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Scaling in lattice QCD: Glueball masses and string tension
Volume 205, number 4
PHYSICS LETTERS B
5 May 1988
SCALING IN LATTICE QCD: GLUEBALL MASSES AND S T R I N G T E N S I O N Ape Collaboration P. BACILIERI, L. FONTI, E. R E M I D D I l INFN-CNAF, 1-40126 Bologna, Italy
M. BERNASCHI, S. CABASINO, N. CABIBBO, L.A. FERNANDEZ 2,3,4, E. MARINARI, P. PAOLUCCI 5, G. PARISI, G. SALINA INFN, Sezione di Roma, gruppo collegato di Roma II, and Universit~ di Roma 11, "Tor Vergata'; 1-00173 Rome, Italy
G. F I O R E N T I N I 5,6, S. GALEOTTI, M.P. LOMBARDO, D. PASSUELLO, R. T R I P I C C I O N E INFN, Sezione di Pisa, 1-56100 Pisa, Italy
P. MARCHESINI CERN, CH-1211 Geneva 23, Switzerland
F. MARZANO, F. RAPUANO INFN, Sezione di Roma, and Universitil di Roma I, "'La Sapienza", 1-00185 Rome, Italy
and R. RUSACK Rockefeller University, New York, N Y 10021, USA Received 27 January 1988
We study the scaling behaviour of lattice quantum chromodynamics by comparing the fl dependence of the string tension and the 0 ÷ + glueball mass. We use a source method at/~= 5.7, fl= 5.9 and fl= 6.1, on lattices from 93-24 to 163. 32. Assuming a string tension of about (420 MeV) 2, the lattice spacing ranges from 0.16 to 0.08 fm. In order to separate finite volume from scaling violation effects we have compared data from lattices having approximately the same overall physical size at the different values of ft. We find deviations from scaling to be very small.
The study of the continuum limit in lattice quantum chromodynamics is a crucial point to understand the degree of reliability of the calculations we Also at Universith di Bologna, 1-40126 Bologna, Italy. 2 MEC (Spain) fellow. 3 On leave from Universidad Complutense de Madrid, E-28040 Madrid, Spain. 4 Partially supported by CAICYT (AE86-0029). 5 INFN-Digital Equipment Corporation fellow. 6 Also at Universith di Cagliari, 1-09100 Cagliari, Italy.
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
carry out at a finite fl on a finite lattice. A lot of effort has been devoted to this subject, but to carry out a precise calculation is not an easy task. In principle one has to compute the value of some physical quantity at increasing values offl, which correspond to a decreasing physical size of the lattice cell. It is important to minimize the effects which arise from the finite size of the lattice. This can be obtained if the decrease in the size of the lattice cell is accompanied by an increase in the number of points 535
Volume 205, number 4
PHYSICS LETTERS B
in such a way that the overall size of the lattice is kept constant. To reduce the cell size by a factor two the number of points must thus be increased by a factor eight (assuming that the number of points in the time direction is kept fixed and "very large"). In practice the exact physical value of the cell size is only known a posteriori after one has measured the value of some dimensional quantity, so that the above requirement can only be met with a measure of approximation. It is then important to verify that effects due to the finite overall size of the lattice are small. This can be done comparing the simulation for lattices with a different number of points, but with the same cell size, i.e. with the same value of ft. After the work described in refs. [ 1-3 ] we decided to use 2000 h of our 256 Mflops Ape computers (see ref. [4] ) in order to investigate this problem. The results presented here have been obtained by simulating pure gauge theory at three values of//on different lattices: //=5.7,
93.24,
//=5.9,
103.32, 123.32, 163.32,
//=6.1 ,
163.32.
5 May 1988
Table 1 m ( 0 ++) and ~for fl=5.9 from the 103.32, 123.32, and 163.32 lattices. All values are computed by determining the crossing points. The data for the 103.32 and 123.32 lattices are from the configurations obtained in ref. [ 1 ] and for 163.32 we have used both the data from ref. [ 1 ] and our new data. L
m(O++)a
aa 2
10 12 16
0.65(3) 0.75(4) 0.76(3)
0.047(3) 0.056(2) 0.0535(10)
Table 2 m(0 ++ ) and aforfl= 5.7, 5.9 and 6.1 on 93.24, 123.32 and 163.32 lattices. The physical units have been obtained by fixing the string tension to (420 MeV) 2.
