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Stability of instantons on the lattice and the renormalized trajectory
Volume 147B, number 1,2,3
PHYSICS LETTERS
1 November 1984
STABILITY OF INSTANTONS ON THE LA"I"FICE AND THE RENORMALIZED TRAJECTORY S. ITOH, Y. IWASAKI and T. YOSHII~ Institute o f Physics, University o f Tsukuba, Ibaraki 305, Japan
Received 22 May 1984 Revised manuscript received 16 July 1984
Studying the stability of instantons on a lattice for the two-dimensional 0(3) sigma model and the four-dimensional SU(2) lattice gauge model, we find the following: (i) a parameter space, each point of which corresponding to a lattice action, is divided into two regions: in one of them instantons exist, while in the other instantons do not exist, and (ii) the boundary between the two regions is identical with the line which is the closest to the renormalized trajectory of the block spin renormalization group.
In the continuum euclidean path integral formalism of the two-dimensional 0 ( 3 ) sigma model and the four-dimensional SU(2) gauge model, instantons [1,2] play an essential role for non-perturbative effects. The question which naturally arises is: where are instantons in the lattice formalism? In previous papers we have shown that the existence of instantons on the lattice crucially depends on the form of the lattice action [3,4]. Let us consider first the two-dimensional (2d) 0 ( 3 ) sigma model in detail. Let us take the following action [5,3] parametrized by a2 and a3:
A = 1--~ (~ [v.s(~) v.s(~)] g
+ a2 [v,v~S(n)] [v,v.S(n)]
(1)
+ ~3 {[v~s(,)] [Vx2S(n)] + [Vy2S(,)] [V~S(,)]}). where S is a 3-component unit vector, n is a lattice site and Vtaf(n ) = f ( n + ~ ) - f ( n ) .
(2)
F o r each action in the two-dimensional space spanned by a 2 and a3, instantons either exist or do not on a lattice with finite size. We investigate the existence of instantons on a lattice by a numerical m e t h o d which was employed in refs. [6,3]. The m e t h o d is to start from a discretized instanton
co(n) = [ S l ( n ) + i S 2 ( n ) ] / [ 1 + S3(n)] = c(z - a)/(z - b),
(3)
and to lower the energy (action) of the discretized instanton systematically by replacing the spin in such a way as to minimize its local energy. Here z = n 1 + in 2 and a, b, c are complex numbers. We take c = 1, a = 8 + i12, b = 17 + i12 on a 25 X 25 lattice. In ref. [7], we have found from a block spin renormalization group analysis [5,8] that the line 1.34a 2 + a3 = 0.113
(4)
is the closest (in the two-dimensional parameter space) to the renormalized trajectory of the block spin renormalization group [9]. (We are interested in the region with ol2/> 0 and a 3 ~> 0 here and hereafter.) Further we have given a plausible argument that the line (4) is the boundary of two regions: in one of them instantons exist on the lattice, while in the other instantons do not exist. (We will also give the argument below.) Let us now check this conjecture by the numerical m e t h o d described above. We report the results for the actions shown in fig. 1 and given in table 1. When an instanton exists on the lattice, the value of the energy approaches a constant which is roughly equal to 4rr, the energy o f an instanton of the continuum theory (see ref. [3]). On the other hand when an instanton does not exist, the value of the energy decreases sud141
Volume 147B, number 1,2,3
PHYSICS LETTERS
~3A
1 November 1984 T "A "B
10 4
3o
"C
0.104 32
1.34Cl2 + Q3= 0.113 103
0.05
102 >
i0 -a
Ca 0
Aa
0
A_2AjX~Ao
0.05
O. I0
Q2
10-2
iO-I
r"
Fig. 2. Lifetime of an instanton versus the distance from a lattice action to the renormalized trajectory r.
Fig. 1. Various lattice actions investigated.
denly f r o m a c o n s t a n t around 47r to a constant around zero after some iterations (see ref. [3]). We call the n u m b e r o f iterations where the value o f the energy becomes less than five the " l i f e t i m e " o f the instanton. The " l i f e t i m e " for the various actions are given in table 1.
Table 1 Lifetime of an instanton for various lattice actions.
