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The scalar and tensor glueball masses in lattice gauge theory
Volume 185, number 1,2
PHYSICS LETTERS B
12 February 1987
T H E SCALAR A N D T E N S O R G L U E B A L L M A S S E S I N L A T T I C E G A U G E T H E O R Y M. T E P E R 1
Department of TheoreticalPhysics, Universityof Oxford, 1 Keble Road, Oxford OX[ 3NP, UK Received 5 November 1986; revised manuscript received 27 November 1986 We calculate 0 + and 2 + correlation functions using an improved method. On small physical volumes we reproduce the result, m(0 + ) - m ( 2 + ), recently obtained by other methods. However, we also obtain evidence that for larger physical volumes the 2 + is heavier than the 0+: m(2 + )/m(0 + ) - 5/3.
On the lattice the mass of the lightest glueball in a particular channel can be calculated ,1 from the large-time behaviour of the correlation function of an operator q~ with the quantum numbers of that channel:
C( t ) = ( ~ a ( t ) ~ ( O ) ) / ( O ( O ) ~ ( O ) ) 1(4In) ]2
exp(-E,t)
1(41G)I 2 t---~¢O ~'m I(ePl m ) 12 e x p ( - m o t ) ,
(1)
where G is the glueball of interest and we ignore various complications. One must obviously calcul a t e C(t) for large enough t that it becomes dominated by the asymptotic exponential in eq. (1): a minimal criterion for a successful calculation is that at least three neighbouring (in time) values of C(t) should fall on a simple exponential and that the errors there should be small enough for this to be a statistically convincing fit. Even this criterion is hard for a numerical calculation to meet, given the typical rapid fall-off of C(t), so that the only reliable large-volume calculation is for the 0 ++ glueball in a narrow window of couplings just into the weak-coupling region [2]. Typical JPC~O++ correlation functions fall too a ResearchFellow at All Souls College, University of Oxford, Oxford, UK. ,1 For a recent review and references, see ref. [1].
rapidly for C(t) to be measured beyond t = 2a and this has led to the inference [1] that these states are considerably heavier than the 0 ÷+. The situation is aggravated by the fact that if ~ is formed from simple Wilson loops the overlap [ ( 0 IG ) [ 2 decreases rapidly as/3 increases. In a recent paper [3] we suggested a simple technique to help overcome this last problem: the basic idea being to use a simple iterative algorithm for constructing complex (in terms of lattice variables) "spaghetti-like" wave-functionals of roughly the right physical size. We have tested the method with rather limited statistics: tables 1 and 2 summarise the measurements that shall be relevant here. It turns out that the improvements obtained are so great that, despite the meagre statistics, the information we obtain on the large-volume behaviour of the 2 ÷ glueball mass is probably by far the best available. As we remarked above typical 2 + correlation functions fall much faster with t than 0 + ones. If we extract an effective glueball mass from C(t) at finite t by
a m ( t ) = - l n [ C( t ) / C ( t - a ) ] ,
(2)
then the largest t accessible on large volumes is t = 2 a where one finds [1] m(2 + : t = 2a) m(O + : t = 2 a ) = 2 " 2
0370-2693/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
=2.5
SU(2) SO(3).
(3) 121
Volume 185, number 1,2
PHYSICS LETTERS B
Any conclusions from eq. (3) about the lightest glueball masses are obviously not particularly trustworthy. For this reason the recent discovery [4] that on very small physical volumes both the 2 + and 0 + can be reliably calculated is of great interest as is the conclusion [4]
m(2+)/m(O +) _< 1,
(4)
which contradicts the usual level-ordering based on eq. (3). The calculation of ref. [4] is made possible by the fact that Wilson lines - loops winding through the boundaries of the periodic lattice - turn out to provide 0 + and 2 + wavefunctionals with a large projection onto the respective lowest mass states and by the fact that neither of these states is heavy in lattice units. The fact that the ratio in eq. (4) varies slowly with lattice size for the range of small physical volumes considered - despite the rapid variation of the individual masses - provides some evidence [4] that ,the physical 2 + may be close in mass to the 0+. However, there is also reason to be cautious about drawing such a conclusion. The fact that on such small lattices Wilson fines provide much better wave-functionals than small Wilson loops indicates that on these lattices the lowest mass glueball states themselves wind through the boundaries. This is entirely reasonable: a local state on a small lattice would pick up a large energy from its rapidly varying wave-function. There is therefore a physically important transition when the lattice becomes large enough for the glueball states to become local rather than global. O f course at this point any ~ based on Wilson lines will have a small overlap with G and the mass calculation will have become difficult and unreliable. So the calculations of ref. [4] are necessarily confined, by the nature of the techniques employed, to lattice sizes where the glueballs are global rather than local objects. This leaves open t h e possibility that the 2 + to 0 + mass ratio increases in the transition to large volumes. This concern is heightened by the observation [5] that the 2 + and 0 + are degenerate in the simplest of string pictures. In this letter we shall construct wave-functionals based on Wilson loops which have a good projection onto the lightest glueball on small lattices. Unlike a wave-function based on 122
12 February 1987
Wilson lines such a wave-function does not change as the lattice size is changed: hence we will be able to watch how C(t), calculated with a fixed 4, changes when we go to lattice sizes too large to be accessible to the techniques of ref. [4]. We form wave-functionals q~ as follows [3]. The first step is to form composite space-like paths U/l(n) joining pairs of lattice sites separated by 2a. We define U,.l(n, t) as the sum of the direct path and the "staples":
U?(n, t)= Ui(n , t)Ui(n+ ~, t) 3
+ E g(., t)<(.+L t) +_j= 1 jq=i
× < ( . + :+f, t)g*(. + 2L t).
