Monopole condensation and confinement in SU (2) QCD

I ~ tlIll l -'1';I ti '.11i'l,'l [15"1 "t

PROCEEDINGS SUPPLEMENTS

Monopole

condensation

Nuclear Physics B (Proc. Suppl.) 34 (1994) 182-188 North-Holland

and confinement

in S U ( 2 )

QCD

Hiroshi Shiba * and Tsuneo Suzuki t ~ D e p a r t m e n t of Physics, Kanazawa University, Kanazawa 920-11, J a p a n An effective monopole action is derfved from vacuum configurations after abeliau projection in the maximally abelian gauge in SU(2) QCD. Entropy dominance over energy of mouopole loops is seen on the renormalized lattice with the spacing b > b~ ~_ 5.2 x 10-3AL 1 when the physical volume of the system is large enough. Monopole and photon contributions to Wilson loops are calculated also. The string tensions of SU(2) QCD are well reproduced by extended monopole contributions alone.

1. I n t r o d u c t i o n The 'tHooft idea of abelian projection[l] is very interesting to understand confinement mechanism in QCD. A lot of numerical studies have been done to test the idea[2-5]. It was found that confinement phenomena seem to be well reproduced by abelian link fields alone in the maximally abelian (MA) gauge in SU(2) QCD[3, 4]. The abelian dominance suggests the existence of an effective U(1) theory describing confinement. In the first part of this note, we derive an effective U(1) action on the dual lattice as done in compact QED[6] from monopole distributions given by Monte-Carlo simulations and show that the monopole condensation occurs in the confinement phase in SU(2) QCD. If the monopoles alone are responsible for the confinement mechanism, the string tension which is a key quantity of confinement must be explained by monopole contributions. This is realized in compact QED as shown recently by Stack and Wensley[7]. In the second part of this note, we show that the same thing happens also in SU(2) QCD by means of evaluating monopole and photon contributions to Wilson loops. 2. T h e m o n o p o l e a c t i o n

the following partition function

s,~ k.(s)=-e¢

s

x e x p ( - E fiSi[k]), i where 0' is a backward derivative on a dual lattice and k~(s) is the conserved integer-valued monopole current, fi is a coupling constant of an interaction 5~[k]. Since the dynamical variables here are kl~(s ) satisfying the conservation rule, it is necessary to extend the original Swendsen method by considering a plaquette (s ', #~, z/) instead of a link. Introducing a new set of coupling constants {f/}, define

Si[k] = E M = - ~ Si[k']

E i fiSi[k']) fiSi[k']) ' (1) where k; ( s) = k~(s) + M ( Ss,,,6,,, + 5 ~ . s ' + # , 5 , ~ , 5~,s,+~,5,~,- 6,,~,5,,,). When all .~ are equal to fl, one can prove an equality (Si) = (Si), where EM=-~

exp(exp(- ~i

the expectation values are taken over the above original action with the coupling constants {fi}. Otherwise, one may expand the difference as follows

(L - si> = Z < s i & -

-

(2)

J Swendsen[8] developed a method of determining an action from a given ensemble of configurations. The method can be applied to our case. A theory of monopole loops is given in general by *Speaker of the former part ?Speaker of the latter part with the title 'Abelian monopoles explain the SU(2) string tension

where only the first order terms are written down. This allows an iteration scheme for determination of the unknown constants fi. Practically we have to restrict the number of interaction terms. We a d o p t e d 12 types of quadratic interactions listed in Table 1 in most of these studies.

0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved.

