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Variant source methods
Volume 148B, number 1,2,3
PHYSICS LETTERS
22 November 1984
VARIANT SOURCE METHODS B. B E R G CERN, Geneva, Switzerland and II. Institut fiJr Theoretische Physik, Hamburg, Fed. Rep. Germany
Received 30 July 1984
Hot and cold wall source methods for calculating the mass spectrum of lattice gauge theories are investigated. Sources for SU(2) E+ and A1- states are introduced. Preliminary Monte Carlo results at B = 2.25 are compared with previous Monte Carlo variational results.
Source methods for investigating the mass spectrum of 4d lattice gauge theories were suggested in refs. [1-3]. To be definite, let us consider the SU(2) gauge group. The cold wall method [3] simply consists in fixing on a spacelike plane all SU(2) matrices to the identity. This is mainly useful for investigating the A1 + representation. C o m p a r e d to other representations of the cubic group, M o n t e Carlo (MC) investigations of the A I + state are in good shape *t. Sources with strong projections on other representations, corresponding to excited states [4], are therefore desirable. Sources specialized to the E + and A 1 states are introduced in the following. First numerical results are also presented. Let me first discuss the fixed wall method (hot or cold) in general. It consists in fixing a wave function I~p) on a spacelike plane:
where we have chosen the normalization T[0) = 10) of the vacuum. Let r = (R, P, C) specify a representation of the cubic group and W ~ be a Wilson loop in this representation (for details, see ref. [4]). Let us define O r=
W r-
<01wq0>.
(3)
The matrix element is fixed by the source and b y means of an MC calculation we m a y obtain the matrix elements
(~lTtOrlO),
t=1,2 . . . . .
(4)
(If (0[Wrl 0> 4:0 this requires also an MC calculation without source.) We define leor > = Oq0> and expand in eigenstates of T: oo
I+) =l-IS(u(b)--A(b))lO),
(1)
b
I~)r) = E
b.ln).
(5)
n=l
with the A(b) given and the product goes over all bonds of the spacelike plane. We may expand I~p) in terms of eigenstates of the transfer matrix T:
Eqs. (2) and (5) imply oo
<~plZ'Orl 0> = ~ c, exp(-tE,)
withc.=-a.b..
n=l
Iq') = ~
a,ln),
TI n)
= exp(-E,),
(6)
n=0
= o, 1 . . . . .
,1 For a review see ref. [4]. 140
(2)
The coefficients c n can be positive or negative real numbers. In contrast to the MC variational method (MCV) [4] the upper bound property of
0 3 7 0 - 2 6 9 3 / 8 4 / $ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
3
E+
SOURCE
Table 1 E+ source: distance t = 0 correlation on an N,S3 lattice. In parentheses: enhancement as compared with the MCV method on this lattice.
I Fig. 1.
mass spectrum results is lost. An appropriately chosen source may, however, si~ific~tly increase the signal and eventually allow us to go to larger distances. For increasing the signal of the E+ and the Al - representation, I will introduce two different sources (hot walls). The first source consists in putting on a spacelike plane all link variables V,= 1, except on lines in B, direction indicated in fig. 1. On these lines, we set the U,= - 1. Periodic boundary conditions are taken and the extension of the lattice in the spatial directions has to be even. The contribution of this source to the E+ (and other) representations of 28 Wilson loops was considered. The Wilson loops are numerated in a corresponding figure of refs. [4,5]. Using this notation, some of the operators contributing to the E+ representations are listed in table 1. On an N3N1 lattice, we would like to carry out a MC calculation of momentum zero correlation functions p(t) = (+lN0lO).
22 November 1984
PHYSICS LETTERS
Volume 148B, number 1,2,3
(7)
Here 0 is one of the Wilson loop operators in the Et representation to which the source contributes. Without source, we have for each operator (0) = 0. With source, the distance t = 0 correlation (GO(O)) is a measure for the strength of the source and proportional to the spatial volume. For definite comparison with MCV results [5], we consider an S3N, lattice. In table 1, the numbers in parentheses give the ratio (~0(0))/(~(0)2), in other words the enhancement of the signal as compared with the MCV method on the f.S3.N, lattice. Using the source, I have obtained preliminary MC results at p = 2.25, which are based on rather moderate statistics of about 6000 triple sweeps
OP.