(5)
We call this the first level of blocking. At the N t h blocking level we create composite paths u/V(n, t) between n and n + 2Nf using eq. (5) with U/v in place of Ui1 and UjN-1 in place of Uj. We form ~ = 0 wave-functionals CON(t) from the superplaquettes U~y(n): U/~(n, t) = ( 1 / N c ) tr{ U/U(n,
X UiN*(n+ 2uf, t)gN*(n,
t)g.N(n + 2Nt, t)
t)}
(6)
in the usual fashion:
~,u[O+; t]= E {Ug(n, t)+ U~(n, t) H
+ v2(., t)}, *N[2+; tl = • {U~(n, t)- U~(n, t)},
(7)
R
etc. We then form correlation functions as usual
Cs(t ) =- (q~u(t)q~N(O))/(ou(O)q~N(O)),
(8)
using ~ N -
Volume 185, number 1,2
PHYSICS LETTERS B
it does so in a fast and economic fashion. In fig. 1 we show twice-blocked SU(2) correlation functions, C2(t), obtained on 4316 and 6316 lattices at /3 = 4 / g 2 = 2.5. (We use the standard Wilson action and periodic b o u n d a r y conditions throughout.) We see that the 2 + correlation function on the 4316 lattice is measured reasonably accurately out to t = 4a. That one can do so with only 6000 measurements (see table 1) represents a vast improvement over the unblocked calculation. Indeed comparing [3] we find
C2(2a)/Co(2a ) = 100
~ : 2.5
6316
C2(t)
0.1
+
+
+
0.01
o
0+
•
2+
I
r
a
2a
3a
4n
Table 1 m(2+), m(0 +) from C2(a)/C2(2a) at fl = 2.3. Lattice "(num. meas.)
m(2 + )
m(0 + )
rn(2+)/m(O+)
4316 (10500) 6316 (440O) 8" (3200) 10 4 (1375)
1.17(6)
1.28(8)
0.91(7)
1.81(14)
1.15(8)
1.57(16)
2.02(~)
1.13(14)
1.79(46)
2.08(413)
1.34(~)
1.55(38)
(9)
a n d in fact we surely expect that the asymptotic exponential decay of C O will not set in before t = 3a where it will have fallen by about another factor of 10. Taking into account that in a M o n t e Carlo calculation errors fall as the square root of the n u m b e r of measurements we see that we have a saving in compUter time that is - (10: × 10)2. 10 6. Factors like this transform our low statistics
4316
12 February 1987
5o
0 f
I
I
I
r
a
2c
30
Ca
results into the best available 2 + calculations on large volumes. We see in fig. 1 that on the smaller 4316 lattice the 2 + glueball is significantly lighter than the 0 +. O n the larger but still small 6316 lattice their masses appear to be comparable. This is very similar to the pattern observed - albeit m u c h more accurately,and with m u c h higher statistics in ref. [4]. The m e t h o d o f r e f . [4] cannot be extended to larger lattices because the wave-functional used is based on Wilson lines and hence has a ,size equal to that of the lattice; so as the lattice size becomes bigger than that of the glueball - which is of course h o w we want it to be - this growing mismatch i n size will lead t o a rapidly decreasing coefficient o f the asymptotic exponential in eq. (1). The m e t h o d [3] we are using herein o n the other hand, produces wave-functionals independent of lattice size and hence any variation with lattice size of CN(t ) should be (primarily) due to changes in the glueball mass. I n fig. 2 we show C2(t ) for 0 + and 2:- states on a variety of lattice sizes at/3 = 2.3. To set the scale of physics at this /3 we remark that the deconfining temperature has a value - ( 4 a ) -1, so that only the 4316 lattice is a really small lattice. C o n sider n o w the data shown for the 2 +. O n the 4316 lattice C2(t ) is consistent with a simple exponential from t = a onwards. The overlap of 42 with [G) is large
5a
Fig. 1. Twice-blocked 0 + and 2 + correlation functions on 4316 and 6316 lattices at fl = 2.5 in SU(2).
