Dual string from lattice Yang-Mills theory

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Dual string from lattice Yang-Mills theory V .I.Zakharov Max-Planck Institut für Physik Werner-Heisenberg Institut F?hringer Ring 6,80805,Munich E-mail:xxz@mppmu.mpg.de Abstract.We review properties of lower-dimension vacuum defects observed in lattice simulations of SU(2)Yang-Mills theories.One-and two-dimensional defects are associated with ultraviolet divergent action.The action is the same divergent as in perturbation theory but the ?uctuations extend over submanifolds of the whole 4d space.The action is self tuned to a divergent entropy and the 2d defects can be thought of as dual strings populated with particles.The newly emerging 3d defects are closely related to the con?nement https://www.sodocs.net/doc/2412731873.html,ly,there is a kind of holography so that information on the con?nement is encoded in a 3d submanifold.We introduce an SU(2)invariant classi?cation scheme which allows for a uni?ed description of d =1,2,3defects.The scheme ?ts known data and predicts that 3d defects are related to chiral symmetry breaking.Relation to stochastic vacuum model is brie?y discussed as well.INTRODUCTION Studies of the con?nement mechanism have become since long a prerogative of the lattice simulations,for a recent review see [1].The continuum theory provided in fact little guidance for search of the con?nement mechanism.Equations which one borrows from the continuum physics refer mostly to U(1)Higgs models or instantons,see,e.g.,[2].However,these hints from the continuum theory could be used at a qualitative level at best.Painstaking analysis of the lattice simulations did allow to extract vacuum ?uctuations which are actually responsible for the con?nement.These are so called monopoles and

central vortices,for review see,e.g.,[3]and [1,4].By construction,monopoles are in?nitely thin closed trajectories while the central vortices are in?nitely thin closed 2d surfaces 1.Separation of the two types of the defects is actually super?cial.Rather,one observes vortices populated with monopoles.Monopoles live on 2d surfaces,not in the whole 4d space and there can be no vortices without monopoles,[6,7].

In?nitely thin (with size of the lattice spacing a ),percolating trajectories and surfaces look very different from,say,instantons which are bulky ?elds,with size of order Λ?1QCD .

Thus,one is tempted to say that lattice simulations uncovered existence of objects of

lower dimensions in the vacuum state of Yang-Mills theories.However,a prevailing viewpoint,for a recent presentation see,e.g.,[2],is that apparent point-likeness of the monopoles and vortices is an artifact of their de?nition and in fact they only mark some

bulky?eld?uctuations.

It is only rather recently that it was understood that the monopoles and vortices might still be physical lower-dimensional defects.The basic observation which brings about such a conclusion is the ultraviolet divergence in the corresponding non-Abelian action2associated with the monopoles[8]and vortices[6].The power of the ultraviolet divergence in the action is the same as for pointlike particles and in?nitely thin strings, respectively.To explain the survival of the monopoles and vortices on theΛQCD scale–despite of their ultraviolet divergent action–one is forced to postulate[9]self-tuning of the ultraviolet divergent action and of ultraviolet divergent entropy3.Moreover,the d=2 defects appear to be nothing else but dual strings with excitations of scalar?eld living on them.

The ultraviolet divergences in the action are the earliest evidence in favor of relevance of singular?elds to con?nement.There exist further observations[11,12,13]indicating that lower dimensional defects are of physical signi?cance4.

When the lattice studies were undertaken?rst,there was no theory of extended objects at all.However,more recently the idea that strings are relevant to QCD has become quite common.More speci?cally,one expects that if a dual formulation of YM theories exists, it would be a string theory[14]5.

Thus,there appears a possibility that the languages of lattice and continuum theories would get uni?ed again,this time in terms of theory of extended objects.A possible feedback from lattice studies to the continuum theory is that topological excitations observed within a‘direct’formulation might become fundamental variables of the dual formulation of the same theory,see,e.g.,[15].Thus,if the strings are indeed observed as excitations in lattice simulations of YM theories,this is an indication that there exists a dual formulation in terms of fundamental strings[16].