1. E+ representation
2. Et representation
1 7 22 26
2048 (= 8192 ( = 4096 ( = 2048 ( =
682.6 ( = 2730.6 (= 1365.5 (= 628.6 ( =
1409) 695) 626) 407)
470) 232) 209) 136)
with meas~ements* Before taking the measurements, 1350 sweeps were done for reaching equilibrium. Let us use the finite distance mass definitions
dt2, t*>= - (5 - t,)--‘lnMb)/P(h)l*
(8)
In table 2, the results are compared with high statistics MCV results of ref. [5]. Within our present limited statistics, consistency is found. We are now able to obtain E+ results up to distance t = 3, whereas previously one was not able to go beyond distance t = 2, even with much higher statistics. This is an improvement by a factor 100 in CPU time. On the other hand, however, the asymptotic large t behaviour seems to be approached slightly more slowly. In the strong coupling limit ]6] and in the finite (spatial) volume weak coupling limit [7], one finds a mass ratio m(E+ )/m(Al+ ) = 1.0. In contrast, MCV calculations apply to the region between these limits and give much larger values [4,5]. Using the introduced source, a considerable extension of our present numerical work is in progress [8], and we may confirm a large m(E+ )f m(Al+ ) ratio up to distance t = 3. My second source is constructed for giving a large Al- signal. The Al- state is of phenomenological interest, because a low-lying experimental candidate exists. Previous MCV copulations gave, however, a high mass value. As this seems to be to some extent controversial (for a review, see ref. [4]), further confirmation is desirable. The Alsource is defined as follows: let n = (n,,n,,n,) and take all ni (i=1,2,3) odd. At each such point n, the matrices V, on links of the shape of fig. 2 are set equal to U,= - 1. All other 141
Volume 148B, number 1,2,3
PHYSICS LETTERS
22 November 1984
Table 2 Op.
Rep.
m(0,1)
m(0,2)
1 1 7 7 22 22
1 2 1 2 1 2
3.72 -+3.71 + 3.90 -+3.92 + __ 3.91 ± 3.92 + --
0.01 0.01 0.01 0.01 0.01 0.01
26
1
5.49
+
0.03
26
1
5.72 ___0.09
m(0,3)
3.58 + 0.04 3.45 ± 0.09 3 ' 12 -+0 . 21.66 4 3.67 + 0.15 3.62 ± 0.32 -3.49 + 0.06 3.07 ± 0.14 2 " 8 7 -+0.28 -0,16 3.43 + 0.16 n ae+0.31 __ +0.41 3.51 + 0.07 3.12 ± 0.15 2.92_0.19 3.44+ 0.12 2 o~+o.31 ~.,. 7 . . . "
0.43
-'/0+0.23
m(2,3)
") ~1 + 2.50 ~'u~ - 0.35 --
2 " 1~ + 5.07 0-0.78 -1"~a+°"98 v J0.60
,~ -J~+0.43 0.24 z"" ° "~ -
.
- -
+0.62 2.42_0.28
.
.
- -
1
-7 2
-070 +1"38
.
+0.50
.,..0_0.16
.
m(1,3)
- -
-7u-0.24
•~
MCV AI+ a) M C V E + a)
m(1,2)
2.08_0.35
.
--
.
.
1.69 + 0.01 1.53 + 0.03 1.40 ± 0.08 3.04+0.04 2.27+0.20 240 +°'51 . . . . 0.35
--
--
.
1.41 + 0.07 --
1.27 + 0.10 1.15 _+0.24 --
--
a) Ref. [5].
Table 3 A 1 - source: distance t = 0 correlations on an Nt83 lattice. In parentheses: enhancement as compared with the MCV method on this lattice.
A1- SOURCE
2
¸n D-
A 1 - representation
8 9 21
0.75 (~- 0.22) 216 ( --- 96) 126.75 (-~ 37)
1
Fig. 2.