[(@2 1G)2/(@2 1~2) - 0.7,
(10)
indicating that 42 10) is a good wave-function for 123
V o l u m e 185, n u m b e r 1,2
PHYSICS LETTERS B
summarised in table 1 and the conclusion is that
L3 = 2.3
lo
_2
12 F e b r u a r y 1987
i¸
m(2+)/m(O +)
0÷
5/3,
(11)
B=2.3 so the 2 + is indeed lighter then previous estimates (eq. (3)) but still considerably heavier than the 0 ÷. All the above calculatons have been performed with /5 = 0 wave-functionals. Non-zero momentum wave-functionals can also be easily constructed [6], e.g.
° 0.1
0N[0+; p ; t]
1
0.01
I a
2fct
I
I , a
30
I 2a
30
f Fig. 2. T w i c e - b l o c k e k d 0 + a n d 2 + correlation functions for a range of lattice sizes a t / 3 = 2.3 in SU(2):
a glueball. If we now focus on the 6316 lattice we observe a substantial steepening of C2(t); so much so that we are unable to obtain a useful measurement at t = 3a. There is a further, less severe, steepening as we move to the 84 lattice. Increasing the lattice size to 10 4 produces no significant further change. Since we are using in all cases the same wave-functional, ~2' as on the 4316 lattice, it is reasonable to assume that the asymptotic exponential continues to dominate from t = a onwards. This is consistent assumption in that if we extrapolate the exponential interpolation between C2(t = a) and C2(t = 2 a ) to t = 0 we obtain an overlap close to that in eq. (10). We then conclude that as the lattice is increased in spatial size from 43 to 83 the 2 + glueball mass becomes rapidly heavier. It appears that this glueball m a s s only stabilises for lattices larger than 84. The 0 + correlation functions in marked contrast, show only a small variation with lattice size. The results are 124
large lattice
Cz [t)
=
E e i J " " ( U N ( n , t) + UN(n, t) n
+ U3~(n, t ) } .
(12)
The reasons for considering the corresponding correlation functions, CN( p, t), are usually to test for the restoration of the continuum dispersion relation [6], E 2 = p 2 + m 2 , and to obtain more accurate measurements of masses by using the measured energies and assuming the continuum dispersion relation [7]. Here we have a further reason: to distinguish between local glueball states and those which extend throughout the lattice. Once a(fl) is small enough the local states should satisfy E 2 = p 2 + mZ; the extended states on the other hand have no a priori reasons to do so and hence their presence may be detected b y an "anomalous" e n e r g y - m o m e n t u m dispersion relation. In the limit where an extended state becomes completely translation invariant in the z direction it will have no projection onto Pz "¢ 0 and so E(p~ ~ 0). This is of course an extreme case. It does suggest, however, that the signal for an extended non-local glueball state is a n anomalously large energy for non-zero momentum. In fig. 3 we plot the variation of the 2 + glueball masses with blocking level, lattice size and momentum. The mass is extracted from C u ( a ) / CN(O). For such small t there will be an admixture~ of higher mass glueballs, especially for Co(t ). This. will obviously introduce its own (small [6,7])p anomaly into the energy-momentum dispersion relation, and so we should only assign significance to large anomalies. In fig. 3 we plot the 2 + mass as extracted directly from CN( p = 0; t = a) or u s i n g m 2 = E 2 _ p 2 from Cw( p = 2"~/L; t = a). (Note that for the 2 + we must consider maximal
Volume 185, number 1,2
PHYSICS LETTERS B
2+at ~3 = 2 . 3 : m ( t = a l
afrom pa=0 • 2re o pa = --ff
rn(1 N
3
I
Table 2 m(2 + ) / m ( O ÷ ) from C ( t ) / C ( t -
1) at fl = 2.5.