We see f r o m table 1 that the instanton is really stable on the line (4) and becomes unstable w h e n a lattice action moves towards the origin (~2 = a3 = 0) f r o m the line (4). We have also checked that when a lattice action moves to the opposite side the instanton certainly remains stable. We show in fig. 2 h o w the " l i f e t i m e " o f the instant o n increases as a lattice action becomes close to the line (4). We define the distance b e t w e e n two actions simply by [(a~ - a2) 2 + (a~ - ~3) 2] 1/2. The " l i f e t i m e " o f the instanton is parametrized as ~- = 13.8 X r -0"94
(5a)
Point
a2
a3
Distance
"Lifetime"
Ao A1 A2 A3 O
0.0843 0.0772 0.0700 0.0500 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0057 0.0115 0.0275 0.0676
stable 1876 836 422 174
z = 22.2 × r - ° ' 7 7
Bo B1 B3 O
0.0 0.0 0.0 0.0 0.0
0.1130 ~1070 0.1000 0.0671 0.0
0.0 0.0036 0.0078 0.0275 0.0676
stable 1788 886 382 174
Co C1 C2 C3 O
0.05975 ~05544 0.05069 ~03558 0.0
0.03250 0.03016 0.02757 0.01935 0.0
0.0003 0.0051 0.0105 0.0275 0.0676
~able 1955 856 411 174
for the sequence CO, C1, C2, C3, O. Here ~- is the "lifet i m e " and r is the distance b e t w e e n a lattice action and the line (4). Thus as r -+ 0, r -~ oo for all sequences. T h a t is, as a lattice action approaches the line (4), the " l i f e t i m e " o f the instanton becomes longer and it becomes infinite w h e n a lattice action reaches the line (4). Because we have checked this for three typical sequences o f actions, we conclude that the line (4) is the b o u n d a r y o f two regions: in one o f t h e m the instanton is stable, while in the other it is unstable.
B2
142
for the sequence A0, A1, A2, A3, O; (5b)
for the sequence B0, B1, B2, B 3, O; 7- = 14.8 × r -0"91
(5c)
Volume 147B, number 1,2,3
PHYSICS LETTERS
Although we have investigated one discretized instanton on a 25 × 25 lattice, what will happen when the scale is enlarged (the size of the lattice and the size of the instanton is enlarged)? As far as we have investigated, the stability of the instanton for a given lattice action does not change when the scale is enlarged. Only the "lifetime" of the instanton increases. Thus as far as the size of a discretized instanton is finite, the "lifetime" of the instanton is finite for the lattice actions on one side of the boundary. For twoinstanton solutions and so on, we find the same result. Thus we have confirmed our conjecture that the line (4) is the boundary of the two regions: in one of them instantons exist, while in the other instantons do not exist. Because our numerical method is not exhaustive, we may admit some uncertainty in eq. (4) such as (1.34 + 0.05)a2 + a 3 = 0.113 + 0.01. It is very remarkable that two independent analyses give the same line with the uncertainty admitted above: One is the block spin renormalization group analysis in the perturbation theory and the other is an analysis of instantons on a lattice. This may imply some uniqueness of the renormalized trajectory. So far the analysis has been for the 2d 0 ( 3 ) sigma model. We now make a similar analysis for the 4d SU(2) lattice gauge model, but not so extensively as in the case of the 2d 0 ( 3 ) sigma model, because the analysis is time consuming. We take the action A =
co ~ T r ( s i m p l e plaquette loop)
+ Cl ~ T r ( r e c t a n g u l a r loop) + c2 ~ T r ( c h a i r - t y p e loop) + c3 ~
Tr(three-dimensional loop) + c o n s t a n t ) . l
(6)
In ref. [10] we have found from a block spin renormalization group analysis in the perturbation theory that the line (1.5 + 0.2) c 1 + (c2 + ca) = - ( 0 . 5 -+ 0.01)
(7)
is very close to the renormalized trajectory. Our method to analyze the stability of instantons on the lattice is the same as in ref. [4]. We have investigated in some detail the case c2 = c3 = 0, because we are interested in
1 November 1984
this case in Monte Carlo calculations. We have found the following: For c 1 ~ - 0 . 3 3 instantons are stable, while instantons are unstable for Cl 2 - 0 . 2 9 . The region - 0 . 3 3 ~ Cl ~ - 0 . 2 9 is critical. For Cl = - 0 . 3 3 7 5 we have instantons with topological number q = 1,2,3, 4 for ten random starts (see ref. [4]), for Cl = - 0 . 3 3 1 we have instantons with q = 1, 2, 3 and for Cl = - 0 . 