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Gluon condensate from SU(2) lattice gauge theory
Volume 101B, number 3
PHYSICS LETTERS
7 May 1981
GLUON CONDENSATE FROM SU(2) LATTICE GAUGE THEORY J. KRIPFGANZ 1
CERN, Geneva, Switzerland Received 27 January 1981
Recent Monte Carlo results are analyzed in terms o f an operator product expansion for Wilson loops of small size. The value o f (o~s G ~v a G~u) a is f o u n d to be consistent with previous phenomenological estimates.
Monte Carlo techniques [1-5] have opened new ways of studying non-perturbative effects in gauge theories. By putting these theories on a lattice one may calculate ratios of renormalization group invariant quantities (like masses) for small but f'mite lattice spacing a. These dimensionless numbers should easily survive in the continuum limit. So far, most of the discussion concentrated on the string tension o. The mass of the lowest-lying ghieball state has also been studied [5,6]. For SU(2) pure gauge theory the most accurate result for the string tension is [5] : a = (0.011 -+ 0.002)-2)t 2,
(1)
where )tO is the QCD [SU(2)] scale parameter on the lattice Xo = a-1 (6 n2/~)sl/121 exp ( - ~ n2/3),
(2)
fl is defined as fl = 4/g2(a) where go(a) is the bare coupling constant. 2to and therefore a are manifestly renormalization group invariant. In this way confinement and asymptotic freedom are shown to be features of one and the same theory. )to is related to more conventional scale parameter (like XMOM , defined by momentum subtraction in the continuum theory) by [7] xMOM/x0 = 57.5.
(3)
Another quantity of significant interest is the gluon condensate (a s Guy G~v) [strictly speaking, the renormalization group invariant expression is ([fl(g)/g] Guy I Permanent address: Karl-Marx-Universit~t, Leipzig, DDR.
X Guy) ] . Through the trace anomaly [8] it is related to the vacuum energy density and is therefore an important parameter characterizing the QCD vacuum. It also appears as a "higher twist" contribution in various current correlation functions. Exploiting this fact, the 1TEP group [9] estimated a value N -c l < ( a /sr c ) G /.a) a G /~v a >~(O.012/3)GeV 4,
N c = 3 , (4)
from heavy quark spectroscopy. For later use we note that the combination N - 1 c × (asGuvGuv> is not expected to show a sizeable dependence on the number of colour N c [10]. The determination of (a s Guy Guy) through Monte Carlo lattice calculations is an important but not quite trivial task. The obvious first guess would be to use the average action per plaquette since this quantity reduces to Guy Guu a 4 in the classical continuum limit. On the lattice, however, this quantity receives perturbative contributions completely dominating the exponentially (in 3) falling non-perturbative term which is the only one of interest from the point of view of continuum theory. A different approach is therefore required. We show in this paper that a considerable enhancement of the "signal-to-background" ratio can be achieved through the study of larger Wilson loops (large in lattice units). A clear separation of perturbative and non-perturbative components will in fact be possible. The technique we will use is again the operator product expansion. Considering small Wilson loops in the continuum theory we may expand the field strength tensor at the origin of the loop [11 ]. The
0 031-9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company
169
Volume 101B, number 3
PHYSICS LETTERS
minimal area of the loop then appears as a natural expansion parameter, and one obtains a series in terms of vacuum matrix elements of local composite gluon operators. For a rectangular Wilson loop of size ( L , T ) one finds in this way [11] W ( L , T ) "" 1 - 4 r r 2 L 2T2 N-I<(o~ /rr)G a G a ) + ~ c S ,up ,up "/68 X L 2 T 2 ½ ( L 2 + T2"
(5) Higher order terms in this expansion explicitly depend on the shape of the loop. In eq. (5) we have neglected quantum fluctuations. They have also been studied to some extent [ 1 2 - 1 4 ] . Linear divergences are found which can be absorbed into a mass renormalization factor for the test particle [ 12,13 ]. Otherwise logarithmic divergences introduce the usual coupling constant renormalization. For contours with sharp bends (like we are considering) there are also additional logarithmic singularities. One may get rid of those, as well as the linear divergences, by considering the ratio of two rectangular Wilson loops of same perimeter. Such a quantity has indeed been studied in the lattice Monte Carlo approach [5]. It is x(/) - - log[ W(I,1)/w(r + 1,I-
l)].