Ue in this s p a t i a l p l a n e a r e set e q u a l to + 1. W e n o w h a v e c o n t r i b u t i o n s to t h e A 1 - r e p r e s e n t a t i o n s o f o p e r a t o r s 8, 9 a n d 21 (in t h e n o t a t i o n o f refs. [4,5]). F o r a n 83Nt lattice, t h e s t r e n g t h o f the s o u r c e a n d t h e e n h a n c e m e n t o f t h e signal as c o m p a r e d t o M C V c a l c u l a t i o n s a r e listed in t a b l e 3. W e r e c o g n i z e t h e A 1 - s o u r c e to b e less efficient t h a n t h e i n t r o d u c e d E + source. F u r t h e r o p t i m i z a t i o n is d e s i r a b l e . B a s e d o n 2 5 0 0 t r i p l e s w e e p s at fl = 2.25 o n a n 838 lattice, I h a v e o b t a i n e d p r e l i m i n a r y M C r e s u l t s . T h e A 1 - m a s s c o m e s o u t l a r g e again. B e y o n d d i s t a n c e t = 1 the signal is noise, at least w i t h t h e p r e s e n t statistics. T h i s h a p p e n s s i m i l a r l y i n t h e h i g h statistics M C V c a l c u l a t i o n s [4,5]. In conclusion, I have demonstrated that variant s o u r c e s a r e u s e f u l for g l u e b a l l s p e c t r o s c o p y in n o n - a b e l i a n l a t t i c e g a u g e theories. F i n a l l y , a combination of sources with multi-hit methods [9,1] s e e m s t o b e n a t u r a l . 142
Op.
I w o u l d l i k e to t h a n k G . M f m s t e r for a useful d i s c u s s i o n a n d H . K a m e n z k i f o r v a l u a b l e h e l p in carrying out the numerical calculation. Support with computer time by DESY and by Hamburg U n i v e r s i t y is g r a t e f u l l y a c k n o w l e d g e d .
References [1] K.H. Mi~tter and K. Schilling, Phys. Lett. l17B (1982) 75. [2] M. Falciorti, E. Marinari, M.L Paeiello, G. Parisi, B. Taglienti and Zhang Yi-Cheng, Nucl. Phys. B215 [FS7] (1983) 265; C. Michael and I. Teasdale, Nucl. Phys. B215 [FS7] (1983) 433. [3] G. Parisi, private communication; Ph. de Forcrand, to be published. [4] B. Berg, Carg6se lectures (1983), DESY preprint 84-012 (1984). [5] B. Berg, A. Billoire, S. Meyer and C. Panagiotakopoulos, Saclay preprint PHT 84-24 (1984), to be published in Commun. Math. Phys. 179. [6] G. Mi~mster, Nucl. Phys. B190 [FS3] (1981) 439; B205 [FS5] (1982) 648(E). [7] M. Li~scher and G. Mi~mster,Nucl. Phys. B232 (1984) 445. [8] B. Berg and H. Kamenzki, in preparation. [9] H. Meyer-Ortmamas and I. Montvay, DESY preprint 84-34 (1984).
PHYSICS LETTERS
22 November 1984
VARIANT SOURCE METHODS B. B E R G CERN, Geneva, Switzerland and II. Institut fiJr Theoretische Physik, Hamburg, Fed. Rep. Germany
Received 30 July 1984
Hot and cold wall source methods for calculating the mass spectrum of lattice gauge theories are investigated. Sources for SU(2) E+ and A1- states are introduced. Preliminary Monte Carlo results at B = 2.25 are compared with previous Monte Carlo variational results.
Source methods for investigating the mass spectrum of 4d lattice gauge theories were suggested in refs. [1-3]. To be definite, let us consider the SU(2) gauge group. The cold wall method [3] simply consists in fixing on a spacelike plane all SU(2) matrices to the identity. This is mainly useful for investigating the A1 + representation. C o m p a r e d to other representations of the cubic group, M o n t e Carlo (MC) investigations of the A I + state are in good shape *t. Sources with strong projections on other representations, corresponding to excited states [4], are therefore desirable. Sources specialized to the E + and A 1 states are introduced in the following. First numerical results are also presented. Let me first discuss the fixed wall method (hot or cold) in general. It consists in fixing a wave function I~p) on a spacelike plane:
where we have chosen the normalization T[0) = 10) of the vacuum. Let r = (R, P, C) specify a representation of the cubic group and W ~ be a Wilson loop in this representation (for details, see ref. [4]). Let us define O r=
W r-
<01wq0>.
(3)
The matrix element
(~lTtOrlO),
t=1,2 . . . . .
(4)
(If (0[Wrl 0> 4:0 this requires also an MC calculation without source.) We define leor > = Oq0> and expand in eigenstates of T: oo
I+) =l-IS(u(b)--A(b))lO),
(1)
b
I~)r) = E
b.ln).
(5)
n=l
with the A(b) given and the product goes over all bonds of the spacelike plane. We may expand I~p) in terms of eigenstates of the transfer matrix T:
Eqs. (2) and (5) imply oo
<~plZ'Orl 0> = ~ c, exp(-tE,)
withc.=-a.b..
n=l
Iq') = ~
a,ln),
TI n)
= exp(-E,),
(6)
n=0
= o, 1 . . . . .
