Dual Ginzburg-Landau Theory with QCD-Monopoles for Dynamical

Progress of Theoretical Physics, Vol. 94, No.3, September 1995

Dual Ginzburg-Landau Theory with QCD-Monopoles for Dynamical Chiral-Symmetry Breaking

Shoichi SASAKI,* Hideo SUGANUMA* and Hiroshi TOKI*·**

*Research Center for Nuclear Physics (RCNP), Osaka University, lbaraki 567

**Institute for Physical and Chemical Research (RIKEN), Wako 351-01

(Received February 17, 1995)

We study dynamical chiral-symmetry breaking of non-perturbative QCD in the dual Ginzburg

Landau theory, where the QCD-monopole field is introduced as an essential field for color confinement

stemming from the choice of the abelian gauge fixing a !a 't Hooft. In this theory, QCD-monopole

condensation causes the dual Meissner effect, which changes the gluon propagator. The dynamical

chiral-symmetry breaking is investigated using the Schwinger-Dyson equation with the modified

gluon propagator in the QCD-monopole condensed vacuum. We introduce the low momentum cutoff

on the modified gluon propagator by considering the effects of the q- q pair creation and/or the

quarks being confined in hadrons. We find that dynamical chiral-symmetry breaking is largely

enhanced by QCD-monopole condensation, which suggests the close relation between the color

confinement and the chiral symmetry breaking. The dynamical quark mass, the pion decay constant

and the quark condensate are reproduced consistently with the confining mechanism in this theory.

§ 1. Introduction

373

It is a challenge in modern physics to understand the confinement of colors (e.g.,

quarks and gluons) and chiral symmetry breaking in terms of QCD. The early lattice

QCD simulations indicate these phenomena, but we are not sure how these phenomena

take place in nature.1> Recently, the Kanazawa group proposed the dual Ginzburg

Landau (DGL) theory with abelian QCD-monopoles as an infrared effective theory of

QCD to describe the quark confinemene> in line with the idea of the abelian gauge

fixing presented by 't Hooft.3> The successive lattice QCD simulations provided

strong evidence of the role of QCD-monopoles for quark confinement.4> It seems then

very plausible that the dual Ginzburg-Landau theory provides also chiral symmetry

breaking, since the lattice QCD simulations show a strong correlation between quark

confinement and chiral symmetry breaking.!) This led several authors to study chiral

symmetry breaking using the Schwinger-Dyson (SD) equation in terms of the DGL

theory5> or the dual QCD theory.6> Some of the authors firstly claimed to find the

close relation between chiral symmetry breaking and color confinement,6>.7> but fatal

errors in their calculations were pointed out by the Kanazawa group later. (See

Erratum in Ref. 6).) Their revised calculations showed that QCD-monopole conden

sation weakens chiral symmetry breaking.s>.s> These results seem to be against the

common belief that chiral symmetry breaking and confinement are intimately

related.!) Very recently, the present authors found a physically natural way to deal with an

infrared divergence associated with QCD-monopole condensation in the DGL

theory.8>·9> We were able to show that the DGL theory provides a linear confining

potential between heavy quarks, which includes the saturation effect in the long

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374 S. Sasaki, H. Suganuma and H. Toki

distance, in the same scheme. We could then show that chiral symmetry breaking is enhanced with the strength of QCD-monopole condensation by explicitly solving the SD equation.8>'9>

In this paper, we would like to study in detail the role of QCD monopole condensation on chiral symmetry breaking in the DGL theory. In particular, we shall discuss how the different conclusions on chiral symmetry breaking among various groups came about. We emphasize here that it is essential to introduce the low momentum cutoff parameter in order to take into account the fact that the dynamical quarkantiquark pair creation and/ or the quarks being confined in the hadrons. Further, we investigate the light quark confinement in terms of the absence of physical pole in the quark propagator by examining the solution of the SD equation. We also calculate the quark condensate, the dynamical quark mass and the pion decay constant.