2 9 3 we have instantons with q = 2, 3, 4. Thus the result supports the conclusion that the line (7) is the boundary of two regions: in one of them instantons exist, while in the other instantons do not exist. We have also investigated [4] other cases such as Wilson parameter [8] and Weisz's parameters [ 11]. The results are in accordance with the above conclusion. In both cases of the 2d 0 ( 3 ) sigma model and the 4d SU(2) lattice gauge model instantons do not exist for the standard model (a2 = ~3 = 0; Cl = c2 = c3 = 0). F o r the tree-level "improved" lattice action in Symanzik's approach [ 11,12] (or2 = 0, ot3 = ~4 ; Cl = - i~2, c2 = c3 = 0) instantons do not exist: the "lifetimes" of instantons are short as for the standard model. Let us now discuss the physical implications of this result. Even if instantons do not exist on the lattice for a lattice action, the effective action approaches gradually the renormalized trajectory. Therefore the effect of instantons will appear ultimately, but only at large distance. On the other hand even if instantons exist on the lattice but if the lattice action is far from the renormalized trajectory, the rate of contributions from the one-instanton configuration and the two-instanton configuration and so on are not proper in general: for example, the contribution from the one-instanton configuration might be too high compared with that in the continuum limit. However the effective action for long distance behavior again gradually approaches the renormalized trajectory and consequently the effect of instantons is properly taken into account. Thus when a lattice action is near the renormalized trajectory, not only instantons exist on the lattice but also the effect of instantons is properly taken into account, even on a lattice with small size. If there are configurations other than instantons which give dominant contributions in the continuum limit, the effect of such configurations is also properly taken into account, when a lattice action is close to the renormalized trajectory. Next let us give an argument for the fact that the boundary is identical with the line (4) or (7) which is 143
Volume 147B, number 1,2,3
PHYSICS LETTERS
very close to the renormalized trajectory. When instantons do not exist on a lattice for a lattice action, we may argue that instantons do not exist even for the lattice actions obtained by block spin transformations from the action, because the block spin transformation is a scale transformation and there is no intrinsic scale parameter for instantons. Let us assume this. We also assume that if instantons exist on the lattice for a lattice action, instantons exist also for the lattice actions obtained by the renormalization group. Then it is easy to prove that the renormalized trajectory is located at the boundary which divides the parameter space into two parts: In one of them instantons exist, while in the other instantons do not exist. This, combined with the fact that the line (4) or (7) is very close to the renormalized trajectory, implies that the line is at least approximately the boundary o f the two parts. F r o m the above argument it is expected that if we take a lattice action close to the renormalized trajectory, the effects of instantons and other possible configurations which dominate in the continuum limit are properly taken into account, even on a relatively small size. Their effects will be significant for physical quanrities such as the string tension and the hadron spectrum. Monte Carlo calculations for the string tension and the hadron spectrum using a lattice action close to the renormalized trajectory are given elsewhere [13,14].
144
1 November 1984
The numerical calculation has been performed with the FACOM M200 computer at the University of Tsukuba and the HITAC M200H at KEK. We would like to thank Hirotaka Sugawara and other members of KEK for kind hospitality.
References [i] A.A. Belavin and A.M. Polyakov, Zh. Eksp. Teor. Fiz. Pis'ma 22 (1975) 503. [2] A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu.S. Tyupkin, Phys. Lett. 59B (1975) 85. [3] Y. Iwasaki and T. Yoahi6, Phys. Lett. 125B (1983) 197. [4] Y. Iwasaki and T. Yoshi~, Phys. Lett. 131B (1983) 159. [5] S.H. Shenker and J. Tobochnik, Phys. Rev. B22 (1980) 4462. [6] B. Berg and M. Liischer, Nucl. Phys. B190 (1981) 412. [7] Y. Iwasaki, preprint UTHEP-117. [8] K.G. Wilson, in: Recent development in gauge theories, eds. G. 't Hooft et al. (Plenum, New York, 1980). [9] See also A. Hasenfratz and A. Margaritis, Phys. Lett. 133B (1983) 211. [10] Y. Iwasaki, preprint UTHEP-118. [11] P. Weisz, NucL Phys. B212 (1983) 1. [ 12] K. Symanzik, in: Mathematical problems in theoretical physics, eds. R. Schrader et al. (Springer, Berlin, 1982) ; NucL Phys. B226 (1983) 187, 205. [13] Y. Iwasaki and T. Yoshi6, Phys. Lett. 143B (1984) 449. [14] S. Itoh, Y. Iwasaki, Y. Oyanagi and T. Yoshi~, preprint UTHEP-127.