Lattice Blocking p a = 0 (num. meas.) level t~a
t=2a
t=a
t=2a
4316 . (6000) 6316
2
12 February 1987
pa = 2~r/L
2
0.79(2)
0.63(6)
-
-
2
1.14(5)
0.95(10) -
-
2
1.11(10) 0.76(3~)
3
1.14(9)
(2oo0) I
I
0
124 (360) 124
6316
;316 I
i
I
1.46(7)
1.47(24)
0.65(223) 1.65(7) 1.73(485)
+ rlii2 N
3 2 I 8/,
0
I
I
1
2
r
3 ; N, brooking [ever
I
r
1
2
Fig. 3. Effective 2 + mass extracted from CN(a)/CN(O ) for various blocking levels, lattice sizes and momenta for SU(2) at fl=2.3.
helicity states only, to avoid 0 + contamination.) The pattern we observe - a large anomaly on the smallest 43 lattice, with the anomalous behaviour rapidly disappearing at any given blocking level as we increase the lattice size - strongly reinforces our interpretation that the lowest mass glueball state changes from being extended to being local as the lattice size is increased from 4 3 to 8 3 at this /3. The 0 4 results show the same pattern of behaviour. The implication of all the above is that N = 2 on 8 4 and 10 4 lattices at f l - - 2 . 3 projects reasonably well onto the desired local glueball states. Accordingly we extract the masses from C2(P; 2a)/C2(P; a), average the 84 and 10 4 r e s u l t s and obtain: m ( 2 +)
1 t;"/+30 --'~
m ( 0 +) /?=2.3
. . . . .
26'
= 1/qK+32 .vo _ 22 ' /~=2.3
p = 0
p= 2v/L,
(13)
which again displays the consistency of our analysis. In table 2 we show the results for the 2 + to 0 4
mass ratio at fl = 2.5. The entries for the smallest lattices are extracted from the correlation functions displayed in fig. 1. We had! hoped that the 124 lattice would already have been large enough to contain a local glueball state. Unfortunately the disparity between the masses extracted with p = 0 and p + 0 shows this not to be the case: at fl = 2.5 a 124 lattice is still too small. Finally some remarks on our calculations a t /3 = 5.7 and 5.9 in SU(3). These are very low statistics results but they show exactly the behaviour found in SU(2). The ratio ( m ( 2 + + ) / m(0 ++) increases from about 1 o n t h e 4312 lattice at/3 = 5.7 to about 2 on the 638 lattice. A t / 3 = 5.9 this ratio is substantially less than 1 on the 4312 lattice. Anomalies in the e n e r g y - m o m e n t u m dispersion relation appear just as in SU(2). By improving the method for calculating glueball masses we have been able to conclude quite a lot from rather little in the way of siatistics. There are limits to this of course and the iaext step is to replace some of our plausible arguements with accurate measurements at larger t. This can be achieved by replacing our - 1 0 3 measurements with the - 10 6 which is typical of current efforts [4]. In particular we remark that because the 2 + is heavier than the 04 it will probably be easier to obtain reliable 2 + m e a s u r e m e n t s on, say, 164 lattices at /3 = 2.5 than on 84 latti+es at /3 = 2.3; this unconventional situation arises because our composite path algorithm appears [3] to provide good wave-functionals at this large/3 value. Some of this work was performed while I was at Southampton University and during visits to 125
Volume 185, number 1,2
PHYSICS LETTERS B
C E R N a n d the R u t h e r f o r d A p p l e t o n L a b o r a t o r y . I a m g r a t e f u l to these i n s t i t u t i o n s for their hospitality.
References [1] M. Teper, Proc. Workshop on Non-perturbative methods (MontpeUier, 1985) ed. S. Narison (World Scientific, Singapore, 1985). [2] Ph. de Forcrand, G. Schierholz, H. Schneider and M. Teper, Phys. Lett. B 152 (1985) 107;
126
12 February 1987
A. Patel, R. Gupta, G. Guralrtik, G. Kilcup and S. Sharpe, Harvard preprint HUTP-86/A035. [3] M. Teper, Phys. Lett. B 183 (1987) 345. [4] B. Berg, A. Billiore and C. Vohwinkel, Phys. Rev. Lett. 57 (1986) 400. [5] C. Michael, Illinois preprint (1986). [6] G. Schierholz and M. Teper, Phys. Lett. B 136 (1984) 69. [7] G. Schierholz and M. Teper, Phys. Lett. B 136 (1984) 64; Ph. de Forcrand, G. Schierholz, H. Schneider and M. Teper, Z. Phys. C 31 (1986) 87.