§ 2. Dual Ginzburg-Landau theory

We start with the DGL Lagrangian including the QCD-monopole field, 2>.s>

_[DGL=.£gauge+ q(i~ -eA. • H- mq)q

1 1 .£gauge=- 2n2 [n·(a /\A)]"[ n· *(a 1\ B)]v+--znr[n·(a 1\ B)]"[n· *(a 1\A)]v

1 1 - 2n2 [n·(aAA)]2- 2n2 [n·(aAB)]2

(1)

(2)

in the Zwanziger form. 10> Here, A~-'=(A~-'3, A~-'8) denotes the diagonal part of the gluon

field and B~-' the dual gauge field to handle the QCD-monopoles.4>' 10> H=(T3, Ts) with Ta being the generator of SU(3)c. The quark field q with current mass mq couples with the gluon field AJ.<. The QCD-monopole fields Xa (a=1, 2, 3) couple with the dual gauge field BJ.<. The color-magnetic charge of Xa is given by the root vector Ea: £1 =(1, 0), e2=( -1/2, -/3/2) and £3=( -1/2, /3/2). The self-interaction with coupling ,\ and the expectation value v of lxal are introduced phenomenologically in the same spirit as the Ginzburg-Landau theory of superconductivity. The gauge coupling constant e provides a color-electric charge unit, and satisfies the Dirac condition, eg=47r, with the color-magnetic charge unit g.4

> Except for the self-interaction of Xa, all the rest can be derived from QCD by using the abelian gauge fixing a la 't Hooft, which brings the non-abelian gauge theory into the abelian gauge theory with QCD-monopoles.3

> In this scheme, there remains only the abelian gauge symmetry as the essential degrees of freedom at low energy, and the off-diagonal part would be irrelevant besides the appearance of QCD-monopoles. This idea of abelian dominance was also discussed by Ezawa and Iwazaki. 11

> The DGL Lagrangian has the dual gauge symmetry together with the residual abelian gauge symmetry, [ U(1)2]e X [ U(1)2]m.

The QCD vacuum becomes non-trivial by QCD-monopole condensation due to the self-interaction of xa. In this vacuum the dual gauge field B~-' gets a mass, similar to

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Dual Ginzburg-Landau Theory with QeD-Monopoles 375

the mass generation of the photon due to the Cooper pair condensation in supercon

ductivity. The DGL Lagrangian then becomes at tree level for the Xa field as

..fDGL-MF=..fgauge+ q(iiJ- eJ.. · H- mq)q+ ~ mB2B/, (3)

where the mass of the dual gauge field is mB=f3gv and the mass of the QCD-monopole

field is mx.=2/X v.2>.s>.s> To proceed, we integrate out the dual gauge field BP. in the

partition functional with the result,

X P.J/=_1_&: p.a{J_..pvr8n n a a - n2 "-P t a r fJ 8 • (5)

In the Lorentz gauge ..fG.F.=-(1/2ae)(a~'Ap.)2 , the gluon propagator in the QeD

monopole condensed vacuum is given by

Dp.v_ 1 [ I'll+( 1)aP.aJ/J 1 mB2

n2

xp.v -7j2 g ae- -r -7j2 a2 +mi (n·a) 2 (6)

This form clearly shows the effect of the QCD-monopole condensate as the appear

ance of the mB dependent term and coincides with the perturbative gluon propagator

in the ultraviolet limit. This mB dependent part originates from the second term in

the Lagrangian (4), which keeps the residual gauge symmetry by the condition,

X~'vap.=O.

One gets the quark color-electric current-current correlation by integrating out

AP. in the DGL Lagrangian (4),

ro - 1 · np.v • -Lj-j- -'[JiUJ Jv

(7)

where jp. is the quark color-electric current. It should be noted that the current

correlation is independent of the gauge fixing parameter ae owing to the current

conservation, a~'jp.=O. We emphasize that there arises the long-range current corre

lation coming from the second term in Eq. (6), because X~'vjp.*O. The long-range

confining potential between static heavy quarks is provided by the nonlocal factor

<xi(n2 /(n· a)2)iy>cx:(xn- Yn)O(xn- Yn)+ C1(xn- Yn)+ Cz, where C1 and C2 are arbitrary

constants.8> The perturbative gluon propagator in the axial gauge also includes the

similar spurious singularity, (n·a}-2.12

> The term containing (n·a}- 2, however, has

ap.av in the numerator in the perturbative gluon case instead of X~'v and therefore it

vanishes due to the color-electric current conservation in the calculation of the current

correlation. The static potential is led within the quenched approximation by considering the

action of the current correlation (7) in the static system.2>.s> We take a static

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quark-antiquark pair with opposite color-charge located at a and b,

j,..= Qg,..o{83(x- b)-83(x-a)}, (8)

where Q=(Q3, Qs) is the color charge of the static quark. Inserting Eq. (8) into Eq. (7), we obtain the static quark potential.