(6)
For large I = L/a, but not too large L, we may apply the loop expansion (5) to this quantity X(/) ~ Xpert(/) + ~ rr2(212 - 1) (a2o) 2 x
- 10'4
+ 12 - 1)
X (a2cy'13&y-3N-l) + v " ~" c a oefl 13"I "l'ce " ....
(7)
The dimensionless numbers in curly brackets are the ones we wish to estimate. For convenience, we have chosen the string tension o to set the scale. When translating into units of GeV we use o = 1/2rr~',
~ ' = 1 GeV -2.
(8)
On the lattice one can expand the perturbative contribution Xpert(/) in terms of the unrenormalized coupiing Xpert(/) ~ X 1(/)/fl
+ 0(1/(32).
(9)
We should, however, always consider the limit of large I since we are using concepts derived from the 170
7 May 1981
continuum theory. For large I the bare coupling expansion is not very appropriate. This is obvious from the point of view of the continuum limit. In higher orders there will be large logarithms of the type [g2(a) log (L/a)ln,
(I0)
which just transform the bare coupling go(a) into the renormalized coupling gR(L). Expanding in the renormalized coupling therefore already means a partial summation of the bare coupling expansion. Hence we use
Xpert(/)--~X1 (/)//~R(/)
+ O(1//32R)
(1 1)
instead of eq. (9). In the one-loop approximation fiR =- 4/g2R is given by
fiR(/) = fl -- (11/37r2)log(IXR/XO),
(12)
XR depends on the renormalization prescription. Changes in XR are correlated to changes in the nextto4eading order contribution to Xpert- As long as we do not take into account next-order effects explicitly XR is a more or less arbitrary parameter which should be of the order of a typical continuum scale parameter, however [compare eq. (3)]. We have calculated the first order coefficient Xl (/) following a method and using results of ref. [15]. The method consists of applying a saddle point approximation to the corresponding group integrals. Results are shown in table 1. Some care is required in computing ×1 (/) since it is obtained as a small difference between two increasingly large numbers. We now turn to a discussion of the Monte Carlo "data" of ref. [5]. ×(/) has been studied for I up to 5 (compare fig. 1). I = 5 is obviously not a very big number. We might therefore encounter certain problems in applying our formalism which assumes I is large. In order to obtain a reasonable description of
Table l xl(/) and heff/k o f o r / u p to 5. I
×1 (/)
heff/Xo
1 2 3
0.75 0.251 0.100
6 15 30
4 5
0.049 0.029
(38) (42)
Volume 101B, number 3 I0.0
I
PHYSICS LETTERS 1
II
[
~
7 May 1981 I0
I
--
I ~1\\'5\I \~\ ",
• I=I
~
~ i|
!i
x
I= 2
o
I= 3
I
I
I
I
+' :' a I= 5
II I fO
r¢3 n
-0.1
!
0.1(
_
\\
\
\ \
\
\
\ \ \\
O.O
0
I
2
5
4
5
#
Fig. 1. Comparison of the first two terms of the expansion (7) with Monte Carlo data of ref. [5 ]. Eq. (4) is used for (o~sGtauG#u).Also shown is a2o (dashed lines) with upper and lower limits taken from eq. (1). the perturbative background we allow XR to be an effective I dependent parameter ~kR -+ •eff(/), which is obtained b y analyzing the perturbative tail of X (/). Results for Xeff/k0 are shown in table 1. For I = 4 and 5 the data only allow an order of magnitude estimate. Xeff is found to depend significantly on I for small values o f / w i t h this I dependence flattening off for larger I. Here the expected magnitude is reached. Such behaviour is not surprising. It indicates that higher order effects (which Xeff tries to simulate) become less important with increasing I. This is consistent with what we expect from our previous discussion. The curves shown in fig. 1 also include the contribution due to (asGuvG~v). We did not try to fit this quantity but just used the value given in eq. (4). It gives a very good description o f the sudden transition from a power-like (perturbative) to some exponential (non-perturbative) behaviour. In order to see the effects o f the different types o f contributions
O.Ol
175
f ~
2,00
_
I
_
2.25
250 275
I\\
I
3.00 525 3.50
# Fig. 2. Comparison with the high statistics data of ref. [5 ] for I = 3. Two values for (C~sG.vG#v) are used as explained in the text, The dash-dotted ~ne gives the perturbative background alone. The dotted lines are obtained ff the third term in the expansion (7) is included, with eq. (13) for the new matrix element. Again shown is a2o (dashed lines).
