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GLUEBALL CALCULATIONS IN LATTICE GAUGE THEORIES
B. Berg
To cite this version:
B. Berg. GLUEBALL CALCULATIONS IN LATTICE GAUGE THEORIES. Journal de Physique
Colloques, 1982, 43 (C3), pp.C3-272-C3-277. �10.1051/jphyscol:1982355�. �jpa-00221910�
JOURNAL DE PHYSIQUE
CoZZoque ~ 3 , supptgment au n o 12, Tome 43, de'cembre 1982 page C3-272
GLUEBALL CALCULATIONS I N L A T T I C E GAUGE THEORIES
B. Berg
*
CERN, Geneva, SwitzerZand
1. Introduction.- Within the last year there has been considerable progress in cal- culating the glueball spectrum for SU(2) and SU(3) lattice gauge theories by means of Monte Carlo (MC) methods. In these notes I will concentrate on the main results:
reliable SU(2) and SU(3) mass gap (m(O++)) estimates now exist and some very prelimin- ary results have been obtained for various spin excitations. A more detailed discus- sion can be found in my recent concise review1).
Most of the results are based on an MC variational (MCV) method, which has turned out to be .rather powerful. The method was suggested by wilson2) and first applied in- dependently by three groups3)-5) to the SU(2) theory. It is an adaptation of the Ritz variational principle of quantum mechanics to MC calculatiors in Euclidean lat- tice gauge (or spin) theories. Let us start from the variational definition of the mass gap
(la)
(T = transfer matrix), and make a truncated ansatz
for the states entering Eq. (1). Here,@. (i = l,...,n) is a "suitableff selected set of gauge invariant operators (Wilson loop; in arbitrary representations). The relevant expectation values can now already be obtained by means of an MC calculation of cor- relations coi(0)Oj(t)> (i, j = 1,
...,
n) at distance t = 1, and an upper bound for the glueball mass is obtained by means of Eq. (1). In favourable cases the bound may be very close to the real value. If MC statistics allow it, there are various obvious consistency checks (which improve the obtained bound if they fail) by incorporating correlations from distance t = 2, 3,... .
For details, see Refs. 3)-5). By restric- ting the trial statesI$>
in Eq. (2) to the irreducible representationI$>R,
R~~ = ATC, A,'! E ~TTC, ~ ,
~z~
(P = parity, C = C-parity) of the symmetry groups of the cube, connection with spin eigenstates J~~ is made in the continuum limit. Details and a construction of all irreducible representations R~~ on Wilson loops up tolength 8 are given in Ref. 6). The result is that for J = 0, 1, 2, 3 all possible P = +- C = i: combinations are obtained: in particular, ftoddballsfl which cannot mix with flavourless q{ bound states of the quark model.
*on leave from I1 Institut fcr Theoretische Physik, Universitzt Hamburg, F.R.G.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982355
B. Berg
2. The SU(2) Mass Gap.- The MCV mass gap e s t i m a t e s o f Ref. 3 ) and Ref, 5 ) a r e n e a r l y i d e n t i c a l , and a s i m i l a r v a l u e i s a l s o obtained by t h e Rome group4). A d i r e c t com- p a r i s o n is n o t p o s s i b l e , because i n Refs. 3 ) and 5 ) t h e Wilson a c t i o n i s used, whereas Ref. 4 ) makes use of t h e Manton a c t i o n .
E a r l y MC e s t i m a t e s 7 ) o f t h e SU(2) mass gap r e l i e d on a s i g n a l f o r t h e o n s e t o f s c a l i n g . These e s t i m a t e s were n o t s e l f - c o n s i s t e n t i n t h e s e n s e t h a t s c a l i n g could be followed over a f i n i t e r e g i o n i n t h e c o u p l i n g 6 ( s c a l i n g window). S i m i l a r l y , t h e e s t i m a t e from matching a t a n g e n t t o t h e e i g h t h o r d e r s t r o n g c o u p l i n g (SC) expansion o f ~ i i n s t e r ~ ) i s n o t s e l f - c o n s i s t e n t i n t h i s s e n s e . Therefore i t i s p a r t i c u l a r l y s a t i s f y i n g t h a t i n t h e MCV i n v e s t i g a t i o n o f Ref. 3 ) a s c a l i n g window was found.
F i g u r e 1 reproduces t h e main r e s u l t o f Ref. 3 ) , Addendum, a f t e r minimization. The f i n a l mass gap e s t i m a t e i s
The r a t h e r s m a l l s c a l i n g r e g i o n ( a s compared w i t h t h e s t r i n g t e n s i o n 9 ) ) is explained by a d i f f e r e n c e i n t h e s p i n wave behaviour.
M = ( 1 7 0 . t 3 0 . ) A L
2
.-
t- - - . .
. . . . . .
1.625 1.875 2.125 2.375 2.625
: S U ( 2 ) mass gap from R e f . 1 3 ) and Addendum
Another convincing s c a l i n g curve was p r e s e n t e d by Miitter and ~ c h i l l i n g l ' ) and i s reproduced i n F i g . 2. T h e i r e s t i m a t e
i s based on s t u d y i n g t h e dependence of t h e p l a q u e t t e a c t i o n on boundary c o n d i t i o n s . The MCV method h a s t h e advantage o f p r o j e c t i n g ( i d e a l l y ) o u t t h e lowest e i g e n s t a t e above'the vacuum and having only z e r o momentum c o n t r i b u t i o n s . It i s t h e r e f o r e expec- t e d t o g i v e a somewhat lower value t h a n t h e method o f Ref. 1 0 ) .
JOURNAL DE PHYSIQUE
lo4 Lattice
Fig.? : SU(2) mass gap from Miitter and schilling1').