(9)

where n is a unit vector representing the direction of the Dirac string2>.s> and r= b- a is the relative coordinate between the quark-antiquark pair. The linear potential is obtained from the second term of Eq. (9) by taking n//r, following the energy minimum condition and the axial symmetry of the system.

VitnearC r) Q2mi1"" d 2Qr 1m,• 2 1

8 2 --2-{1-cos(qrr)} dqr 2+ 2+ 2 7i -co Qr 0 Qr QT mB

Q2m 2 (m 2+m 2) ---=::--.=2

8'--ln 8 2 x • r = a· r 87i mB '

(10)

where r =I rl and Qr denotes the momentum component perpendicular to r. A physical ultraviolet cutoff mx appears in the Qr-integral similar to the argument of the vortex in the superconductivity,8>'13> because the QCD-monopole condensate almost vanishes (mB::::O) in the central region of the hadron flux tube (:Smx-1). It should be noted that the infrared divergence does not appear owing to the condition, n//r. The string tension a obtained from Eq. (10) is similar to the energy per unit length of the Abrikosov vortex in type-II superconductors.8>'13> Thus, the long-range confining potential is provided by the second term in the gluon propagator (6), which is generated by QCD-monopole condensation.

In the phenomenological argument, the gluon propagator with the infrared behavior ex: q-4 14

> is used to realize the linear potential for the quarks. In this case, the color-electric field is not confined, because there appears a long range Van der Waals force of the r-3 behavior, which is against experiment.15

> In our case, the color-electric field is confined in the finite region by constructing the flux tube between the color sources in fair agreement with the lattice QCD simulations16

> and the long-range residual force between hadrons does not come about.

§ 3. Dynamical chiral-symmetry breaking

We are now ready to investigate the effect of the confining mechanism for chiral symmetry breaking in terms of the SD equation by the gluon propagator (6) in the QCD-monopole condensed vacuum, which causes the strong and long-range correlation. In other words, our strategy is to substitute the gluon propagator of the DGL theory for the full gluon propagator of QCD on the SD equation in order to include the non-perturbative effect in infrared region.

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The SD equation in the rainbow approximation and in the chirallimit is given by

sq-1(PE)=iJiE+ !&~~ Q2 rllSQ(kE)rvD~'11(PE-kE); Q2 =CF N::1 (11)

in the Euclidean metric after the Wick rotation in the ko-plane. SiPE) in the SD

equation denotes the quark propagator in the non-trivial QCD vacuum. Here, CF

=(Nc2-1)/2Nc for SU(Nc). The Euclidean variables are simply denoted as p or k

hereafter. We assume a simple form for the quark propagator

(12)

although we ought to take Sq -I=- iAJJ + B- ( CJJ- iD) it for completeness and the

argument of A, B, C and D should be dependent on P2 and n · p. We take this simple

form to investigate chiral symmetry breaking to see the effect of the QCD-monopole

condensate semi-quantitatively. This assumption was made also by several

groups.5H> The SD equation for the dynamical quark mass M(p2) is obtained by

taking the trace of Eq. (11),

M( 2)-f d4k Q2 M(k2) D ~'(k ) P - (27r)4 k2+ M2(k2) ~' - P . (13)

The trace of the gluon propagator is

2 + 1+ae+ 2 mi k2+mi -----p- (n·k)2 • k 2 +mB2

• (14)

It is notable that the gluon propagator includes the infrared divergence in the form of

the double pole factor 1/(n·k)2. This singularity causes the infrared divergence in

the SD equation (13).8>

At this place, the Kanazawa group considered that this infrared divergence

should be removed by the principal-value prescription as will be discussed in the

