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Quantum chromodynamics on a simplical lattice
Volume 128B, number 3,4
PHYSICS LETTERS
25 August 1983
QUANTUM CHROMODYNAMICS ON A SIMPLICIAL LATTICE R.W.B. ARDILL and J.-P. CLARKE Department of Mathematics, Royal Holloway College, Englefield Green, Surrey, 714120OEX, UK J.-M. DROUFFE 1 CERN, Geneva, Switzerland and K.J.M. MORIARTY Department of Mathematics, Royal Holloway College, Engtefield Green, Surrey, TW20 OEX, UK Received 21 June 1983
Monte Carlo simulations on a 64 simplicial lattice are used to calculate Wilson loops, and hence the string tension, for pure SU (3) gauge theory. The asymptotic freedom scale parameter A is measured for both triangles and squares and universality is established.
Lattice gauge theories are usually formulated on a hypercubical lattice [ 1]. Recently, a number of alternative lattice formulations have been put forward. These include the random lattice [2], the body-centered tesseract lattice [3] and the simplicial lattice [4,5]. One of the motivations for studying gauge theory on at least one of these alternative lattices is to test the principle of universality which asserts that the continuum physics is independent of the lattice on which it is formulated. Previously, we studied SU(2) [4] and U(2) [5] gauge theory on a simplicial lattice and found no surprises over the results found on a hypercubical lattice. However, the gauge group for quantum chromodynamics (QCD), the theory of strong interactions, is SU(3). It thus seems reasonable to extend our previous analysis to SU(3). In the present paper, we study SU(3) on a simplicial lattice, the lattice having six sites in each space-time direction. The Wilson loops for both triangles and squares are calculated, the associated string tensions are extracted and the asymptotic-freedom scale parameter A is evaluated. i Permanent address: S.Ph.T., CEN Saclay, 91190 Gif-surYvette, Cedex, France. 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
We define our system on a four-dimensional euclidean space-time simplicial lattice [4]. Nearest-neighbour lattice sites are denoted by i a n d / a n d are joined to form a link denoted by (i, j), on which sits an N X N unitary-unimodular matrix Uii E SU(N) such that =
1
Our partition function is defined by
(;,/} where dUij is the normalized invariant Haar measure for SU(N) and/~ is the inverse coupling constant squared defined by
~= 12/~'-f g 2 , where go is the bare coupling constant and the action S is the sum over all unoriented triangles such that
S[U] =13 ~ E A = ~
(1 - N - 1
Re tr
UA),
where UA is the parallel transporter around a triangle. Periodic boundary conditions are used throughout our 203
Volume
128B, number
PHYSICS
3,4
LETTERS
25 August
1983
calculations. The method of Metropolis et al. [6] is used to equilibriate our lattice. Following the calculational techniques outlined in refs. [4] and [S] and using standard techniques [7-91, we find the average action for (E,)=1-$3-&32-2&33 -
15 lt569544f15 -
+ +
979::;::
63
120f17
4,‘l??‘i?17
_25fl4 279
906
+
760
;“9;
26
f19
640
p6
12f535S8
+
;I ,,,,
936
0.0
032
0’
(1)
‘@“)
and the average action for squares to be
We also obtain the strong-coupling expansions string tension for triangles and squares to be
for the
]ln($CI) + AI-3 + &
f&P3-
P4 +
3.0
~,~,~,; 4 0
5.0
6.0
7.0
8.0
9.0
10.0
Fig. 1. The average action per pbquette (EA) for pure SU(3) gauge theory on a simplicial lattice as a function of the inverse coupling constant squared p. The curve (A) represents the strong-coupling expansion of eq. (I) and curve (B) represents the (4/4] Pad6 approximation to the expansion.
(2)
(E,)=l-2(~~)4-16(~~)6+O(~7).
