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Monopole distribution in momentum space in SU(2) lattice gauge theory
Physics Letters B 285 (1992) 343-346 North-Holland
PHYSICS LETTERS B
Monopole distribution in momentum space in SU (2) lattice gauge theory S. Hioki
a S. Kitahara b y . Matsubara c, O. Miyamura d S, Ohno b and T. Suzuki b
a Department of Management Information, Fukuyama University, Fukuyama 729-02, Japan b Department of Physics, Kanazawa University, Kanazawa 920, Japan c Nanao Junior College, Nanao, Ishikawa 926, Japan Department of Physics, Hiroshima University, Higashi-Hiroshima 724, Japan Received 27 February 1992; revised manuscript received 4 May 1992
In this paper, the monopole defined in the maximally abelian gauge is investigated by the use of the cooling technique. We find that the scaling behavior of extended monopoles appears after cooling, while it is not seen before cooling. The monopole distribution in momentum space begins to broaden as the temperature increases in the deconfinement phase; this feature is clearer for extended and cooled monopoles.
Although there have been several definitions of color magnetic m o n o p o l e s in QCD, we have not the definite one which plays a crucial role in color confinement mechanism. In this paper ~1 we a d o p t maximal abelian gauge fixing to extract the monopole field from Q C D [ 1 ]. In this gauge, the string tension measured from the abelian link variables seems consistent with that o b t a i n e d for the original S U ( N ) link variables [ 2 ]. This fact suggests that there is abelian d o m i n a n c e in SU (N) color confinement. In SU ( 2 ) lattice gauge theory the abelian m o n o p o l e becomes dilute in the d e c o n f i n e m e n t phase whereas it looks dense in the confinement phase [1 ]. Recently Brandstaeter et al. [ 3 ] have suggested that the same scenario holds also in the SU ( 3 ) case and they found that the m o n o p o l e becomes static as the t e m p e r a t u r e increases in the deconfinement phase. In this paper we will investigate how the abelian m o n o p o l e s are related to the mechanism o f color confinement focusing on both the extended m o n o p o l e s and the monopole distribution in m o m e n t u m space. Let us consider the pure SU ( 2 ) gauge system. To define the U ( 1 ) variable, first fix the gauge such that *~ The preliminary version of this work was reported at "LATTICE 91" (International Symposium on Lattice field theory ).
R = ~ Tr(0"3 L~.~03 U~.,)
( 1)
n,p
is maximized. Then extract the abelian field u,., from the original link variable as
fun,.
U...=An..~ 0
0 )
u*.
'
(2)
where A,,.u represents the charged matter field whereas u,, 4, = exp (i~o,.,) is the gauge field with respect to the residual U ( 1 ) gauge transformation. In lattice gauge simulations, this gauge fixing can be done only by an iterative procedure. ( A b o u t the ambiguity o f this gauge fixing, see ref. [ 4 ]. ) Locally the m a x i m i z a t i o n o f R at site n in eq. ( 1 ) is equivalent to the diagonalization o f a 2 × 2 matrix X ( n ) which can be written as -X
X(n)-
( cos2a R\ei:,sin2 a
e-i:' sin 2a'~ - - c o s 2 a }"
(3)
The corresponding gauge transformation function
g,O is then { cos oJa g~ = ~, - e i:, sin oJa
e - i : ' sin coa~ cos ~oa / ,
(4)
where o~ is an over-relaxation p a r a m e t e r which con-
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
343
Volume 285, number 4
PHYSICS LETTERS B
trols the speed o f convergence [4,5]. (In this paper we adopt to= 1.7 [4 ]. ) In terms of u,,, u we define the plaquette angle 0n,,~ just as in the U( 1 ) case through [6]
0,,... = A r g ( u..1,u.+/,.~u*+o.~u*.) .