more clearly we show them separately in fig. 2 for the case I = 3. The two sets o f curves correspond to different values o f (asGuvGuv>, i.e., we use the phenomenological value given in eq. (4) times ½ and 1, respectively. We also include here the contribution due to the third term in the expansion (5) using
(g 3f abcGodja G b#~GC'rc~) =
0.04 GeV 6 .
(13)
The actual transition to the non-perturbative behaviour is not very sensitive to the value o f this matrix element as long as it does not become larger than about 0.3 GeV 6. A value as given in eq. (13) has been estimated on the basis o f some instanton gas approximation [9]. Such an approximation is, however, not very reliable, and is in fact not expected to give the dominant contribution to m a t r ~ elements like (c~s Guy × Gu~,) [16]. Still, the order o f magnitude is certainly consistent with the lattice results o f fig. 2. To summarize, we have found that the sudden 171
Volume 101B, number 3
PHYSICS LETTERS
transition from the perturbative to some non-perturbative behaviour in the Monte Carlo lattice data of ref. [5] is well described b y the first few terms o f the small loop operator product expansion (5). This allows an estimate (with a subjective error of about 50%) o f the gluon condensate (OLsGuvGuv).Its value is found to be consistent with previous phenome. nological estimates [9] [eq. ( 4 ) ] . One should keep in mind, however, that we considered SU(2) instead of SU(3) and, more importantly, did not include light fermions. The separation o f perturbative and non-perturbatire contributions was possible b y considering larger loops on the lattice. In this way, the transition to the non-perturbative behaviour is clearly separated from the transition to the strong coupling r6gime. This is not the case for the plaquette term. I would like to thank R. Kirschner, J. Ranft, C. Rebbi, E. Shuryak and G. Veneziano for interesting discussions, and the CERN Theory group for its kind hospitality.
172
7 May 1981
References [1 ] K. Wilson, Carg~se Lecture Notes (1979). [2] M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. Lett. 42 (1979) 1390; Phys. Rev. D20 (1979) 1915. [31 M. Creutz, Phys. Rev. Lett. 43 (1979) 553; Phys. Rev. D21 (1980) 2308. [4] C. Rebbi, Phys. Rev. D21 (1980) 3350. [5] G. Bhanot and C. Rebbi, CERN preprint TH. 2979 (1980). [6] B. Berg, DESY preprint 80/82 (1980). [7] A. Hasenfratz and P. Hasenfratz, Phys. Lett. 93B (1980) 165. [8] R.J. Crewther, Phys. Rev. Lett. 28 (1972) 1421 ; M.S. Chanowitz and J. Ellis, Phys. Lett. 40B (1972) 397; J.C. Collins et al., Phys. Rev. DI6 (1977) 438. [9] M. Shifman, A. Vainshtein and V. Zakharov, Nucl. Phys. B147 (1979) 385,448. [10] W. Bardeen and V. Zakharov, Phys. Lett. 91B (1980) 111. [11 ] M.A. Shifman, Nucl. Phys. B173 (1980) 13. [121 A.M. Polyakov, Nucl. Phys. B164 (1980) 171. [13 ] V.S. Dotsenko and S.N. Vergeles, Landau Institute preprint (1980). [14] N.S. Craigie and H. Dorn, Trieste preprint IC/80/167 (1980). [15] V.F. Mfiller and W. Rfihl, Univ. of Kaiserslautern preprints (1980). [16] E.M. Ilgenfritz and M. MueUer-Preussker, Phys. Lett. 99B (1981) 128.