SSD1 0920-5632(94)00237-P

183

H. Shiba, T. Suzuki/Monopole condensation and confinement in SU(2) QCD

Table 1 The form of monopole action adopted. All terms in which the relation of the two currents is equivalent must be added. i

interaction

1

k~(o, O, O, 0)='

2 3

~1(0, O, O, 0)]~1 (1, O, O, O) /~1(0, 0, 0, 0)/~1(0, l, 0, 0)

4 5

kl (0, O, O, O)kl (1, 1, O, O) kl(O, O, O, O)kl(O, 1, 1, O)

6 7

/~1(0, 0, 0, 0)kl (2, 0, 0, 0) kl (0, O, O, O)kl(1, 1, 1, O)

8 9

kl(O, O, O, O)kl(O, l, 1, 1) kl (0, O, O, O)kl(1, 1, 1, 1)

10

1¢1(0, O, O, 0)/~1 (2, 1, O, 0 )

11 12

kl(O, O, O, 0)k2(2, 1, O, O) kl(O, O, O, O)kl (2, 1, 1, O) 0

[

0.6

First we applied the method to the Villain form of U(1) lattice gauge theory[9], since the Villain model is reformulated exactly as a theory of monopole loops having an action

i

a,s~,tt

where D(s) is the lattice Coulomb Green function[6]. Using the Villain action, we generated 100 gauge field configurations separated by 40 sweeps after a thermalization of 4000 sweeps at /3~ = 0.60,0.64, and 0.68 on 84 lattice. We used the DeGrand-Toussaint scheme[10] for locating monopole currents in the lattice gauge field configurations to obtain an ensemble of monopole configurations. The coupling constants ./'1 ~ f6 determined are plotted in Fig. 1 in comprison with the theoretical values, f l agree well with those of the theoretical values, whereas there are small discrepancies with respect to f~ ~ f6. The discrepancies come from the truncation of the terms of the action taken. Really, if we include more quadratic terms, they disappear. Next we applied the method to SU(2) lattice gauge theory. Monte-Carlo simulations were

done on 124 , 144 , 164 , 184 , 204 , and 244 lattices for

o fl

,, f4

[] f 2

•

•

o f6

f3

I

I

In7

f5

I

I

I

0.64

I

I

I

0.68

BETA

Figure 1. Coupling constants fi versus /3 in the Villain model. Theoretical curves are plotted by solid (fl), dotted (f2,f3), dashed ( 2 ' 4 , £ ) and dash-dotted (.f6) lines.

2.3 _
184

H. Shiba, T. Suzuki /Monopole condensation and confinement in SU(2) QCD

,, . . . .

-i

....

-=- . . . .

i .....

| ....

~- ....

o fl r, f2

" f3 o f4 = f5

°

}

{

-o

/ / In7

244

2.5

B ETA =2.6

fl

In7

1~

m

023

0

I

2

| I

I

14

! I

I

16

}

• =

;

}

T I

I

18

]

20

=33

1.5 *

! I

}

I

I

22

I

I

I

m I

24

2.4

2.5

LATTICE SIZE

43

A

I

2.6

I

I

2.7 2.8 BETA

I

I

2.9

3

Figure 2. Coupling constants fi versus lattice size at./3 = 2.6 for 2 a extended monopole.

Figure 3. Coupling constants f l versus /3 for 2a,33, and 4 a extended monopoles on 244 lattice.

from Fig. 2. We see fl is dominant and the coupling constants decrease rapidly as the distance between the two monopole currents increases. 2)The energy of a monopole loop (with a unit, charge Ik,(s)l = 1) of length L may be approximated by a. self-energy part flL, whereas its entropy grows like Lln7 when L is large[6]. If ./'1 < lnT, the entropy dominates over the energy and the monopole condensation occurs. We plot .fl versus /3 for various extended monopoles on 244 lattice in comparison with the entropy value ln7 for the infinite volume in Fig. 3. Each extended monopole has its own /3 region where the condition fl < ln7 is satisfied. When the extendedness is bigger, larger/3 is included in such a region. 3)It is interesting to see the relation between the monopole condensation and the deconfinement transition in a finite volume. In ref.[12], Polyakov loops are measured on symmetric lattices and the value T~ = (25.8 4- 1.1)AL is given

for the critical 'temperature'. The critical coupling /3~. is fixied from the value for each lattice volume. For example, /3~ is 2.60 on 144 and 2.81 on 244 lattices, respectively. Our data indicate the following features irrespective of the original lattice volume. If the renormalized lattice is larger than 74, the fl value is above the ln7 line at/3¢. It seems to cross the ln7 line just at. /3,. when the renormalized lattice is about. 7 4. See Fig. 2 for 23 extended monopoles a.t ,/3 = 2.6 which is just /3c of 144 lattice. There fl takes the value about ln7. Also ,in the Villain case shown in Fig. 1, the energy-entropy arguments give the correct critical coupling on a lattice as small as 84 . We may assert the above energy-entropy arguments can be used when the renormalized lattice is larger than 74. Then we see condensation of some extended monopoles occurs in the confinement phase.