Here we address a problem of reformulating some of the lattice results in terms of the continuum theory.The point is that many results,especially on the lattice strings,are obtained originally in terms of so called projected?elds,see,e.g.,[1].We will discuss a classi?cation scheme of the defects in explicitly SU(2)invariant terms.Also we will comment on possible relation to the stochastic picture of the vacuum[17,18].

LATTICE STRINGS

We have reviewed recently the properties of the two-dimensional defects,or lattice strings[19]and will be brief here.

Magnetic monopoles

Theoretically,the most dif?cult point about the monopoles is their de?nition on the lattice.Monopoles are topological excitations of the compact U(1)[20].To de?ne them in non-Abelian case one uses projection of the original YM?elds onto the‘closest’Abelian con?guration.The physical idea behind considering the monopoles is that con?nement is mostly due to Abelian degree of freedom[21].

While the de?nition of the monopoles is not so transparent,many observed properties are beautiful and formulated in perfectly SU(2)invariant way.Monopoles are observed as clusters of trajectories.In?nite,or percolating cluster corresponds to the classical expectation value,<φM>of a magnetically charged?eldφM.Short,or ultraviolet clusters correspond to quantum?uctuations of the?eldφM.The total length of the clusters is trivially proportional to the total volume of the lattice,V4:

L tot=4ρtot·V4=4(ρperc+ρf inite)·V4.(1) According to the data[5]:

ρtot≈1.6(f m)?3+1.5(f m)?2·a?1,(2) where a is the lattice spacing.The a?1term is entirely due to the?nite clusters.For the percolating cluster the density is a constant in the physical units.

One can translate(2)into the standard?led theoretic language by observing that[9]:

|φM|2 =(const)a·ρtot.(3) Thus,we have

|φM|2 ～Λ2QCD.(4) Theoretically,the estimate(4)can be derived as a constraint implied by the asymptotic freedom of YM theories[10].

Point-like facet of the monopoles

The monopoles action diverges with a→0and the power of the divergence is the same as for point-like particles[8]:

S mon≡M·L mon,M(a)≈ln7·a?1,(5) where M(a)corresponds to the radiative mass and is found by measuring extra non-Abelian action associated with the monopoles.We quote the data in a way which allows for a straightforward theoretical https://www.sodocs.net/doc/2412731873.html,ly,in?eld theory(see,e.g.,[22])if one starts with the classical action of a particle,S=M·L the propagating mass is not M

but:

m2prop=(const)

a

,(6)

where the constants const,ln7are of pure geometrical origin and depend on the lattice used.In particular,ln7corresponds to the hypercubic lattice.Note that in Euclidean space a physical mass of a point-like particle can appear only as a result of tuning between divergent action and entropy.

Thus,the data(5)correspond to a small monopole mass.Moreover,data(2)imply that globally monopoles live on a2d surface.For ordinary point-like particlesρtot～a?3.

Closed strings

Closed surfaces are topological defects of the Z2gauge theory.In simulations of SU(2) theory these surfaces are de?ned in terms of the closest Z2projection which replaces the original YM?elds with Zμ(x)=±1.The central vortices are de?ned as uni?cation of all the plaquettes on the dual lattice which pierce negative plaquettes in the Z2projection, for review see[1,4].

Two most striking properties of the central vortices is that their total area scales in physical units,for review see[1,4]while non-Abelian action is ultraviolet divergent[6]:

A tot

A tot≈4(f m)?2V4,S tot≈0.54

frequent.One can,however,minimize the number of negative https://www.sodocs.net/doc/2412731873.html,ing remaining Z2 invariance.Physicswise,one?xes the gauge by localizing large potentials on as a small number of links as possible.Since link values correspond to potentials and are gauge dependent,one can wonder what is the objective meaning of such minimization.The point is that minimizing,say,potential squared one arrives at a gauge invariant quantity [24].Minimizing number of negative links is a variation of such a procedure.

