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ITEP-LAT/2008-01 Z electric strings and center vortices in SU (2) lattice gauge theory 2 M. I. Polikarpov1,∗ and P. V. Buividovich2,1,† 1ITEP, B. Cheremushkinskaya str. 25, 117218 Moscow, Russia 2JIPNR, National Academy of Science, 220109 Belarus, Minsk, Acad. Krasin str. 99 (Dated: January 1, 2008) WestudytherepresentationsofSU(2)latticegaugetheoryintermsofsumsovertheworldsheets of center vortices and Z2 electric strings, i.e. surfaces which open on the Wilson loop. It is shown thatincontrasttocentervorticesthedensityofelectricZ2 stringsdivergesinthecontinuumlimitof thetheory independentlyof thegauge fixing,however, their contribution to theWilson loop yields physicalstringtensionduetonon-positivityoftheirstatistical weightinthepathintegral, whichis in turn related to thepresence of Z2 topological monopoles in thetheory. PACSnumbers: 12.38.Aw;11.25.Pm 8 0 0 It is often believed that Yang-Mills theory can be en- center vortices and their geometric properties were ex- 2 tirely reformulated in terms of string degrees of free- tensively studied in [8]. Detection of center vortices in n dom, since the basic property of non-Abelian gauge the- lattice configurations of gauge fields is based on the sep- a ories – quark confinement – is explained by the emer- aration of SU(N) link variables into SU(N)/Z vari- J N gence of confining string which stretches between test ablesandthevariableswhichtakevaluesinthe centerof 1 quark and antiquark. Even the simplest string models the gauge group, Z . Configurations of these Z vari- N N ] for Yang-Mills theory turn out to be very successful in ablescanbeexactlymappedontoconfigurationsofclosed t reproducing the spectrum of bound states of the theory. self-avoiding surfaces, correspondingly, summation over a -l One of the most successful recent developments is the ZN variablescanbe representedasa sumoverallvortex p AdS/QCD,thedescriptionofQCDboundstatesinterms configurations [9]. e of string theory on five-dimensional anti-de Sitter space For Z lattice gauge theories there exists a duality h or its modifications [1, 2]. In this description the Wilson N [ transformation[10]whichallowstorepresenttheobserv- loop behaves as W [C] ∼ exp(−S [C]), where S [C] 1 is the area of the minimal surfa5cDe in five-dimen5sDional ables either in terms of center vortices (ZN magnetic strings) or in terms of surfaces which open on Wilson v space spanned on the loop C. The loop C is assumed loops. Since Wilson loop represents the worldline of an 2 to lie on the boundary of this five-dimensional space. 6 However, up to now there is no exact representation of electric charge, it is reasonable to call such surfaces ZN 2 electric strings. The aim of this paper is to investigate continuumYang-Millstheoryintermsofelectricstrings, 0 therepresentationofSU(2)latticegaugetheoryinterms i.e. the strings which open on Wilson loops. In fact, the . of a sum over all surfaces of such Z electric strings and 1 only string one usually encounters in non-Abelian gauge 2 0 to compare the properties of electric strings and center theories is a chromoelectric string of finite thickness at 8 vortices. It turns out that in contrast to center vortices, finitelatticespacing,whichisobservedinnumericalsim- 0 electric Z strings are gauge-independent, thus the area ulations as a cylindric region with higher energy density 2 v: between two test colour charges [3, 4]. ofsuchstringscannotbeminimizedbyaproceduresim- i ilar to the maximal center projection, which ensures the X Recentlyadifferenttypeofstringshasbeendiscovered physical scaling of the density of center vortices [6, 7]. ar inlattice gaugetheories,namely,ZN magneticstringsor The density