H. Shiba, T Suzuki /Monopole condensation and confinement in SU(2) QCD

2.5 o28

a 33

A4 3

In7

1.5

1:

I

I

I

I

I I I I I [ I I 5 I0 b

Figure 4. Coupling constants fl versus b(10-3(A)L) . The solid curve is the prediction of the action (4) with A = 42(A)L, D(0) = 0.11,m0 = 60(A)L.

On the other hand, when the renormalized lattice is smaller than 74, such simple arguments may not apply as seen froln small discrepancy between the ln7 cross point of f i in the 48 case and fl~ in Fig. 3. There is a possible entropy decreasing effect due to the periodic boundary condition on a small lattice. Such entropy calculations are very important to know if the confining phase and the monopole-condensed phase are exactly the same as expected. 4)The behaviors of the coupling constants are different for different extended monopoles. But if we plot t h e m versus b , we get a unique curve as in Fig. 4. The coupling constants seem to depend only on b, not on the extendedness nor /3. This suggests the existence of the continuum limit and the monopole action in the limit may be similar to that, given here.

185

5)From Fig. 4, we see the self-energy per monopole loop length fl on the renormalized lattice decreases as the length scale b increases. A critical length b,: ~ 5.2 x 10-a(AL) -1 exists at. which the fl value crosses the ln7 line. The b dependence and the lattice-size independence of A suggest the following picture of the QCD vacuum on the standpoint of the nlonopole condensation. When the physical volnme of the system l 4 (l = Na(,[]) on N 4 lattice) is large enough, that is, l >> b,:, one can take a. large renormalized lattice with a spacing b > b~. The entropy of monopole loops dominates the energy on such a renormalized lattice and the monopole condensation occurs. On the other hand, when the physical volume is small, the renormalized lattice must have a. small lattice size or a small lattice spacing or both. A small lattice size would lead to a possible entropy decreasing and a small spacing makes the self-energy large. Thus the monopole condensation does not occur in a small physical volume. The transition from the monopolecondensed phase to the normal phase takes place on a finite physical volume l 4. L. must be several times of b,. If it corresponds to the deconfining phase transition, we get l~ ~ 7b,. The existence of l,. means that there are always both monopolecondensed and uncondensed phases on a finite (in a lattice unit) lattice. 6) The monopole action may be fitted by ,

= 1

47r

2

(4)

where 9(b) is the SU(2) running coupling constant with a scale parameter A. D(s) is a modified lattice Coulomb propagator. This form of the action is predicted theoretically by Smit and Sijs[13]. The existence of the bare monopole mass m0 and the running coupling constant g(b) is characterisitic of the action in comparison with that of compact QED. The scale parameter determined is A ~ 42AL. The solid line is the prediction given by the action with the parameters written in the figure caption of 4.

H. Shiba, T Suzuki/Monopole condensation and confinement in SU(2) QCD

186

it is rewritten for such a simple Wilson loop in terms of an antisymmetric variable Mr,(s ) as .Jv(s) = O~,M,v(s), where d)' is a backward derivative on a lattice. M~,,(s) takes +1 on a surface with the Wilson loop boundary. Although we call choose any surface of such a type, we adopt a minimal surface here. We get .

BETA=2.6

0.1 0.08 o

•

abelian

•

monopole

0.06

/

l/V = exp { - i E

0.o4 0.02

III

•

-1

•

I

I

i

-o.02 -0.04 I

I

210 40 6;

80 1 ; 0 1 2 0

I*J

Mr"(S) = - E Figure 5. Creutz ratios fl'om abelian and monopole Wilson loops at. /3 = 2.6. The Monopole Creutz ratio values are devided by 4, being adjusted to those in unit a(,3).

