- Email: [email protected]
Estimate of the relation between scale parameters and the string tension by strong coupling methods
Volume 96B, number 1,2
PHYSICS LETTERS
20 October 1980
ESTIMATE OF THE RELATION BETWEEN SCALE PARAMETERS AND THE STRING TENSION BY STRONG COUPLING METHODS Gernot MI~INSTER and Peter WEISZ 11. lnstitut fiir Theoretische Physik der UniversitiTtHamburg, Hamburg, Fed. Rep. Germany Received 24 June 1980
We estimate the relation between the scale parameter AL and the string tension c~in pure SU(3) gauge theory using strong coupling expansions for euclidean lattice gauge theory up to 12th order. The result is AL = 3.7 X 10-3x/~. For the more conventional scale parameter AMOM this gives AMOM = 0.31 x/~ by use of the proportionality factor of Hasenfratz. Results for the string tension of Z3 lattice gauge theory are also discussed.
It is generally believed that in non-abelian pure gauge theories far separated static quarks are confined by a linear potential V(r) = a'r. The string tension a is not computable by perturbation theory. Due to dimensional transmutation [1 ] there exists no free dimensionless parameter in asymptotically free gauge theory. The only parameter is a scale parameter A which has the dimension of a mass. Every other physical quantity such as a is proportional to some power of A with a numerical factor which is fixed by the theory and can in principle be calculated by non-perturbative methods. The natural framework for such problems is lattice gauge theory [2,3]. It provides us with a non-perturbative cut-off which respects local gauge symmetry. Strong coupling pure lattice gauge theories are known to confine static quarks [4]. The main question is whether the theory possesses a continuum limit with persisting confinement property. Due to asymptotic freedom [5] the continuum limit of SU(2) or SU(3) lattice gauge theory is supposed to be a weak coupling limit [6,7], where the lattice spacing a and the bare coupling g simultaneously go to zero. Recent investigations suggest that this picture is correct [ 8 - 1 1 ]. In this paper we use strong coupling expansions to calculate the relation between a and A. We consider euclidean lattice gauge theories on a hypercubical lattice. Our main interest is the case of the gauge group SU(3) in v = 4 dimensions, but we also investigate the
gauge group Z 3 and dimension v = 3. The gauge field U(b) E SU(3) is attached to links b of the lattice. The ordered product of the variables U(b) on the boundary of an elementary plaquette p is called U(p). The action is L =-~3 ~ Re Tr U(p). p
(1)
The sum extends over all unoriented plaquettes of the lattice and 3 is related to the usual coupling constant g by
3 = 6/g 2.
(2)
We use the methods described in detail by one of us in ref. [12], where SU(2) was considered, to calculate the cluster expansion for the string tension of pure SU(3) and Z 3 [13] lattice gauge theories up to 12th order. The series for SU(3) has been calculated up to 10th order previously by Kogut et al. [8]. The natural expansion parameters are the Fourier coefficients 0 ~< ar (3) < 1 in the series exp (~/3 Re Tr U) = N ~ ) [ 1 + r~ ~ 1 drar(3)Xr(U)l' (3) where the sum extends over all inequivalent nontrivial irreducible representations r of SU(3) with dimension d r, and ×r are the corresponding primitive characters. Representations are denoted in the usual way: r = 1, 119
V o l u m e 9 6 B , n u m b e r 1,2
PHYSICS LETTERS
3, 3, .... N is an irrelevant normalization factor. For Z 3 the natural expansion parameter is [13] : x = [1 - e x p ( - ~ / 3 ) ] / [ 1 + 2 e x p ( - ~ / 3 ) ] .
(4)
Our results for the string tension are given in table 1 for v = 3 and v = 4 dimensions. Expanding the series for SU(3), u = 4 in terms of u = a 3 the result is 3918 a2a = - l n u - - 4 0 4 - 1205 + 10u 6 + 3 6 u 7 - --T-u 1131U910
2 5 5 10 28 03 7 / , / 1 0 + ~-.52182871/11
(5)
20 October 1980
totic freedom in the form predicted by perturbation theory is true for the physical theory and confinement persists in the continuum limit, the string tension should behave like [ 14,15] ot ..~ Ca-2(/30g2)-(~l/~) exp ( - / 3 0 1 g - 2 ) ,
(6)
in the weak coupling limit of the lattice theory, t30 and t31 are the lowest coefficients in the Gell-MannLow function with 130 = 11/(167r2),
131 = 102/(167r2)2,
for SU(3),
273755099 UI2 61440
This is in agreement with the 10th order result of Kogut et al. [8]. The relation between a and A in the continuum limit can be estimated in the following way. If asymp-
(7)
and C is a constant. The theory can be renormalized by holding the string tension fixed, while a and g go to zero [8]. Then a renormalization scale parameter A L is given by A 2 --
c -1 .