And,indeed,one?nds[12]that volume of negative links scales as a physical3d defect:

V3=c3ΛQCD·V4.(8) Note that by construction the volume is bound by the central vortices.This volume can be called Dirac volume[1].Eq(8)then states that the minimal Dirac volume scales in physical units or,alternatively,has a zero fractal dimension.

Holography and con?nement

Relation of the volume discussed above to the con?nement is revealed through a re-markable observation of the authors of Ref[25].One replaces the original link matrices Uμ(x)by?Uμ(x)where

?U

μ(x)≡Uμ(x)·Zμ(x),(9) where Zμ(x)is the projected value of the same link.Next,one evaluates the Wilson loop and quark condensate ˉqq in terms of the modi?ed links?U.The result[25]is that both the con?ning potential and spontaneous breaking of the chiral symmetry disappear. Originally[25]the change(9)affected approximately half of the total number of links.Now,we see that it is enough to perform the change(9)on a3d submanifold to kill the con?nement and chiral symmetry breaking.In other words,substitution(9) is an ad hoc modi?cation in the ultraviolet of the?elds on a3d volume plus pure gauge transformations.Thus,we observe a kind of holography,with information on the con?nement being encoded on a submanifold of the whole space.

In more detail,consider a plane on which we will draw a Wilson line.Consider, furthermore,a particular con?guration of the gauge?elds generated with the standard SU(2)action.Determine then the3d volume described in the preceding section.Inter-sections of this volume with the plane considered are segments of1d lines.Now,we can draw any Wilson line on the plane.The statement is that the sign of the Wilson line can be determined by counting the number of intersections with segments of1d defects.It is a highly non-trivial observation,challenge to interpret.Note that there is no logical contradiction,though.Indeed,there are gauges where the con?ning?elds are soft,of order Aμ～ΛQCD.Apparently,one can use gauge invariance to choose a gauge where the con?ning?elds are of order Aμ～1/a but occupy a3d volume6.

Chiral symmetry breaking

There is a series of observations,not directly related to each other that indicate relevance of some3d defects to the spontaneous breaking of the chiral symmetry7: (a)procedure of Ref[13]described above makes also the quark condensate vanish:

ˉqq ?U≈0(10) Now,we know[12]that the change(9)affects not a?nite part of the4d space but only a3d submanifold.

(b)there is evidence in favor of long range topological structure in QCD vacuum which is related to chiral symmetry breaking[11].The search process for the topological structure is formulated in terms of eigenfunctions of the Dirac operator and explicitly gauge invariant8.

(c)One introduces the so called inverse participation ratio,see in particular[27], de?ned in terms of eigenfunctions of the Dirac operator:

I=NΣxρ2i(x),(11) where N is the number of lattice sites x,ρi(x)=ψ?iψi(x),andψi(x)is the i-th normal-ized Σxρi(x)=1 lowest eigenvector of the Dirac operator.

Dependence of the inverse participation ratio on the lattice spacing a was studied in Ref[13].The result is:

I =c1+c2·a?γ,(12) with a non-vanishing exponentγ:

1≤γ≤2.

Note that the valueγ=1would correspond,in the limit a→0to localization of the eigenfunctions on a3d volume.It is worth emphasizing that the a dependence observed refers to an explicitly gauge invariant quantity.

To summarize,there are indications that the chiral symmetry breaking is determined by gauge?elds living on a subspace.Since the con?nement itself also seems to be related to a3d volume(see above),it is not clear whether we deal with a phenomenon speci?c for chiral symmetry breaking or with an effect common to con?nement.

CLASSIFICATION SCHEME

Invariants

There is no theory of the defects in the non-Abelian case.However,even in the absence of such a theory one can try to?nd a SU(2)invariant classi?cation scheme.