(5)
The monopole density p is
p/A3 =
3~n.,,p,,4,
(6)
1
~n,l, 1 (aAL) 3 '
where p,,~,= t K,,~,] with Kn,u the monopole current defined by K,,z= e~,,poA,¢,,zo/47r, a is the lattice spacing and Ae is the lattice A parameter. If we want to see whether a quantity is a physical one or not, we have to check the scaling property o f the quantity. For the monopole density p, evidence of the scaling behavior has been reported by Bornyakov et al. [7 ] and Hioki et al. [8 ], suggesting that the monopole density in the maximal abelian gauge is a physical quantity. Color confinement is, in itself, a long range property of QCD. This is recognized easily by considering the potential between heavy quarks, V(r) = ar+ a / r . The first term (the confinement force) is d o m i n a n t in the long range region. So this feature should not be affected by short range quantum fluctuations. In this paper we adopt two different methods to remove the short range effects. One way is to use the extended definition of the monopole which was first introduced by Ivanenko et al. [ 9 ]. The other is to use the so called cooling method which has been successful for extracting the topological charge in Q C D [ 10 ]. It is noted that the string tension survives rather long under cooling whereas the action decreases monotonically, suggesting that the confinement is caused by some semiclassical object which survives under cooling [ 1 1 ]. The extended monopole is defined on an l × l × l cube. [ 9 ] First we define the extended plaquette angle O(/) where " ( / ) - n.pP -- -a~ gt ~, L ' t /t,,(/),,(/) t , p ~ n + l / ~ t , p t a ~ t +,,(/)* h ) . l a t 4 t t ,,,(l).~ p ]~ ~*n,p
bln,lzl~ln+/J.p'"~ln+
--
16 July 1992
rice for fl=2.2-2.5 and a 164 lattice for fl=2.4-2.6. 200 independent configurations are accumulated for each ft. We measured the monopole density p(1) for l = 1-4. Cooling was performed up to 50 cooling sweeps where the action is about v~6 of the action before cooling. The result for the monopole density p ( l ) before cooling was reported in ref. [ 8 ]. As stated in ref. [ 8 ], the scaling behavior of p( 1 ) ( = p ) has been observed, but the scaling behavior for p ( l ) seems to disappear as l increases. The same quantity but after 10 cooling sweeps is displayed in fig. 1. In this case p ( l ) for large ! begins to scale. These results suggest that after cooling the extended monopole is a physical quantity whereas it does not before cooling. To realize the dual Meissner effect in QCD, condensation of the magnetic monopole is needed. In superconductors, condensation of the Cooper pair takes place. This means that a macroscopic number of Cooper pairs is in one quantum state. So in QCD, the condensation should occur in the m o m e n t u m (or energy) space rather than in the coordinate space. To obtain the information of m o m e n t u m space, here we propose to use the Fourier transformation from the three-dimensional coordinate space r to the corresponding m o m e n t u m space k. Consider the monopole field ~u(r) at a time slice
lO s
43,
10 ~
8~
•
p
[]
•
p (2)
•
p (3)
•
p(7)
(I-- I )li,M"
Then the monopole density p ( l ) is defined by
p ( l ) / A 3 _ ~',,~, IK~(], I 1 Z,,,/, 1 ( laAe ) 3 ,
(7)
where K~(], is the monopole current defined just as in the l = i case by using ..-(/),,.~ instead of u,, u. The simulation was done for SU (2) on an 84 lat344
16 ~
©
212
z'.3
)D
1)
O
2'.4
21
Fig. 1. Monopole density after 10 coolingsweeps. Open and black symbols are the results on 84 and 164 lattices, respectively. The errors are within the symbols.
Volume 285, number 4
PHYSICS LETTERS B
whose square equals the monopole density P,.a. Generally ~,(r) is complex and can be written as ~,(r) = I ~'(r)[e i°(r) ,
(8)
where O(r) is a phase factor. Since what we can obtain is p, we do not have enough information to determine this phase factor. Here we assume that O(r) is independent from r, i.e.