K, = -(4/d)
1.0 2.0
L ,,,,
O(P)1 >
(3)
Fig. 1 shows the average action for triangles (E, > as a function of the inverse coupling constant squared. We also show the 8th and 10th order strong-coupling expansion of eq. (1) in fig. I. The Monte Carlo results for the triangular Wilson loops are shown in fig. 2. The numbering scheme is
and K,
= -4{ln[I’(~)/3/361’(~)]
+ & /3+ &$”
SU(3)
SIMPLICIAL
LATTICE
0.9 i
+ 0c03)]
9
(4)
respectively. From the renormalization group, we obtain a relationship between the asymptotic string tension u, the lattice string tension K and the asymptotic-freedom scale parameter A of
To perform our calculations we carried out a total of 136 iterations through the simplicial lattice with 20 Monte Carlo updates per link. The first 16 iterations were used for equilibriating the lattice. We then averaged over the next 120 iterations through our lattice. However, in order to reduce correlations between events every second lattice configuration was eliminated. Thus we used 60 lattice configurations in our averages. The crossover point between strong and weak-coupling is 0 N 4.8. As a result we used disordered starting lattices for /I < 4.8 and ordered starting lattices for fl 25.0. Equilibriation was enhanced by storing the lattice configurations from the run with the previous f3 value and starts were initiated at f3= 9.0 (ordered) and 0 = 1.O (disordered). 204
t
:: o
0.6
s
P Fig. 2. The Wilson loops for pure SU(3) gauge theory on a simplicial lattice as a function of the inverse coupling constant squared p. The full upward triangles represent WA, the open circles represent WP, the full circles represent w’p, the crosses represent w,“, the open upward triangles represent IU,“, the full downward triangles represent W,” and the open downward triangles represent Wf3.
Volume 128B, number 3,4
PHYSICS LETTERS
i
'°!'C '\I, xj ~
roo ~_
x X3~x
,oI,
0.8
°°el
O 6i -
/ -]
"~
25 August 1983
1
0.4~-
"
•
0.2 }-~_SO(5) SIMPLICIAL LATTICE
~
0.0[.~ 0.0
J__l~ Ll~_ 2.0
4.0
I I~L_UJ_._t~L~L 6.0 80 B
I IJ I0.0
Fig. 4. The averageaction per plaquette (E B ) for pure SU(2) gauge theory on a simplicial lattice as a function of the inverse coupling constant squared ft. The curve (A) represents the strong-coupling expansion of eq. (2). I°-I
0.0
I.O
2.0
3.0
4.0
,e
5.0
6.0
7.0 8.0
9.0
Fig. 3. The string tension in trianguhr planes for pure SU(3) gauge theory on a simplicial lattice as a function of the inverse coupling constant squared/3. The full upward triangles represent x~,, the full circles represent × f and the open circles represent x3~. Also shown in the diagram axe the behaviours of eqs. (3) and (5).
of size up to 3 X 3 are shown in fig. 5 where we have followed the numbering scheme o f fig. 3 o f ref. [4]. We also show the leading-order strong-coupling behaviour o f e q . (2). Fig. 6 shows the quantities X~ and X~ defined by
x? = lnl#/W
X~ = ½1n]W22 /W~] W~ 1.0
(7)
I I I-- J ~ i SU(5) SIMPLICIAL LATTICE
0.71
(6)
0 0 J
wF
0.6
z 0.5 0
and
= (2/vg)l wf
I
0.8
,
x~ = ( ~/.,/~) ~n Iw¢/w~31,
I
],
0.9
taken from fig. 3 o f ref. [1]. In fig. 2, we also show the leading-order strong-coupling expansion of eq. (1). We present the results for the logarithmic ratios
=(2/v'5)lnlW~'/w~'l
t,
/wCw¢l,
in fig. 3. We show the truncated strong-coupling expansion o f eq. (3) and its Pad6 approximant. We also display a curve which corresponds to the behaviour o f e q . (5) with
4
-~ 0.4 •
0"3 f 0.2
•
which is about one half of the hypercubical lattice result [10]. In fig. 4 we show the average action for squares ( E ~ ) for pure SU(3) gauge theory as a function of the inverse coupling constant squared ~. Also shown is the strong-coupling expansion of eq. (2). All Wilson loops
0.0 1 0.0
o
• o°°~ xa
O.I
1.0 2.0
W 3 × H
×
x~'
•
o x~. •
•
o
A = (3.0 + 0 . 5 ) X 10-3 x / ~ ,
." •
W4
t I
"wg t •
Cl
W9
3.0
4.0 5.0 6.0 7.0 8.0 9.0 B Fig. 5. The Wilson loops for pure SU(3) gauge theory on a simplicial lattice as a function of the inverse coupling constant squared/3. The full upw~d triangles represent W~, the open circles represent W~, the crosses represent W~, the open upward triangles represent I*'~, the full downward triangles represent It,'~ and the full circles represent I¢~. 205
Volume 128B, number 3,4
PHYSICS LETTERS
25 August 1983
We would like to thank the Science and Engineering Research Council of Great Britain for the award of research grants (Grants NG-0983.9, NG-1068.2 and NG-13300) to buy time on the CRAY-IS computer at Daresbury Laboratory where part of this calculation was carried out and the University of London Computer Centre for the granting of discretionary time on their CRAY-IS where the remainder of this calculation was performed. The calculations of the present paper took 48 hours of CRAY-IS CPU time to perform.