(8)
Table 1
SU(3)
u = a 3 , ul = a 6 , u 2 = a 8 , w l = a l o , w2 = a l 5 m
I)=3: a2ot = - I n u - 2 u 4 - 6 u 5 + 3 2 u 6 - 12U4Ol - 1 6 u 4 o 2 - 4608 + 2109 + 4801° + 48u801 - 80802 - 7206Ol 2 _ 9 6 u 6 0 2 2 - 1 2 o l s - 1 6 o 2 s + 4 1 4 0 1 1 + 90u9~)1 + 1 4 4 u 9 0 2 - 432070102 - 3201s020 -1 _ 3202501 u-1 _ 8s52012 + 116401°01 + 1552u1°02 _ 30U9Wl 3 - 1 3 5 u 9 w 2 + 6 4 0 8 0 ~ + 384u80102 - 3 6 0 u 6 0 1 2 0 2 - 224u6023 - 4804014 - 640404
+ 1 4 4 0 1 6 + 128026
_ 40o15Wl u-1 - 80025w20 -1 _ 60015w2u -1 u=4:
a2a = --In u - 4 u 4 -- 1 2 u 5 + 6 4 0 6 - 2 4 0 4 U l - 3 2 u 4 v 2 -- 1 2 8 0 8 _ 1 8 6 u 9 + 8 7 6 u 1° _ 3 3 6 u 8 0 1 -- 5 9 2 u 8 0 2 -- 1 4 4 u 6 0 1 2 -- 1 9 2 0 6 0 2 2 _ 2 4 0 1 s -- 3 2 0 2 5 + 4 8 3 6 0 1 1 -- 1 3 3 2 u 9 0 1 - 1 6 3 2 u 9 0 2 - 864u70102 --6401502u -1 _ 64025010-1 59656 u 1 2 + 1 0 3 4 4 0 1 o 0 1 + 1 3 7 9 2 u 1 ° 0 2 3 -- 1 8 0 U 9 W l -- 8 1 0 u 9 w 2 - 1 2 9 6 0 8 0 1 2 _ 2 1 7 6 u 8 0 2 2 -- 2688u80102 - 720u601202 --448u6023 - 9 6 u 4 0 1 4 _ 1 2 8 u 4 0 4 + 2 8 8 0 1 6 + 2 5 6 0 2 6 -- 80015Wl u - 1
-- 160025w2u -1 -- 120UlSW2U -1 Z3
x = [1 - e x p ( - 3 f 3 ) ] / [ 1
Monte Carlo calculations [9,11,15] indicate that the above picture is correct. They show a rather small intermediate coupling region in which a changeover from strong coupling to weak coupling asymptotic freedom behaviour takes place. High temperature expansions for the SU(2) string tension up to 12th order [16,12] also show such an abrupt breakaway. They yield values for the string tension, which are in good agreement with the Monte Carlo data at strong and intermediate coupling. If one adjusts the constant C such that the weak coupling function (6) touches the high temperature curve, the obtained value for C agrees with the result of the Monte Carlo computation. This experience suggests applying the same procedure to the case of the gauge group SU(3). In fig. 1 we have plotted the expansion of ~ according to table 1 up to 10th, 1 l t h and 12th order. Fitting the weak coupling function to the 12th order curve we get A L = 3.7 X 10-3 x/-~.
(9)
This is in agreement with Creutz's result for SU(3) [15]:
+ 2exp(-3~)]
u=3: a2c( =
-Inx
-
2x 4 -
2x s -
22x 8 + 7x 9 -
A L ---(5.0 -+ 1.5) × 10 -3 V ~
29x I ° - 6 x 11
4 2 8 . 12 u=4: a2~=-Inx-4x 5776X12 3
120
4 -4x s-80x
8-62x
9-130x
l°-20x
I1
(Creutz).