16 July 1992 p(1) R f 0"4
0.2
L/2
L/2
L/2
211
212
213
~
0 cool
214
215
p(7)
R•.06 (b)
t
0.05
20 c o o l
0.04
1
I
-L/2
-L/2
0.03
-L/2
/2m ) Xexp~,--~- (k,x+k,,y+k_-z) ,
0.02
(10)
where L is the spacial extent and V( - L 3) is the threedimensional spacial volume. To see the distribution shape explicitly, we define the ratio Rrby If(1)l
Mr= If(0) I '
(11)
which corresponds to the width of the distribution in m o m e n t u m space. If the monopole condensation takes place, we expect that the width is small in the confinement phase whereas it begins to broaden in the deconfinement phase. We made finite temperature simulations on a 163 × 4 lattice for fl= 2.1-2.5. 100 independent samples have been accumulated for each ft. The result shows that the distribution f u n c t i o n f ( k ) has its peak at k ( - Lkl ) = 0 and begins to decrease rapidly and monotonically as k increases. Therefore, to use If(/<) [ / I f ( 0 ) I for k = 2,3... makes no qualitative difference compared to using R , The result about the 3 dependence of RI for p( 1 ) and p (7) after 0, 10, 20 cooling sweeps are displayed in figs. 2a and 2b, respectively. In this case, the critical point is at f l ~- 2.3. The difference between the confinement and the deconfinement becomes clear as the cooling goes on. In the p ( 7 ) case, a drastic change of the ratio Rris seen after 10 cooling sweeps whereas no fl dependence can be seen before cooling. These results suggest that the
10 cool i
0.1
(9)
It is noted that this is the case in usual superconductors over a macroscopic length scale. The distribution f u n c t i o n f ( k ) in m o m e n t u m space is then
r
0.3
0.0
0 (r) = constant.
20 cool
(a)
0.01
i z
I
t 10 cool 0 cool i
2'.1 1.2 213 214 215
Fig. 2. The width of the monopole distribution in momentum space for (a) p ( 1) and (b) p ( 7 ) at 0,10 and 20 cooling sweeps. cooling appears to bring out some structure of the extended monopole in high temperature deconfinement phase. Using the lattice gauge Monte Carlo simulations, we have investigated the density of monopoles defined on several length scales using the cooling technique. The scaling behavior is seen for the extended monopole after several cooling sweeps. The monopole distribution in m o m e n t u m space is also investigated. In the deconfinement phase, broadening of the width of the distribution is obtained. This feature seems clearer for the extended and cooled monopoles. There is a possibility that this type of monopole condenses into one quantum state in the confinement phase. The result of this paper is not the final answer to the dual Meissner effect, but it is one of the necessary conditions of monopole condensation. It would be worthwhile challenging the dual Meissner picture of color confinement. We expect that some progress will be made in the near future. S.H. would like to thank the members of Saga University for useful discussions. The simulation of this work has been done on the HITAC $820/80 at the 345
Volume 285, number 4
PHYSICS LETTERS B
c o m p u t a t i o n a l center o f N a t i o n a l L a b o r a t o r y for High Energy Physics ( K E K ) . T.S. is f i n a n c i a l l y s u p p o r t e d by G r a n d - i n - A i d for Scientific R e s e a r c h ( c ) ( N o . 02640220).
References [ 1 ] A.S. Kronfeld, M.L. Laursen, G. Schierholz and U.-J. Wiese, Phys. Lett. B 198 (1987) 516. [2] T. Suzuki and I. Yotsuyanagi, Phys. Rev. D 42 (1990) 4257, [3] F. Brandstaeter, G. Schierholz and U.-J. Wiese, Phys. Lett. B272 (1991) 319.
346
16 July 1992
[4] S. Hioki et al., Phys. Lett. B 271 ( 1991 ) 201. [5] J.E, Mandula and M. Ogilvie, Phys. Lett. B 248 (1990) 156. [6] T.A. DeGrand and D. Toussaint, Phys. Rev, D 22 (1980) 2478. [7] V.G. Bornyakov et al., Phys. Lett. B 261 ( 1991 ) 116. [8] S. Hioki et al., Phys, Lett. B 272 ( 1991 ) 326, [ 9] T.L. Ivanenko, A.V. Pochinsky and M.1. Polikarpov, Phys. Lett. B 252 (1990) 631. [ 10] M. Teper, Phys. Lett. B 162 (1985) 357; M.I. Polikarpov and A.I. Veselov, Nucl. Phys. B 297 (1988) 34. [ 11 ] A. DiGiacomo, M. Maggiore and S. Olenjlk, Nucl. Phys. B 347 (1990) 441.