PHYSICS LETTERS B
12 February 1987
T H E SCALAR A N D T E N S O R G L U E B A L L M A S S E S I N L A T T I C E G A U G E T H E O R Y M. T E P E R 1
Department of TheoreticalPhysics, Universityof Oxford, 1 Keble Road, Oxford OX[ 3NP, UK Received 5 November 1986; revised manuscript received 27 November 1986 We calculate 0 + and 2 + correlation functions using an improved method. On small physical volumes we reproduce the result, m(0 + ) - m ( 2 + ), recently obtained by other methods. However, we also obtain evidence that for larger physical volumes the 2 + is heavier than the 0+: m(2 + )/m(0 + ) - 5/3.
On the lattice the mass of the lightest glueball in a particular channel can be calculated ,1 from the large-time behaviour of the correlation function of an operator q~ with the quantum numbers of that channel:
C( t ) = ( ~ a ( t ) ~ ( O ) ) / ( O ( O ) ~ ( O ) ) 1(4In) ]2
exp(-E,t)
1(41G)I 2 t---~¢O ~'m I(ePl m ) 12 e x p ( - m o t ) ,
(1)
where G is the glueball of interest and we ignore various complications. One must obviously calcul a t e C(t) for large enough t that it becomes dominated by the asymptotic exponential in eq. (1): a minimal criterion for a successful calculation is that at least three neighbouring (in time) values of C(t) should fall on a simple exponential and that the errors there should be small enough for this to be a statistically convincing fit. Even this criterion is hard for a numerical calculation to meet, given the typical rapid fall-off of C(t), so that the only reliable large-volume calculation is for the 0 ++ glueball in a narrow window of couplings just into the weak-coupling region [2]. Typical JPC~O++ correlation functions fall too a ResearchFellow at All Souls College, University of Oxford, Oxford, UK. ,1 For a recent review and references, see ref. [1].
rapidly for C(t) to be measured beyond t = 2a and this has led to the inference [1] that these states are considerably heavier than the 0 ÷+. The situation is aggravated by the fact that if ~ is formed from simple Wilson loops the overlap [ ( 0 IG ) [ 2 decreases rapidly as/3 increases. In a recent paper [3] we suggested a simple technique to help overcome this last problem: the basic idea being to use a simple iterative algorithm for constructing complex (in terms of lattice variables) "spaghetti-like" wave-functionals of roughly the right physical size. We have tested the method with rather limited statistics: tables 1 and 2 summarise the measurements that shall be relevant here. It turns out that the improvements obtained are so great that, despite the meagre statistics, the information we obtain on the large-volume behaviour of the 2 ÷ glueball mass is probably by far the best available. As we remarked above typical 2 + correlation functions fall much faster with t than 0 + ones. If we extract an effective glueball mass from C(t) at finite t by
a m ( t ) = - l n [ C( t ) / C ( t - a ) ] ,
(2)
then the largest t accessible on large volumes is t = 2 a where one finds [1] m(2 + : t = 2a) m(O + : t = 2 a ) = 2 " 2
0370-2693/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
=2.5
SU(2) SO(3).
(3) 121
Volume 185, number 1,2
PHYSICS LETTERS B
Any conclusions from eq. (3) about the lightest glueball masses are obviously not particularly trustworthy. For this reason the recent discovery [4] that on very small physical volumes both the 2 + and 0 + can be reliably calculated is of great interest as is the conclusion [4]
m(2+)/m(O +) _< 1,
(4)
which contradicts the usual level-ordering based on eq. (3). The calculation of ref. [4] is made possible by the fact that Wilson lines - loops winding through the boundaries of the periodic lattice - turn out to provide 0 + and 2 + wavefunctionals with a large projection onto the respective lowest mass states and by the fact that neither of these states is heavy in lattice units. The fact that the ratio in eq. (4) varies slowly with lattice size for the range of small physical volumes considered - despite the rapid variation of the individual masses - provides some evidence [4] that ,the physical 2 + may be close in mass to the 0+. However, there is also reason to be cautious about drawing such a conclusion. The fact that on such small lattices Wilson fines provide much better wave-functionals than small Wilson loops indicates that on these lattices the lowest mass glueball states themselves wind through the boundaries. This is entirely reasonable: a local state on a small lattice would pick up a large energy from its rapidly varying wave-function. There is therefore a physically important transition when the lattice becomes large enough for the glueball states to become local rather than global. O f course at this point any ~ based on Wilson lines will have a small overlap with G and the mass calculation will have become difficult and unreliable. So the calculations of ref. [4] are necessarily confined, by the nature of the techniques employed, to lattice sizes where the glueballs are global rather than local objects. This leaves open t h e possibility that the 2 + to 0 + mass ratio increases in the transition to large volumes. This concern is heightened by the observation [5] that the 2 + and 0 + are degenerate in the simplest of string pictures. In this letter we shall construct wave-functionals based on Wilson loops which have a good projection onto the lightest glueball on small lattices. Unlike a wave-function based on 122
12 February 1987
Wilson lines such a wave-function does not change as the lattice size is changed: hence we will be able to watch how C(t), calculated with a fixed 4, changes when we go to lattice sizes too large to be accessible to the techniques of ref. [4]. We form wave-functionals q~ as follows [3]. The first step is to form composite space-like paths U/l(n) joining pairs of lattice sites separated by 2a. We define U,.l(n, t) as the sum of the direct path and the "staples":
U?(n, t)= Ui(n , t)Ui(n+ ~, t) 3
+ E g(., t)<(.+L t) +_j= 1 jq=i
× < ( . + :+f, t)g*(. + 2L t).