PHYSICS LETTERS
1 November 1984
STABILITY OF INSTANTONS ON THE LA"I"FICE AND THE RENORMALIZED TRAJECTORY S. ITOH, Y. IWASAKI and T. YOSHII~ Institute o f Physics, University o f Tsukuba, Ibaraki 305, Japan
Received 22 May 1984 Revised manuscript received 16 July 1984
Studying the stability of instantons on a lattice for the two-dimensional 0(3) sigma model and the four-dimensional SU(2) lattice gauge model, we find the following: (i) a parameter space, each point of which corresponding to a lattice action, is divided into two regions: in one of them instantons exist, while in the other instantons do not exist, and (ii) the boundary between the two regions is identical with the line which is the closest to the renormalized trajectory of the block spin renormalization group.
In the continuum euclidean path integral formalism of the two-dimensional 0 ( 3 ) sigma model and the four-dimensional SU(2) gauge model, instantons [1,2] play an essential role for non-perturbative effects. The question which naturally arises is: where are instantons in the lattice formalism? In previous papers we have shown that the existence of instantons on the lattice crucially depends on the form of the lattice action [3,4]. Let us consider first the two-dimensional (2d) 0 ( 3 ) sigma model in detail. Let us take the following action [5,3] parametrized by a2 and a3:
A = 1--~ (~ [v.s(~) v.s(~)] g
+ a2 [v,v~S(n)] [v,v.S(n)]
(1)
+ ~3 {[v~s(,)] [Vx2S(n)] + [Vy2S(,)] [V~S(,)]}). where S is a 3-component unit vector, n is a lattice site and Vtaf(n ) = f ( n + ~ ) - f ( n ) .
(2)
F o r each action in the two-dimensional space spanned by a 2 and a3, instantons either exist or do not on a lattice with finite size. We investigate the existence of instantons on a lattice by a numerical m e t h o d which was employed in refs. [6,3]. The m e t h o d is to start from a discretized instanton
co(n) = [ S l ( n ) + i S 2 ( n ) ] / [ 1 + S3(n)] = c(z - a)/(z - b),
(3)
and to lower the energy (action) of the discretized instanton systematically by replacing the spin in such a way as to minimize its local energy. Here z = n 1 + in 2 and a, b, c are complex numbers. We take c = 1, a = 8 + i12, b = 17 + i12 on a 25 X 25 lattice. In ref. [7], we have found from a block spin renormalization group analysis [5,8] that the line 1.34a 2 + a3 = 0.113
(4)
is the closest (in the two-dimensional parameter space) to the renormalized trajectory of the block spin renormalization group [9]. (We are interested in the region with ol2/> 0 and a 3 ~> 0 here and hereafter.) Further we have given a plausible argument that the line (4) is the boundary of two regions: in one of them instantons exist on the lattice, while in the other instantons do not exist. (We will also give the argument below.) Let us now check this conjecture by the numerical m e t h o d described above. We report the results for the actions shown in fig. 1 and given in table 1. When an instanton exists on the lattice, the value of the energy approaches a constant which is roughly equal to 4rr, the energy o f an instanton of the continuum theory (see ref. [3]). On the other hand when an instanton does not exist, the value of the energy decreases sud141
Volume 147B, number 1,2,3
PHYSICS LETTERS
~3A
1 November 1984 T "A "B
10 4
3o
"C
0.104 32
1.34Cl2 + Q3= 0.113 103
0.05
102 >
i0 -a
Ca 0
Aa
0
A_2AjX~Ao
0.05
O. I0
Q2
10-2
iO-I
r"
Fig. 2. Lifetime of an instanton versus the distance from a lattice action to the renormalized trajectory r.
Fig. 1. Various lattice actions investigated.
denly f r o m a c o n s t a n t around 47r to a constant around zero after some iterations (see ref. [3]). We call the n u m b e r o f iterations where the value o f the energy becomes less than five the " l i f e t i m e " o f the instanton. The " l i f e t i m e " for the various actions are given in table 1.
Table 1 Lifetime of an instanton for various lattice actions.