,1 For a review see ref. [4]. 140
(2)
The coefficients c n can be positive or negative real numbers. In contrast to the MC variational method (MCV) [4] the upper bound property of
0 3 7 0 - 2 6 9 3 / 8 4 / $ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
3
E+
SOURCE
Table 1 E+ source: distance t = 0 correlation on an N,S3 lattice. In parentheses: enhancement as compared with the MCV method on this lattice.
I Fig. 1.
mass spectrum results is lost. An appropriately chosen source may, however, si~ific~tly increase the signal and eventually allow us to go to larger distances. For increasing the signal of the E+ and the Al - representation, I will introduce two different sources (hot walls). The first source consists in putting on a spacelike plane all link variables V,= 1, except on lines in B, direction indicated in fig. 1. On these lines, we set the U,= - 1. Periodic boundary conditions are taken and the extension of the lattice in the spatial directions has to be even. The contribution of this source to the E+ (and other) representations of 28 Wilson loops was considered. The Wilson loops are numerated in a corresponding figure of refs. [4,5]. Using this notation, some of the operators contributing to the E+ representations are listed in table 1. On an N3N1 lattice, we would like to carry out a MC calculation of momentum zero correlation functions p(t) = (+lN0lO).
22 November 1984
PHYSICS LETTERS
Volume 148B, number 1,2,3
(7)
Here 0 is one of the Wilson loop operators in the Et representation to which the source contributes. Without source, we have for each operator (0) = 0. With source, the distance t = 0 correlation (GO(O)) is a measure for the strength of the source and proportional to the spatial volume. For definite comparison with MCV results [5], we consider an S3N, lattice. In table 1, the numbers in parentheses give the ratio (~0(0))/(~(0)2), in other words the enhancement of the signal as compared with the MCV method on the f.S3.N, lattice. Using the source, I have obtained preliminary MC results at p = 2.25, which are based on rather moderate statistics of about 6000 triple sweeps
OP.
1. E+ representation
2. Et representation
1 7 22 26
2048 (= 8192 ( = 4096 ( = 2048 ( =
682.6 ( = 2730.6 (= 1365.5 (= 628.6 ( =
1409) 695) 626) 407)
470) 232) 209) 136)
with meas~ements* Before taking the measurements, 1350 sweeps were done for reaching equilibrium. Let us use the finite distance mass definitions
dt2, t*>= - (5 - t,)--‘lnMb)/P(h)l*
(8)
In table 2, the results are compared with high statistics MCV results of ref. [5]. Within our present limited statistics, consistency is found. We are now able to obtain E+ results up to distance t = 3, whereas previously one was not able to go beyond distance t = 2, even with much higher statistics. This is an improvement by a factor 100 in CPU time. On the other hand, however, the asymptotic large t behaviour seems to be approached slightly more slowly. In the strong coupling limit ]6] and in the finite (spatial) volume weak coupling limit [7], one finds a mass ratio m(E+ )/m(Al+ ) = 1.0. In contrast, MCV calculations apply to the region between these limits and give much larger values [4,5]. Using the introduced source, a considerable extension of our present numerical work is in progress [8], and we may confirm a large m(E+ )f m(Al+ ) ratio up to distance t = 3. My second source is constructed for giving a large Al- signal. The Al- state is of phenomenological interest, because a low-lying experimental candidate exists. Previous MCV copulations gave, however, a high mass value. As this seems to be to some extent controversial (for a review, see ref. [4]), further confirmation is desirable. The Alsource is defined as follows: let n = (n,,n,,n,) and take all ni (i=1,2,3) odd. At each such point n, the matrices V, on links of the shape of fig. 2 are set equal to U,= - 1. All other 141
Volume 148B, number 1,2,3
PHYSICS LETTERS
22 November 1984
Table 2 Op.
Rep.
m(0,1)
m(0,2)
1 1 7 7 22 22
1 2 1 2 1 2
3.72 -+3.71 + 3.90 -+3.92 + __ 3.91 ± 3.92 + --
0.01 0.01 0.01 0.01 0.01 0.01
26
1
5.49
+
0.03
26
1
5.72 ___0.09
m(0,3)
3.58 + 0.04 3.45 ± 0.09 3 ' 12 -+0 . 21.66 4 3.67 + 0.15 3.62 ± 0.32 -3.49 + 0.06 3.07 ± 0.14 2 " 8 7 -+0.28 -0,16 3.43 + 0.16 n ae+0.31 __ +0.41 3.51 + 0.07 3.12 ± 0.15 2.92_0.19 3.44+ 0.12 2 o~+o.31 ~.,. 7 . . . "
0.43
-'/0+0.23
m(2,3)
") ~1 + 2.50 ~'u~ - 0.35 --
2 " 1~ + 5.07 0-0.78 -1"~a+°"98 v J0.60
,~ -J~+0.43 0.24 z"" ° "~ -
.