3 . SU(2) Excited G l u e b a l l S t a t e s . - The s i t u a t i o n has t o be c l a r i f i e d . The Rome group41 f i n d s s p i n 2+ and 3+ s t a t e s i n t h e range m(0+) f &n, where &n i s roughly 20% of m(0+). Thus w i t h i n t h e i r r e s o l u t i o n , they cannot d i s t i n g u i s h t h e s e e x c i t e d s t a t e s from t h e mass gap m ( O + ) .
On t h e o t h e r hand, Ishikawa, Teper and .Schierholz5) o b t a i n
Apart from t h e use of d i f f e r e n t a c t i o n s , t h e d i s c r e p a n c y may very w e l l be explained by a l a c k of s c a l i n g .
B. Berg C3-275
4. The SU(3) Mass Gap.- Finite size scaling indicated1') the rather large value
m(0*) = (720
+
100)k
(4a)Using the MCV method a self-consistent analysis has been done by Billoire and myself12). In Figure 3 our original curve from correlations at distance t = 1 is reproduced. The required amount of computer time was surprisingly small. This might be explained by the large number of statistical variables of a single SU(3)
H = [ 350. ? so. I
3
4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2
B
Fig.3 : SU(3) mass gap estimate from Ref. 12).
matrix. Since Ref. 12) was submitted for publication, we have taken new data 6
,
which allow us to include correlations at distance t = 2.(and in part t = 3) into the analysis. This lowers the estimate to the value
Independently, three investigations 8),13) have now converged to identical SC expan- sions for the SU(3) mass gap. Using a naive "tangent method" m(0++) n 310 is ob- tained. Further, the order of magnitude (4b) has been confirmed by another MCV investigation14)
.
C3-276 JOURNAL DE PHYSIQUE
5. SU (3) Excited Glueball States
.-
So far, Billoire and I ~ ) have carried out an MCV calculation which is based on Wilson loops up to length six. With a statistics of 11 200 sweeps at each point we have analyzed data at the values 6 = 5.2 (close to the SC region) and 6 = 5.6. The results are summarized in the Table. They are preliminary and we do not see scaling. This suggests extending the analysis to aTABLE
-
Preliminary resrllts for excited SU(3)-
glueball states at 6 = 5.2 and f3 = 5.6 in units of m(OCC) 22) 124).State 0--
1+- 1-+
2+' 2+- 2-+
2-- 3++
3+-
larger lattice and including all Wilson loops up to length eight in the MC investiga- tion. Results are presently obtained. Some preliminary results for excited SIJ(3) glueball states have also been obtained by the Hamburg group14). They were reported by G. Schierholz at this conference.
In contrary to SC predictions15)916) all MCV results for excited SU(3) glueball states predict a large gap between the m(O++) state and all other (so far investigated) glueball states. The SC extrapolations may fail due to a too complicated singularity structure. However, the MCV results are also still far from being on solid ground.
6 = 5.2 22.5 2.4 f0.2
22.4 1.72t0.05 2.6 2.3 t0.2 2.4 50.2
22.3
22.5
6. Summary.- Scaling curves 3)210) ,12) for the SU(2) and the SU(3) mass gap have been obtained. We can hardly expect much more from present-day MC investigations; it would, however, be interesting to investigate the problem of universality in more detail.
Conceptual problems concerning the investigation of excited glueball states by means of MC methods on a finite lattice have been settled6). Also, preliminary MC results for some states were obtained6) 914). A lot of work remains to be done to get these results on a similar footing, as we have the mass gap results now.
8 = 5.6 +0.5 3*9 -0.4 4.1 M.3 +0.9 4'7 -0.5 2.96f0.06
24.7 3.4520.11 4.6
1;:;
3.74ro.25 3.8
::::
+
REFERENCES
1) BERG B., Lecture notes presented at the John Hopkins Workshop, Florence, (2-4 June 1982) ; CERN preprint TH-3327 (1982).
2) WILSON K., Communication at the Abingdon Meeting on Lattice Gauge Theories (March 1981).
3) BERG B., BILLOIRE A. and REBBI C., Ann. Phys. (N .Y. ) 142 (1982) to appear and Addendum, to be published.
4) FALCIONI M., MARINARI E., PACIELLO M.L., PARISI G., RAPUANO E., TAGLIENTI B.
and ZHANG-YI-CHENG, Phys. Lett. (1982) 295.
5) ISHIKAWA K., TEPER M. and SCHIERHOLZ G., Phys. Lett. (1982) 399.
6) BERG B. and BILLOIRE A., Phys. Lett.
-
114B (1982) 324, and in preparation.7) BERG B., Phys. Lett. (1980) 401;
BHANOT G. and REBBI C., Nucl. PhyS. B180 rF~2] (1981) 469.
8) MUNSTER G.
,
Nucl. Phys. B190 [FS~] (1981) 439; 1E: [FS~] (1982) 536; 2E:B205 [FS~] (1982) 648-
9) CREUTZ M., Phys. Rev.
g
(1980) 2308.10) MUTTER K.H. and SCHILLING K., CERN preprint TH-3246 (1982).
11) HAMBER H. and PARISI G., Phys. Rev. Lett.
47
(1981) 1792.12) BERG 8 . and BILLOIRE A., Phys. Lett,
2
(1982) 65.13) KIMURA N. and UKAWA A., unpublished;
SEO K., University of Chicago preprint EFI 82-10 and Erratum (1982).
14) ISHIKAWA K., TEPER M. and SCHIERHOLZ G., DESY preprint 82-24 (19821, and in preparation.
15) KOGUT J.
,
SINCLAIR D .K. and SUSSKIND L.,
Nuc~. Phys. (1976) 199.16) SMIT j,