(10)
We would like to add some critical remarks concerning our procedure. Above/3 ~ 6 the contributions of the higher orders are relatively large. In particular the 1 l t h order curve differs significantly from the curves representing the series up to 10th and 12th or-
Volume 96B, number 1,2
PHYSICS LETTERS SU(3),v=&
,
,
,
,
i
i
20 October 1980
very different from what Pads approximants produce. Furthermore from consideration of graphs we expect that in higher orders in the expansion there are relatively more contributions with positive sign than in the low orders, while Pads approximants merely extrapolate the low order terms. Therefore we do not see any reason why Pads approximants should be closer to the truth. The relation between A L and the better known scale parameter A MOM [17] has been investigated by Hasenfratz and Hasenfratz [14]. They find
,
AL:3.7 xi0_3]/~ 1.0
aza
A M°M = 83.5 A L.
(12)
Using this relation one gets with formula (9) A MOM = 0.31%/a.
(13)
Inserting the "experimental" value x/~ ~ 450 MeV [18] one obtains
0.1 lit
1
2
3
[3- 6
Z
g
A MOM ,~ 140 MeV.
6-7
Fig. 1. The string tension a times the lattice spacing squared as a function of# = 6/g 2 for SU(3) lattice gauge theory in v = 4 dimensions. The solid lines represent results of strong coupling expansions up to 10th (I), 11 th (II), and 12th (III) order. The dashed lines are the lowest order strong coupling curve and the fitted weak coupling function (6).
This number cannot y e t be taken too seriously, because dynamical quarks have been neglected. Finally we would like to discuss the expansion for the string tension o f Z 3 lattice gauge theory in 4 dimensions. The theory is self-dual [13] and is supposed to undergo a first order phase transition at the point
31
der and is situated even above the 8th and 9th order curves. Therefore one has to be cautious in using the series at these values of/3. On the other hand the sequence o f even order curves moves down uniformly and appears to converge even in the intermediate region. Remembering that for SU(2) only even orders contribute, one gets the impression that the sequence of even order curves behaves better than the odd order ones, and we propose to use the even order curves to extract numerical results. For comparison we quote the numbers extracted from some lower order curves: A L • a - 1 / 2 = 2.4 X 10 - 3 ,
up to 8th o r d e r ,
= 3.4 × 10 - 3 ,
upto 10thorder,
( = 1 . 7 × 10 - 3 ,
up t o l l t h o r d e r ) .
(14)
2
I
I
I
Z3 , V : 4 l I--I
~ I
p
i
I
a2C~
(xg) : O(xl2)~
(11)
One might think of an extrapolation of the strong coupling series for a, e.g. by Pads methods. But the expected weak coupling behaviour is qualitatively
O} 0.2 0.3 0.4 0.5 0.6 10.7 0.8 0.9
13¢
Fig. 2. The string tension times the lattice spacing squared for Z 3 lattice gauge theory in v = 4 dimensions. The lines represent results of strong coupling expansions up to 9th and 12th order. The critical coupling tic is indicated. 121
Volume 96B, number 1,2
PHYSICS LETTERS
/3c = 0.670 [19]. For/3 >/3 c the string tension is zero. Because of the first order nature of the transition a may have a discontinuity at/3 c. In fig. 2 we plot the high temperature series for ot up to 9th and 12th order. At/3c it appears to converge to some finite value around 0.83. This supports the expected first order nature of the transition. We would like to thank G. Mack, M. Lfischer and B. Berg for discussions. One of us (P.W.) thanks the Deutsche Forschungsgemeinschaft for financial support.
Note added. After completion of this work, Drouffe sent us an unpublished paper (Stony Brook preprint ITP-SB-78-35) in which he calculates the high temperature series for the string tension of several three-dimensional models up to 16th order, including SU(3) and Z 3. Our results agree with his. References [ 1] S. Coleman and E. Weinberg,Phys. Rev. D7 (1973) 1888; D. Gross and A. Neveu, Phys. Rev. D10 (1974) 3235. [2] K. Wilson, Phys. Rev. D10 (1974) 2445. [3] J. Kogut and L. Susskind, Phys. Rev. D11 (1975) 395. [4] K. Osterwalder and E. Seiler, Ann. Phys. (NY) 110 (1978) 440. [5] G. 't Hooft (1972), unpublished; H. Politzer, Phys. Rev. Lett. 30 (1973) 1346; D. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343.