L = 9 , ]3=5.7 L = 12,]3=5.9 L = 16,fl=6.1
m(O÷+)a
aa 2
m(0 ++) (MeV)
0.84(2) 0.75(4) 0.54(4)
0.1221(18) 0.056(2) 0.0274(8)
1200(30) 1330(70) 1370(100)
3ool-
200
The three lattices at fl=5.9 (103.32, 123.32 and 163. 32) are used to study finite volume effects. The possible existence of such effects at intermediate// values on fairly large lattices was left open after refs. [ 1-3], and happily enough we can now claim that according to expectations these effects are very small for lattices larger than 1.2 fm. The three lattices (93.24,//=5.7, 123.32,//=5.9 and 163.32,//=6.1) have, with good approximation, the same physical size so that the results they yield are relevant for an investigation of the scaling problem. In this paper we use the inverse of the square root of the string tension as the fundamental physical unit; when we quote more familiar units such as the fm, we assume a string tension of (420 MeV) 2. Before entering into the details of the computation, let us discuss the main results of this letter, which are summarized in tables 1 and 2, where we show the results for the 0 + + mass and for the string tension a (see ref. [ 5 ] for a recent study on the same subject). We give respectively in figs. 1 and 2 the plot of the 536
lOO
o
I
I
I
I
I
5.7
5.8
5.9
6
6.1
Fig 1. Scaling plot for m ( 0 + + ) (solid line ) and ~ in units of the two loops AL.
(dashed line )
0 + + mass, of a (in units OfAL computed at two loops) versus fl and of the ratio versus the string tension in lattice units ~
m(O++)/g/~
~t We recall that the value of A L computed at two loops [ALa= (8n2fl/33) 5~/~2~ exp(--4n/]3/33), a being the lattice spacing] differs from the exact one by terms which are of order 1/]3, and in our range of]3 such terms cannot normally be neglected (the 1/J3 expansion is well known to be poorly convergent when ]3/=6). Alternatively we could say that the experimentally measured AMom is equal to 83.4AL+ O( 1/]3).
Volume 205, number 4 -
'
'
'
'
PHYSICS LETTERS B I
. . . .
I
. . . .
I
'
'
'
I
. . . .
' ~
v
. . . .
0.05
0. I
] 0 15
Qa8
Fig. 2. The dimensionless ratio m(0 +÷)/x/~ as a function of o-a 2"
As far as a is concerned there is a clear deviation, of the order of 10%, from the two-loop prediction. This deviation appears to be a real effect, since it is much larger than the error in the measurement o f the string tension, which is about 2%. For the glueball mass we do not see such an effect. If any, we see the opposite trend when going from r = 5.7 to 5.9. The failure o f the high fl expansion is likely related with the presence o f a singularity (either in the complex fl plane or in the plane fllhnd .... tal--fladjoint) that lies close to the point at fl~ 5.6; there the glueball mass goes to zero, and eventually it has rise again for higher fl values. The crucial test o f scaling is the fl independence o f the mass ratios. In this case we expect that a generic mass ratio R, as function of the lattice spacing a should be given (neglecting logs) by
R ( a ) = R ( 0 ) + R2a2 + O ( a 4) ,
( 1)
when the lattice spacing is sufficiently small. In other words a mass ratio plotted as function o f aa 2 should extrapolate to the physical value at zero and should be an approximately linear function of aa2 near o ' a 2 - 0 . The mass ratio shown in fig. 2 (mass of the spin 0 glueball over the square root o f the string tension) is weakly dependent on tra 2 near zero. The coefficient o f the term proportional to tra 2 seems to be small, but we cannot determine it precisely due to the error ,2. In conclusion our present data strongly ~2 The dominant contribution to the error is given by the error in the glueball mass.
