27 May 1999

Ž .Physics Letters B 455 1999 239–245

Monopole loop distribution and confinementž /in SU 2 lattice gauge theory

Michael GradyDepartment of Physics, SUNY College at Fredonia, Fredonia, NY 14063, USA

Received 28 July 1998; received in revised form 26 March 1999Editor: H. Georgi

Abstract

The abelian-projected monopole loop distribution is extracted from maximal abelian gauge simulations. The number ofloops of a given length falls as a power nearly independent of lattice size. This power increases with bs4rg 2, reachingfive around bs2.85, beyond which, it is shown, loops any finite fraction of the lattice size vanish in the infinite latticelimit. q 1999 Published by Elsevier Science B.V. All rights reserved.

A strong correlation has been established betweenconfinement and abelian monopoles extracted in the

Ž .maximal abelian gauge. The entire SU 2 string ten-sion appears to be due to the monopole portion of

Ž . w xthe projected U 1 field 1,2 . Confinement appearsto require the presence of large loops of monopolecurrent of order the lattice size, possibly in a perco-lating cluster. As bs4rg 2 is raised, the finitelattice theory undergoes a deconfining phase transi-tion which is coincident with the disappearance of

w xlarge monopole loops 3 . This transition is inter-preted as a finite-temperature phase transition, onewhich exists if one of the four lattice dimensions iskept finite as the others become infinite, but whichdisappears in the 4-d symmetric infinite lattice limit.

Ž .In the U 1 lattice gauge theory itself, monopoleshave also been identified as the cause of the phase

w xtransition 4 . However, in this theory, monopolesare interpreted as strong-coupling lattice artifactswhich do not survive the continuum limit. The con-tinuum limit is non-confining.

Assuming the presence of large monopole loops isa necessary condition for confinement, in order for

Ž .the SU 2 theory to confine in the continuum limit itis necessary for some abelian monopoles to survivethis limit, i.e. to exist as physical objects. Evidencehas been presented of scaling of the monopole den-

w xsity which would make this so 5 . However, it is notsufficient just to have some monopoles survive thislimit; one needs large loops of a finite physical sizeto survive. Small loops of finite size on the lattice,which are by far the most abundant, will shrink tozero physical size as the lattice spacing goes to zero,becoming irrelevant. It is believed that to causeconfinement, monopole loops must be at least aslarge as the relevant Wilson loops of nuclear size,and may need to span the entire space.

Consider a large but finite universe, representedas an N 4 lattice with lattice spacing a. The stronginteractions should not care whether the universe isfinite or infinite, so long as it is much larger than ahadron. One can then take the continuum limit as

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 00449-9

( )M. GradyrPhysics Letters B 455 1999 239–245240

a™0, N™`, simultaneously, holding Na constantat the universe size. It is then clear that the size ofany object that is to remain of finite physical size inthe continuum limit must also become infinitelylarge in lattice units, with linear dimension propor-tional to N, i.e., some finite fraction of the fulllattice size. It would therefore appear that at the veryleast, monopole loops some finite fraction of thelattice size must survive the continuum limit if it isto be confining, and probably loops at least as largeas the lattice itself. Recently, it was shown that if theplaquette is restricted to be greater than 0.5, the loopdistribution function falls so fast that no monopoleloops any finite fraction of the lattice size survive in

w xthe large lattice limit for any value of b 6 . Here itwill be shown that for the standard Wilson action thesame is true if b)2.85.

By studying the monopole loop size distributionfunction, one can tell how quickly the probability offinding loops of increasing size decreases with loopsize. This function turns out to have very little finitelattice size dependence for loops of length less thantwice the lattice size. Thus one can fairly confidentlyextrapolate this function, which appears to be asimple power law, to the infinite lattice, from whichthe probability of finding loops of finite physical sizecan be determined. Due to the dependence of thepower on b , it is shown below that the probability offinding loops whose length is a finite fraction of thelattice size vanishes in the large lattice limit for allb)2.85.

