Noncompact simulations of SU(2) gauge theory at strong coupling

240 Nuclear Physics B (Proc. Suppl.) 20 (1991) 240-244 North-Holland

NONCOMPACT SIMULATIONS OF SU(2) GAUGE THEORY AT STRONG COUPLING

Kevin Cahill

Department of Physics and Astronomy University of New Mexico Albuquerque, New Mexico 87131 USA

Wilson loops have been measured on a 124 lattice in noncompact s~mutations of pure SU(2) without gauge fixing at strong coupling, g = 0.5, and at moderate coupling, j~ = 2. There is no sign of quark contin~men!t in the loops that have been measurecl accurately.

THE QUESTION

In 1974 Wilson showed that in the strong- coupling limit of his compact formulation of lattice gauge theory, static quarks are confined by a lin-

ear potential for both abdian and nonabdian gauge groups. 1 A few years later, Crentz and others dis- played quark confinement at moderately strong cou-

. piing in wilsonian lattice simulations of both abdian and nonabdian gauge theories. 2 Since the confine- ment that occurs in abdian simulations is dearly an artifact of Wilson's formulation, it has never been entirely clear whether the confinement seen the non- abdian simulations is another artifact of Wilson's formulation or an actual property of QCD.

The basic variables in Wilson's formulation are

dements of a compact gauge group rather than real numbers as in continuum gauge theory. Both abdian

and nonabdian theories confine at strong coupling in this formulation because the Wilson action is a function of the product of the group dements of the

links around the dementary squares of the lattice. The Wilson action consequently possesses multiple minima, some of which correspond to false vacua not present in the continuum theory. 3 These extra vacua

contribute importantly to the string tension, as has been shown by Mack and Pietarinen 4 and by Grady. 5

In their simulations of SU(2), they modified Wilson's action gauge invariantly by erecting, in effect, infinite potential barriers between the true vacuum and the

false vacua. Mack and Pietarinen found a sharp drop in the string tension; Grady found no string tension at all,

NONCOMPACT METHODS

To avoid this artifact of the Wilson action° some physicists have devdo'~ed MGnte Ca~o simulation methods 3"0-13 that do not have confinemeflt built in_ These methods are called ~noncompac~" b~zause their basic variables are fields rather than group merits as in Wilson's ~c~ompacC method. For U(J) these noncompact methods seem accurate ~or gen- era! coupling strengths: 9 for SU(2) they agree- wall ¢¢ith perturbation theory at very. weak coupling. 1-1 ~12

In this report I describe the results of measur- ing Wilson loops on a 12 ~ lattice in a noncompact simulation of SU(2) gauge theory without gauge fix- ing or fermions at strong coupling, ;~ ~ 4 / g ~ = 0.5.

and at moderate coupling, ~ = 2_ Crenl2 ratios of

large Wilson loops provide a lattice estimate d the

qf/-force for heavy quarks. There is no sign of quark confinement at either coupling in the loops that have been measured accurately, tn the largest data set at

= 0.5, however, the force at six lattice spacings is stronger than that at four and five lattice spacings.

which does suggest confinement_ But the errors here are very large, and the relatively large value of the force at six lattice spadngs is probably a trans/ent effect.

Patrascioiu. Seller, and Stamatescu 6 performed

the first noncompact simulations of SU(2) using a simple discretization of the classical action. They

fixed the gauge to avoid the effects of zero modes and saw a force rather like Coulomb's. Seller, Sta- matescu, Wolff, and Zwanziger 7 used a more sym-

metrical discretization of the classical action with

~-~:::~ 5632/9!/$3.50 © Elsevier Science Publishers B.V. (North-Holland)

K. Oal~ill/SC-(2) gauge theory at strong coupling

fewer zero modes+ but stiff had to fix the gauge. They also saw a coulombic force.

THIS NONCOMPACT METHOD

I have described etse~here I2 the code that I

used in the present simulations_ The t idds are con-

s tant on the links of length ~0 the latt ice ~pacing. hut are interpolated linearly throughout the s~x plaque- ttes that are transverse to each link. in the p|aquette

field is

and the ~etd s t r eng th / s g/~en by the c~ati~uum for- mula

+2;r The act++. ++ is the sum o~er zl] plaquette~ of the m~tegrals over each plaqu~tte of the squared fie~l strength,

S = ~ -3- "i'~.a-'~F2- +~-+ ~3}

The mean-value in the vacuum I_Q} of a eudideae~ time-ordere~] operator O~.4) may the~ be apl~,ox- imated ~-]f a n~.mat~zed muff/~e integral over the

. ~ : ( . ) ' s

f '+-st+'̀ ~) l]..~_~ d A ~ . )

which one may estimate by Crentz's Monte Carlo techniques. 2

CREUTZ RATIOS

The quanti ty normally used to study confine-

ment in quarkless gauge theories is the Wilson loop.

lt:(r~ t), which is the mean-value in the vacuum of

the path-and-time-ordered exponential

|'|'(,'01) = ~-(-ql~'~c-i#'+'r°~"IO~ (5)

divided by the dimension d of the d x d matrices

"/~ that represent the generators of the gauge group_

241

Although Wilson loops vanish 14 in the exact the- ory, Creutz ratios \ ( r+ ! i of Wilson loops defined 2 as double differences of logarithms of Wiison loops