122
20 October 1980
[6] J. Kogut, Rev. Mod. Phys. 51 (1979) 659. [7] G. Mack, Properties of lattice gauge theory models at low temperatures, DESY-report 80/03 (Jan. 1980), to be published in: Recent developments in gauge theories, eds. G. 't Hooft et al. (Plenum, New York, 1980). [8] J. Kogut, R. Pearson and J. Shigemitsu, Phys. Rev. Lett. 43 (1979) 484. [9] M. Creutz, Solving quantized SU(2) gauge theory, Brookhaven preprint BNL-26847 (Sep. 1979). [10] G. Mack, Predictions of a theory of quark confinement. DESY 80/21 (March 1980). [11 ] G. Mack and E. Pietarinen, Phys. Lett. 94B (1980) 397. [12] G. MOnster, High temperature expansions for the free energy of vortices respectively the string tension in lattice gauge theories, DESY 80/44 (June 1980). [13] T. Yoneya, Nucl. Phys. B144 (1978) 195; C. Korthals-Altes, Nucl. Phys. B142 (1978) 315. [14] A. Hasenfratz and P. Hasenfratz, Phys. Lett. 93B (1980) 165. [15] M. Creutz, Asymptotic freedom scales, Brookhaven preprint (April 1980). [16] G. Miinster, Phys. Lett. 95B (1980) 59. [17] See A. Buras, Rev. Mod. Phys. 52 (1980) 199. [18] E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane and T.M. Yan, Phys. Rev. D21 (1980) 203; P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Nucl. Phys. B56 (1973) 109. [19] S. Elitzur, R. Pearson and J. Shigemitsu, Phys. Rev. D19 (1979) 3698; A. Ukawa, P. Windey and A. Guth, Phys. Rev. D21 (1980) 1013 ; M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. D20 (1979) 1915.
PHYSICS LETTERS
20 October 1980
ESTIMATE OF THE RELATION BETWEEN SCALE PARAMETERS AND THE STRING TENSION BY STRONG COUPLING METHODS Gernot MI~INSTER and Peter WEISZ 11. lnstitut fiir Theoretische Physik der UniversitiTtHamburg, Hamburg, Fed. Rep. Germany Received 24 June 1980
We estimate the relation between the scale parameter AL and the string tension c~in pure SU(3) gauge theory using strong coupling expansions for euclidean lattice gauge theory up to 12th order. The result is AL = 3.7 X 10-3x/~. For the more conventional scale parameter AMOM this gives AMOM = 0.31 x/~ by use of the proportionality factor of Hasenfratz. Results for the string tension of Z3 lattice gauge theory are also discussed.
It is generally believed that in non-abelian pure gauge theories far separated static quarks are confined by a linear potential V(r) = a'r. The string tension a is not computable by perturbation theory. Due to dimensional transmutation [1 ] there exists no free dimensionless parameter in asymptotically free gauge theory. The only parameter is a scale parameter A which has the dimension of a mass. Every other physical quantity such as a is proportional to some power of A with a numerical factor which is fixed by the theory and can in principle be calculated by non-perturbative methods. The natural framework for such problems is lattice gauge theory [2,3]. It provides us with a non-perturbative cut-off which respects local gauge symmetry. Strong coupling pure lattice gauge theories are known to confine static quarks [4]. The main question is whether the theory possesses a continuum limit with persisting confinement property. Due to asymptotic freedom [5] the continuum limit of SU(2) or SU(3) lattice gauge theory is supposed to be a weak coupling limit [6,7], where the lattice spacing a and the bare coupling g simultaneously go to zero. Recent investigations suggest that this picture is correct [ 8 - 1 1 ]. In this paper we use strong coupling expansions to calculate the relation between a and A. We consider euclidean lattice gauge theories on a hypercubical lattice. Our main interest is the case of the gauge group SU(3) in v = 4 dimensions, but we also investigate the
gauge group Z 3 and dimension v = 3. The gauge field U(b) E SU(3) is attached to links b of the lattice. The ordered product of the variables U(b) on the boundary of an elementary plaquette p is called U(p). The action is L =-~3 ~ Re Tr U(p). p
(1)
The sum extends over all unoriented plaquettes of the lattice and 3 is related to the usual coupling constant g by
3 = 6/g 2.
