)
Studying the transformation properties of these fields and quarks under U(l) x U(l)
with the gauge fields .,tL=(Ap 3, A p S ), we easily see their charges as shown in Table I.
In terms of these new variables, the QeD Lagrangian is expressed as
'£QCD='£1'+'£2+'£G.F. ,
(2·4)
(2:5)
.£2=-
~ ~ll(fDa/\ca)pv+
e2
3
:;c
a
2
m Cp*PCv*71 +
i; ~l(fpv.
It
ta)(c*a/\cayv
_
+4[~1 €a(c*a /\C )pv)2+ ¢(irJ - M<t+ e41· H)¢
(2·6)
where (C/\D)pv=CJ)v-CvD p and fpv=(f2v'/2v). H~re fDpa=(Jp+ie(€a·Ap).
Without monopoles, fpv= a~v - avAp.
There exist monopoles in general. Then the abelian Bianchi identities with
respect to fw are violated. Using the quantization rule between electric and
magnetic charges, we find that magnetic charges are distributed on the root lattice.
Namely, they are expressed as
with integers ~a and ~~=l €a=O.
identities is written by
Here g=47re- 1 • The violation of the Bianchi
(Jp* fpv=kV(x)
(2·8)
S. Maedan and T. Suzuki
232
where
* !Pl/=6Pl/p,,!P<J'j2.
The field strength !Pl/ in (2·5) can be given by5)
(2·9)
where
'HA6(X) = fdY hA(x- y)k,,(y)-==(hy ® k,,)(x) ,
(2·10)
hix)=-nln·a)-!(x) .
(2·11)
Here nA is an arbitrary constant four vector. Substituting (2·9) into (2·5), we see that
.£{ becomes non-local due to the non-local operator in (2·11). But if one introduces
another abelian vector potential 13p, .£ { can be rewritten in a local form: 4 )
(2·12)
(2·13)
(2·14)
where (a· C)l/ -== apCPl/. We adopt the Lagrangian of this form, because magnetic
monopole interactions are simpler and also because electric (X) and magnetic (B)
parts are symmetric:
.£QCD=.£! +.£2+.£3+ '£G.F..
(2·15)
Really the Lagrangian (2·15), without additional abelian gauge-fixing terms, has
magnetic U(l) x U(l) symmetries characterized by 13p in addition to the original
electric U(l) x U(l). Note that iL and 13 P are not all independent. 4 )
§ 3.
Summation over the monopole trajectories
and an infrared effective Lagrangian
We have written the QeD Lagrangian (2 ·15) with the monopoles as particle
trajectories. To get the partition function, one has to sum up all possible monopole
trajectories. We first follow the method of Bardakci and Samuel. 6 )
In SU(3), the monopole charges are distributed on the root lattice as shown in
(2·7). We assume that the monopoles with a smallest magnetic charge, that is, with
~a= ± 1 alone contribute to the sum. We identify a trajectory travelling backward in
time with a forward travelling trajectory with the opposite magnetic charge.
All three directions in the root diagram are not independent, since ~~=! 6a=0. If
we take the summation using any set of two independent monopole-pairs with ~a= ± 1,
the Weyl invariance is broken. In order to keep the invariance in the course of the
summation, we introduce a trick. We regard the system as having magnetic U(l?
and add an auxiliary vector potential Bpo using JDBpOo(B pO)=1. Also we define 13'
-==(B 3 , B 8 , BO) and 6/=(6a 3 , 6}, 6aO) with 6ao=1. All three directions can be regarded
as independent. After the summation is over, we eliminate the extra degree with a
An Infrared Effective Theory of Quark Confimement
233
constraint.
Making use of an equation derived from Ref. 6),
~ 1
/;;0 N!
fns=1n-sexp [.zg /;;1P
~ rd (~' B~ ,)dxs'l(r
rs 1· p.
drs
N
X
€
s )]
(3·1)
we obtain a new Lagrangian
(3·2)
where Xa is a complex scalar field with magnetic charges g € a.
Since all three sets of monopole trajectories are not really independent, there is
a constraint among the new scalar' fields. Let us write the Xa field in terms of two real
fields as
xa=IXalexp(i9'a) .