Generically,the?rst example of such a scheme for monopoles was proposed long time ago[28].In pure YM theory,there are no classical monopole solutions.However, imagine that there exists a scalar?eld,vector in the color space H a,a=1,2,3.Then one could?x the gauge by rotating vector H a to the third direction at each point.This ?xation of the gauge would fail however at the points where

H a=0.(13) Condition(13)can be viewed as three equations de?ning1d defects in the4d space which can be identi?ed with monopole trajectories[28].It is crucial that(13)is SU(2) invariant.

For various reasons,this idea does not seem to work in the realistic case,for review and references see[29].Rather,monopoles are associated with singular non-Abelian ?elds(see above).Let us try to adjust the classi?cation scheme to this set up[19,29,30]. Begin with YM theory in three dimensions and assume that monopoles violate the Bianchi identities.If the Bianchi identities

D?G=0,(14) hold,the potential A can be expressed in terms of the?eld strength tensor,see,e.g.,[31]:

1

A=

which determine2d defects.Moreover,if both conditions(18)are satis?ed,there is no inversion of the Bianchi identities similar to(15).

Finally,zeros of a second order of the determinant would de?ne1d defects.They automatically fall onto the2d defects as well.

Classi?cation scheme vs data

The classi?cation scheme proposed above is based on symmetry alone and is not unique.But,nevertheless,let us try to identify the2d and1d defects arising within this scheme with the central vortices and monopoles.There are a few quite remarkable con?rmations of such an identi?cation:

(a)the2d defects are associated,according to the scheme,with singular?elds and, possibly,violations of the Bianchi identities And,indeed,the central vortices carry a singular action[6].Moreover,monopoles live on the vortices,on one hand,and may well signify violation of the Bianchi identities,on the other;

(b)non-Abelian?elds associated with the2d defects are aligned with the surface.This is con?rmed by the measurements,according to which the excess of the action vanishes already on the plaquettes next to the central vortices[6];

(c)the monopole trajectories are predicted to lie on the central vortices,in agreement with the data[23,6];

(d)‘monopoles’appear to be Abelian?elds since zero of second order of the deter-minant constructed on three independent(within a3d defect)?elds implies that there is only a single independent color vector.Thus,monopoles can well be detected through the U(1)projection.

(e)on the other hand,the non-Abelian?eld of the monopoles is not spherically sym-metrical but rather aligned with the surface.This collimation of the?eld was observed in measurements,[23].

It is worth emphasizing that all the properties(a)-(e)are gauge invariant.Thus,the data so far do con?rm that through projections one detects gauge invariant objects. Finally,the scheme predicts that breaking of the chiral symmetry is associated with 3d defects.The corresponding lattice data were summarized in the preceding section.

STOCHASTICITY

In the continuum limit,association of the con?ning?elds with lower-dimension defects implies stochastic-type of correlators9.Indeed,the3d volumes,e.g.,are‘not visible’in the continuum limit.a→0.Denote byˉA the con?ning potential obtained in the gauge minimizing the number of negative links(see above).Then

ˉA(x),ˉA(y) =ΛQCD·ΛUV f sing(x?y)+(regular terms),(19)

where

f sing(0)=1,f sing(x=0)=0.

The singular nature of the con?ning potential could explain observed dependence of the localization of zero modes on the lattice spacing,see above.

It is worth emphasizing,however,that reduction of the con?ning potential to the ‘white noise’would be a great oversimpli?cation10.Indeed,the3d nature of the do-mains assumes also non-trivial correlators for the derivatives of the potential.The issue deserves further consideration.

Consider now contribution of strings into an explicitly gauge invariant correlator:

G2(x),G2(y) strings=(const)Λ4QCDΛ4UV f sing(x?y)+(const)Λ8QCD f phys(x?y),

(20) where f phys depends on the physical mass scale.Note that appearance of the extra factor Λ4QCD in front of f phys is of pure geometrical origin and re?ects relative suppression of the2d volumes compared to a4d volume.On the other hand,appearance of the ultraviolet cut off in a non-local term would contradict the asymptotic freedom.It is one more example of consistency of the lattice strings with the asymptotic freedom,see also [10].