'~+r+fi = -Inll-(r.~i-lnII'i~--../-~

+ In [ | ' ( r - - . . Q + In l I ' (r+/ -- ,~+(6)

are fin/to; and foe large ! pro'aide an estimate of ,~-+

times the force between a q~ark and an antio~a~rk ~eparated ~y the di~ance ++_

For a compact E+e group wfth .V generatocs 71+ no~maliz~ as Tr! !,,t+ ~ = &'~ ~,,, the [ o ~ t - ~ d ~ per+ turbat~ve focm.u|a f~r the Cre~tz rat~ ~s

w~e~e the f~n~_/o~ f(~: t i: is

- ~ + {s}

and ~ is the i n v e ~ squared c ~ l ~ . g .+~ = ,f/r £'~+;=~]-

MEASUREMENTS A~D RESULTS To meagre ~ ' / ~ e rca~ps at~ t ~ e C ~ z e~

tios ~fr.~+ I used a t:~= 1~6od/c |art/co and a f~,a~

bath. and made t E ~ ~ t . ~ g r u ~ . eack

beginmug mfl~ a cold starL i . ~ all ~ ~ e

init/aEzed to zero. Tim firs~ ran b~lgan ~ 2S.00~ therma~zing smeegs at :J = ~ on the ~ 7000 o~ which i made 350 measurements. I t = ~ smgtched .~ from 2-0 to 0-S and d-a] SO00 ~ / a g I then did 60+000 mm~=ps a t . ; = 0~-3 m ~ a g 3000

measurements. The second and third rum both be-

gan vv~h 20~000 ~ i z i n g sweeps at ::~ = 0.-~ and remained at +~ = 0_5. I c~.~nued the ~ mn

doing 60.000 s=meps, making 3000 mea~remeats ,

and the third run by doing 25.000 s=~eps~ maidng

12S0 measurements. Successive measurements were

always separated I~ twenty thermal=zing s~tm~I~.

In all three runs. I measured Wilson loops 1~_ using Parisi's trick 15 in an impiem~atat~ that r~

speets the dependencies in the corners of the .~oaps

The values of the Crentz ratios so obtainc~ from

the 7,250 measurements at -J - tLS a~d the ~5~

242 K. Cah~H / 5U(2) gauge theory at strong coupling

at f l = 2.0 are listed in the table. The error es- timates listed in the table within parentheses are the averages of the asymmetric errors given 5y the bootstrap method 16 under the assumption that suc- cessive measurements were independent. Binning in groups of 2, 4, and 8 made little difference, but there may be longer correlations.

,Noncompact Creutz ratios.

x t. i ~=0 .5

2 x 2 0.2311(1)

3 x 3 0.03594(31)

4 x 4 0.00511(84)

5 x 5 0.00216(237)

6 x 6 0.00979(654)

,'3= 2.0 I

0_06930(8)

0 0~84800) , i 0.00395(12) ,l

o.ooo88(18) I 0.00036(26) :I

If the static force between heavy quarks is inde-

pendent of distance, corresponding to a linear con- fining potential, then the Creutz ratios x(r~ ~) should be independent o f r and ~ at least for large t. There es no sign o f confinement in these noncompact simu- lations. Even at ~ = 0.5, the loops that are reason- ably well measured are all smaller than those given by the lowest-order perturbative formula (T) and fall o~ff faster with r and ~_

One might wonder whether there is a sign of confinement in the fact that the (i × G ratio is bigger than the4 × 4 and 5 x 5 ones. But the error in

X(ga,6a) is huge, and the large value o f X(6a~6a) is likely to be a transient effect that would disappear in a longer simulation.

The noncompaet lattice spacing . : v : . i ,~i is prob- ably much smaller than the compact .one .r.I' ~9- ~ so, then confinement might appear iF* no,compact simulations done on much larger lattices o~ at much stronger coup]i~lg_ Both possibilities ~,~ou~d be ex- pensive to test.

Possibly as Grihov has sugg~te~ ~7 pure SU(3) does not confine~ .confinement he~g a treatufe on~, of QCD with figkt quarks. The~ the confinement seen in compact lattice SO(3) ~,¢~d be as m~a.rJ~ a~ artifact of ~J~son's me~hod as ~s lhe confineme~t see.v in compact lattice U(1)~

~erhaps the textbook quantiza~io~ o~ QCD. which th~s noncompact method em~ates. ~ e s not cor~ectb] implement Gausses la~v. PoKm~,~ ~as suggested !8 that Gauss's law should ~: e eafo~ced b~

an integration over the group man,old weighted by the invar~ant Haar measure, rather than by the usual integration over cop~es of the re~l ~i~e_ This pc6s~- bility is very inter~fing_

Pos~bly QCD is wrong and does not confine. Then the correct continuum theory might be one with an acllon doser to ~lilson's lattice action than to the action o f continuum Q C D

ACKNOWLEDGEMENTS I am grateful to H Bryant, M_ Creutz. G_ Kilcup.

J_ Pc4onyi0 and D_ Topa for useful conversations_ This research was supported by the Department of Energy under grant DE-FG04-84ER40166 and was

done on a DEC_station 3100 computer_

PLAUSIBLE INTERPRETATIONS

Why don't noncompact simulations display quark confinement? Here are six dhTerent answers:

Noncompact methods lack an exact lattice gauge invariance, although they have approximate forms of continuum gauge invariance. The methods may thus be inaccurate at strong coupling, and their failure to display confinement may he due to their lack of exact gauge invariance.

In Nature the confining gauge theory is the one associated with the group SU(3) not SU(2). Perhaps continuum SU(2) gauge theory is not a confining theory.

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