(3·3)
Then the last term of (3·2) becomes
(3·4)
In the unitary gauge adopted, we can eliminate all 9'a:
(3·5)
(3·6)
(3·7)
We can safely integrate out Bp.° without any modification with a constraint
(3·8)
where we have adopted a suitable boundary condition. It is easy to see that the
quantity ~~=19'a is invariant under the original U(l)2. It is an extra degree that must
be abandoned.
In the above summation, the backtracking or overlapping trajectories with
opposite magnetic charges are included. Since magnetic charges are abelian, such
cases must be excluded. There must exist a repulsive force of a delta function type
between monopole trajectories, which is known to lead to Alxl4(A >0) interaction. 7),8)
Also a mass term may arise naturally.7),8) Other higher-order self-interactions, if any,
are infrared irrelevant. Radiative corrections may also be important. The exact
result of the summation needs an extensive study of the properties of monopoles in '
QeD. Here we only assume that such interactions really .appear after the summation
S. Maedan and T. Suzuki
234
is over. We propose a Lagrangian
(3·9)
with the constraint (3·8) as our infrared effective model of confinement. An interaction (~aIXaI2)2 is also possible.. But we neglect it for simplicity, since no essential
differences occur in the following discussion. When v2> 0, the two independent rpa
become unphysical Goldstone bosons due to the Higgs mechanism. The gauge fields
B p 3 and B p 8 are massive with the degenerate mass mB=.j3gv. The three neutrallXal
fi~lds have also the degenerate mass ms=2/X v. The degeneracy reflects the fact that
the theory is invariant under the action of the Weyl group of SU(3).
§ 4.
Static potential
Let us evaluate a potential between static quarks or between static gluons. We
. neglect all dynamical charged particles on the assumption that the abelian
components play the essential role in quark confinement. Adding abelian gaugefixing terms,4)
(4 ·1)
we can integrate out
A
p•
Then the Lagrangian is reduced to
(4·2)
where
~
~
~
(A 7(1)
H pJJ = JpBJJ-JJJB p + €PJJA(1 h ® J ex
.
(4·3)
Here J%x are the external charged current. They are expressed as
J:x=QgPO{o(x-a)-o(x-b)} ,
(4·4)
where, e.g., Q =(0, e/.j3) for ¢J3¢3 and (e,O) for Cp *ICJJ 1. It is noted that the
Lagrangian (4·2) is just the dual form of the Ginzburg-Landau theory with a pair of
static magnetic monopoles.
Only numerical calculations can treat the above non-linear system exactly. To
calculate the potential analytically, we utilize the knowledge obtained in the analyses
of the vortex solution of the usual Ginzburg-Landau theory.
1. We assume mS>./2mB, since the vortex solution exists only in the case .
. 2. Physical quantities such as the vacuum energy we are calculating are np.P
independent.~) When we consider the axial symmetric case choosing n along
the direction between the static sources, we may set the phase rpa of the Xa fields
vanishing. 10)
3. The region inside the inner core with a radius ./2ms -1 of the vortex does not give
a large contribution to the vacuum energy.ll) We neglect the contribution.
Then we cannot choose np exactly parallel to r=a- b due to an apparent
235
An Infrared Effective Theory of Quark Confimement
infrared singularity from the center of the vortex. We take
Ir-LI~y'Zms-l
(4°5)
,
where r-L=r-(ron)n with nJl=(O, n) and n Z=1.
4. The Xa fields take the vacuum expectation value v everywhere except inside the
inner core of the vortex, if y'ZmS- 1 4:;.mB- 1 • We assume therefore Xa=V.
With these assumptions, we can carry out the functional integration with respect
to 13 Jl. Neglecting an irrelevant infinite constant, we obtain
+ kZ-mi
(- mBZ)
Z
n (
nJlnv)} 7 V(k)]}
(n ok)2 gJlV---:nz- J ex
(4°6)
,
where a is a gauge parameter and ]exCk) is a Fourier transform of ]ex(X).
use an infrared prescription of (nok)-Z such that
Here we
(4°7)
The justification is discussed in Appendix A.