PHYSICS LETTERS B
Monopole distribution in momentum space in SU (2) lattice gauge theory S. Hioki
a S. Kitahara b y . Matsubara c, O. Miyamura d S, Ohno b and T. Suzuki b
a Department of Management Information, Fukuyama University, Fukuyama 729-02, Japan b Department of Physics, Kanazawa University, Kanazawa 920, Japan c Nanao Junior College, Nanao, Ishikawa 926, Japan Department of Physics, Hiroshima University, Higashi-Hiroshima 724, Japan Received 27 February 1992; revised manuscript received 4 May 1992
In this paper, the monopole defined in the maximally abelian gauge is investigated by the use of the cooling technique. We find that the scaling behavior of extended monopoles appears after cooling, while it is not seen before cooling. The monopole distribution in momentum space begins to broaden as the temperature increases in the deconfinement phase; this feature is clearer for extended and cooled monopoles.
Although there have been several definitions of color magnetic m o n o p o l e s in QCD, we have not the definite one which plays a crucial role in color confinement mechanism. In this paper ~1 we a d o p t maximal abelian gauge fixing to extract the monopole field from Q C D [ 1 ]. In this gauge, the string tension measured from the abelian link variables seems consistent with that o b t a i n e d for the original S U ( N ) link variables [ 2 ]. This fact suggests that there is abelian d o m i n a n c e in SU (N) color confinement. In SU ( 2 ) lattice gauge theory the abelian m o n o p o l e becomes dilute in the d e c o n f i n e m e n t phase whereas it looks dense in the confinement phase [1 ]. Recently Brandstaeter et al. [ 3 ] have suggested that the same scenario holds also in the SU ( 3 ) case and they found that the m o n o p o l e becomes static as the t e m p e r a t u r e increases in the deconfinement phase. In this paper we will investigate how the abelian m o n o p o l e s are related to the mechanism o f color confinement focusing on both the extended m o n o p o l e s and the monopole distribution in m o m e n t u m space. Let us consider the pure SU ( 2 ) gauge system. To define the U ( 1 ) variable, first fix the gauge such that *~ The preliminary version of this work was reported at "LATTICE 91" (International Symposium on Lattice field theory ).
R = ~ Tr(0"3 L~.~03 U~.,)
( 1)
n,p
is maximized. Then extract the abelian field u,., from the original link variable as
fun,.
U...=An..~ 0
0 )
u*.
'
(2)
where A,,.u represents the charged matter field whereas u,, 4, = exp (i~o,.,) is the gauge field with respect to the residual U ( 1 ) gauge transformation. In lattice gauge simulations, this gauge fixing can be done only by an iterative procedure. ( A b o u t the ambiguity o f this gauge fixing, see ref. [ 4 ]. ) Locally the m a x i m i z a t i o n o f R at site n in eq. ( 1 ) is equivalent to the diagonalization o f a 2 × 2 matrix X ( n ) which can be written as -X
X(n)-
( cos2a R\ei:,sin2 a
e-i:' sin 2a'~ - - c o s 2 a }"
(3)
The corresponding gauge transformation function
g,O is then { cos oJa g~ = ~, - e i:, sin oJa
e - i : ' sin coa~ cos ~oa / ,
(4)
where o~ is an over-relaxation p a r a m e t e r which con-
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
343
Volume 285, number 4
PHYSICS LETTERS B
trols the speed o f convergence [4,5]. (In this paper we adopt to= 1.7 [4 ]. ) In terms of u,,, u we define the plaquette angle 0n,,~ just as in the U( 1 ) case through [6]
0,,... = A r g ( u..1,u.+/,.~u*+o.~u*.) .