PHYSICS LETTERS
7 May 1981
GLUON CONDENSATE FROM SU(2) LATTICE GAUGE THEORY J. KRIPFGANZ 1
CERN, Geneva, Switzerland Received 27 January 1981
Recent Monte Carlo results are analyzed in terms o f an operator product expansion for Wilson loops of small size. The value o f (o~s G ~v a G~u) a is f o u n d to be consistent with previous phenomenological estimates.
Monte Carlo techniques [1-5] have opened new ways of studying non-perturbative effects in gauge theories. By putting these theories on a lattice one may calculate ratios of renormalization group invariant quantities (like masses) for small but f'mite lattice spacing a. These dimensionless numbers should easily survive in the continuum limit. So far, most of the discussion concentrated on the string tension o. The mass of the lowest-lying ghieball state has also been studied [5,6]. For SU(2) pure gauge theory the most accurate result for the string tension is [5] : a = (0.011 -+ 0.002)-2)t 2,
(1)
where )tO is the QCD [SU(2)] scale parameter on the lattice Xo = a-1 (6 n2/~)sl/121 exp ( - ~ n2/3),
(2)
fl is defined as fl = 4/g2(a) where go(a) is the bare coupling constant. 2to and therefore a are manifestly renormalization group invariant. In this way confinement and asymptotic freedom are shown to be features of one and the same theory. )to is related to more conventional scale parameter (like XMOM , defined by momentum subtraction in the continuum theory) by [7] xMOM/x0 = 57.5.
(3)
Another quantity of significant interest is the gluon condensate (a s Guy G~v) [strictly speaking, the renormalization group invariant expression is ([fl(g)/g] Guy I Permanent address: Karl-Marx-Universit~t, Leipzig, DDR.
X Guy) ] . Through the trace anomaly [8] it is related to the vacuum energy density and is therefore an important parameter characterizing the QCD vacuum. It also appears as a "higher twist" contribution in various current correlation functions. Exploiting this fact, the 1TEP group [9] estimated a value N -c l < ( a /sr c ) G /.a) a G /~v a >~(O.012/3)GeV 4,
N c = 3 , (4)
from heavy quark spectroscopy. For later use we note that the combination N - 1 c × (asGuvGuv> is not expected to show a sizeable dependence on the number of colour N c [10]. The determination of (a s Guy Guy) through Monte Carlo lattice calculations is an important but not quite trivial task. The obvious first guess would be to use the average action per plaquette since this quantity reduces to Guy Guu a 4 in the classical continuum limit. On the lattice, however, this quantity receives perturbative contributions completely dominating the exponentially (in 3) falling non-perturbative term which is the only one of interest from the point of view of continuum theory. A different approach is therefore required. We show in this paper that a considerable enhancement of the "signal-to-background" ratio can be achieved through the study of larger Wilson loops (large in lattice units). A clear separation of perturbative and non-perturbative components will in fact be possible. The technique we will use is again the operator product expansion. Considering small Wilson loops in the continuum theory we may expand the field strength tensor at the origin of the loop [11 ]. The
0 031-9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company
169
Volume 101B, number 3
PHYSICS LETTERS
minimal area of the loop then appears as a natural expansion parameter, and one obtains a series in terms of vacuum matrix elements of local composite gluon operators. For a rectangular Wilson loop of size ( L , T ) one finds in this way [11] W ( L , T ) "" 1 - 4 r r 2 L 2T2 N-I<(o~ /rr)G a G a ) + ~ c S ,up ,up "/68 X L 2 T 2 ½ ( L 2 + T2"
(5) Higher order terms in this expansion explicitly depend on the shape of the loop. In eq. (5) we have neglected quantum fluctuations. They have also been studied to some extent [ 1 2 - 1 4 ] . Linear divergences are found which can be absorbed into a mass renormalization factor for the test particle [ 12,13 ]. Otherwise logarithmic divergences introduce the usual coupling constant renormalization. For contours with sharp bends (like we are considering) there are also additional logarithmic singularities. One may get rid of those, as well as the linear divergences, by considering the ratio of two rectangular Wilson loops of same perimeter. Such a quantity has indeed been studied in the lattice Monte Carlo approach [5]. It is x(/) - - log[ W(I,1)/w(r + 1,I-
l)].