Using the static current (4 °4) and subtracting the Coulomb self-energy, we get the
static potential,
(4°8)
-+Z -mBT
Q-+z Z
mB rn&
= __Q.__e_ _ +
47l"
r
47l"
0
(f()2
) Q-+z mBZI(r)
~ +
ms
2
'
(4°9)
where rn=l(ro n)1 and (4°5) is used. Ko(x) is the modified Bessel function and
I(
)-1
r -
00
-00
dZk-L exp( - rnJk-LZ+ mi) -ikl·rl
(k-LZ+mBZ)3/Z
e
.
(27l")2
(4 °10)
The function I(r) is regular and bounded by
(4°11)
Hence it can be neglected numerically. We show the detailed calculations in
236
S. Maedan and T. Suzuki
Appendix B. Since
Ir-LI
is small, we approximate rn by r, so that
~2
-mBT
Q
V(r)= ___
e__
+
4/T
r
Q~2
2
mB K.
4/T
0
(f1i2
)
~ r.
ms
(4-12)
The potential is composed of the Yukawa and the linear term as expected. Our
potential (4-12) is applicable only for r > 12ms -1. Around r=O, all charged gluons as
well as neutral gluons contribute equally. Then the static quark potential becomes
that of the Coulomb interaction with the coefficient four times larger than that of the
Yukawa term in (4-12). When v 2 <O, mB=O, so that the potential (4-12) is reduced to
the Coulomb potential. There is no confinement in the case. Therefore we see that
the vacuum expectation values of the fields Xa play the role of the order parameter.
§ 5_
Comparison with experiments and predictions
In this section, we compare our results with experimental data. First, the string
tension (J determined from (4 -12) is proportional to C? Thus the string tensions of
static quarks and static gluons have the ratio/ 2)
(JqUark
-l
(Jg!uon -
3 .
(5-1)
If the Pomeranchuk trajectory is that of gluon exchange and if the string-model
connection between string tensions and Regge slopes holds, we get the ratio of the
Regge slopes of the Pomeranchuk and the p trajectories:
(5-2)
When a/=O.9 GeV- 2 is used, ap'=O.3 GeV- 2 which is consistent with the experimene3 )
Next, we get a relation from (4-12),
(5-3)
where VQQ and VQQ are the static potentials of quark-quark and quark·antiquark.
Assuming the relation (5-3), Basdevant and Boukraa14 ) showed that the ground state
of baryons can be calculated with good accuracy.
N ow let us determine the values of the parameters of our model using (1) the
static poten~ial determined from the quark onium data 15 )
(5-4)
with as=(O.19~O.36) and a=(2.22~2.34) GeV- 1 and (2) the deconfinement temperature
Tc determined from the Moilte Carlo simulation/) i.e., Tc~250 MeV.
To calculate T c , we have to evaluate both quantum and thermal fluctuations in
general. However our model is a low· energy effective theory. Hence we may
neglect quantum fluctuations, since radiative corrections are expected to be included
already. This is analogous to the situations of the one-pion' exchange model in
237
An Infrared Effective Theory of Quark Confimement
nuclear force.
The thermal fluCtuations are calculated by Caldi and N ussinov: 16 )
(5·5)
Considering the fact that the coefficient of the Coulomb interaction is modified at r
we get
~O,
e=J47ws =(1.55~2.13) ,
g=k=(5.91 ~8.11),
e
v= Tc =125 MeV
2
'
mB=J3gv=(1.28~1.76)
ms=(3.93~6.29)
GeV,
GeV ,
;1=(5.1 x 102~ 1.26 x 103 )
•
(5·6)
Some comments are in order.
1. Our model predicts the existence of an axial-vector and a scalar glueball-like
states around (1.3~ 1.8) GeV and (3.9~6.3) GeV, respectively.
2. The ratio mS/mB in 5U(3) is larger than one and then is consistent with our
assumption. But the ratio is fairly smaller than that in 5U(2).9) Hence other
contributions we have neglected may have a sizable effect, although the main
features are not modified. Numerical calculations are interesting.
3. The value of ;1 is very large. This may be consistent with the assumption that
Xa=V almost everywhere. When the other interaction (~aIXaI2)2 is also taken
into account in (3·9), however, the value of;1 is not determined in (5·6).
§ 6.