(5)
We call this the first level of blocking. At the N t h blocking level we create composite paths u/V(n, t) between n and n + 2Nf using eq. (5) with U/v in place of Ui1 and UjN-1 in place of Uj. We form ~ = 0 wave-functionals CON(t) from the superplaquettes U~y(n): U/~(n, t) = ( 1 / N c ) tr{ U/U(n,
X UiN*(n+ 2uf, t)gN*(n,
t)g.N(n + 2Nt, t)
t)}
(6)
in the usual fashion:
~,u[O+; t]= E {Ug(n, t)+ U~(n, t) H
+ v2(., t)}, *N[2+; tl = • {U~(n, t)- U~(n, t)},
(7)
R
etc. We then form correlation functions as usual
Cs(t ) =- (q~u(t)q~N(O))/(ou(O)q~N(O)),
(8)
using ~ N -
Volume 185, number 1,2
PHYSICS LETTERS B
it does so in a fast and economic fashion. In fig. 1 we show twice-blocked SU(2) correlation functions, C2(t), obtained on 4316 and 6316 lattices at /3 = 4 / g 2 = 2.5. (We use the standard Wilson action and periodic b o u n d a r y conditions throughout.) We see that the 2 + correlation function on the 4316 lattice is measured reasonably accurately out to t = 4a. That one can do so with only 6000 measurements (see table 1) represents a vast improvement over the unblocked calculation. Indeed comparing [3] we find
C2(2a)/Co(2a ) = 100
~ : 2.5
6316
C2(t)
0.1
+
+
+
0.01
o
0+
•
2+
I
r
a
2a
3a
4n
Table 1 m(2+), m(0 +) from C2(a)/C2(2a) at fl = 2.3. Lattice "(num. meas.)
m(2 + )
m(0 + )
rn(2+)/m(O+)
4316 (10500) 6316 (440O) 8" (3200) 10 4 (1375)
1.17(6)
1.28(8)
0.91(7)
1.81(14)
1.15(8)
1.57(16)
2.02(~)
1.13(14)
1.79(46)
2.08(413)
1.34(~)
1.55(38)
(9)
a n d in fact we surely expect that the asymptotic exponential decay of C O will not set in before t = 3a where it will have fallen by about another factor of 10. Taking into account that in a M o n t e Carlo calculation errors fall as the square root of the n u m b e r of measurements we see that we have a saving in compUter time that is - (10: × 10)2. 10 6. Factors like this transform our low statistics
4316
12 February 1987
5o
0 f
I
I
I
r
a
2c
30
Ca
results into the best available 2 + calculations on large volumes. We see in fig. 1 that on the smaller 4316 lattice the 2 + glueball is significantly lighter than the 0 +. O n the larger but still small 6316 lattice their masses appear to be comparable. This is very similar to the pattern observed - albeit m u c h more accurately,and with m u c h higher statistics in ref. [4]. The m e t h o d o f r e f . [4] cannot be extended to larger lattices because the wave-functional used is based on Wilson lines and hence has a ,size equal to that of the lattice; so as the lattice size becomes bigger than that of the glueball - which is of course h o w we want it to be - this growing mismatch i n size will lead t o a rapidly decreasing coefficient o f the asymptotic exponential in eq. (1). The m e t h o d [3] we are using herein o n the other hand, produces wave-functionals independent of lattice size and hence any variation with lattice size of CN(t ) should be (primarily) due to changes in the glueball mass. I n fig. 2 we show C2(t ) for 0 + and 2:- states on a variety of lattice sizes at/3 = 2.3. To set the scale of physics at this /3 we remark that the deconfining temperature has a value - ( 4 a ) -1, so that only the 4316 lattice is a really small lattice. C o n sider n o w the data shown for the 2 +. O n the 4316 lattice C2(t ) is consistent with a simple exponential from t = a onwards. The overlap of 42 with [G) is large
5a
Fig. 1. Twice-blocked 0 + and 2 + correlation functions on 4316 and 6316 lattices at fl = 2.5 in SU(2).