- -
+0.62 2.42_0.28
.
.
- -
1
-7 2
-070 +1"38
.
+0.50
.,..0_0.16
.
m(1,3)
- -
-7u-0.24
•~
MCV AI+ a) M C V E + a)
m(1,2)
2.08_0.35
.
--
.
.
1.69 + 0.01 1.53 + 0.03 1.40 ± 0.08 3.04+0.04 2.27+0.20 240 +°'51 . . . . 0.35
--
--
.
1.41 + 0.07 --
1.27 + 0.10 1.15 _+0.24 --
--
a) Ref. [5].
Table 3 A 1 - source: distance t = 0 correlations on an Nt83 lattice. In parentheses: enhancement as compared with the MCV method on this lattice.
A1- SOURCE
2
¸n D-
A 1 - representation
8 9 21
0.75 (~- 0.22) 216 ( --- 96) 126.75 (-~ 37)
1
Fig. 2.
Ue in this s p a t i a l p l a n e a r e set e q u a l to + 1. W e n o w h a v e c o n t r i b u t i o n s to t h e A 1 - r e p r e s e n t a t i o n s o f o p e r a t o r s 8, 9 a n d 21 (in t h e n o t a t i o n o f refs. [4,5]). F o r a n 83Nt lattice, t h e s t r e n g t h o f the s o u r c e a n d t h e e n h a n c e m e n t o f t h e signal as c o m p a r e d t o M C V c a l c u l a t i o n s a r e listed in t a b l e 3. W e r e c o g n i z e t h e A 1 - s o u r c e to b e less efficient t h a n t h e i n t r o d u c e d E + source. F u r t h e r o p t i m i z a t i o n is d e s i r a b l e . B a s e d o n 2 5 0 0 t r i p l e s w e e p s at fl = 2.25 o n a n 838 lattice, I h a v e o b t a i n e d p r e l i m i n a r y M C r e s u l t s . T h e A 1 - m a s s c o m e s o u t l a r g e again. B e y o n d d i s t a n c e t = 1 the signal is noise, at least w i t h t h e p r e s e n t statistics. T h i s h a p p e n s s i m i l a r l y i n t h e h i g h statistics M C V c a l c u l a t i o n s [4,5]. In conclusion, I have demonstrated that variant s o u r c e s a r e u s e f u l for g l u e b a l l s p e c t r o s c o p y in n o n - a b e l i a n l a t t i c e g a u g e theories. F i n a l l y , a combination of sources with multi-hit methods [9,1] s e e m s t o b e n a t u r a l . 142
Op.
I w o u l d l i k e to t h a n k G . M f m s t e r for a useful d i s c u s s i o n a n d H . K a m e n z k i f o r v a l u a b l e h e l p in carrying out the numerical calculation. Support with computer time by DESY and by Hamburg U n i v e r s i t y is g r a t e f u l l y a c k n o w l e d g e d .
References [1] K.H. Mi~tter and K. Schilling, Phys. Lett. l17B (1982) 75. [2] M. Falciorti, E. Marinari, M.L Paeiello, G. Parisi, B. Taglienti and Zhang Yi-Cheng, Nucl. Phys. B215 [FS7] (1983) 265; C. Michael and I. Teasdale, Nucl. Phys. B215 [FS7] (1983) 433. [3] G. Parisi, private communication; Ph. de Forcrand, to be published. [4] B. Berg, Carg6se lectures (1983), DESY preprint 84-012 (1984). [5] B. Berg, A. Billoire, S. Meyer and C. Panagiotakopoulos, Saclay preprint PHT 84-24 (1984), to be published in Commun. Math. Phys. 179. [6] G. Mi~mster, Nucl. Phys. B190 [FS3] (1981) 439; B205 [FS5] (1982) 648(E). [7] M. Li~scher and G. Mi~mster,Nucl. Phys. B232 (1984) 445. [8] B. Berg and H. Kamenzki, in preparation. [9] H. Meyer-Ortmamas and I. Montvay, DESY preprint 84-34 (1984).