Again on SU(3) glueball mass

Nuclear Physics B251 [FSI3] (1985) 624-632 (~ North-Holland Publishing C o m p a n y

A G A I N O N SU(3) G L U E B A L L M A S S M. FALCIONI, 1'2 M. L. PACIELLO, I G. PARISI 3 and B. T A G L I E N T I t Received 10 July 1984 (Revised 24 September 1984) We extract the lowest glueball mass by studying a suitable response function of an SU(3) pure lattice gauge theory. Our estimate is m~ = (305 + 55) At.. We do not find finite size etiects at ,8 = 6.0 for lattice sizes from 53 x 10 to 73 × 10.

1. Introduction

In a recent paper [1] we extended to an SU(2) lattice gauge theory a new method to c o m p u t e large-distance correlation functions. This method is based on a c o m p u t e r simulation o f two statistical gauge systems at different temperatures reaching equilibrium according to the Langevin equation; it permits an increase in accuracy o f several orders o f magnitude over the standard Monte Carlo methods. The main advantage o f this simulation algorithm is a partial cancellation of statistical fluctuations in the measured correlation functions. However two unpleasant behaviours were still present in the method: (i) the a p p r o a c h to the equilibrium o f the system was too slow; (ii) a sort o f "magnetic field" term we a d d e d in the action to reduce the noise associated to gauge invariance makes the a p p r o a c h to the equilibrium even worse. On the other hand it is known that to obtain reliable glueball mass estimates it is m a n d a t o r y to study correlation functions at large distances and they can be efficiently measured with a method allowing for a coherent cancellation o f statistical errors between two highly correlated stochastic processes. The purpose o f this work is to extend to the gauge g r o u p SU(3) the analysis we already performed on the lowest mass state o f the SU(2) theory and get rid of the unpleasant behaviours described above. In the next section we briefly describe the two algorithms used to measure the SU(3) mass gap.

lstituto Nazionale di Fisica Nucleare, Sezione di Roma, Roma, Italy. -" Dipartimento di Fisica, Universita "La Sapienza", Roma, Italy. Universita di Roma II "Tor Vergata "°, Roma, Italy, and INFN-Laboratori Nazionali di Frascati, Frascati, Italy. 624

M. Falcioni et al. / Glueball m a s s

625

2. The heat bath method

As in the SU(2) case we consider two gauge systems K and K'(i) on two lattices of identical size M 3 x L ; K is at temperature T = lift, K ' ( i ) h a s a "time slice" i ( i = 1,2,3 . . . L) at temperature T ' = 1/(/3 + t5/3). We define (i) E ( j ) = average energy of time slice j for K, (ii) W ( i , j ) = average energy of time slice j for K'(i). We use the Wilson form for the action: S = E [I - ~I Re Tr U ( p ) ] .

(I)

p

It has been shown in ref. [l] that the connected correlation function at a distance d of the energy operator C(d) is given by

C ( d ) = (E(j) - Wfi, j))/Sfl

(2)

for any i,j such that d = IJ- il. Then to reduce considerably the noise in the correlation functions it is necessary to generate two "close" sets of configurations and evaluate the corresponding expectation values E (j) and W(i, j ) ; so we need a method to generate configurations that are continuous in the/3-variable. It is known that the most efficient updating algorithms are the heat bath ones [2], [3]; moreover it is very easy to make this algorithm to satisfy the continuity requirement. For the SU(3) gauge theory we have used the method proposed in ref. [4]. The random choice algorithm (which is essentially equivalent to that of ref. [3]) does not quite provide matrices with the right distribution, namely, dP(U) ~ e x p

(-flS(U)) dU.

(3)

However the detailed balance is satisfied, so the method is suitable for our purposes. This method exhibits two relevant improvements with respect to the numerical solution of the Langevin equation: it is about ten times faster in reaching thermal equilibrium and does not need arbitrary parameters arising from the discretized version of the equation. The noise associated to the gauge invariance effects can be reduced by performing on each site o f f h e generated configuration a gauge transformation g built as follows. Let U ( i ) ( V ( i ) ) be the SU(3) 4a~atrices on the links outgoing (incoming) from a given site; we daose g such that the quantity G = Re Tr { g [ U ( l ) + U ( 2 ) + U ( 3 ) + U ( 4 ) + V + ( l ) + V+(2) + V+(3) + V÷(4)]}

(4)

is maxirn~m. It turns.t~at that t~is~:peration greatly reduces the noise as the additional degree o f fr¢odom assockated ,0¢ith tire gauge i~variance is strongly reduced. This procedure does not induce any "stow c o m p o n e n t " in the approach to the equilibrium of the correlation functions as the extra term in the action did in ref. [l].