Gauge configurations from the simulations weretransformed to maximal abelian gauge using the

w xadjoint field technique 7 . Abelian monopole cur-rents were then extracted using the DeGrand–Tous-

w xsaint procedure 8 . Sample sizes ranged from 5004 Ž .configurations for the 20 lattices 1500 at bs3.1 ,

to 5000 for the 84 lattices, always after 1000 equili-bration sweeps.

Ž .Define the loop distribution function, p l , as theprobability, normalized per lattice site, of finding amonopole loop of length l on a lattice of any size.The probability of finding a loop of length N or

4 4 Ž .larger on an N lattice is given by N I N where Iis the integrated loop distribution function

8 N 4

I M s p l d l , 1Ž . Ž . Ž .HM

where, since N will be taken large, the discretedistribution has been replaced by a continuous one.The upper limit comes from the fact that each duallink can contain at most two abelian monopoles. Toget confinement from the monopole mechanism, atleast some finite fraction of lattices would have to

Žcontain loops of length order N or larger. Somewould argue that loops of size N 2 or larger might be

.necessary due to loop crumpling . Conversely, if

lim N 4I N s0 2Ž . Ž .N™`

then there will be no loops of length N or larger onthe N 4 lattice in the large lattice limit, and confine-ment from the monopole mechanism will not bepossible.

Ž . Ž .In Fig. 1, log p l is plotted versus log l for10 10

various b and lattice sizes. The data are consistentŽ . yqwith a power law, p l A l , for loops up to around

Žlength ls1.5N the size 4 loops, which fall well.below the trend are excluded from the fits . This is

w xconsistent with the results of 9 where the loopdistribution was studied over a narrower b range.The larger the lattice, the further the power law isvalid before some deviation occurs at large l. Alsonote that the 124 and 204 data are virtually identicalfor loops up to size 30 or so, and even the 84 arehardly different. Linear fits were made for loop sizes

Ž 4in the range 6 to 1.1 N only the 12 fits are shown.for clarity . For larger loop sizes, occasionally zero

instances of a particular size was observed. Zeroscannot be plotted on the logarithmic scale, but ifignored the data will be biased upward. A movingaverage was used in this circumstance to properlyaccount for the zero observations.

The deviations from linearity for large loops canbe easily understood as a finite size effect coupledwith the periodic boundary condition. For loopslonger than about 1.5N, there is a significant proba-bility of reconnection through the boundary. Thismakes a would-be large loop terminate earlier than itwould on an infinite lattice. Thus, on a finite latticethere must be a deficit of very large loops, and anexcess of mid-size loops due to reconnection. Atbs2.4 and 2.5 this is especially apparent. Thelinear trend continues further for the 204 lattice thanfor the 124, which itself continues further than the84. At some point starting around 1.5N, but not

( )M. GradyrPhysics Letters B 455 1999 239–245 241

Ž . Ž . Ž .Fig. 1. Log–log plots of loop probability versus loop length: a bs2.4 upper graph, right scale , bs2.5 left scale, shifted for clarity ;Ž . 4b from upper to lower, data series are bs 2.6, 2.7, 2.8, 2.9, 3.0, 3.1; linear fits are to upper region of the 12 data, as explained in text.

( )M. GradyrPhysics Letters B 455 1999 239–245242

readily apparent until about 4N, there is a bulge ofexcess probability, followed eventually by a steepdrop that falls below the linear trendline. Often thedata cuts off before it falls below, but the fact that noloops larger than those plotted occurred can be usedto infer that the probability distribution must eventu-ally fall below the trendline. For instance if oneassumes that very large loops follow the trendline forthe 124 data at bs2.4, then one can calculate that3.5 instances of loops longer than 1250 lattice unitsshould have been seen in the sample. The fact thannone occurred implies that the data most likely doesfall below the trend line for very large loops, andcertainly could not remain significantly above it.