References LO-I
iO-2
f
0.0
t
I i I J I I A. t I I 1.0
2,0
5.0
4.0
,5.0
"
6.0
t ~ t 7.0
8.0
l
9.0
/3
Fig. 6. The string tension in quadratic planes for pure SU(3) gauge theory on a simplicial lattice as a function of the inverse coupling constant squared/3. The full upward triangles represent x~ and the full circles represent ×20. Also shown in the diagram axe the behaviours of eqs. (4) and (5).
as a function of the inverse coupling constant squared ~3.We also show in fig. 6 the strong-coupling expansion of eq. (4).
206
[1] K.G. Wilson, Phys. Rev. DI0 (1974) 2445. [2] N.H. Christ, R. Friedberg and T.D. Lee, Nucl. Phys. B202 (1982) 89. [3] W. Celmaster, Phys. Rev. D26 (1982) 2955. [4] J.-M. Drouffe and K.J.M. Moriarty, Nucl. Phys. B220 (FS8) (1983) 253. [5] J.-M. Drouffe and K.J.M. Moriarty, U(2) four-dimensional simplicial lattice gauge theory, CERN preprint (April 1983). [6] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H, Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087. L7] J.-M. Drouffe and C. ltzykson, Phys. Rep. 38C (1975) 133. [8] R. Bailan, J.-M. Drouffe and C, Itzykson, Phys. Rev. D l l (1975) 2104; D19 (1979) 2514. [9] J.-M. Dtouffe, Nucl. Phys. BI70 (FS1) (1980) 91. [10] M. Creutz and K.J.M. Moriarty, Phys. Rev. D26 (1982) 2166.
PHYSICS LETTERS
25 August 1983
QUANTUM CHROMODYNAMICS ON A SIMPLICIAL LATTICE R.W.B. ARDILL and J.-P. CLARKE Department of Mathematics, Royal Holloway College, Englefield Green, Surrey, 714120OEX, UK J.-M. DROUFFE 1 CERN, Geneva, Switzerland and K.J.M. MORIARTY Department of Mathematics, Royal Holloway College, Engtefield Green, Surrey, TW20 OEX, UK Received 21 June 1983
Monte Carlo simulations on a 64 simplicial lattice are used to calculate Wilson loops, and hence the string tension, for pure SU (3) gauge theory. The asymptotic freedom scale parameter A is measured for both triangles and squares and universality is established.
Lattice gauge theories are usually formulated on a hypercubical lattice [ 1]. Recently, a number of alternative lattice formulations have been put forward. These include the random lattice [2], the body-centered tesseract lattice [3] and the simplicial lattice [4,5]. One of the motivations for studying gauge theory on at least one of these alternative lattices is to test the principle of universality which asserts that the continuum physics is independent of the lattice on which it is formulated. Previously, we studied SU(2) [4] and U(2) [5] gauge theory on a simplicial lattice and found no surprises over the results found on a hypercubical lattice. However, the gauge group for quantum chromodynamics (QCD), the theory of strong interactions, is SU(3). It thus seems reasonable to extend our previous analysis to SU(3). In the present paper, we study SU(3) on a simplicial lattice, the lattice having six sites in each space-time direction. The Wilson loops for both triangles and squares are calculated, the associated string tensions are extracted and the asymptotic-freedom scale parameter A is evaluated. i Permanent address: S.Ph.T., CEN Saclay, 91190 Gif-surYvette, Cedex, France. 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
We define our system on a four-dimensional euclidean space-time simplicial lattice [4]. Nearest-neighbour lattice sites are denoted by i a n d / a n d are joined to form a link denoted by (i, j), on which sits an N X N unitary-unimodular matrix Uii E SU(N) such that =
1
Our partition function is defined by
(;,/} where dUij is the normalized invariant Haar measure for SU(N) and/~ is the inverse coupling constant squared defined by
~= 12/~'-f g 2 , where go is the bare coupling constant and the action S is the sum over all unoriented triangles such that
S[U] =13 ~ E A = ~