Appendix. As a consequence, the role of QCD-monopole condensation acts against

the chiral symmetry breaking.5> Moreover, the same result as the Kanazawa group

is obtained by other groups.6>·7> These results mean that chiral symmetry breaking is

not caused by the confinement mechanism. On the contrary, we coped with the

unphysical infrared divergence on the basis of the phenomenological argument and

found a large enhancement for chiral symmetry breaking due to QCD-monopole

condensation in our previous work.8>.9> We shall discuss how the difference among

various groups comes about in the Appendix.5>-9>

Our remedy for the issue of the infrared divergence is to introduce the infrared

cutoff a, corresponding to the size of hadrons,8>.9>'17> to the double pole factor in D~'v(k)

as

1 1 (n· kY ... (n· kY+ a2

• (15)

We note that this type of the infrared cutoff scheme can keep both the Lorentz

covariance and the invariance of the residual gauge symmetry.8> This cutoff scheme

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leads the compact formula for the linear potential including the saturation property in the long distance,

( 1 -ar) Vifrfear( r) = (J. -: , (16)

where a is the string tension.8> The infrared screening effect on the quark potential

has been observed in the lattice QCD simulation with the dynamical quarks. 0 •18> It

seems that the long-range correlation between quark and anti-quark is softened due to the pair creation of dynamical quarks. Thus, we deduce that the existence of the phenomenological cutoff to the double pole naturally removes the infrared divergence in the momentum integration by the screening effect of the q- if pair creation. From the theoretical argument based on the Schwinger formula for the q- ij pair creation, a is estimated to be of the order of 100 MeV.8

> In addition, this formula (16) for the quark potential was used for the phenomenological analysis of the hadron decay. 19>

We discuss further a possible source of the infrared cutoff a by studying the results of the lattice QCD simulation. Particularly, in the quenched approximation, the logarithm of the vacuum expectation value of the Polyakov line diverges as the volume increases, whereas the Wilson loop stays finite. This is caused by quark confinement; i.e., the Wilson loop is color singlet, while the Polyakov line is nonsinglet at each time. Furthermore, a large size Wilson loop is suppressed by the weight of e-5

, where S is the action. The chiral condensate, < ijq)cx.J d 4p trSq(p), is a loop-like configuration of a quark line and behaves similarly as the Wilson loop. The contribution of large scale quark loops to the chiral condensate is suppressed by the e-s factor as the case of the Wilson loop. This feature ought to be included in our calculation of the full quark propagator in the SD equation, Eq. (15). We may therefore interpret the infrared cutoff a as representing the suppression factor, e-5

, of large scale quark loops.

In the scheme of the use of the effective theory, we may have to deal with whole color singlet systems as ij q, q3 or more complex systems in order to describe the behavior of quark. 0 This is a difficult task. Instead, we adopt the SD equation in the ladder approximation to calculate the full quark propagator, but introduce one parameter (infrared cutoff a) to cure the infrared divergence, in the consideration similar to the Pauli exclusion operator in the Brueckner Theory of nuclear manybody problems. At this level, we may say that this parameter a is merely to effectively introduce the terms dropped in the ladder approximation. In this respect, we are presently investigating the vacuum polarization diagram associated with the quark loop contribution on the gluon propagator.20

> We may rather say, as discussed in the previous paragraph, that the infrared cutoff a represents the suppression factor, e- 5

, of large scale quark loops. This fact indicates that the quarks are confined in the small region of the hadronic scale, and hence the infrared momentum cutoff ( ~ R-1

)

corresponds to the size ( ~ R) of hadrons. This infrared momentum cutoff would lead naturally the moderation of the infrared divergence in the SD equation, of the order of the hadronic size, for chiral symmetry breaking, which is present even in the case of the quenched approximation in the lattice QCD simulation.