Summary and concluding remarks
We have constructed an infrared effective theory of quark confinement, developing the 't Hooft's idea of abelian projection. After summing up the monopole trajectories, we have complex scalar monopole fields coupled to the dual vector potential.
The assumption that the monopoles condense leads to confinement of quarks and
off-diagonal gluons. The slope ratio of the Pomeranchuk and the p trajectories is
evaluated to be in good agreement with the data. Also our model predicts the
existence of an axial-vector and a scalar glueball-like states around (1.3~1.8) GeV
.
and (3.9~6.3) GeV, respectively. We hope they will be discovered in near future.
In comparison with other infrared effective theories,17)-19) our model is llUlch more
attractive, because
1. it explains what are essential dynamical quantities in quark confinement,
namely, they are the magnetic monopoles and the dual abelian gluons,
2. it explains what is the mechanism of quark confinement, namely, the monopole
condensation is the mechanism,
3. it respects the chiral flavor symmetry when the quark mass vanishes and
238
S. Maedan and T. Suzuki
4.
it does not use any results of perturbative QCD such as the behavior of the
effective coupling constant.
There are many problems to be studied in our model. Let us list some of the
important problems.
1. It is important to explain why the abelian components (with the monopoles) are
dominant in the confining phase, although the non-abelian gluons are essential
in the deconfining phase. This will be shown in a succeeding paper. 20l
2. To calculate <iF¢;) and to study the spontaneous breaking of the chiral flavor
symmetry are also interesting. In this connection, we believe, it may be easier
to derive the low-energy chiral Lagrangian starting from our model than
directly from QCD.
3. It is interesting to perform computer simulations of our model and to check if
our picture (monopole condensation and abelian dominance) is true. A DESY
group2Il has shown recently, using numerical calculations, that the monopole
condensation is responsible for confinement in SU(2). We believe therefore
that our model (4-2) or (3-9) neglecting off-diagonal gluons may explain th~
hadronic world in low-energy.
4. The most important is to derive our parameters A and v directly from QCD.
Thi~ will need an extensive study of the properties of the monopoles in QCD,
but it is very challenging.
Acknowledgements
The authors are thankful to Professor E. Yamada for useful discussions, and one
of the authors (S.M.) wishes to thank Professor Y. Matsubara for useful conversations. The other (T.S.). was financially supported by JSPS in Japan and CNP q in"
Brazil. He also thanks Professor C. Aragao de Carvalho and other members of the
theory group at PUC/RJ for their kindness extended to him during his stay.
Appendix A
We discuss the infrared prescription of (n° k)-2. Usually (n° k)-I is defined as the
principal part P/(n_k).3)-5l But the product of the principal part is ill-defined. We
have adopted the prescription (4-7) which is the partial finite part and is well-defined.
Comparing the calculations in coordinate space and in momentum space, we show
the definition (4 ° 7) is consistent. The integral in problem takes the form,
(A-I)
where we take nJL=(O, 0, 0, 1) and then
Also p(X)= 0'(x3-a)-0'(x3+ a) and the ® product is defined in (2-10). In coordinate
space,
An Infrared Effective Theory of Quark Confimement
J=
239
!dX3{!dY3~ [e(X3- Y3)- e(Y3-X3)][o(Y3-a)- O(Y3+ a)]}
X
{!dZ3
~ [e(X3- Z3)- e(Z3-X3)][O(Z3- a)- o(z3+a)]}
(A02)
=2a.
In momentum space,
(A03)
Usually we cannot change the order of integrations in this case.
definition (4 ° 7) after integrating first with respect to X3, we get
00
1-0027r
dk l[(_l_.)2 _(_1_.)2J [2 _
2 k+
k-
J=
z€
But if we use the
e-2ika _ e2ika]
z€
(A04)
=2a.
This justifies the definition (4 ° 7).
Appendix B
Let us evaluate the integral
A=-
f
(_1)2 .
d3k e- ik · r
(27r)3 k 2 +mi nok
(Bo!)
Using (4 7), we get
0
(B02)·
Now we easily obtain
(B03)
Introducing a new variable
(B04)
we find
S. Maedan and T. Suzuki
240
1
II(r)I~-2-
JrmB
1'"du (e-mBTnU)
1
U
2
(B·5)
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