Glueball masses and the loop-loop correlation functions

Volume 197, number 3

PHYSICS LETTERS B

29 October 1987

GLUEBALL M A S S E S A N D THE L O O P - L O O P C O R R E L A T I O N F U N C T I O N S Ape Collaboration M. ALBANESE, F. C O S T A N T I N I l, G. F I O R E N T I N I 2, F. FLORE, M.P. LOMBARDO, R. T R I P I C C I O N E I N F N - Sezione di Pisa, 1-56100 Pisa, Italy

P. BACILIERI, L. F O N T I , E. R E M I D D I 3 I N F N C n a f 1-40126 Bologna, Italy

M. BERNASCHI, N. CABIBBO, L.A. F E R N A N D E Z 4,5,6, E. M A R I N A R I , G. PARISI, G. SALINA I N F N - Gruppo collegato di Roma II, and Universitd di Roma II "'Tor Fergata'; 1-00161 Rome, Italy

S. CABASINO, F. M A R Z A N O , P. P A O L U C C I 7, S. P E T R A R C A , F. R A P U A N O I N F N - Sezione di Roma, and Universitd di Roma "'La Sapienza'; 1-00185 Rome, Italy

P. M A R C H E S I N I CERN, CH-1211 Geneva 23, Switzerland

P. G I A C O M E L L I a n d R. R U S A C K Rockefeller University, New York, N Y 10021, USA

Received 6 August 1987

We analyse the pure gauge lattice QCD by measuring loop-loop correlation functions on a 123 )<32 lattice at fl = 5.9. We select a set of operators given by the smearing procedure. We obtain a good estimate of the mass of the 0 ++ state and for the string tension, and upper bounds for the masses of the 2 + + and the 1+ - states.

In ref. [ l ] we have studied the glueball spectrum in lattice Q C D by using a source method. Here we have used the Ape computer [2] in order to compute directly the loop-loop correlation functions. In this case we get finite-time upper b o u n d s (in a statistical sense) to the true mass of the ground state, Also Universit~tdi Pisa, 1-56100 Pisa, Italy. AlsoUniversitfidi Cagliari, 1-09100Cagliari, Italy. AlsoUniversitfidi Bologna, 1-40126 Bologna, Italy. AlsoMEC (Spain) fellow. On leaveof absencefrom UniversidadComplutensede Madrid, Madrid, Spain. 6 Partially supported by CAICYT (AE86-0029). 7 INFN-Digital Equipment Corporation fellow. 2 3 4 5

400

b u t the signal-over-noise ratio for correlation functions is considerably smaller than for the case of the source method. With the present results we clarify the behavior of the smeared operators [ 1 ], and reach for example a better u n d e r s t a n d i n g of the way in which one can choose optimal operators (with a maximal projection over the glueball state). We think this is a crucial goal during the process of approaching the c o n t i n u u m limit. W h e n we go to higher fl values a n d larger lattices naive operator (a small plaquette, for example) become physically smaller a n d smaller objects, a n d smoothing techniques become crucial.

0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)

Volume 197, number 3

PHYSICS LETTERS B

The results we have obtained show that the two pictures (source and loop-loop method) describe the same situation, and we get mass estimates that are coherent with the ones o f ref. [ l ]. Due to the worse noise-to-signal ratio the mass values obtained here are less accurate than those of ref. [ 1 ], but they have the advantage of representing upper bounds. An extended analysis of the data presented here, will be given elsewhere [ 3 ]. We have used a 5 hit Metropolis algorithm which performs a link update in 50 Ixs, measuring the observables every 20th sweep of the whole lattice. We have performed 22000 iterations on a 123×32 lattice at fl = 5.9 by analysing 20 smeared operators, and 2400 iterations with 100 smeared operators. Before all runs we have discarded 2000 thermalization iterations. We have used about 150 hours of the 256 Mflops 4 unit Ape computer [2]. We consider the connected correlation function between zero m o m e n t u m operators at a time separation t,

C.a(t) = ( O*.(t)Op(O) ) - ( O~(t) ) ( O a ( 0 ) ) ~ ~ i ~i

~.

~p e x p ( - m i t ) .