As for the gauge coupling constant, we use the Higashijima-Miransky approxima-

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Dual Ginzburg-Landau Theory with QCD-Monopoles 379

tion, Q2=4nCF•asett(max{P2, k2}), as the hybrid type of the running coupling,m'21

>

1271" (17)

where Pc approximately divides the momentum scale into the infrared region and the ultraviolet region, Pc2=A&oexp[(48Jr2/e2)·(Nc+1)/(11Nc-2Nf)].8>·9> By using this effective running coupling, the SD equation of the dual Ginzburg-Landau theory is reduced to the usual one of the QCD-like theory in ultraviolet limit.17l'21

>

We provide now the final expression of the SD equation to be solved numerically after performing explicitly the angle integrations,

X[< 2_ 2)1 { ~n2 +(k..L + p_d+ mi }+ 21 { ~n2 +(kj_ + PJ_)2 }]) (1S)

ms a n kn2+(h- pj_)2+ mi a n knz+(kj_- pj_)z

with kn=kn-Pn and kj_ =(k2 -kn2)

112• This expression was provided in previous

publications. a>

We also discuss the case of the angular average on the direction of the Dirac string. We consider that the light quarks in hadrons move in various directions and hence the costituent quark mass should be regarded as the dynamical mass in the angle-averaged case. The angular average over the direction of the Dirac string9

> is

2 1 a· a+J k2+a2 •

(19)

We note that it is positive definite where a is the phenomenological parameter representing the q- ij pair creation effect and/ or the infrared cutoff at the size of hadrons. The fact that the angular average, Eq. (19), of the double pole factor is positive, is the crucial difference from the other groups5>.7> as discussed in the Appen

dix. With the use of the angular average over the direction of the Dirac string,9

> we get the final expression for the SD equation as

z -1"" dk2 Q

2M(k2) ( 4k

2

M(p )- o 1671"2 kz+ M2(kz) kz+ pz+ mi+J(k2+ pz+ ms2)2-4k2p2

+ (1+ae)k2

+8k21"de sin

28

max(k2, P2

) Jra o a+J k2 + p2 +a2 -2kPcosB

[ mi-a

2 a

2 ])

x k2 +p2 +ms2 -2kpcos8 + k2+P2-2kPcos8 (20)

It should be noted that the right-hand side of Eq. (20) is always non-negative. In other words, the dynamical mass generation of the quark is enhanced due to the finite value

of the QCD-monopole condensate.

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~

"' s

2.0r---.---...,----.------,

~ 0.5

•• •••. m• =0.3GeV

00 ·•••••·•············ ••··•···••·• ··········· ·o w ~ w

2 2 p [AQCD]

Fig. 1. The dynamical quark mass M(p2) as a

function of the Euclidean momentum squared P2 for ms=0.3, 0.4 and 0.5 GeV. The other parameters are e=5.5 and a=85 MeV, while the QCD scale parameter is fixed at AQco=200 MeV.

§ 4. Numerical results

We solve the SD equation (20) numerically in the Landau gauge ae=O,

where the wave function renormalization is not necessary for mB=O. We show in Fig. 1 the results on the mass function M(JI) as a function of P2 for several values of the dual gauge mass mB, which indicates the strength of the QCD-monopole condensate. Only the trivial solution exists for small mB :S0.2 GeV. The quark mass M(p2

),

which is obtained as a non-trivial solu-tion, increases rapidly with mB.

We show in Fig. 2 the parameter dependence of M (p2 = 0). The dynamical quark mass M(O) increases

-+- m8 = 2.5 AQCD

2.5 -.t.- m• = 2.0 AQCD

0 ··•·· m8 =l.SAQCD g 2.0 •O· ms =0 s L...-___:~-.....1

1.5

8 .._.. 1.0

0.5

e Fig. 2. The dynamical quark mass M(p2 =0) of the

Schwinger-Dyson equation for the angleaveraged case is calculated for various dual gauge mass ms as a function of the QCD inter· action strength e. M(O) indicates the strength of chiral symmetry breaking, which increases with e and also with ms.