(5)
The monopole density p is
p/A3 =
3~n.,,p,,4,
(6)
1
~n,l, 1 (aAL) 3 '
where p,,~,= t K,,~,] with Kn,u the monopole current defined by K,,z= e~,,poA,¢,,zo/47r, a is the lattice spacing and Ae is the lattice A parameter. If we want to see whether a quantity is a physical one or not, we have to check the scaling property o f the quantity. For the monopole density p, evidence of the scaling behavior has been reported by Bornyakov et al. [7 ] and Hioki et al. [8 ], suggesting that the monopole density in the maximal abelian gauge is a physical quantity. Color confinement is, in itself, a long range property of QCD. This is recognized easily by considering the potential between heavy quarks, V(r) = ar+ a / r . The first term (the confinement force) is d o m i n a n t in the long range region. So this feature should not be affected by short range quantum fluctuations. In this paper we adopt two different methods to remove the short range effects. One way is to use the extended definition of the monopole which was first introduced by Ivanenko et al. [ 9 ]. The other is to use the so called cooling method which has been successful for extracting the topological charge in Q C D [ 10 ]. It is noted that the string tension survives rather long under cooling whereas the action decreases monotonically, suggesting that the confinement is caused by some semiclassical object which survives under cooling [ 1 1 ]. The extended monopole is defined on an l × l × l cube. [ 9 ] First we define the extended plaquette angle O(/) where " ( / ) - n.pP -- -a~ gt ~, L ' t /t,,(/),,(/) t , p ~ n + l / ~ t , p t a ~ t +,,(/)* h ) . l a t 4 t t ,,,(l).~ p ]~ ~*n,p
bln,lzl~ln+/J.p'"~ln+
--
16 July 1992
rice for fl=2.2-2.5 and a 164 lattice for fl=2.4-2.6. 200 independent configurations are accumulated for each ft. We measured the monopole density p(1) for l = 1-4. Cooling was performed up to 50 cooling sweeps where the action is about v~6 of the action before cooling. The result for the monopole density p ( l ) before cooling was reported in ref. [ 8 ]. As stated in ref. [ 8 ], the scaling behavior of p( 1 ) ( = p ) has been observed, but the scaling behavior for p ( l ) seems to disappear as l increases. The same quantity but after 10 cooling sweeps is displayed in fig. 1. In this case p ( l ) for large ! begins to scale. These results suggest that after cooling the extended monopole is a physical quantity whereas it does not before cooling. To realize the dual Meissner effect in QCD, condensation of the magnetic monopole is needed. In superconductors, condensation of the Cooper pair takes place. This means that a macroscopic number of Cooper pairs is in one quantum state. So in QCD, the condensation should occur in the m o m e n t u m (or energy) space rather than in the coordinate space. To obtain the information of m o m e n t u m space, here we propose to use the Fourier transformation from the three-dimensional coordinate space r to the corresponding m o m e n t u m space k. Consider the monopole field ~u(r) at a time slice
lO s
43,
10 ~
8~
•
p
[]
•
p (2)
•
p (3)
•
p(7)
(I-- I )li,M"
Then the monopole density p ( l ) is defined by
p ( l ) / A 3 _ ~',,~, IK~(], I 1 Z,,,/, 1 ( laAe ) 3 ,
(7)
where K~(], is the monopole current defined just as in the l = i case by using ..-(/),,.~ instead of u,, u. The simulation was done for SU (2) on an 84 lat344
16 ~
©
212
z'.3
)D
1)
O
2'.4
21
Fig. 1. Monopole density after 10 coolingsweeps. Open and black symbols are the results on 84 and 164 lattices, respectively. The errors are within the symbols.
Volume 285, number 4
PHYSICS LETTERS B
whose square equals the monopole density P,.a. Generally ~,(r) is complex and can be written as ~,(r) = I ~'(r)[e i°(r) ,
(8)
where O(r) is a phase factor. Since what we can obtain is p, we do not have enough information to determine this phase factor. Here we assume that O(r) is independent from r, i.e.