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M o r e o v e r we know that for local operators the noise-to-signal ratio diverges as we a p p r o a c h the c o n t i n u u m limit as discussed in detail in ref. [5]; so that, in order to c o m p u t e masses, it is convenient to measure correlation functions for operators which are a functional o f the field smeared in space and not in time. To this end we construct the following block gauge field with a simple gauge invariant procedure. For each link o f a generated configuration we consider the p r o d u c t o f the other three link variables defining with it a plaquette, and we sum these products over 4 choices which define plaquettes orthogorral to the time axis; the resulting matrix, projected on the SUE3) group, will be the new link variable. This procedure can be iterated to obtain a field more and more smeared in space at fixed time as it is convenient to get information on the low-energy spectrum. The main disadvantage o f this method is that we have not succeeded in the purpose o f eliminating the increase o f the noise for large upgrading "times". We can see in figs. 1 and 2 that this p h e n o m e n o n may be dramatic: an increase in the

O.OO6

I

DISTANCE 3 OPERATOR Of l )

DISTANCE 3 OPERATOR 0 (2)

00O4

O002

A

00050."

~

,,

- O.005 O 0.012

20

40

20

60 STEPS 80

DISTANCE 3 OPERATOR 0(3)

0.01

0.008

40

60 STEPS 80

40

60 STEPS 80

DISTANCE 3 OPERATOR 0(4)

0005

OIlM 0

'v

- 0 004 .... O

, .... l . . . . . . . . . ~. . . . . . . . . I ..... ,, ..! 20 40 60 STEPS 80

0

20

Fig. I. Correlation functions at distance 3,/3 = 6, in a 53 × 10 lattice, relative to the operators O( I ), 0 ( 2 ) , O(3), 0 ( 4 ) computed with the heat bath method.

M. Falcioni et al. / Glueball mass

f

OISTANCE 2 001

OPERATOR 0 Ell

001 ~r

627

DISTANCE 2 OPERATOR 0 ( 2 )

0.005

-°::I o .........

0.01

.........

l

;o"

-O. Ol

--0.02 . . . . . . . . .

'00

~ ......... 20

DISTANCE 2

DISTANCE

2

OPERATOR 0 ! 3 )

OPERATOR

0 (4)

~ ..... ,, , , 40 STEPS 60

0008 0.00~

0 000d - 0005

--0.015

........................ill 20

40

STEPS

60

20

40

STEPS

60

Fig. 2. Correlation functions at distance 2,/3 = 6, in a 7 J x 10 lattice, relative to the operators O( 1), 0 ( 2 ) , O(3), 0 ( 4 ) c o m p u t e d with the heat bath method.

noise by a factor o f 10 or more often occurs in a few iterations. In this case the two configurations b e c o m e completely different after the onset o f the strong noise. We have tried to blame our gauge fixing method, using a different procedure: the first system is not gauged at all and the second one is gauged at each step in such a way as to bring it as near as possible to the first one. The noise has remained nearly the same, so we c o n c l u d e that its origin is independent o f gauge invariance. We also noticed that the n u m b e r o f steps for the onset o f the noise strongly increases with/3, so it may be possible that it goes to infinity with/3 faster than the n u m b e r o f steps needed to thermalize the system.

3. The source method

The second m e t h o d used in this work is based on the study o f the response o f the expectation value o f an operator on several space slices at time = 2, 3 , . . . L,

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M. Falcioni et al. / Glueball m a s s

when the space slice at time = 1 is kept at T = 0 (all link variables = 1). In this case too the procedure of smearing the gauge field in space turns out to be useful. In order to extract the mass gap from the expectation value o f operators defined on space slices we assume an exponential behaviour for these values as follows: (O(n))~A+

B exp ( - M n ) + C exp (Mn).

(5)

If we apply periodic b o u n d a r y conditions, A, B and C must be chosen such that (O(n)) = (O( L - n + 2))

(L -- lattice time size).