Because the power law trend continues further thelarger the lattice and the deviations always occur forl)N, it seems quite reasonable to assume that onthe infinite lattice one would have a pure power law.It is difficult to imagine what length other than thelattice size could set the scale for a change inbehavior at extremely large loop sizes beyond thosemeasured here. In addition, since the small loop dataare nearly independent of lattice size, it would seemreasonable that the power, which is completely deter-mined by the small loops, must also be essentiallythe same on the infinite lattice as observed here for

4 4 Žthe 12 or 20 lattices some slight variations are4.visible on the 8 . Assuming this, one can easily

Ž .predict the point at which condition 2 becomessatisfied, namely q)5. For q)5 the probability ofhaving a monopole loop with length equal to anyfinite fraction of the lattice size N, or larger, van-ishes in the large lattice limit, whereas for q-5 thesame becomes overwhelmingly likely. This is be-

Ž .cause for q)5 not only is condition 2 satisfied,but also

lim N 4I NrM s0 3Ž . Ž .N™`

for any finite M.The power q is plotted as a function of b in Fig.

2. A definite rising trend is observed, in sharp con-w xtradiction to the conclusion of Ref. 9 , where a

constant value of q was inferred from runs over aŽ .rather narrow b range 2.3 to 2.5 . It is seen that q

apparently passes 5 around bs2.85. For any b

beyond this, including the continuum limit, b™`,there will be no loops any finite fraction of the

Fig. 2. Power, q, characterizing the decrease of loop probabilitywith loop length, versus b. Lines are drawn to guide the eye.

lattice size in the infinite lattice limit. A small resid-ual finite size dependence in q is possible, whichcould shift the infinite lattice critical b to 2.9 orpossibly 3.0, but it is hard to picture how this couldprevent q from ever reaching 5, which would benecessary to have large loops survive the continuumlimit.

Error bars in Fig. 2 were obtained from the leastsquares fits. If data for different loop sizes werehighly correlated, then these errors could be underes-timated. As a test, correlation coefficients were com-puted between the numbers of each size loop occur-ring on a lattice, for samples of 1000 124 lattices atboth bs2.6 and 2.9. The degree of correlation wasin all cases very small, with coefficients, r, averag-

Ž .ing 0.02 with a maximum of 0.04 at bs2.6 and0.003 at bs2.9. These fall within the expectednoise level of 0.03 for these sample sizes, and arecertainly small enough to justify ignoring correla-tions in the fits. The autocorrelation in Monte Carlotime was also computed and found to fall belownoise after two Monte Carlo sweeps. One might alsoask with what confidence one can reject the hypoth-esis qF5 at bs3.0 or 3.1. At bs3.1 on the 204

lattice, q was found to be 7.65"0.32. If one as-sumes qs5 instead, then the number of loops of

( )M. GradyrPhysics Letters B 455 1999 239–245 243

size 12 links and larger that should have occurred inŽ .the sample based on the number of six-link loops is

285, whereas the actual number was 35. The possi-bility of this occurring is, by raw Poisson statistics,less than 10y76. At bs3.0 the same analysis rejectsthe hypothesis at the level 10y59. This shows that onthese lattices it is virtually certain that q exceeds 5,even if some latitude is allowed for the modestMonte Carlo time correlations.

It is interesting to note that the above conclusionsare insensitive to the precise behavior of the distribu-

Ž .tion for l)N. If p l follows a different power lawwith exponent qX for l)N, qX could be as small as

Ž .unity and still condition 2 will hold. For instance, ifqs5qe and qX s1 then

ly5ye , lFNp l s , 4Ž . Ž .½ y4ye y1N l , l)N

so

8 N 4y4ye y1I N AN l d l , 5Ž . Ž .H

N

4 Ž . ye Ž 3.and N I N AN ln 8 N which still vanishes asN™`. Thus the presence of large enough loops to

cause confinement is, paradoxically, controlled al-most entirely by the distribution function for loops oflength smaller than N, because this determines thebase probability for the case lsN. This observationmakes the above result far more robust, since thedistribution for l-N is seen to always follow a purepower law which is almost independent of latticesize. It was the distribution for l)N which wassomewhat less certain, but all that one needs to knowabout this part of the distribution is that it falls atleast as fast as ly1, not a very stringent requirement.As before, the loop size cutoff for the condition canbe taken to be NrM instead of N, where M is somefinite number. This further reduces the likelihood ofthe finite lattice size affecting the important small-loop part of the distribution.