(1 - N - 1
Re tr
UA),
where UA is the parallel transporter around a triangle. Periodic boundary conditions are used throughout our 203
Volume
128B, number
PHYSICS
3,4
LETTERS
25 August
1983
calculations. The method of Metropolis et al. [6] is used to equilibriate our lattice. Following the calculational techniques outlined in refs. [4] and [S] and using standard techniques [7-91, we find the average action for (E,)=1-$3-&32-2&33 -
15 lt569544f15 -
+ +
979::;::
63
120f17
4,‘l??‘i?17
_25fl4 279
906
+
760
;“9;
26
f19
640
p6
12f535S8
+
;I ,,,,
936
0.0
032
0’
(1)
‘@“)
and the average action for squares to be
We also obtain the strong-coupling expansions string tension for triangles and squares to be
for the
]ln($CI) + AI-3 + &
f&P3-
P4 +
3.0
~,~,~,; 4 0
5.0
6.0
7.0
8.0
9.0
10.0
Fig. 1. The average action per pbquette (EA) for pure SU(3) gauge theory on a simplicial lattice as a function of the inverse coupling constant squared p. The curve (A) represents the strong-coupling expansion of eq. (I) and curve (B) represents the (4/4] Pad6 approximation to the expansion.
(2)
(E,)=l-2(~~)4-16(~~)6+O(~7).
K, = -(4/d)
1.0 2.0
L ,,,,
O(P)1 >
(3)
Fig. 1 shows the average action for triangles (E, > as a function of the inverse coupling constant squared. We also show the 8th and 10th order strong-coupling expansion of eq. (1) in fig. I. The Monte Carlo results for the triangular Wilson loops are shown in fig. 2. The numbering scheme is
and K,
= -4{ln[I’(~)/3/361’(~)]
+ & /3+ &$”
SU(3)
SIMPLICIAL
LATTICE
0.9 i
+ 0c03)]
9
(4)
respectively. From the renormalization group, we obtain a relationship between the asymptotic string tension u, the lattice string tension K and the asymptotic-freedom scale parameter A of
To perform our calculations we carried out a total of 136 iterations through the simplicial lattice with 20 Monte Carlo updates per link. The first 16 iterations were used for equilibriating the lattice. We then averaged over the next 120 iterations through our lattice. However, in order to reduce correlations between events every second lattice configuration was eliminated. Thus we used 60 lattice configurations in our averages. The crossover point between strong and weak-coupling is 0 N 4.8. As a result we used disordered starting lattices for /I < 4.8 and ordered starting lattices for fl 25.0. Equilibriation was enhanced by storing the lattice configurations from the run with the previous f3 value and starts were initiated at f3= 9.0 (ordered) and 0 = 1.O (disordered). 204
t
:: o
0.6
s
P Fig. 2. The Wilson loops for pure SU(3) gauge theory on a simplicial lattice as a function of the inverse coupling constant squared p. The full upward triangles represent WA, the open circles represent WP, the full circles represent w’p, the crosses represent w,“, the open upward triangles represent IU,“, the full downward triangles represent W,” and the open downward triangles represent Wf3.
Volume 128B, number 3,4
PHYSICS LETTERS
i
'°!'C '\I, xj ~
roo ~_
x X3~x
,oI,
0.8
°°el
/ -]
"~
25 August 1983
1
0.4~-
"
•
0.2 }-~_SO(5) SIMPLICIAL LATTICE
~
0.0[.~ 0.0
J__l~ Ll~_ 2.0
4.0
I I~L_UJ_._t~L~L 6.0 80 B
I IJ I0.0
Fig. 4. The averageaction per plaquette (E B ) for pure SU(2) gauge theory on a simplicial lattice as a function of the inverse coupling constant squared ft. The curve (A) represents the strong-coupling expansion of eq. (2). I°-I
0.0
I.O
2.0
3.0
4.0
,e
5.0
6.0
7.0 8.0
9.0
Fig. 3. The string tension in trianguhr planes for pure SU(3) gauge theory on a simplicial lattice as a function of the inverse coupling constant squared/3. The full upward triangles represent x~,, the full circles represent × f and the open circles represent x3~. Also shown in the diagram axe the behaviours of eqs. (3) and (5).