(2)
We use the methods described in detail by one of us in ref. [12], where SU(2) was considered, to calculate the cluster expansion for the string tension of pure SU(3) and Z 3 [13] lattice gauge theories up to 12th order. The series for SU(3) has been calculated up to 10th order previously by Kogut et al. [8]. The natural expansion parameters are the Fourier coefficients 0 ~< ar (3) < 1 in the series exp (~/3 Re Tr U) = N ~ ) [ 1 + r~ ~ 1 drar(3)Xr(U)l' (3) where the sum extends over all inequivalent nontrivial irreducible representations r of SU(3) with dimension d r, and ×r are the corresponding primitive characters. Representations are denoted in the usual way: r = 1, 119
V o l u m e 9 6 B , n u m b e r 1,2
PHYSICS LETTERS
3, 3, .... N is an irrelevant normalization factor. For Z 3 the natural expansion parameter is [13] : x = [1 - e x p ( - ~ / 3 ) ] / [ 1 + 2 e x p ( - ~ / 3 ) ] .
(4)
Our results for the string tension are given in table 1 for v = 3 and v = 4 dimensions. Expanding the series for SU(3), u = 4 in terms of u = a 3 the result is 3918 a2a = - l n u - - 4 0 4 - 1205 + 10u 6 + 3 6 u 7 - --T-u 1131U910
2 5 5 10 28 03 7 / , / 1 0 + ~-.52182871/11
(5)
20 October 1980
totic freedom in the form predicted by perturbation theory is true for the physical theory and confinement persists in the continuum limit, the string tension should behave like [ 14,15] ot ..~ Ca-2(/30g2)-(~l/~) exp ( - / 3 0 1 g - 2 ) ,
(6)
in the weak coupling limit of the lattice theory, t30 and t31 are the lowest coefficients in the Gell-MannLow function with 130 = 11/(167r2),
131 = 102/(167r2)2,
for SU(3),
273755099 UI2 61440
This is in agreement with the 10th order result of Kogut et al. [8]. The relation between a and A in the continuum limit can be estimated in the following way. If asymp-
(7)
and C is a constant. The theory can be renormalized by holding the string tension fixed, while a and g go to zero [8]. Then a renormalization scale parameter A L is given by A 2 --
c -1 .
(8)
Table 1
SU(3)
u = a 3 , ul = a 6 , u 2 = a 8 , w l = a l o , w2 = a l 5 m
I)=3: a2ot = - I n u - 2 u 4 - 6 u 5 + 3 2 u 6 - 12U4Ol - 1 6 u 4 o 2 - 4608 + 2109 + 4801° + 48u801 - 80802 - 7206Ol 2 _ 9 6 u 6 0 2 2 - 1 2 o l s - 1 6 o 2 s + 4 1 4 0 1 1 + 90u9~)1 + 1 4 4 u 9 0 2 - 432070102 - 3201s020 -1 _ 3202501 u-1 _ 8s52012 + 116401°01 + 1552u1°02 _ 30U9Wl 3 - 1 3 5 u 9 w 2 + 6 4 0 8 0 ~ + 384u80102 - 3 6 0 u 6 0 1 2 0 2 - 224u6023 - 4804014 - 640404
+ 1 4 4 0 1 6 + 128026
_ 40o15Wl u-1 - 80025w20 -1 _ 60015w2u -1 u=4:
a2a = --In u - 4 u 4 -- 1 2 u 5 + 6 4 0 6 - 2 4 0 4 U l - 3 2 u 4 v 2 -- 1 2 8 0 8 _ 1 8 6 u 9 + 8 7 6 u 1° _ 3 3 6 u 8 0 1 -- 5 9 2 u 8 0 2 -- 1 4 4 u 6 0 1 2 -- 1 9 2 0 6 0 2 2 _ 2 4 0 1 s -- 3 2 0 2 5 + 4 8 3 6 0 1 1 -- 1 3 3 2 u 9 0 1 - 1 6 3 2 u 9 0 2 - 864u70102 --6401502u -1 _ 64025010-1 59656 u 1 2 + 1 0 3 4 4 0 1 o 0 1 + 1 3 7 9 2 u 1 ° 0 2 3 -- 1 8 0 U 9 W l -- 8 1 0 u 9 w 2 - 1 2 9 6 0 8 0 1 2 _ 2 1 7 6 u 8 0 2 2 -- 2688u80102 - 720u601202 --448u6023 - 9 6 u 4 0 1 4 _ 1 2 8 u 4 0 4 + 2 8 8 0 1 6 + 2 5 6 0 2 6 -- 80015Wl u - 1
-- 160025w2u -1 -- 120UlSW2U -1 Z3
x = [1 - e x p ( - 3 f 3 ) ] / [ 1