3. A b e l i a n m o n o p o l e s string tension

explain

'It (s)or(s)}'

(5)

where Jr(s) is a.n external current, taking +1 along the Wilson loop. Since J,(s) is conserved,

D ( s - s')[O2(OrM~, - O,M,r)(S' ) 1

we get W

=

wt • w.~

W~

=

exp{-i ~

W2

=

exp{2rri E

t h e SU(2)

After the a.belian projection in the MA gauge, a diagonal matrix u(s, #) can be extracted uniquely fl'om the original SU(2) link field. The diagonal matrix u(s,p.) con'esponds to a U(1) gauge field written by an angle variable O~,(s). We show an abelian Wilson loop operator (which we consider after the abelian projection) is rewritten by a product of monopole and photon contributions. Here we take into account only a simple Wilson loop, say, of size I x J. Then such a.n abelian Wilson loop operator is expressed as W = exp{i E

(6)

where .ft,(s) = O,O,(s)- #),Or(s ) and 0 r is a forward derivative on a lattice. The gauge plaquette variable can be decomposed into f~,(s) = ft, u(s) +2rrn~u(s) where f~,(s) corresponds to a field strength and nr,(S ) is an integer-valued plaquette variable denoting the Dirac string. Since Mr,(S ) and nr~(S ) are integers, the latter does not contribute to Eq. (6). Hence .f~(s) in Eq. (6) is replaced by ft,(s). Using a decomposition rule

o

0

Mr~(S).fr"(s)}'

(7) 0; f;,~(s)D(s - s')J~(s')}

k cj(s)D(s - s')

1

kl~(s ) is defined (1/4rr)er~/>~O~f~.r(s) following DeGrand-Toussaint[10]. D(s) is the lattice where a monopole current

as k,(s)

=

Coulomb propagator. Since .ft,(s) corresponds to the field strength of the photon field, WI(W2) is the photon (the monopole) contribution to the Wilson loop. To study the features of both contributions, we evaluate the expectation values (l/V1) and (W2) separately and compare them with those of W. The Monte-Carlo simulations were done on 244 lattice from ~ = 2.4 to )t = 2.8. All measurements were done every 30 sweeps after a thermalization of 1500 sweeps. We took 50 configurations totally for measurements. Using gauge-fixed configurations, we evaluated monopole currents. As

H. Shiba, T Suzuki/Monopole condensation and confinement in SU(2) QCD

187

0.31 • monopole+photon

{

o monopole

3000 - photon

0.2

=

{

[] a b e l i a n

= 2000

[]

0.1

½

,, °

i 1000

iii

| []

t

t

t

abelian

o monopole •

photon

,

,

'I

•

|

6

1o

1

R/a

I

2.4

! ,

,

I

x ,

2.5 BETA

,

,

,2

I.

6

,

Figure 6. Static potentials a V ( R ) versus R / a at fl = 2.6. The values are shifted by a constant.

Figure 7. String tensions ¢/(A)~ at./3 = 2.4, 2.5, and 2.6.

shown in the previous section, type-2 extended monopole loops with b > be "-~ 5.2 x 10-3(AL) -1 condense, where b = na(fl) for n a extended monopoles and a(/3) is the lattice constant. So we measured 23 extended monopole with b = 2a(/?) of the type-2[11]. Then the effective (renormalized) lattice volume becomes 124. Since the original lattice is 244, 23 extended monopoles are the largest from which we can get useful data of the static potentials from Wilson loops. For fl = 2.7 and 2.8, the value b = 2a(fl) becomes less than be and so the monopoles may not reproduce the string tension. We have evaluated the averages of W using abelian link variables (called abelian), of W1 • W2 (called total), of W1 (photon part), and I/V2 (monopole part), separately. The results are summarized as follows. 1)The monopole contributions to Wilson loops are obtained with relatively small errors. Surpris-