(6)
For large I = L/a, but not too large L, we may apply the loop expansion (5) to this quantity X(/) ~ Xpert(/) + ~ rr2(212 - 1) (a2o) 2 x
- 10'4
+ 12 - 1)
X (a2cy'13&y-3N-l
(7)
The dimensionless numbers in curly brackets are the ones we wish to estimate. For convenience, we have chosen the string tension o to set the scale. When translating into units of GeV we use o = 1/2rr~',
~ ' = 1 GeV -2.
(8)
On the lattice one can expand the perturbative contribution Xpert(/) in terms of the unrenormalized coupiing Xpert(/) ~ X 1(/)/fl
+ 0(1/(32).
(9)
We should, however, always consider the limit of large I since we are using concepts derived from the 170
7 May 1981
continuum theory. For large I the bare coupling expansion is not very appropriate. This is obvious from the point of view of the continuum limit. In higher orders there will be large logarithms of the type [g2(a) log (L/a)ln,
(I0)
which just transform the bare coupling go(a) into the renormalized coupling gR(L). Expanding in the renormalized coupling therefore already means a partial summation of the bare coupling expansion. Hence we use
Xpert(/)--~X1 (/)//~R(/)
+ O(1//32R)
(1 1)
instead of eq. (9). In the one-loop approximation fiR =- 4/g2R is given by
fiR(/) = fl -- (11/37r2)log(IXR/XO),
(12)
XR depends on the renormalization prescription. Changes in XR are correlated to changes in the nextto4eading order contribution to Xpert- As long as we do not take into account next-order effects explicitly XR is a more or less arbitrary parameter which should be of the order of a typical continuum scale parameter, however [compare eq. (3)]. We have calculated the first order coefficient Xl (/) following a method and using results of ref. [15]. The method consists of applying a saddle point approximation to the corresponding group integrals. Results are shown in table 1. Some care is required in computing ×1 (/) since it is obtained as a small difference between two increasingly large numbers. We now turn to a discussion of the Monte Carlo "data" of ref. [5]. ×(/) has been studied for I up to 5 (compare fig. 1). I = 5 is obviously not a very big number. We might therefore encounter certain problems in applying our formalism which assumes I is large. In order to obtain a reasonable description of
Table l xl(/) and heff/k o f o r / u p to 5. I
×1 (/)
heff/Xo
1 2 3
0.75 0.251 0.100
6 15 30
4 5
0.049 0.029
(38) (42)
Volume 101B, number 3 I0.0
I
PHYSICS LETTERS 1
II
[
~
7 May 1981 I0
I
--
I ~1\\'5\I \~\ ",
• I=I
~
~ i|
!i
x
I= 2
o
I= 3
I
I
I
I
+' :' a I= 5
II I fO
r¢3 n
-0.1
!
0.1(
_
\\
\
\ \
\
\
\ \ \\
O.O
0
I
2
5
4
5
#
Fig. 1. Comparison of the first two terms of the expansion (7) with Monte Carlo data of ref. [5 ]. Eq. (4) is used for (o~sGtauG#u).Also shown is a2o (dashed lines) with upper and lower limits taken from eq. (1). the perturbative background we allow XR to be an effective I dependent parameter ~kR -+ •eff(/), which is obtained b y analyzing the perturbative tail of X (/). Results for Xeff/k0 are shown in table 1. For I = 4 and 5 the data only allow an order of magnitude estimate. Xeff is found to depend significantly on I for small values o f / w i t h this I dependence flattening off for larger I. Here the expected magnitude is reached. Such behaviour is not surprising. It indicates that higher order effects (which Xeff tries to simulate) become less important with increasing I. This is consistent with what we expect from our previous discussion. The curves shown in fig. 1 also include the contribution due to (asGuvG~v). We did not try to fit this quantity but just used the value given in eq. (4). It gives a very good description o f the sudden transition from a power-like (perturbative) to some exponential (non-perturbative) behaviour. In order to see the effects o f the different types o f contributions
O.Ol
175
f ~
2,00
_
I
_
2.25
250 275
I\\
I
3.00 525 3.50
# Fig. 2. Comparison with the high statistics data of ref. [5 ] for I = 3. Two values for (C~sG.vG#v) are used as explained in the text, The dash-dotted ~ne gives the perturbative background alone. The dotted lines are obtained ff the third term in the expansion (7) is included, with eq. (13) for the new matrix element. Again shown is a2o (dashed lines).