5 May 1988
suggest that starting from r = 5.9 we are in the region where eq. ( 1 ) safely applies. To improve the situation one would have to improve substantially the statistics in the region fl=5.9-6.1, and extend the measurement towards higher ft. In our simulations we used the cold wall method, as described in ref, [ 1 ], where one measures the expectation value o f suitable operators as a function o f the distance in time from a spacelike hyperplane at t = 0 where all the x and y links have been set to the unit matrix. Our experience indicates that this method is more performing on larger lattices than the one based on the study o f loop-loop correlation functions. L o o p - l o o p correlations seem to have a clear advantage since they can be used to construct mass estimators which give upper bounds to the true asymptotic mass on the lattice. Unfortunately an upper bound is not enough: in a typical calculation the signal disappears after 3-4 time distances, and it is very difficult to be sure that the upper bound one obtains is a good one [2,3]. A very useful tool to approach the continuum limit is the smearing procedure introduced in ref. [ 1 ]. This procedure allows the construction of families of progressively larger operators of given q u a n t u m numbers. Each family descends from an simple operator (e.g. the plaquette in the case of the 0 ÷ + state), and the elements of the family vary smoothly as a function o f the smearing number. The continuous nature o f the smearing procedure (see ref. [ 1 ] for a discussion of this point) is crucial in that we get a smooth behavior as a function o f the number of smearing steps. In order to extract the value of the mass we have use various stategies in order to check the consistency of the results. We study the dependence of the relevant logarithms o f ratios of expectation values,
rhs(t) = l o g [ F ' ( t ) / F ' ( t +
1) ] ,
(2)
as a function o f the smearing n u m b e r s. The plots of these functions (see fig. 3, where we show the time and smearing dependent mass: the upper side refers to the spin 0 glueball mass, and the lower side refers to the string tension as measured from the Polyakov loops) show a very interesting behavior which can be interpreted in a simple model where the time dependence is ascribed to a combination o f two states o f mass m~ and rn2: 537
Volume 205, number 4
PHYSICS LETTERS B
0.8 0.6 . . . . .~
' ~' l,'
I . . . .~ . .
0.4 0.6 0.5 0.4 ~3 2 0.3
~
0
,i ,
"
,
,
L
l
0.05
,
0.1
(Smearing)
,
, 0.15
0.2
-t
Fig. 3. Thc effective mass rh(0 ++ ) (upper sidc) and aL (lower side) versus (smearing number) - ~at r = 6.1. The different curves correspond to different euclidean times.
F~(t)=a~ exp(-m~t)+a~ e x p ( - m 2 t ) ,
(3)
r h ' ( t ) = l o g ( [a] exp( - m~ t) + a ~ e x p ( - m 2 t ) ] × { a ] e x p [ - m , ( t + 1)] + a S exp[ - m 2 ( t + 1 ) 1} - ' ) .
(4)
In this model a decrease o f r ~ ( t ) as a function o f t indicates that the coefficients a~ and aS have the same sign, while an increase indicates opposite signs. Furthermore, at a value o f s where rhS(t) = the(t+ 1 ), a crossing point, one of the two coefficients, say aS, vanishes, and r~ s (t) = m i. Looking at our fig. 3 we see that the curves for different t cross, but the crossings of different curve pairs, although close, do not coincide. This indicates that the simple model is too crude, and that higher mass states are relevant. We can improve the situation by considering not the individual crossing between curve pairs, but the sequence o f crossings between curves of contiguous time as a function o f increasing time, a direction in which the effect of higher mass states should vanish exponentially. Unfortunately with increasing time the error also increases, so that we have to choose an optimal point where to stop. We have chosen to evaluate the masses using in each ease a crossing point such that, if we use crossing points at higher euclidean times, we obtain a compatible value with a larger error. We are confident, 538
5 May 1988
from our analysis (that can be partially repeated by looking at the figures of this paper), that the systematic error due to the choice o f the time distances is smaller or in the worst case of the same order of the statistical error. In order to learn more about the finite size effects, we have improved the measures of ref. [1] on a 163.32 lattice at fl=5.9. Because o f the anomalous behavior on the 103 lattice (on which finite size errors are quite sizeable [ 1 ] ) we wanted to be sure that the 123 lattice and the 163 one give the same results. We have carded out 1500 measures by using a pseudo heat-bath algorithm, and measuring every 10 sweeps. To get a better resolution of the crossing points we have repeated the smearing procedure 40 times (instead o f the 20 times ofref. [ 1 ] ). The results (see table 1 ) show that the possible finite size effects are of the order of magnitude of our (small) statistical error. The glueball mass (that was suffering a