Monte Carlo calculations [9,11,15] indicate that the above picture is correct. They show a rather small intermediate coupling region in which a changeover from strong coupling to weak coupling asymptotic freedom behaviour takes place. High temperature expansions for the SU(2) string tension up to 12th order [16,12] also show such an abrupt breakaway. They yield values for the string tension, which are in good agreement with the Monte Carlo data at strong and intermediate coupling. If one adjusts the constant C such that the weak coupling function (6) touches the high temperature curve, the obtained value for C agrees with the result of the Monte Carlo computation. This experience suggests applying the same procedure to the case of the gauge group SU(3). In fig. 1 we have plotted the expansion of ~ according to table 1 up to 10th, 1 l t h and 12th order. Fitting the weak coupling function to the 12th order curve we get A L = 3.7 X 10-3 x/-~.
(9)
This is in agreement with Creutz's result for SU(3) [15]:
+ 2exp(-3~)]
u=3: a2c( =
-Inx
-
2x 4 -
2x s -
22x 8 + 7x 9 -
A L ---(5.0 -+ 1.5) × 10 -3 V ~
29x I ° - 6 x 11
4 2 8 . 12 u=4: a2~=-Inx-4x 5776X12 3
120
4 -4x s-80x
8-62x
9-130x
l°-20x
I1
(Creutz).
(10)
We would like to add some critical remarks concerning our procedure. Above/3 ~ 6 the contributions of the higher orders are relatively large. In particular the 1 l t h order curve differs significantly from the curves representing the series up to 10th and 12th or-
Volume 96B, number 1,2
PHYSICS LETTERS SU(3),v=&
,
,
,
,
i
i
20 October 1980
very different from what Pads approximants produce. Furthermore from consideration of graphs we expect that in higher orders in the expansion there are relatively more contributions with positive sign than in the low orders, while Pads approximants merely extrapolate the low order terms. Therefore we do not see any reason why Pads approximants should be closer to the truth. The relation between A L and the better known scale parameter A MOM [17] has been investigated by Hasenfratz and Hasenfratz [14]. They find
,
AL:3.7 xi0_3]/~ 1.0
aza
A M°M = 83.5 A L.
(12)
Using this relation one gets with formula (9) A MOM = 0.31%/a.
(13)
Inserting the "experimental" value x/~ ~ 450 MeV [18] one obtains
0.1 lit
1
2
3
[3- 6
Z
g
A MOM ,~ 140 MeV.
6-7
Fig. 1. The string tension a times the lattice spacing squared as a function of# = 6/g 2 for SU(3) lattice gauge theory in v = 4 dimensions. The solid lines represent results of strong coupling expansions up to 10th (I), 11 th (II), and 12th (III) order. The dashed lines are the lowest order strong coupling curve and the fitted weak coupling function (6).
This number cannot y e t be taken too seriously, because dynamical quarks have been neglected. Finally we would like to discuss the expansion for the string tension o f Z 3 lattice gauge theory in 4 dimensions. The theory is self-dual [13] and is supposed to undergo a first order phase transition at the point
31
der and is situated even above the 8th and 9th order curves. Therefore one has to be cautious in using the series at these values of/3. On the other hand the sequence o f even order curves moves down uniformly and appears to converge even in the intermediate region. Remembering that for SU(2) only even orders contribute, one gets the impression that the sequence of even order curves behaves better than the odd order ones, and we propose to use the even order curves to extract numerical results. For comparison we quote the numbers extracted from some lower order curves: A L • a - 1 / 2 = 2.4 X 10 - 3 ,
up to 8th o r d e r ,
= 3.4 × 10 - 3 ,
upto 10thorder,
( = 1 . 7 × 10 - 3 ,
up t o l l t h o r d e r ) .