M'(p')=-p' \ 0 .. -10 -5 0

p2

m8 =0.5 GeV e = 5.5 AQCo = 200 MeV a =85 MeV

Fig. 3. The mass function M(P2) is extrapolated

analytically from the space-like region to the time-like region by the fourth-order polynomial and is shown as a function of the Euclidean momentum. The extrapolated curve does not cross the on-shell relation line (M 2(P2

) =-P2),

which may indicate confinement of light quarks.

monotonically with e and mB. This figure indicates that chiral symmetry breaking is largely enhanced by QCD-monopole condensation; i.e., when mB=O, M(O) is zero for e:S7. We comment that mB=O corresponds to the case of the ordinary SD approach without considering color confinement,22 >'23> in which they obtain chiral symmetry breaking without mB by taking e~IO and AQCo~l GeV. These values, however, contradict with the QCD scale parameter AQCo~200 MeV and the gauge coupling e.

Hereafter, we further examine the results under one parameter set, e=5.5, AQCo

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=200 MeV, ma=0.5 GeV and a=85 MeV. This is the case where the linear potential between heavy quark-antiquark agrees with the phenomenological inter-quark potential.24> We discuss now the light quark confinement by extrapolating the mass function M(P2

) into the time-like region and comparing with the on-shell condition M 2(p2)=-p2

• Although this argument may be rather simple for the discussion of quark confinement, it should certainly provide some crude idea of light quark confinement. The results of smooth extrapolation, by using the fourth-order polynomial fit in P2 to the space-like mass function, into the time-like region are shown in Fig. 3. We find that, due to the sudden rise of the mass function around P2 ~0, the extrapolated mass curve does not cross the on-shell line; M2(P2)=-P2

• This result may suggest that the light quarks are also confined due to QCD-monopole condensation.

We find that monopole condensation makes the slope of M(P2) around P2 ~0 larger

as can be seen by comparing the ma=O case with the ma=t=O case. This tendency can be physically explained as follows.8> The strong confining force between q and q appears due to QCD-monopole condensation, and this attractive force should promote q- q pair condensation similar to the Nambu-Jona-Lasinio model. Since such an effect of the confining force becomes stronger in the infrared region P2 ~ 0, the corresponding dynamical mass generation of quarks is much enhanced there. Thus, the strong confining force in the infrared region provides the large slope of M(P2

), which leads to the confinement of light quarks.8

>

We also calculate the quark condensate < qq) and the pion decay constant A. The quark condensate is given by

where the ultraviolet cutoff A is introduced to regularize the ultraviolet divergence of the integral. As was mentioned, the quark mass M(P2

) has the perturbative asymptotic form in the ultraviolet limit, since the SD equation (20) is reduced to the usual one of the QCD-like theory in this limit. Therefore, one gets the quark condensate in the renormalization-group invariant form for Nc=Nf=3, (ijq)RGr=(ijq)A·{ln (A 2 /A&:o)} -419 in the asymptotic region, A~AQCo.25> The pion decay constant Un: =93 MeV, Exp.) is given by the PagelsStokar formula, 26

>

[GeV2

]

2.s0r---i---,...---,----,

4 o.s [GeVJ

2 2 P [AQcnJ

Fig. 4. The mass function M(j/) of the SchwingerDyson (SD) equation as a function of the Eu·

clidean momentum squared P2• The results

without the angular average using Eq. (18) are

shown by dashed curves for various angles n · P =PcosB. The effect of the QCD-monopole condensate increases with the angle B. The result with the angular average using Eq. (20)

is shown by solid curve. The unit AQCn=200

MeV is used to calculate the SD equation.

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(22)

These quantities are well reproduced as< ijq)RGI~ -(222 MeV)3 and /rr~88 MeV under one parameter set, e=5.5, mB=0.5 GeV and a=85 MeV, which is consistent with the quark confinement potential (the string tension (J~ 1 GeV /fm) and the flux tube radius ~o.4 fm.8>·9> Here, the QCD scale parameter is fixed at Aoco=200 MeV.

We have discussed up to here the case with angular average on the direction of the Dirac string. Here, we would like to compare the present results with the previous results,8

> where the Dirac string angle is explicitly left in the integral equation, Eq. (18). We show in Fig. 4 the results on the mass functions M(P2

) as a function of P2 for the two cases. The result with angular average is shown by solid curve, while results at various angles are shown by dashed curves. We find the mass functions M(#) for the two cases are qualitatively similar and the angular averaged one seems to be the averaged mass function over those at various angles. The quarks in hadrons are moving in various directions and hence the constituent quark mass should be regarded as the angle-averaged dynamical mass.