Finally,for a stochastic model of the con?nement(see,e.g.,[17,18])it is the correla-tor of two non-Abelian?elds connected by a‘Dirac-string’operator,

G aμν(x)Φab(x?y)G bμν =D(x?y),

which is crucial.The contribution of the string,discussed above,to this correlator is of the form:

D string(x?y)=(const)·f sing(x?y)ΛQCD·Λ2UV.(21) Moreover,using standard approximations of the stochastic model11one obtains for the string tensionσdetermining the heavy quark potential at large distances:

1

σ≈θstring?S string≈

10Actually,the‘white noise’would not con?ne.

11Using the minimal area spanned on the Wilson line is the most sensitive point,dif?cult to justify theoretically[17].

because the Dirac string,Φ(x?y)is a color object and has in?nite self energy12.Thus, the singular nature of the con?ning?elds,see(7)is the only mechanism which can make the stochastic model relevant.

CONCLUSIONS

Physics of con?nement might undergo quite a dramatic change soon.There have been emerging data indicating relevance to con?nement of lower-dimension defects,or singu-lar?elds.Two-dimensional defects with divergent action and entropy,which selftuned to each other are naturally interpreted as the dual string,observed as a vacuum excitation. The string possesses many SU(2)invariant properties but is detected through projec-tions.Other emerging phenomena,a kind of holography and localization of modes on a submanifold shrinking to zero with a→0,are observed in explicitly SU(2)invariant terms.The price is that the structure of the?elds responsible for these observational phenomena is less transparent.

ACKNOWLEDGMENTS

I am thankful to W.Bardeen,M.N.Chernodub,A.DiGiacomo,A.Gorsky,F.V.Gubarev, I.Horvath,J.Greensite,M.I.Polikarpov,A.Polyakov,L.Stodolsky,L.Susskind,T. Suzuki,A.Vainshtein for discussions.

This mini review is based on the talks presented at the conferences“Quark Con?ne-ment and the Hadron Spectrum VI”(Village Tanka,September2004)and“QCD and String Theory”(Santa Barbara,November2004).

This research was supported in part by the National Science Foundation under Grant No.PHY99-07949and by the Grant INTAS00-00111.

REFERENCES

1.J.Greensite,Prog.Part.Nucl.Phys.511(2003)(hep-lat/0301023).

2.M.Engelhardt,“Generation of con?nement and other nonperturbative effects by infrared gluonic

degrees of freedom”,hep-lat/0409023.

3.M.N.Chernodub,F.V.Gubarev,M.I.Polikarpov,A.I.Veselov,Progr.Theor.Phys.Suppl.131,309

(1998);

A.Di Giacomo,Progr.Theor.Phys.Suppl.131,161(1998)(hep-lat/9802008);

T.Suzuki,Progr.Theor.Phys.Suppl.131,633(1998).

https://www.sodocs.net/doc/2412731873.html,ngfeld,et.al.,“Vortex induced con?nement and the IR properties of Green functions”,

hep-lat/0209173.

5.V.G.Bornyakov,P.Yu.Boyko,M.I.Polikarpov,V.I.Zakharov,Nucl.Phys.B672222(2003)

(hep-lat/0305021);

6. F.V.Gubarev,et al.,Phys.Lett.B574(2003)136,(hep-lat/0212003)

7. A.V.Kovalenko,M.I.Polikarpov,S.N.Syritsyn,V.I.Zakharov,“Interplay of monopoles and P-

vortices”,hep-lat/0309032.

8.V.G.Bornyakov,et al.,Phys.Lett.B537(2002)291.

9.V.I.Zakharov,“Hidden mass hierarchy”,(hep-ph/0202040);

M.N.Chernodub,V.I.Zakharov,Nucl.Phys.B669,233(2003)(hep-th/0211267);

V.I.Zakharov,Usp.Fiz.Nauk4739(2004).