[(@2 1G)2/(@2 1~2) - 0.7,
(10)
indicating that 42 10) is a good wave-function for 123
V o l u m e 185, n u m b e r 1,2
PHYSICS LETTERS B
summarised in table 1 and the conclusion is that
L3 = 2.3
lo
_2
12 F e b r u a r y 1987
i¸
m(2+)/m(O +)
0÷
5/3,
(11)
B=2.3 so the 2 + is indeed lighter then previous estimates (eq. (3)) but still considerably heavier than the 0 ÷. All the above calculatons have been performed with /5 = 0 wave-functionals. Non-zero momentum wave-functionals can also be easily constructed [6], e.g.
° 0.1
0N[0+; p ; t]
1
0.01
I a
2fct
I
I , a
30
I 2a
30
f Fig. 2. T w i c e - b l o c k e k d 0 + a n d 2 + correlation functions for a range of lattice sizes a t / 3 = 2.3 in SU(2):
a glueball. If we now focus on the 6316 lattice we observe a substantial steepening of C2(t); so much so that we are unable to obtain a useful measurement at t = 3a. There is a further, less severe, steepening as we move to the 84 lattice. Increasing the lattice size to 10 4 produces no significant further change. Since we are using in all cases the same wave-functional, ~2' as on the 4316 lattice, it is reasonable to assume that the asymptotic exponential continues to dominate from t = a onwards. This is consistent assumption in that if we extrapolate the exponential interpolation between C2(t = a) and C2(t = 2 a ) to t = 0 we obtain an overlap close to that in eq. (10). We then conclude that as the lattice is increased in spatial size from 43 to 83 the 2 + glueball mass becomes rapidly heavier. It appears that this glueball m a s s only stabilises for lattices larger than 84. The 0 + correlation functions in marked contrast, show only a small variation with lattice size. The results are 124
large lattice
Cz [t)
=
E e i J " " ( U N ( n , t) + UN(n, t) n
+ U3~(n, t ) } .
(12)
The reasons for considering the corresponding correlation functions, CN( p, t), are usually to test for the restoration of the continuum dispersion relation [6], E 2 = p 2 + m 2 , and to obtain more accurate measurements of masses by using the measured energies and assuming the continuum dispersion relation [7]. Here we have a further reason: to distinguish between local glueball states and those which extend throughout the lattice. Once a(fl) is small enough the local states should satisfy E 2 = p 2 + mZ; the extended states on the other hand have no a priori reasons to do so and hence their presence may be detected b y an "anomalous" e n e r g y - m o m e n t u m dispersion relation. In the limit where an extended state becomes completely translation invariant in the z direction it will have no projection onto Pz "¢ 0 and so E(p~ ~ 0). This is of course an extreme case. It does suggest, however, that the signal for an extended non-local glueball state is a n anomalously large energy for non-zero momentum. In fig. 3 we plot the variation of the 2 + glueball masses with blocking level, lattice size and momentum. The mass is extracted from C u ( a ) / CN(O). For such small t there will be an admixture~ of higher mass glueballs, especially for Co(t ). This. will obviously introduce its own (small [6,7])p anomaly into the energy-momentum dispersion relation, and so we should only assign significance to large anomalies. In fig. 3 we plot the 2 + mass as extracted directly from CN( p = 0; t = a) or u s i n g m 2 = E 2 _ p 2 from Cw( p = 2"~/L; t = a). (Note that for the 2 + we must consider maximal
Volume 185, number 1,2
PHYSICS LETTERS B
2+at ~3 = 2 . 3 : m ( t = a l
afrom pa=0 • 2re o pa = --ff
rn(1 N
3
I
Table 2 m(2 + ) / m ( O ÷ ) from C ( t ) / C ( t -
1) at fl = 2.5.