large finite size effect on the 103 lattice) has now a value that is constant within our statistical error. For the string tension we only remark that, since we cannot see any finite size effect, we are not able on our larger lattices to either confirm or exclude the n/3 effect [i.e. a(L) = a~o-n/3L2]. The values of the string tension reported in the tables are not corrected for this effect; because of our choice of lattice sizes the correction would, however, be approximatively constant (in percentage), so that our conclusions on the scaling would not be affected by it. Finite size effects at p = 5.9 are sizeable when going from L = 10 to L = 12 lattices, but the 123.32 lattice seems already asymptotic enough for our goals. We have performed a high statistics calculation at r = 6.1 on a 163. 32 lattice. For updating the gauge fields we have mainly used a pseudo heat-bath algorithm. We have carried out 4000 measurements (separated in part by 20 sweeps and in part by 5 sweeps). We have also done some simulations on a 93.23 lattice at r = 5.7. In this case we have carried out 1500 measurements, separated by 20 sweeps of a 5 hits Metropolis algorithm, as well as 1500 steps measuring every sweep (that were also used in order to check the efficiency of different updating methods, see ref. [ 6] ). The results we report in this paper have been always obtained looking at the crossing points according to the procedure defined above. This is important
Volume 205, number 4
PHYSICS LETTERS B
in o r d e r to a v o i d the inclusion o f different systematic errors due to different measuring procedures. Nevertheless, to check the consistency o f our results we have also carried out global fits, with results that are very similar ( a n d clearly consistent within our statistical errors) to the ones we reported. F o r the f l = 5.9 case see ref. [ 1 ]. As far as p = 6.1 is concerned, with a one mass fit we have to discard the times from t = 1 to t = 4 in o r d e r to get a quasi-constant b e h a v i o r (at about the 20% level) in the smearing n u m b e r (see fig. 4, u p p e r side). But the error is larger than with the p r o c e d u r e based on crossing points. A two mass fit with u n c o n s t r a i n e d masses is unstable, but two mass fits with a fixed mass ratio (see fig. 4, lower side) work better, i f we assume (see ref. [ 3 ] ) that the second mass is close to twice the first one, we o b t a i n a ver good fit i f we discard two time points. We do rem a r k that the fit would surely not work with a mass ratio of, let us say, 3 or 1.5. As far as the 2 + + state is concerned, the situation is by far m o r e complicated, so that we are not able to
0.8
0.6
0.4
$ o
5 May 1988
d r a w any conclusion as to the scaling b e h a v i o u r o f this state. At f l = 5.7 indeed is in fact simple, since we do not get any useful signal. The best we can do here is looking at the ratio o f the signals for t = 1 and t = 2 , r~S( 1 ). This curve corresponds to very high effective masses, a result which cannot be accepted lacking the corrob o r a t i o n o f either crossing or signals at higher t. The smearing is very useful in rejecting this result as spurious, since it is strongly d e p e n d e n t on the smearing number. At p = 5.9 on the 163.32 lattice we can look at the crossing between ~ s ( 2 ) and nSs(3), which gives m (2 ÷ ÷ ) = 0.8 3 ( 12 ) (the 123.32 yields a c o m p a t i b l e result with a larger error). We r e m a r k that, if we were using the rh s (2) / rh ~( 3 ) crossing to estimate m (0 ÷ ÷ ) we would get a result which is 5% lower than the one we get from ffzs(2)/ffzs(4) (the one we used). I f we assume that we have the same b e h a v i o r for the spin 2 (this is not granted but not unreasonable) we would conclude that m ( 2 + + ) / m ( 0 ÷ + ) = 1.15(25) at f l = 5.9. The f l = 6.1 case is the most difficult. We have a signature with a large error for a state with a mass ranging in the interval from 0.5 to 1.0 (with a spin 2 over spin 0 ratio ranging between one a n d t w o ) . It is possible that the p o o r performance o f our m e t h o d s for the 2 + + state is in part due to our choice o f the t = 0 cold wall. We will a t t e m p t to i m p r o v e the situation by experimenting with different prescriptions.
o.8
0.6
References
0.4
,
I 0.05
,
, 0.1 (Smearing) -t
,
I 0.15
0.2
Fig. 4. Upper side: m (0 ÷ +) from a one mass fit at fl= 6.1, discarding two (solid) three (dashed) and four (dotted) euclidean times. Lower side: m ( O ++ ) from a two mass fit at fl=6.1, with the mass ratio fixed to two, discarding one (solid) and two (dashed) times.
[ 1] Ape Collab., M. Albanese et al., Phys. Lett. B 192 (1987) 163. [21 Ape Collab., M. Albanese et al., Phys. Lett. B 197 (1987) 400. [3] L.A. Fernandez and E. Marinari, Nucl. Phys. B, to be published. [ 4 ] Ape Collab., M. Albanese et al., Comput. Phys. Commun. 45 (1987) 345. [5] C. Michael and M. Teper, Oxford preprint 72/87 (1987). [ 6 ] M, Bernaschi and L.A. Fernandez, to be published.