§ 5. Conclusions

We have studied the role of QCD-monopole condensation on chiral symmetry breaking using the dual Ginzburg-Landau (DGL) theory. With the inclusion of the quark-antiquark pair creation effect and/ or the infrared momentum cutoff at the hadron size, we have formulated the Schwinger-Dyson (SD) equation and found chiral symmetry breaking with the parameters of the inter-quark confining potential. Here, QCD-monopole condensation plays an essential role of dynamical chiral-symmetry breaking.

We have found also some indication of light quark confinement by the extrapolation of the mass function into the time-like region. We have calculated other physical variables indicating chiral symmetry breaking.9

> They are M(0)~354 MeV, < ijq)RGI= -(222 MeV)3 and /rr=88 MeV, which compare nicely with the corresponding empirical values. These results indicate the evidence for the close relation between color confinement and dynamical chiral-symmetry breaking through QCD-monopole condensation in the DGL theory. In relation with our works using the DGL theory, it is very interesting to investigate the possible close relation between QCD-monopole condensation and dynamical chiral-symmetry breaking using the lattice QCD simulation.27l

Acknowledgements

We would like to acknowledge fruitful discussions with Professor T. Suzuki (Kanazawa University) and Professor 0. Miyamura. We have used the program D7 NLFRHM of TLIB in Tokyo Computational Center made by K.-1 Kondo, H. Nakatani and H. Mino in solving the Schwinger-Dyson equation. One of the authors (H.S.) is supported by the Special Researchers' Basic Science Program at RIKEN.

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Appendix

We would like to clarify how the differences in the results on chiral symmetry breaking come about among various groups.s>-9> The Kanazawa group made a

replacement, which is used in the axial gauge in order to avoid the infrared double pole, as2>.s>

(A1)

where 7J is an arbitrary finite constant. In the framework of the perturbative theory, this regularization is conventional and ueful to remove the infrared singularity in the axial gauge. This replacement provides the angular average as5>

( ( n : k )2 ) average= - ; 2 • (A2)

The apparent divergence is removed, but the sign is changed by throwing out the

divergent term. As a consequence, the role of QCD-monopole condensation acts against chiral symmetry breaking, which is opposite to the finding of Ref. 8).

Baker, Ball and Zachariasen6> used the same formula as the Kanazawa group, although their derivation is slightly different. They used the partial integration in the SD equation as

. { [ 1 ]00

[ 1 ]-~ 100

dkn , =hm - -k f(kn) - -k f(kn) + -k f (kn) '1-0 n 1J n -oo TJ n

+ 1-~ dkn j'(kn)} ' -oo kn

(A3)

and dropped the first and the second terms in (A3). We note however that the surface terms at kn= ± 7J never cancel each other. They dropped simply the infrared singularity. Krein and Williams7> also dealt with the infrared singularity as in Ref. 6) and

used the same formula for the angle average of the double pole factor 1/(n·k)2 as Eq. (A2).

The double pole provides divergence and this divergence was removed by using some methods in the works of these groups. We, instead, introduce the dynamical effect of the quark-antiquark pair creation and/ or quarks being confined in hadrons on the double pole factor (long range linear correlation), and find that this effect

naturally moderates the infrared divergence. The expression (A2) should be compared with our expression (19). The signs are

opposite between the two expressions and therefore the final results on chiral symmetry breaking are totally opposite. Here, we would like to state again the meaning of the divergence coming from the infrared double pole. This divergence is caused by a physical process. Quarks are confined due to the linearly rising potential, which

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diverges at a long distance and therefore at small momentum. Hence, this physical divergence should not be removed by any mathematical method. Instead, we have to moderate it by a physical process. We consider it as caused by the quark dynamics and argue it as the q- q pair creation and/ or quark confinement itself. In this respect, it is interesting to mention that the vacuum polarization diagram associated with the quark loop contribution on the gluon propagator seems to provide naturally the softening of the double pole divergence.20

>

References

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Dual Ginzburg-Landau Theory with QCD-Monopoles for Dynamical