(1)

In order to get a good determination of the groundstate mass mo we need a signal at large euclidian time separation. We will be interested in operators corresponding to the irreducible representations o f the cubic group. We consider (x and y are chosen as all the possible couples o f spatial directions)

29 October 1987

Table 1 Mass and string tension values. Mass (best operator) time distance 0 ÷+ 2++ 1+~ a

the different coefficients

1 0.92(3) 1.47(2) 2.03(3) 0.0655(11)

states

contribute

2 0.81(5) 1.30(9) 1.98(19) 0.0616(16)

with

positive

C , , ( t ) = ~ Ic~ I 2 e x p ( - m i t ) ,

(4)

is an upper b o u n d to the true mass, r~(t)>~m, V t. This statement only holds in the limit of vanishing statistical error. The smearing procedure has been introduced in ref. [ 1 ]. By iteratively substituting all the links o f the lattice with a gauge-invariant average of the first neighboring links, we transform the elementary plaquette into operators of larger and larger spatial extent. In this way we suppress the unphysical and unwanted short wavelength fluctuations, and try to maximize the projection o f the measured operator over the lowest state in each channel. The m a x i m u m of such projection will be found after g smearing steps, where g is a number that is somehow related to the size o f the given state. The symmetry o f the original operator is preserved under smearing. In table l we give our results for the 0 + +, the 2"- +,

O~A~)++ = Re[O~S)(x, y) + O ~ (y, z) + O~S)( z, x ) ] , O[~2 + = R e [ O ~ ( x ,

1

y) - O~)(y, z)] ,

0~)+_ =Im[OtS~(x, y)] ,

(2)

where s labels the different operators, selected by smearing [1 ], and Ot~)(n, m) is the plaquette operator built with s times smeared links in the n - m plane. We assum~ that the lowest-mass states in each channel are respectively the 0 + +, the 2 ++ and the 1 +We

monitor the behavior of the elective mass defined as

rh(t) = l o g [ C . , ~ ( t ) / C . . ( t + 1 ) ] ,

(3)

which, since for the diagonal correlation functions (as opposed to the case o f the cold wall method [ 1 ])

2.0 -

A

1

........

1

........

I

........

i1

1,5

1.0

J-

........

5

10 Smearing

50

I

100

Fig. 1. Effective mass of the 0 +÷ glueball state, for O(t = 1)/O(t-- 2) and O(t = 2)/O(t = 3) versus the smearing time. 401

Volume 197, number 3

PHYSICS LETTERS B

3.5

29 October 1987

The large smearing step part o f figs. 1 a n d 2 comes from a low statistics run, a n d it is not very useful in d e t e r m i n i n g the best mass u p p e r bound, but it is crucial in o r d e r to show the typical b e h a v i o r o f the smearing operator. O u r best value for the O- + mass is

3.0

2.5

B

m(0++) =0.81(5)

(5)

8.0

from distance 2 over distance 3. F o r the 2 + ÷ mass we can quote

1.5

m(2÷+)< I

. . . . . . .

I 5

10 8moarlag

. . . . . . . 50

100

Fig. 2. As in fig. 1 but for the 2 ÷ + state. the 1 ÷ - a n d the string-tension masses. We consider all the s m e a r e d o p e r a t o r diagonal correlation functions at two different euclidean times. The u p p e r b o u n d to the mass will correspond to the m i n i m u m o f the effective mass as a function o f the n u m b e r o f smearing steps. In figs. 1 a n d 2, which refer respectively to the 0 ÷ ÷ a n d the 2 ÷ ÷ state, one can identify a m a r k e d m i n i m u m close to ~ 20. Clearly the results o b t a i n e d from the ratio o f correlation functions at time 1 over 2 are a worse u p p e r b o u n d than the ones from 2 over 3, but are affected b y a smaller statistical error.

402

1.3(1) •

(6)

I

Fig. 2 shows the effectiveness o f the smearing m e t h o d in obtaining a good b o u n d for the 2 ÷ + mass. F o r the string tension we get our best signal at a very high g, a n d the results we quote at distance 2 could be perhaps slightly i m p r o v e d by measuring with high statistics larger operators (with let say s ~ 30).

References [ 1 ] The Ape Collab., M. Albanese et al., Phys. Lett. B 192 (1987) 163. [2] The Ape Collab., P. Bacilieri et al., Conf. Computing in high energy physics (Amsterdam, The Netherlands, 1985 ); The Ape Collab., M. Albanese et al., Conf. Computing in high energy physics (Asilomar, CA, USA, 1987). [ 3 ] A. Fernandez and E. Marinari, in preparation.