539
PHYSICS LETTERS B
5 May 1988
SCALING IN LATTICE QCD: GLUEBALL MASSES AND S T R I N G T E N S I O N Ape Collaboration P. BACILIERI, L. FONTI, E. R E M I D D I l INFN-CNAF, 1-40126 Bologna, Italy
M. BERNASCHI, S. CABASINO, N. CABIBBO, L.A. FERNANDEZ 2,3,4, E. MARINARI, P. PAOLUCCI 5, G. PARISI, G. SALINA INFN, Sezione di Roma, gruppo collegato di Roma II, and Universit~ di Roma 11, "Tor Vergata'; 1-00173 Rome, Italy
G. F I O R E N T I N I 5,6, S. GALEOTTI, M.P. LOMBARDO, D. PASSUELLO, R. T R I P I C C I O N E INFN, Sezione di Pisa, 1-56100 Pisa, Italy
P. MARCHESINI CERN, CH-1211 Geneva 23, Switzerland
F. MARZANO, F. RAPUANO INFN, Sezione di Roma, and Universitil di Roma I, "'La Sapienza", 1-00185 Rome, Italy
and R. RUSACK Rockefeller University, New York, N Y 10021, USA Received 27 January 1988
We study the scaling behaviour of lattice quantum chromodynamics by comparing the fl dependence of the string tension and the 0 ÷ + glueball mass. We use a source method at/~= 5.7, fl= 5.9 and fl= 6.1, on lattices from 93-24 to 163. 32. Assuming a string tension of about (420 MeV) 2, the lattice spacing ranges from 0.16 to 0.08 fm. In order to separate finite volume from scaling violation effects we have compared data from lattices having approximately the same overall physical size at the different values of ft. We find deviations from scaling to be very small.
The study of the continuum limit in lattice quantum chromodynamics is a crucial point to understand the degree of reliability of the calculations we Also at Universith di Bologna, 1-40126 Bologna, Italy. 2 MEC (Spain) fellow. 3 On leave from Universidad Complutense de Madrid, E-28040 Madrid, Spain. 4 Partially supported by CAICYT (AE86-0029). 5 INFN-Digital Equipment Corporation fellow. 6 Also at Universith di Cagliari, 1-09100 Cagliari, Italy.
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
carry out at a finite fl on a finite lattice. A lot of effort has been devoted to this subject, but to carry out a precise calculation is not an easy task. In principle one has to compute the value of some physical quantity at increasing values offl, which correspond to a decreasing physical size of the lattice cell. It is important to minimize the effects which arise from the finite size of the lattice. This can be obtained if the decrease in the size of the lattice cell is accompanied by an increase in the number of points 535
Volume 205, number 4
PHYSICS LETTERS B
in such a way that the overall size of the lattice is kept constant. To reduce the cell size by a factor two the number of points must thus be increased by a factor eight (assuming that the number of points in the time direction is kept fixed and "very large"). In practice the exact physical value of the cell size is only known a posteriori after one has measured the value of some dimensional quantity, so that the above requirement can only be met with a measure of approximation. It is then important to verify that effects due to the finite overall size of the lattice are small. This can be done comparing the simulation for lattices with a different number of points, but with the same cell size, i.e. with the same value of ft. After the work described in refs. [ 1-3 ] we decided to use 2000 h of our 256 Mflops Ape computers (see ref. [4] ) in order to investigate this problem. The results presented here have been obtained by simulating pure gauge theory at three values of//on different lattices: //=5.7,
93.24,
//=5.9,
103.32, 123.32, 163.32,
//=6.1 ,
163.32.
5 May 1988
Table 1 m ( 0 ++) and ~for fl=5.9 from the 103.32, 123.32, and 163.32 lattices. All values are computed by determining the crossing points. The data for the 103.32 and 123.32 lattices are from the configurations obtained in ref. [ 1 ] and for 163.32 we have used both the data from ref. [ 1 ] and our new data. L
m(O++)a
aa 2
10 12 16
0.65(3) 0.75(4) 0.76(3)
0.047(3) 0.056(2) 0.0535(10)
Table 2 m(0 ++ ) and aforfl= 5.7, 5.9 and 6.1 on 93.24, 123.32 and 163.32 lattices. The physical units have been obtained by fixing the string tension to (420 MeV) 2.