10.V.I.Zakharov,“Fine tuning in lattice SU(2)glyuodynamics versus continuum theory constraints”,

hep-ph/0306262;

V.I.Zakharov,“Nonperturbative match of ultraviolet renormalon”,hep-ph/0309178.

11.I.Horvath,et al.,Phys.Rev.D68(2003)114505(hep-lat/0302009);

I.Horvath,“The analysis of space-time structure in QCD vacuum:localization vs global behaviour

in local observables and Dirac eigenmodes”,hep-lat/0410046.

12.M.I.Polikarpov,S.N.Syritsyn,V.I.Zakharov,“A novel probe of the vacuum of the lattice gluody-

namics”,hep-lat/0402018;

A.V.Kovalenko,M.I.Polikarpov,S.N.Syritsyn,V.I.Zakharov,“Three dimensional vacuum domains

in four dimensional SU(2)gluodynamics”,hep-lat/0408014.

https://www.sodocs.net/doc/2412731873.html,C Collaboration(C.Aubin et al.),“The scaling dimension of low lying Dirac eigenmodes of the

topological charge density”hep-lat/0410024

14.J.M.Maldacena,Adv.Theor.Math.2,231(1998),(hep-th/9711200);

A.M.Polyakov,Int.J.Mod.Phys.A14,645(1999),(hep-th/9809057).

15.R.Savit,Rev.Mod.Phys.52(1980)453.

16.V.I.Zakharov,“Hints on dual variables from lattice SU(2)gluodynamics”,hep-ph/0309301.

17.J.Ambjorn,P.Olesen,Nucl.Phys.B170(1980)60;

J.M.Cornwall,Phys.Rev.D26(1982)1453;

T.I.Belova,Yu.M.Makeenko,M.I.Polikarpov,A.I.Veselov,Nucl.Phys.B230(1984)473;

H.G.Dosch,Phys.Lett.B190(1987)177;

H.G.Dosch,Yu.A.Simonov,Phys.Lett.B205(1988)339.

18.A.Di Giacomo,H.G.Dosch,V.I.Shevchenko,Yu.A.Simonov,Phys.Rept.,372(2002)

(hep-ph/0007223).

19.V.I.Zakharov,“Lower-dimansion vacuum defects in lattice Yang-Mills theory”,hep-ph/0410034.

20.A.M.Polyakov,Phys.Lett.B59(1975)82.

21.T.Suzuki,I.Yotsuyanagi,Phys.Rev.D42(1990)4257.

22.J.Ambjorn,“Quantization of geometry”,hep-th/9411179.

23.J.Ambjorn,J.Giedt,J.Greensite,JHEP0002:033,(2000),(hep-lat/9907021).

24.F.V.Gubarev,L.Stodolsky,V.I.Zakharov,Phys.Rev.Lett.86,2220(2001)(hep-ph/0010057).

25.Ph.de Forcrand,M.D’Elia,Phys.Rev.Lett.82,458(1999)(hep-lat/9901020).

26.G.’t Hooft,“The hidden information in the standard model”,ITP-UU-02/68.

27.Ch.Gattringer,et al.,Nucl.Phys.B617(2001)101,(hep-lat/0107016);

J.Gattnar,et al.,“Center vortices and Dirac eigenmodes in SU(2)lattice gauge theory”.

hep-lat/0412023.

28.G.t’Hooft,Nucl.Phys.,B190,455(1981).

29.F.V.Gubarev,V.I.Zakharov,“Interpreting the lattice monopoles in the continuum terms”

hep-lat/0211033.

30.F.V.Gubarev,V.I.Zakharov,Int.J.Mod.Phys.A17(2002)157(hep-th/0004012).

31.M.B.Halpern,Phys.Rev.D19(1979)517;

O.Ganor,J.Sonnenschein,Int.J.Mod.Phys.A11(1996)5701,(hep-th/9507036).

32.A.Di Giacomo,H.Panagopoulos,Phys.Lett.,B285(1992)133;

A.Di Giacomo,E.Meggiolaro,H.Panagopoulos,Nucl Phys.B483(1997)371.