We see f r o m table 1 that the instanton is really stable on the line (4) and becomes unstable w h e n a lattice action moves towards the origin (~2 = a3 = 0) f r o m the line (4). We have also checked that when a lattice action moves to the opposite side the instanton certainly remains stable. We show in fig. 2 h o w the " l i f e t i m e " o f the instant o n increases as a lattice action becomes close to the line (4). We define the distance b e t w e e n two actions simply by [(a~ - a2) 2 + (a~ - ~3) 2] 1/2. The " l i f e t i m e " o f the instanton is parametrized as ~- = 13.8 X r -0"94
(5a)
Point
a2
a3
Distance
"Lifetime"
Ao A1 A2 A3 O
0.0843 0.0772 0.0700 0.0500 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0057 0.0115 0.0275 0.0676
stable 1876 836 422 174
z = 22.2 × r - ° ' 7 7
Bo B1 B3 O
0.0 0.0 0.0 0.0 0.0
0.1130 ~1070 0.1000 0.0671 0.0
0.0 0.0036 0.0078 0.0275 0.0676
stable 1788 886 382 174
Co C1 C2 C3 O
0.05975 ~05544 0.05069 ~03558 0.0
0.03250 0.03016 0.02757 0.01935 0.0
0.0003 0.0051 0.0105 0.0275 0.0676
~able 1955 856 411 174
for the sequence CO, C1, C2, C3, O. Here ~- is the "lifet i m e " and r is the distance b e t w e e n a lattice action and the line (4). Thus as r -+ 0, r -~ oo for all sequences. T h a t is, as a lattice action approaches the line (4), the " l i f e t i m e " o f the instanton becomes longer and it becomes infinite w h e n a lattice action reaches the line (4). Because we have checked this for three typical sequences o f actions, we conclude that the line (4) is the b o u n d a r y o f two regions: in one o f t h e m the instanton is stable, while in the other it is unstable.
B2
142
for the sequence A0, A1, A2, A3, O; (5b)
for the sequence B0, B1, B2, B 3, O; 7- = 14.8 × r -0"91
(5c)
Volume 147B, number 1,2,3
PHYSICS LETTERS
Although we have investigated one discretized instanton on a 25 × 25 lattice, what will happen when the scale is enlarged (the size of the lattice and the size of the instanton is enlarged)? As far as we have investigated, the stability of the instanton for a given lattice action does not change when the scale is enlarged. Only the "lifetime" of the instanton increases. Thus as far as the size of a discretized instanton is finite, the "lifetime" of the instanton is finite for the lattice actions on one side of the boundary. For twoinstanton solutions and so on, we find the same result. Thus we have confirmed our conjecture that the line (4) is the boundary of the two regions: in one of them instantons exist, while in the other instantons do not exist. Because our numerical method is not exhaustive, we may admit some uncertainty in eq. (4) such as (1.34 + 0.05)a2 + a 3 = 0.113 + 0.01. It is very remarkable that two independent analyses give the same line with the uncertainty admitted above: One is the block spin renormalization group analysis in the perturbation theory and the other is an analysis of instantons on a lattice. This may imply some uniqueness of the renormalized trajectory. So far the analysis has been for the 2d 0 ( 3 ) sigma model. We now make a similar analysis for the 4d SU(2) lattice gauge model, but not so extensively as in the case of the 2d 0 ( 3 ) sigma model, because the analysis is time consuming. We take the action A =
co ~ T r ( s i m p l e plaquette loop)
+ Cl ~ T r ( r e c t a n g u l a r loop) + c2 ~ T r ( c h a i r - t y p e loop) + c3 ~
Tr(three-dimensional loop) + c o n s t a n t ) . l
(6)
In ref. [10] we have found from a block spin renormalization group analysis in the perturbation theory that the line (1.5 + 0.2) c 1 + (c2 + ca) = - ( 0 . 5 -+ 0.01)
(7)
is very close to the renormalized trajectory. Our method to analyze the stability of instantons on the lattice is the same as in ref. [4]. We have investigated in some detail the case c2 = c3 = 0, because we are interested in
1 November 1984
this case in Monte Carlo calculations. We have found the following: For c 1 ~ - 0 . 3 3 instantons are stable, while instantons are unstable for Cl 2 - 0 . 2 9 . The region - 0 . 3 3 ~ Cl ~ - 0 . 2 9 is critical. For Cl = - 0 . 3 3 7 5 we have instantons with topological number q = 1,2,3, 4 for ten random starts (see ref. [4]), for Cl = - 0 . 3 3 1 we have instantons with q = 1, 2, 3 and for Cl = - 0 . 