(14)
2
I
I
I
Z3 , V : 4 l I--I
~ I
p
i
I
a2C~
(xg) : O(xl2)~
(11)
One might think of an extrapolation of the strong coupling series for a, e.g. by Pads methods. But the expected weak coupling behaviour is qualitatively
O} 0.2 0.3 0.4 0.5 0.6 10.7 0.8 0.9
13¢
Fig. 2. The string tension times the lattice spacing squared for Z 3 lattice gauge theory in v = 4 dimensions. The lines represent results of strong coupling expansions up to 9th and 12th order. The critical coupling tic is indicated. 121
Volume 96B, number 1,2
PHYSICS LETTERS
/3c = 0.670 [19]. For/3 >/3 c the string tension is zero. Because of the first order nature of the transition a may have a discontinuity at/3 c. In fig. 2 we plot the high temperature series for ot up to 9th and 12th order. At/3c it appears to converge to some finite value around 0.83. This supports the expected first order nature of the transition. We would like to thank G. Mack, M. Lfischer and B. Berg for discussions. One of us (P.W.) thanks the Deutsche Forschungsgemeinschaft for financial support.
Note added. After completion of this work, Drouffe sent us an unpublished paper (Stony Brook preprint ITP-SB-78-35) in which he calculates the high temperature series for the string tension of several three-dimensional models up to 16th order, including SU(3) and Z 3. Our results agree with his. References [ 1] S. Coleman and E. Weinberg,Phys. Rev. D7 (1973) 1888; D. Gross and A. Neveu, Phys. Rev. D10 (1974) 3235. [2] K. Wilson, Phys. Rev. D10 (1974) 2445. [3] J. Kogut and L. Susskind, Phys. Rev. D11 (1975) 395. [4] K. Osterwalder and E. Seiler, Ann. Phys. (NY) 110 (1978) 440. [5] G. 't Hooft (1972), unpublished; H. Politzer, Phys. Rev. Lett. 30 (1973) 1346; D. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343.
122
20 October 1980
[6] J. Kogut, Rev. Mod. Phys. 51 (1979) 659. [7] G. Mack, Properties of lattice gauge theory models at low temperatures, DESY-report 80/03 (Jan. 1980), to be published in: Recent developments in gauge theories, eds. G. 't Hooft et al. (Plenum, New York, 1980). [8] J. Kogut, R. Pearson and J. Shigemitsu, Phys. Rev. Lett. 43 (1979) 484. [9] M. Creutz, Solving quantized SU(2) gauge theory, Brookhaven preprint BNL-26847 (Sep. 1979). [10] G. Mack, Predictions of a theory of quark confinement. DESY 80/21 (March 1980). [11 ] G. Mack and E. Pietarinen, Phys. Lett. 94B (1980) 397. [12] G. MOnster, High temperature expansions for the free energy of vortices respectively the string tension in lattice gauge theories, DESY 80/44 (June 1980). [13] T. Yoneya, Nucl. Phys. B144 (1978) 195; C. Korthals-Altes, Nucl. Phys. B142 (1978) 315. [14] A. Hasenfratz and P. Hasenfratz, Phys. Lett. 93B (1980) 165. [15] M. Creutz, Asymptotic freedom scales, Brookhaven preprint (April 1980). [16] G. Miinster, Phys. Lett. 95B (1980) 59. [17] See A. Buras, Rev. Mod. Phys. 52 (1980) 199. [18] E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane and T.M. Yan, Phys. Rev. D21 (1980) 203; P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Nucl. Phys. B56 (1973) 109. [19] S. Elitzur, R. Pearson and J. Shigemitsu, Phys. Rev. D19 (1979) 3698; A. Ukawa, P. Windey and A. Guth, Phys. Rev. D21 (1980) 1013 ; M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. D20 (1979) 1915.