(6)

This leads to an expression like ( O ( n ) ) - C o + C exp ( - ½ ( L + 2 ) M ) cosh [ M ( ½ ( L + 2 ) - n ) ] .

(7)

If e n o u g h values o f (O(n)) are available, a more accurate two-mass behaviour can be fitted: (O(n)) ~ Co+ C exp ( - ½ ( L + 2 ) M ) cosh [ M ( ½ ( L + 2) - n)] + C ' exp ( - ½ ( L + 2 ) M ' ) cosh [ M ' ( ½ ( L + 2) - n ) ] .

(8)

4. Results To c o m p u t e the gluball mass in a pure SU(3) gauge theory we have measured correlation functions o f the following operators O ( k ) :

O(1) = E [l -~- R e T r U(p)], LI ,p

0(2) = Y. [1 -½ Re Tr Q(s)],

(9)

Li,s

0(3) = 5" [ l + I R e T r

U(p)],

L2,p

0 ( 4 ) = E [1 - I Re Tr Q ( s ) ] ,

(10)

k2os

where the sum EL, is performed on a once smeared lattice and ~c2 on a twice smeared one. U ( p ) means a plaquette and Q(s) is a square loop made up o f 8 links. The largest loops are considered for the reasons we mentioned at the end o f sect. 2 and because these operators seem to dominate the 0 ÷+ wave function at our temperature as noticed in a previous analysis [6]. We p e r f o r m e d our analysis on 53x l0 and 73x l0 lattices to evaluate the finite size effects and for/3 = 5.8 a n d / ~ = 6.0 to have a rough determination o f a "scaling window". We measured in all cases correlation functions until distance 3 (at least) ; for the 53 x iO size we obtained enough statistics at distance 4 too.

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M. Falcioni et al. / Glueball m a s s

For/3 = 6 we used the heat bath method; the runs were carried out on 53 × 10 and 73 × 10 lattices. For reference we give the mean plaquette energies measured: (E) = 0.4062 ±0.0002

(73 x 10 lattice),

(E) = 0.4059±0.0002

(53 x 10 lattice),

(11)

where the error is the standard deviation. For each step of the system's upgrading we have computed the quantities E(k, i) and W(k, i,j) defined in sect. 2 for the operators O(k). Correlation functions at a distance d are defined step by step as

C( k, d) =-~ ,:,

~fl

~/3

,

where j, j' = 1, 2 . . . . . L such that

j = i + d (mod L ) , j ' = i - d (mod L). To further reduce the fluctuations of the output we have chosen 3 different starting configurations for the 73 x l0 lattice and 5 for the 53 x l0 lattice. In fig. I we plot C(k, 3) as function of the steps (/3 = 6 , s i z e = 5 3 x 10) and in fig. 2 C(k, 2) (/3=6, s i z e = 7 3 x 10). In both cases correlation functions of the 0 ( 4 ) operator are better determined; moreover there is no drift in the approach to the equilibrium of the systems. We performed an average of C(k, d) in the equilibrium regions as long as the two systems .at different temperature looked "close". To determine the glueball mass we solve a non-linear system using correlation functions at distances 0, l, 2, 3, and assuming a contribution of two masses m and M (m < M) as follows:

C( d) = A exp ( - r o d ) + B exp ( - M d ) .

(13)

The final results are (in lattice units) (a) 5 3 x 1 0 , (b) 7 3 × 1 0 ,

m=0.75±0.10, m =0.72±0.18,

M=2.20±0.10, M = 2.30±0.20.

(14)

In case (a) the better statistics permitted us to perform a two-mass fit on correlation functions up to distance 4; the obtained value is m =0.73+0.10,

M =2.20±0.20,

(15)

and it confirms the previous one. The errors quoted are the standard deviations obtained by collecting in several sub-groups the events in the equilibrium regions. In the case of the fit the error is the "parabolic error" given by M I N U I T .