L = 9 , ]3=5.7 L = 12,]3=5.9 L = 16,fl=6.1
m(O÷+)a
aa 2
m(0 ++) (MeV)
0.84(2) 0.75(4) 0.54(4)
0.1221(18) 0.056(2) 0.0274(8)
1200(30) 1330(70) 1370(100)
3ool-
200
The three lattices at fl=5.9 (103.32, 123.32 and 163. 32) are used to study finite volume effects. The possible existence of such effects at intermediate// values on fairly large lattices was left open after refs. [ 1-3], and happily enough we can now claim that according to expectations these effects are very small for lattices larger than 1.2 fm. The three lattices (93.24,//=5.7, 123.32,//=5.9 and 163.32,//=6.1) have, with good approximation, the same physical size so that the results they yield are relevant for an investigation of the scaling problem. In this paper we use the inverse of the square root of the string tension as the fundamental physical unit; when we quote more familiar units such as the fm, we assume a string tension of (420 MeV) 2. Before entering into the details of the computation, let us discuss the main results of this letter, which are summarized in tables 1 and 2, where we show the results for the 0 + + mass and for the string tension a (see ref. [ 5 ] for a recent study on the same subject). We give respectively in figs. 1 and 2 the plot of the 536
lOO
o
I
I
I
I
I
5.7
5.8
5.9
6
6.1
Fig 1. Scaling plot for m ( 0 + + ) (solid line ) and ~ in units of the two loops AL.
(dashed line )
0 + + mass, of a (in units OfAL computed at two loops) versus fl and of the ratio versus the string tension in lattice units ~
m(O++)/g/~
~t We recall that the value of A L computed at two loops [ALa= (8n2fl/33) 5~/~2~ exp(--4n/]3/33), a being the lattice spacing] differs from the exact one by terms which are of order 1/]3, and in our range of]3 such terms cannot normally be neglected (the 1/J3 expansion is well known to be poorly convergent when ]3/=6). Alternatively we could say that the experimentally measured AMom is equal to 83.4AL+ O( 1/]3).
Volume 205, number 4 -
'
'
'
'
PHYSICS LETTERS B I
. . . .
I
. . . .
I
'
'
'
I
. . . .
' ~
v
. . . .
0.05
0. I
] 0 15
Qa8
Fig. 2. The dimensionless ratio m(0 +÷)/x/~ as a function of o-a 2"
As far as a is concerned there is a clear deviation, of the order of 10%, from the two-loop prediction. This deviation appears to be a real effect, since it is much larger than the error in the measurement o f the string tension, which is about 2%. For the glueball mass we do not see such an effect. If any, we see the opposite trend when going from r = 5.7 to 5.9. The failure o f the high fl expansion is likely related with the presence o f a singularity (either in the complex fl plane or in the plane fllhnd .... tal--fladjoint) that lies close to the point at fl~ 5.6; there the glueball mass goes to zero, and eventually it has rise again for higher fl values. The crucial test o f scaling is the fl independence o f the mass ratios. In this case we expect that a generic mass ratio R, as function of the lattice spacing a should be given (neglecting logs) by
R ( a ) = R ( 0 ) + R2a2 + O ( a 4) ,
( 1)
when the lattice spacing is sufficiently small. In other words a mass ratio plotted as function o f aa 2 should extrapolate to the physical value at zero and should be an approximately linear function of aa2 near o ' a 2 - 0 . The mass ratio shown in fig. 2 (mass of the spin 0 glueball over the square root o f the string tension) is weakly dependent on tra 2 near zero. The coefficient o f the term proportional to tra 2 seems to be small, but we cannot determine it precisely due to the error ,2. In conclusion our present data strongly ~2 The dominant contribution to the error is given by the error in the glueball mass.