Lattice Blocking p a = 0 (num. meas.) level t~a
t=2a
t=a
t=2a
4316 . (6000) 6316
2
12 February 1987
pa = 2~r/L
2
0.79(2)
0.63(6)
-
-
2
1.14(5)
0.95(10) -
-
2
1.11(10) 0.76(3~)
3
1.14(9)
(2oo0) I
I
0
124 (360) 124
6316
;316 I
i
I
1.46(7)
1.47(24)
0.65(223) 1.65(7) 1.73(485)
+ rlii2 N
3 2 I 8/,
0
I
I
1
2
r
3 ; N, brooking [ever
I
r
1
2
Fig. 3. Effective 2 + mass extracted from CN(a)/CN(O ) for various blocking levels, lattice sizes and momenta for SU(2) at fl=2.3.
helicity states only, to avoid 0 + contamination.) The pattern we observe - a large anomaly on the smallest 43 lattice, with the anomalous behaviour rapidly disappearing at any given blocking level as we increase the lattice size - strongly reinforces our interpretation that the lowest mass glueball state changes from being extended to being local as the lattice size is increased from 4 3 to 8 3 at this /3. The 0 4 results show the same pattern of behaviour. The implication of all the above is that N = 2 on 8 4 and 10 4 lattices at f l - - 2 . 3 projects reasonably well onto the desired local glueball states. Accordingly we extract the masses from C2(P; 2a)/C2(P; a), average the 84 and 10 4 r e s u l t s and obtain: m ( 2 +)
1 t;"/+30 --'~
m ( 0 +) /?=2.3
. . . . .
26'
= 1/qK+32 .vo _ 22 ' /~=2.3
p = 0
p= 2v/L,
(13)
which again displays the consistency of our analysis. In table 2 we show the results for the 2 + to 0 4
mass ratio at fl = 2.5. The entries for the smallest lattices are extracted from the correlation functions displayed in fig. 1. We had! hoped that the 124 lattice would already have been large enough to contain a local glueball state. Unfortunately the disparity between the masses extracted with p = 0 and p + 0 shows this not to be the case: at fl = 2.5 a 124 lattice is still too small. Finally some remarks on our calculations a t /3 = 5.7 and 5.9 in SU(3). These are very low statistics results but they show exactly the behaviour found in SU(2). The ratio ( m ( 2 + + ) / m(0 ++) increases from about 1 o n t h e 4312 lattice at/3 = 5.7 to about 2 on the 638 lattice. A t / 3 = 5.9 this ratio is substantially less than 1 on the 4312 lattice. Anomalies in the e n e r g y - m o m e n t u m dispersion relation appear just as in SU(2). By improving the method for calculating glueball masses we have been able to conclude quite a lot from rather little in the way of siatistics. There are limits to this of course and the iaext step is to replace some of our plausible arguements with accurate measurements at larger t. This can be achieved by replacing our - 1 0 3 measurements with the - 10 6 which is typical of current efforts [4]. In particular we remark that because the 2 + is heavier than the 04 it will probably be easier to obtain reliable 2 + m e a s u r e m e n t s on, say, 164 lattices at /3 = 2.5 than on 84 latti+es at /3 = 2.3; this unconventional situation arises because our composite path algorithm appears [3] to provide good wave-functionals at this large/3 value. Some of this work was performed while I was at Southampton University and during visits to 125
Volume 185, number 1,2
PHYSICS LETTERS B
C E R N a n d the R u t h e r f o r d A p p l e t o n L a b o r a t o r y . I a m g r a t e f u l to these i n s t i t u t i o n s for their hospitality.
References [1] M. Teper, Proc. Workshop on Non-perturbative methods (MontpeUier, 1985) ed. S. Narison (World Scientific, Singapore, 1985). [2] Ph. de Forcrand, G. Schierholz, H. Schneider and M. Teper, Phys. Lett. B 152 (1985) 107;
126
12 February 1987
A. Patel, R. Gupta, G. Guralrtik, G. Kilcup and S. Sharpe, Harvard preprint HUTP-86/A035. [3] M. Teper, Phys. Lett. B 183 (1987) 345. [4] B. Berg, A. Billiore and C. Vohwinkel, Phys. Rev. Lett. 57 (1986) 400. [5] C. Michael, Illinois preprint (1986). [6] G. Schierholz and M. Teper, Phys. Lett. B 136 (1984) 69. [7] G. Schierholz and M. Teper, Phys. Lett. B 136 (1984) 64; Ph. de Forcrand, G. Schierholz, H. Schneider and M. Teper, Z. Phys. C 31 (1986) 87.