16 July 1992 p(1) R f 0"4
0.2
L/2
L/2
L/2
211
212
213
~
0 cool
214
215
p(7)
R•.06 (b)
t
0.05
20 c o o l
0.04
1
I
-L/2
-L/2
0.03
-L/2
/2m ) Xexp~,--~- (k,x+k,,y+k_-z) ,
0.02
(10)
where L is the spacial extent and V( - L 3) is the threedimensional spacial volume. To see the distribution shape explicitly, we define the ratio Rrby If(1)l
Mr= If(0) I '
(11)
which corresponds to the width of the distribution in m o m e n t u m space. If the monopole condensation takes place, we expect that the width is small in the confinement phase whereas it begins to broaden in the deconfinement phase. We made finite temperature simulations on a 163 × 4 lattice for fl= 2.1-2.5. 100 independent samples have been accumulated for each ft. The result shows that the distribution f u n c t i o n f ( k ) has its peak at k ( - Lkl ) = 0 and begins to decrease rapidly and monotonically as k increases. Therefore, to use If(/<) [ / I f ( 0 ) I for k = 2,3... makes no qualitative difference compared to using R , The result about the 3 dependence of RI for p( 1 ) and p (7) after 0, 10, 20 cooling sweeps are displayed in figs. 2a and 2b, respectively. In this case, the critical point is at f l ~- 2.3. The difference between the confinement and the deconfinement becomes clear as the cooling goes on. In the p ( 7 ) case, a drastic change of the ratio Rris seen after 10 cooling sweeps whereas no fl dependence can be seen before cooling. These results suggest that the
10 cool i
0.1
(9)
It is noted that this is the case in usual superconductors over a macroscopic length scale. The distribution f u n c t i o n f ( k ) in m o m e n t u m space is then
r
0.3
0.0
0 (r) = constant.
20 cool
(a)
0.01
i z
I
t 10 cool 0 cool i
2'.1 1.2 213 214 215
Fig. 2. The width of the monopole distribution in momentum space for (a) p ( 1) and (b) p ( 7 ) at 0,10 and 20 cooling sweeps. cooling appears to bring out some structure of the extended monopole in high temperature deconfinement phase. Using the lattice gauge Monte Carlo simulations, we have investigated the density of monopoles defined on several length scales using the cooling technique. The scaling behavior is seen for the extended monopole after several cooling sweeps. The monopole distribution in m o m e n t u m space is also investigated. In the deconfinement phase, broadening of the width of the distribution is obtained. This feature seems clearer for the extended and cooled monopoles. There is a possibility that this type of monopole condenses into one quantum state in the confinement phase. The result of this paper is not the final answer to the dual Meissner effect, but it is one of the necessary conditions of monopole condensation. It would be worthwhile challenging the dual Meissner picture of color confinement. We expect that some progress will be made in the near future. S.H. would like to thank the members of Saga University for useful discussions. The simulation of this work has been done on the HITAC $820/80 at the 345
Volume 285, number 4
PHYSICS LETTERS B
c o m p u t a t i o n a l center o f N a t i o n a l L a b o r a t o r y for High Energy Physics ( K E K ) . T.S. is f i n a n c i a l l y s u p p o r t e d by G r a n d - i n - A i d for Scientific R e s e a r c h ( c ) ( N o . 02640220).
References [ 1 ] A.S. Kronfeld, M.L. Laursen, G. Schierholz and U.-J. Wiese, Phys. Lett. B 198 (1987) 516. [2] T. Suzuki and I. Yotsuyanagi, Phys. Rev. D 42 (1990) 4257, [3] F. Brandstaeter, G. Schierholz and U.-J. Wiese, Phys. Lett. B272 (1991) 319.
346
16 July 1992
[4] S. Hioki et al., Phys. Lett. B 271 ( 1991 ) 201. [5] J.E, Mandula and M. Ogilvie, Phys. Lett. B 248 (1990) 156. [6] T.A. DeGrand and D. Toussaint, Phys. Rev, D 22 (1980) 2478. [7] V.G. Bornyakov et al., Phys. Lett. B 261 ( 1991 ) 116. [8] S. Hioki et al., Phys, Lett. B 272 ( 1991 ) 326, [ 9] T.L. Ivanenko, A.V. Pochinsky and M.1. Polikarpov, Phys. Lett. B 252 (1990) 631. [ 10] M. Teper, Phys. Lett. B 162 (1985) 357; M.I. Polikarpov and A.I. Veselov, Nucl. Phys. B 297 (1988) 34. [ 11 ] A. DiGiacomo, M. Maggiore and S. Olenjlk, Nucl. Phys. B 347 (1990) 441.