ingly enough, the Creutz ratios of the monopole contributions are almost independent of the loop size as shown partially in Fig. 5. This means that the monopole contributions are composed only of an area, a perimeter, and a constant terms. 2)Assuming the static potential is given by a linear + Coulomb + constant terms, we can determine them from the least square fit to the Wilson loops [14]. We plot their data in Fig. 6 (at fl = 2.6). We find the monopole contributions are responsible for the linear-rising behaviors. The photon part contributes only to the short-ranged region. Similar data are obtained for ,3 = 2.4 and 2.5. 3)This is seen more clearly from the data of the string tensions which are determined from the static potentials. They are shown in Fig. 7. Systematic errors coming from various least square fits are not completely certain and are not plotted in the figure, although they are not negligi-

188

H. Shiba, T Suzuki /Monopole condensation and confinement in SU(2) QCD

-0.1

•U -0.2

. ..o,oo o

214

abelian

'218 '2:7 '2:8 2:9

lowing may be interesting. The photon parts are evaluated on a.n effective lattice with b = 2a(~). Hence they have different values of b for differnt 3. The Coulomb coefficients of the photon parts are well reproduced by the SU(2) running coupling constants g(b) with a scale parameter A and b = 2a(/3). The scale parameter A determined is A -,~ 46AL which is quite near the value A ~ 42AL fixed from the monopole action. Details will be published elsewhere[15, 16]. We wish to acknowledge Yoshimi Matsubara for useful discussions especially on the entropy decreasing effects of the monopole loops. This work is financially supported by JSPS Grant-in Aid for Scientific Research (c)(No.04640289). REFERENCES

BETA

1 2

Figure 8. Coulomb coefficients. The solid curve is the prediction of SU(2) running coupling constant.

ble. The string tensions are well reproduced by the monopoles alone for/3 < 2.6 and the photon part has almost vanishing string tensions. 4)At /3 = 2.7 and 2.8, the monopoles which have b < be do not seem to reproduce the abelian string tensions. However, the smallness of the renormalized lattice volume seems to influence largely the values of the string tensions for large /3 and we can not conclude at present that the monopoles with b > be alone can reproduce the string tensions as expected. 5)We have derived also Coulomb coefficients fi'om the static potentials as shown in Fig. 8. The monopole part has almost vanishing Coulomb coefficients which is in agreement with the constant behaviors of the Creutz rations of the monopole part as shown above. The photon part has large coefficients, but they do not reproduce the coefficients of the abelian static potentials. The fol-

3

4 5 6 7

G. 'tHooft, Nucl. Phys. B190, 455 (1981). A.S. Kronfeld oral., Phys. Lett. 198B, 516 (1987), A.S. Kronfeld et al., Nucl.Phys. B293, 461 (1987). T. Suzuki and I. Yotsuyanagi, Phys. Rev. D42, 4257 (1990); Nucl. Phys. B(Proc. Suppl.) 20,236 (1991). S. Hioki et al., Phys. Lett. 272B, 326 (1991). T. Suzuki, Nucl. Phys. B(Proc. Suppl.) 30, 176 (1993) and references therein. T.Banks et al., Nucl. Phys. B129, 493 (1977). J.D. Stack and R.J. Wensley, Nucl. Phys.

B371, 597 (1992). R.H. Swendsen,Phys. Rev. Lett. 52,1165 (1984); Phys. Rev. B30, 3866, 3875 (1984). 9 J. Villain, J. Phys. (Paris) 36,581 (1975). i0 T.A. DeGrand and D. Toussaint, Phys. Rev. 8

D22, 2478 (1980). 11 T.L. Ivanenko et al., Phys. Lett. 252B, 631 (1990). 12 E. Kovacs, Phys. Lett. l18B, 125 (1982). 13 J. Smit and A.J. van der Sijs, Nucl. Phys. B355, 603 (1991). 14 S.Itoh et al., Phys. Rev. D33 (1986) 1806. 15 H.Shiba and T.Suzuki, Kanazawa University, Report No. Kanazawa 93-09,10, 1993. 16 H.Shiba, T.Suzuki and Y.Matsubara, in preparation.