On our lattices, as in many previous studies,confinement appears to occur at couplings for whichloops that wrap completely around the periodic lat-tice are common, and the theory is deconfined whensuch loops are absent. This is illustrated in Fig. 3,where the probability that the largest loop has anon-zero winding number is plotted versus b , alongwith the Polyakov loop. The rather sudden rise from

Ž . Ž .Fig. 3. Polyakov loop filled symbols and winding probability of largest loop open symbols versus b.

( )M. GradyrPhysics Letters B 455 1999 239–245244

zero in winding probability is coincident with thepoint of largest slope in the Polyakov loop, whichsignals deconfinement. For the 204 lattice, the aver-age Polyakov loop itself is not a very sensitive testof confinement, but moments of the Polyakov loopbecome decidedly non-Gaussian at this point. For

Ž .instance both the first absolute value and fourthmoment are consistent with Gaussian for bF2.7,but not for b)2.7 where the distribution is notpeaked at zero and the theory is deconfined. If loopsin addition to the largest loop were also tested, thenet winding probability would likely become veryclose to unity when deeply into the confining region.Loops which wind the lattice occur in pairs, with aDirac sheet stretched between them. Once a windingpair exists, the loops can drift apart without increas-ing their length, stretching the Dirac sheet to coveran arbitrarily large section of lattice and incurring noadditional energy penalty in doing so. Such a Diracsheet could easily disrupt the values of Polyakovloops passing through it only once, resulting in arandom, i.e. confined, Polyakov loop average. Verylarge non-winding loops would also have large Diracsheets having the same effect. However, the windingconfigurations, having shorter loops, have a lowerenergy, so would appear first at the periodic latticedeconfinement transition.

Since large loops must disappear when q hits 5, ifone accepts the observation that large loops areassociated with confinement, one would have toconclude that the infinite 4-d symmetric lattice mustundergo a deconfining phase transition at a finitevalue of b around 2.9, rather than at bs` as isusually assumed. Of course, one could instead giveup the link between abelian monopoles and confine-ment, despite the strong evidence in favor of aconnection. Since large abelian monopole loops havebeen shown to be responsible for most if not all ofthe string tension, if the theory still confines whenthey are absent it will have a much smaller stringtension. In this case the theory would confine in thecontinuum, but could be quite different in detail fromthe usual lattice theory in the crossover region of theWilson action, since the latter would be highly con-taminated with monopoles, seen here as essentiallylattice artifacts.

Let us return to the possibility that the continuumnon-abelian pure-gauge theory really doesn’t con-

w xfine, an idea which has been suggested before 10,11 .The renormalized coupling could have an infrared-stable fixed point, and stop increasing at some dis-tance scale. The heavy-quark potential would be alogarithmically-modified Coulomb potential, and thegluons would be massless. How is one then toexplain confinement in the real world? It may be thatthe real-world confinement that we see is a manifes-

w xtation of chiral symmetry breaking 10,12 . Even in anonconfining theory, chiral symmetry can breakspontaneously if the coupling is strong enough. In-stantons will also likely play a role here. Once achiral condensate is established, it can produce aconfining force by expelling strong color fields; i.e.if the condensate abhors external color fields, hadronswill expel some condensate from their immediatevicinity, creating a region of higher vacuum energyproportional to their volume, a sort of chiral-expelledbag. Stretching of this bag would result in a linearpotential. The energy density of the bag which wouldfollow the movement of quarks may also be respon-sible for the quarks’ dynamical mass. This picture is

² :supported by the observation that cc is loweredw xin the neighborhood of a color source 13 , indicating

some expulsion of condensate. Similar ideas arew xcontained in chiral quark models 14 where a polar-

ized Dirac sea is responsible for the binding of thequarks in a baryon, and also in the instanton liquid

w xmodel 15 . Both of these models are able to com-pute with fair accuracy a large number of low-energyproperties of hadrons, and neither has an absolutelyconfining potential. Further details on this scenario

w xand justifications are given in Ref. 6 and referencestherein.

Acknowledgements

It is a pleasure to thank Richard Haymaker foradvice on implementing the adjoint field technique.

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