of size up to 3 X 3 are shown in fig. 5 where we have followed the numbering scheme o f fig. 3 o f ref. [4]. We also show the leading-order strong-coupling behaviour o f e q . (2). Fig. 6 shows the quantities X~ and X~ defined by
x? = lnl#/W
X~ = ½1n]W22 /W~] W~ 1.0
(7)
I I I-- J ~ i SU(5) SIMPLICIAL LATTICE
0.71
(6)
0 0 J
wF
0.6
z 0.5 0
and
= (2/vg)l wf
I
0.8
,
x~ = ( ~/.,/~) ~n Iw¢/w~31,
I
],
0.9
taken from fig. 3 o f ref. [1]. In fig. 2, we also show the leading-order strong-coupling expansion of eq. (1). We present the results for the logarithmic ratios
=(2/v'5)lnlW~'/w~'l
t,
/wCw¢l,
in fig. 3. We show the truncated strong-coupling expansion o f eq. (3) and its Pad6 approximant. We also display a curve which corresponds to the behaviour o f e q . (5) with
4
-~ 0.4 •
0"3 f 0.2
•
which is about one half of the hypercubical lattice result [10]. In fig. 4 we show the average action for squares ( E ~ ) for pure SU(3) gauge theory as a function of the inverse coupling constant squared ~. Also shown is the strong-coupling expansion of eq. (2). All Wilson loops
0.0 1 0.0
o
• o°°~ xa
O.I
1.0 2.0
W 3 × H
×
x~'
•
o x~. •
•
o
A = (3.0 + 0 . 5 ) X 10-3 x / ~ ,
." •
W4
t I
"wg t •
Cl
W9
3.0
4.0 5.0 6.0 7.0 8.0 9.0 B Fig. 5. The Wilson loops for pure SU(3) gauge theory on a simplicial lattice as a function of the inverse coupling constant squared/3. The full upw~d triangles represent W~, the open circles represent W~, the crosses represent W~, the open upward triangles represent I*'~, the full downward triangles represent It,'~ and the full circles represent I¢~. 205
Volume 128B, number 3,4
PHYSICS LETTERS
25 August 1983
We would like to thank the Science and Engineering Research Council of Great Britain for the award of research grants (Grants NG-0983.9, NG-1068.2 and NG-13300) to buy time on the CRAY-IS computer at Daresbury Laboratory where part of this calculation was carried out and the University of London Computer Centre for the granting of discretionary time on their CRAY-IS where the remainder of this calculation was performed. The calculations of the present paper took 48 hours of CRAY-IS CPU time to perform.
References LO-I
iO-2
f
0.0
t
I i I J I I A. t I I 1.0
2,0
5.0
4.0
,5.0
"
6.0
t ~ t 7.0
8.0
l
9.0
/3
Fig. 6. The string tension in quadratic planes for pure SU(3) gauge theory on a simplicial lattice as a function of the inverse coupling constant squared/3. The full upward triangles represent x~ and the full circles represent ×20. Also shown in the diagram axe the behaviours of eqs. (4) and (5).
as a function of the inverse coupling constant squared ~3.We also show in fig. 6 the strong-coupling expansion of eq. (4).
206
[1] K.G. Wilson, Phys. Rev. DI0 (1974) 2445. [2] N.H. Christ, R. Friedberg and T.D. Lee, Nucl. Phys. B202 (1982) 89. [3] W. Celmaster, Phys. Rev. D26 (1982) 2955. [4] J.-M. Drouffe and K.J.M. Moriarty, Nucl. Phys. B220 (FS8) (1983) 253. [5] J.-M. Drouffe and K.J.M. Moriarty, U(2) four-dimensional simplicial lattice gauge theory, CERN preprint (April 1983). [6] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H, Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087. L7] J.-M. Drouffe and C. ltzykson, Phys. Rep. 38C (1975) 133. [8] R. Bailan, J.-M. Drouffe and C, Itzykson, Phys. Rev. D l l (1975) 2104; D19 (1979) 2514. [9] J.-M. Dtouffe, Nucl. Phys. BI70 (FS1) (1980) 91. [10] M. Creutz and K.J.M. Moriarty, Phys. Rev. D26 (1982) 2166.