2 9 3 we have instantons with q = 2, 3, 4. Thus the result supports the conclusion that the line (7) is the boundary of two regions: in one of them instantons exist, while in the other instantons do not exist. We have also investigated [4] other cases such as Wilson parameter [8] and Weisz's parameters [ 11]. The results are in accordance with the above conclusion. In both cases of the 2d 0 ( 3 ) sigma model and the 4d SU(2) lattice gauge model instantons do not exist for the standard model (a2 = ~3 = 0; Cl = c2 = c3 = 0). F o r the tree-level "improved" lattice action in Symanzik's approach [ 11,12] (or2 = 0, ot3 = ~4 ; Cl = - i~2, c2 = c3 = 0) instantons do not exist: the "lifetimes" of instantons are short as for the standard model. Let us now discuss the physical implications of this result. Even if instantons do not exist on the lattice for a lattice action, the effective action approaches gradually the renormalized trajectory. Therefore the effect of instantons will appear ultimately, but only at large distance. On the other hand even if instantons exist on the lattice but if the lattice action is far from the renormalized trajectory, the rate of contributions from the one-instanton configuration and the two-instanton configuration and so on are not proper in general: for example, the contribution from the one-instanton configuration might be too high compared with that in the continuum limit. However the effective action for long distance behavior again gradually approaches the renormalized trajectory and consequently the effect of instantons is properly taken into account. Thus when a lattice action is near the renormalized trajectory, not only instantons exist on the lattice but also the effect of instantons is properly taken into account, even on a lattice with small size. If there are configurations other than instantons which give dominant contributions in the continuum limit, the effect of such configurations is also properly taken into account, when a lattice action is close to the renormalized trajectory. Next let us give an argument for the fact that the boundary is identical with the line (4) or (7) which is 143
Volume 147B, number 1,2,3
PHYSICS LETTERS
very close to the renormalized trajectory. When instantons do not exist on a lattice for a lattice action, we may argue that instantons do not exist even for the lattice actions obtained by block spin transformations from the action, because the block spin transformation is a scale transformation and there is no intrinsic scale parameter for instantons. Let us assume this. We also assume that if instantons exist on the lattice for a lattice action, instantons exist also for the lattice actions obtained by the renormalization group. Then it is easy to prove that the renormalized trajectory is located at the boundary which divides the parameter space into two parts: In one of them instantons exist, while in the other instantons do not exist. This, combined with the fact that the line (4) or (7) is very close to the renormalized trajectory, implies that the line is at least approximately the boundary o f the two parts. F r o m the above argument it is expected that if we take a lattice action close to the renormalized trajectory, the effects of instantons and other possible configurations which dominate in the continuum limit are properly taken into account, even on a relatively small size. Their effects will be significant for physical quanrities such as the string tension and the hadron spectrum. Monte Carlo calculations for the string tension and the hadron spectrum using a lattice action close to the renormalized trajectory are given elsewhere [13,14].
144
1 November 1984
The numerical calculation has been performed with the FACOM M200 computer at the University of Tsukuba and the HITAC M200H at KEK. We would like to thank Hirotaka Sugawara and other members of KEK for kind hospitality.
References [i] A.A. Belavin and A.M. Polyakov, Zh. Eksp. Teor. Fiz. Pis'ma 22 (1975) 503. [2] A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu.S. Tyupkin, Phys. Lett. 59B (1975) 85. [3] Y. Iwasaki and T. Yoahi6, Phys. Lett. 125B (1983) 197. [4] Y. Iwasaki and T. Yoshi~, Phys. Lett. 131B (1983) 159. [5] S.H. Shenker and J. Tobochnik, Phys. Rev. B22 (1980) 4462. [6] B. Berg and M. Liischer, Nucl. Phys. B190 (1981) 412. [7] Y. Iwasaki, preprint UTHEP-117. [8] K.G. Wilson, in: Recent development in gauge theories, eds. G. 't Hooft et al. (Plenum, New York, 1980). [9] See also A. Hasenfratz and A. Margaritis, Phys. Lett. 133B (1983) 211. [10] Y. Iwasaki, preprint UTHEP-118. [11] P. Weisz, NucL Phys. B212 (1983) 1. [ 12] K. Symanzik, in: Mathematical problems in theoretical physics, eds. R. Schrader et al. (Springer, Berlin, 1982) ; NucL Phys. B226 (1983) 187, 205. [13] Y. Iwasaki and T. Yoshi6, Phys. Lett. 143B (1984) 449. [14] S. Itoh, Y. Iwasaki, Y. Oyanagi and T. Yoshi~, preprint UTHEP-127.