more clearly we show them separately in fig. 2 for the case I = 3. The two sets o f curves correspond to different values o f (asGuvGuv>, i.e., we use the phenomenological value given in eq. (4) times ½ and 1, respectively. We also include here the contribution due to the third term in the expansion (5) using
(g 3f abcGodja G b#~GC'rc~) =
0.04 GeV 6 .
(13)
The actual transition to the non-perturbative behaviour is not very sensitive to the value o f this matrix element as long as it does not become larger than about 0.3 GeV 6. A value as given in eq. (13) has been estimated on the basis o f some instanton gas approximation [9]. Such an approximation is, however, not very reliable, and is in fact not expected to give the dominant contribution to m a t r ~ elements like (c~s Guy × Gu~,) [16]. Still, the order o f magnitude is certainly consistent with the lattice results o f fig. 2. To summarize, we have found that the sudden 171
Volume 101B, number 3
PHYSICS LETTERS
transition from the perturbative to some non-perturbative behaviour in the Monte Carlo lattice data of ref. [5] is well described b y the first few terms o f the small loop operator product expansion (5). This allows an estimate (with a subjective error of about 50%) o f the gluon condensate (OLsGuvGuv).Its value is found to be consistent with previous phenome. nological estimates [9] [eq. ( 4 ) ] . One should keep in mind, however, that we considered SU(2) instead of SU(3) and, more importantly, did not include light fermions. The separation o f perturbative and non-perturbatire contributions was possible b y considering larger loops on the lattice. In this way, the transition to the non-perturbative behaviour is clearly separated from the transition to the strong coupling r6gime. This is not the case for the plaquette term. I would like to thank R. Kirschner, J. Ranft, C. Rebbi, E. Shuryak and G. Veneziano for interesting discussions, and the CERN Theory group for its kind hospitality.
172
7 May 1981
References [1 ] K. Wilson, Carg~se Lecture Notes (1979). [2] M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. Lett. 42 (1979) 1390; Phys. Rev. D20 (1979) 1915. [31 M. Creutz, Phys. Rev. Lett. 43 (1979) 553; Phys. Rev. D21 (1980) 2308. [4] C. Rebbi, Phys. Rev. D21 (1980) 3350. [5] G. Bhanot and C. Rebbi, CERN preprint TH. 2979 (1980). [6] B. Berg, DESY preprint 80/82 (1980). [7] A. Hasenfratz and P. Hasenfratz, Phys. Lett. 93B (1980) 165. [8] R.J. Crewther, Phys. Rev. Lett. 28 (1972) 1421 ; M.S. Chanowitz and J. Ellis, Phys. Lett. 40B (1972) 397; J.C. Collins et al., Phys. Rev. DI6 (1977) 438. [9] M. Shifman, A. Vainshtein and V. Zakharov, Nucl. Phys. B147 (1979) 385,448. [10] W. Bardeen and V. Zakharov, Phys. Lett. 91B (1980) 111. [11 ] M.A. Shifman, Nucl. Phys. B173 (1980) 13. [121 A.M. Polyakov, Nucl. Phys. B164 (1980) 171. [13 ] V.S. Dotsenko and S.N. Vergeles, Landau Institute preprint (1980). [14] N.S. Craigie and H. Dorn, Trieste preprint IC/80/167 (1980). [15] V.F. Mfiller and W. Rfihl, Univ. of Kaiserslautern preprints (1980). [16] E.M. Ilgenfritz and M. MueUer-Preussker, Phys. Lett. 99B (1981) 128.