5 May 1988
suggest that starting from r = 5.9 we are in the region where eq. ( 1 ) safely applies. To improve the situation one would have to improve substantially the statistics in the region fl=5.9-6.1, and extend the measurement towards higher ft. In our simulations we used the cold wall method, as described in ref, [ 1 ], where one measures the expectation value o f suitable operators as a function o f the distance in time from a spacelike hyperplane at t = 0 where all the x and y links have been set to the unit matrix. Our experience indicates that this method is more performing on larger lattices than the one based on the study o f loop-loop correlation functions. L o o p - l o o p correlations seem to have a clear advantage since they can be used to construct mass estimators which give upper bounds to the true asymptotic mass on the lattice. Unfortunately an upper bound is not enough: in a typical calculation the signal disappears after 3-4 time distances, and it is very difficult to be sure that the upper bound one obtains is a good one [2,3]. A very useful tool to approach the continuum limit is the smearing procedure introduced in ref. [ 1 ]. This procedure allows the construction of families of progressively larger operators of given q u a n t u m numbers. Each family descends from an simple operator (e.g. the plaquette in the case of the 0 ÷ + state), and the elements of the family vary smoothly as a function o f the smearing number. The continuous nature o f the smearing procedure (see ref. [ 1 ] for a discussion of this point) is crucial in that we get a smooth behavior as a function o f the number of smearing steps. In order to extract the value of the mass we have use various stategies in order to check the consistency of the results. We study the dependence of the relevant logarithms o f ratios of expectation values,
rhs(t) = l o g [ F ' ( t ) / F ' ( t +
1) ] ,
(2)
as a function o f the smearing n u m b e r s. The plots of these functions (see fig. 3, where we show the time and smearing dependent mass: the upper side refers to the spin 0 glueball mass, and the lower side refers to the string tension as measured from the Polyakov loops) show a very interesting behavior which can be interpreted in a simple model where the time dependence is ascribed to a combination o f two states o f mass m~ and rn2: 537
Volume 205, number 4
PHYSICS LETTERS B
0.8 0.6 . . . . .~
' ~' l,'
I . . . .~ . .
0.4 0.6 0.5 0.4 ~3 2 0.3
~
0
,i ,
"
,
,
L
l
0.05
,
0.1
(Smearing)
,
, 0.15
0.2
-t
Fig. 3. Thc effective mass rh(0 ++ ) (upper sidc) and aL (lower side) versus (smearing number) - ~at r = 6.1. The different curves correspond to different euclidean times.
F~(t)=a~ exp(-m~t)+a~ e x p ( - m 2 t ) ,
(3)
r h ' ( t ) = l o g ( [a] exp( - m~ t) + a ~ e x p ( - m 2 t ) ] × { a ] e x p [ - m , ( t + 1)] + a S exp[ - m 2 ( t + 1 ) 1} - ' ) .
(4)
In this model a decrease o f r ~ ( t ) as a function o f t indicates that the coefficients a~ and aS have the same sign, while an increase indicates opposite signs. Furthermore, at a value o f s where rhS(t) = the(t+ 1 ), a crossing point, one of the two coefficients, say aS, vanishes, and r~ s (t) = m i. Looking at our fig. 3 we see that the curves for different t cross, but the crossings of different curve pairs, although close, do not coincide. This indicates that the simple model is too crude, and that higher mass states are relevant. We can improve the situation by considering not the individual crossing between curve pairs, but the sequence o f crossings between curves of contiguous time as a function o f increasing time, a direction in which the effect of higher mass states should vanish exponentially. Unfortunately with increasing time the error also increases, so that we have to choose an optimal point where to stop. We have chosen to evaluate the masses using in each ease a crossing point such that, if we use crossing points at higher euclidean times, we obtain a compatible value with a larger error. We are confident, 538
5 May 1988
from our analysis (that can be partially repeated by looking at the figures of this paper), that the systematic error due to the choice o f the time distances is smaller or in the worst case of the same order of the statistical error. In order to learn more about the finite size effects, we have improved the measures of ref. [1] on a 163.32 lattice at fl=5.9. Because o f the anomalous behavior on the 103 lattice (on which finite size errors are quite sizeable [ 1 ] ) we wanted to be sure that the 123 lattice and the 163 one give the same results. We have carded out 1500 measures by using a pseudo heat-bath algorithm, and measuring every 10 sweeps. To get a better resolution of the crossing points we have repeated the smearing procedure 40 times (instead o f the 20 times ofref. [ 1 ] ). The results (see table 1 ) show that the possible finite size effects are of the order of magnitude of our (small) statistical error. The glueball mass (that was suffering a large finite size effect on