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M. Falcioni et al. / Glueball mass

TABLE I Expectation values of the operator 0(4) for different space slices at t = i (source method) E(i) 2 3 4 5 6

0.3784 + 0.0006 0.4557+0.0019 0.4812+0.0019 0.4894 + 0.0042 0.4924 + 0.0054

In both cases we checked that thermal and spatial loops have a mean value consistent with zero to be well inside the low-temperature confining phase. From these determinations the finite size effects seem to be absent, in this range of/3 and lattice size, where thermal and space loops are disordered; that is expected [7] from the E g u c h i - K a w a i [8] effect for a pure gauge theory. The source method runs were carried out on a 73× 10 lattice a t / 3 = 5.8 for 600 sweeps and keeping " c o l d " the space slice at t = 1. Again, the best data correspond to the o p e r a t o r 0 ( 4 ) ; in table 1 we reported its mean values on the different space slices, E ( i ) , obtained by discarding the first 100 sweeps. The errors are standard deviations relative to groups o f 100 sweeps. To extract the mass gap we have looked at the quantities i = 1 , 2 , 3 , 4 , 5.

D(i)=E(6)-E(i),

(16)

We performed a two-mass fit on i = 1, 2, 3, 4, 5 data (case (a)) and i -- 2, 3, 4, 5 data (case (b)), then a one-mass fit on i - - 2 , 3, 4, 5 data (case (c)) and i = 3, 4, 5 data (case (d)) using respectively the expressions one obtains from (7) and (6): D ( i ) = A exp ( - 6 m ) [ c o s h ( ( 6 - i ) m ) -

1]+ B exp ( - 6 M ) [ c o s h ( ( 6 - i ) M ) - I ] , (17)

D ( i ) = A exp ( - 6 m ) [ c o s h ( ( 6 - i ) m ) - 1],

(18)

obtaining (in lattice units) (a) m = 0 . 9 4 ± 0 . 1 0 ,

M = 2.83 + 0 . 2 0 ,

(b) m = 0 . 9 3 + 0 . 1 0 ,

M = 2.58±0.18,

(c) m = 1.05±0.10, (d) m = 0 . 8 9 + 0 . 1 0 .

(19)

The quoted errors are the "parabolic errors" given by M I N U I T . In fig. 3 we give the comparison o f the fits (b) and (d) with Monte Carlo data. From the mass determinations at/3 = 5.8 a n d / 3 = 6.0 (73x 10 lattice) we extract the following final

631

M. Falcioni et al. / Glueball m a s s

D(i)~ .I

.01

.001 1

I 2

i 3

I 4

5

i

Fig. 3. Fits on D(i): full line refers to case (b) and dashed line to case (d).

result: m g = (305-1" 5 5 ) A L ,

(20)

w h i c h is c o n s i s t e n t with all recent estimates r e p o r t e d in ref. [9].

5. Computation times We r e p o r t C P U times used: (i) 30 h o f C D C 7 6 0 0 to p r o d u c e heat b a t h d a t a a t / 3 = 6.0 on a 73 x l0 lattice (60 sweeps for e a c h o f the 3 starting c o n f i g u r a t i o n s ) ; (ii) 23 h o f C D C 7 6 0 0 to p r o d u c e heat b a t h d a t a at fl = 6 . 0 on a 53 × l0 lattice (80 sweeps for each o f the 5 starting c o n f i g u r a t i o n s ) :

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M. Falcioni et al. / Glueball mass

(iii) 100 h o f VAX 11/780 to produce source method data at/3 = 5.8 on a 73 × 10 lattice (600 sweeps). We t h a n k E. M a r i n a r i for useful d i s c u s s i o n s on the source m e t h o d .

References [I] M. Falcioni, E. Marinari, M.L. Paciello, G. Parisi, B. Taglienti and Zhang Yi-Cheng, Nucl. Phys. B215[FST] (1983) 265 [2] M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. Lett. 42 (1979) 1390 [3] M. Creutz, Phys. Rev. Lett. 43 (1979) 553; Phys. Rev. D21 (1980) 2308 [4] N. Cabibbo and E. Marinari, Phys. Lett. 119B (1982) 387 [5] G. Parisi, in Prolegomena to any future computer evaluation of the QCD mass spectrum, Carg~se lectures, 1983, to be published [6] K. Ishikawa, A. Sato, G. Schierholz and M. Teper, Z. Phys. C21 (1983) 167 [7] G. Parisi, The strategy for computing the hadronic mass spectrum, preprint LN F-83/(36)P (May 1983) [8] T. Eguchi and H. Kawai, Phys. Rev. Lett. 48 (1982) 1063 [9] B. Bergs in Proc. The spectrum in lattice gauge theories, preprint DESY 84-012, Carg~se lectures, 1983, to be published, and references therein