the 103 lattice) has now a value that is constant within our statistical error. For the string tension we only remark that, since we cannot see any finite size effect, we are not able on our larger lattices to either confirm or exclude the n/3 effect [i.e. a(L) = a~o-n/3L2]. The values of the string tension reported in the tables are not corrected for this effect; because of our choice of lattice sizes the correction would, however, be approximatively constant (in percentage), so that our conclusions on the scaling would not be affected by it. Finite size effects at p = 5.9 are sizeable when going from L = 10 to L = 12 lattices, but the 123.32 lattice seems already asymptotic enough for our goals. We have performed a high statistics calculation at r = 6.1 on a 163. 32 lattice. For updating the gauge fields we have mainly used a pseudo heat-bath algorithm. We have carried out 4000 measurements (separated in part by 20 sweeps and in part by 5 sweeps). We have also done some simulations on a 93.23 lattice at r = 5.7. In this case we have carried out 1500 measurements, separated by 20 sweeps of a 5 hits Metropolis algorithm, as well as 1500 steps measuring every sweep (that were also used in order to check the efficiency of different updating methods, see ref. [ 6] ). The results we report in this paper have been always obtained looking at the crossing points according to the procedure defined above. This is important
Volume 205, number 4
PHYSICS LETTERS B
in o r d e r to a v o i d the inclusion o f different systematic errors due to different measuring procedures. Nevertheless, to check the consistency o f our results we have also carried out global fits, with results that are very similar ( a n d clearly consistent within our statistical errors) to the ones we reported. F o r the f l = 5.9 case see ref. [ 1 ]. As far as p = 6.1 is concerned, with a one mass fit we have to discard the times from t = 1 to t = 4 in o r d e r to get a quasi-constant b e h a v i o r (at about the 20% level) in the smearing n u m b e r (see fig. 4, u p p e r side). But the error is larger than with the p r o c e d u r e based on crossing points. A two mass fit with u n c o n s t r a i n e d masses is unstable, but two mass fits with a fixed mass ratio (see fig. 4, lower side) work better, i f we assume (see ref. [ 3 ] ) that the second mass is close to twice the first one, we o b t a i n a ver good fit i f we discard two time points. We do rem a r k that the fit would surely not work with a mass ratio of, let us say, 3 or 1.5. As far as the 2 + + state is concerned, the situation is by far m o r e complicated, so that we are not able to
0.8
0.6
0.4
$ o
5 May 1988
d r a w any conclusion as to the scaling b e h a v i o u r o f this state. At f l = 5.7 indeed is in fact simple, since we do not get any useful signal. The best we can do here is looking at the ratio o f the signals for t = 1 and t = 2 , r~S( 1 ). This curve corresponds to very high effective masses, a result which cannot be accepted lacking the corrob o r a t i o n o f either crossing or signals at higher t. The smearing is very useful in rejecting this result as spurious, since it is strongly d e p e n d e n t on the smearing number. At p = 5.9 on the 163.32 lattice we can look at the crossing between ~ s ( 2 ) and nSs(3), which gives m (2 ÷ ÷ ) = 0.8 3 ( 12 ) (the 123.32 yields a c o m p a t i b l e result with a larger error). We r e m a r k that, if we were using the rh s (2) / rh ~( 3 ) crossing to estimate m (0 ÷ ÷ ) we would get a result which is 5% lower than the one we get from ffzs(2)/ffzs(4) (the one we used). I f we assume that we have the same b e h a v i o r for the spin 2 (this is not granted but not unreasonable) we would conclude that m ( 2 + + ) / m ( 0 ÷ + ) = 1.15(25) at f l = 5.9. The f l = 6.1 case is the most difficult. We have a signature with a large error for a state with a mass ranging in the interval from 0.5 to 1.0 (with a spin 2 over spin 0 ratio ranging between one a n d t w o ) . It is possible that the p o o r performance o f our m e t h o d s for the 2 + + state is in part due to our choice o f the t = 0 cold wall. We will a t t e m p t to i m p r o v e the situation by experimenting with different prescriptions.
o.8
0.6
References
0.4
,
I 0.05
,
, 0.1 (Smearing) -t
,
I 0.15
0.2
Fig. 4. Upper side: m (0 ÷ +) from a one mass fit at fl= 6.1, discarding two (solid) three (dashed) and four (dotted) euclidean times. Lower side: m ( O ++ ) from a two mass fit at fl=6.1, with the mass ratio fixed to two, discarding one (solid) and two (dashed) times.
[ 1] Ape Collab., M. Albanese et al., Phys. Lett. B 192 (1987) 163. [21 Ape Collab., M. Albanese et al., Phys. Lett. B 197 (1987) 400. [3] L.A. Fernandez and E. Marinari, Nucl. Phys. B, to be published. [ 4 ] Ape Collab., M. Albanese et al., Comput. Phys. Commun. 45 (1987) 345. [5] C. Michael and M. Teper, Oxford preprint 72/87 (1987). [ 6 ] M, Bernaschi and L.A. Fernandez, to be published.
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