of Z2 electric strings diverges in the con- center vortices. Although the existence of such strings tinuum limit, consequently, they can not be described in Yang-Mills theories was predicted a long time ago [5], in terms of continuum theory. Nevertheless, there is a theyhavebeenactuallyobservedandinvestigatedinlat- mechanism which makes the contribution of the mini- tice simulations only during the last decade [6, 7]. The mal surface dominant – namely, the statistical weight of simulations has shown that center vortices are infinitely electric strings in the partition function is not positively thin and have a finite density in the continuum limit. defineddue to the presenceoftopologicalZ monopoles. 2 Moreover,lattice results suggest that center vortices are the effective degrees of freedom in the infrared domain In order to separate Z2 center variables in SU(2) lat- of Yang-Mills theory [6, 7], since removing center vor- tice gauge theory, each lattice link variable should be represented as g = (−1)mlg˜, m = 0,1. By definition tices from lattice configurations destroys all its charac- l l l the variables g˜ are the elements of the coset manifold teristic infraredproperties,such as confinement or spon- l SU(2)/Z =SO(3). Itwillbeassumedthatmultiplica- taneous breaking of chiral symmetry. Effective action of 2 tionofall”tilded” variablesis a multiplicationinSO(3) group. Note, however, that the product of two SU(2) group elements can not be in general expressed as the ∗Electronicaddress: [email protected] product of (−1)ml and the product of g˜l in SO(3). In †Electronicaddress: [email protected] order to characterize this deviation, it is useful to define 2 the Z -valued function m(g˜ ,g˜ ) as: Note that gauge transformations in SO(3) gauge the- 2 1 2 ory g˜l → ˜hsg˜lh˜s′ affect mp and m[C]: mp → mp+dm˜l, g1g2 =(−1)m(g˜1,g˜2)+m1+m2g˜1g˜2 (1) m[C] → m[C] + m˜ , where m˜ = m h˜ ,g˜ + l l s l l∈C where g1,2 = (−1)m1,2g˜1,2. Such function can easily be m h˜ g˜,h˜−1 . A gaPuge-invariant conserved(cid:16)curren(cid:17)t of generalized for any number of SU(2) variables. For the s l s′ purposes of this paper it is convenient to define the Z2- top(cid:16)ological Z(cid:17)2 monopoles can be defined as follows [13]: valued plaquette function m and the functional m[C] p which characterize the difference of products of SU(2) andSO(3)linkvariablesoverlatticeplaquettesandover jl∗ =∗dmp =δ∗mp, δjl∗ =0 (5) The Wilson loop in the fundamental representationof arbitrary closed loops, respectively: SU(2)gaugegroupcanbeexpressedintermsofthenew mp+Pml variables g˜l and ml as: g =(−1) l∈p g˜ l l l∈p l∈p Y Y m[C]+P ml Z(β)W [C]= gl =(−1) l∈C g˜l (2) lY∈C lY∈C = dglTr gl exp(β/2Trgp)= ! By definition mp and m[C] depend on the SO(3) vari- SUZ(2) Yl lY∈C Yp ables g˜ only. The values of m(g˜ ,g˜ ), m and m[C] are in one-lto-one correspondence wit1h t2he hopmotopy classes = dg˜lTr g˜l (−1)lP∈Cml+m[C] π (SU(2)/Z ) ≃ Z [11]. In order to make this state- ! 1 2 2 XmlSOZ(3) Yl lY∈C ment more precise, consider, for instance, the function m(g˜1,g˜2). The points g˜1 ∈ SO(3) and g˜1g˜2 ∈ SO(3) exp β/2(−1)dml+mpTrg˜p (6) can be connected by two different geodesicsγ1 and γ2 in Yp (cid:16) (cid:17) the following way: γ : g˜(s)=g˜ g˜s, s∈[0,1] (3) where Z(β) is the partition function of the theory, g = 1 1 2 p γ2 : g˜(s)=g˜11−sg˜2, s∈[0,1] (4) gl and g˜p = g˜l. l∈p l∈p Q Q The geodesics γ1 and γ2 form a loop on the group The sum over Z2-valued variables ml in (6) can be manifold, which is characterized by some element of represented as a sum over self-avoiding surfaces in two π (SO(3))≃Z which is precisely (−1)m(g˜1,g˜2). differentways,usingtheoriginalortheduallattice. One 1 2 Furtheranalysisismosteasilyperformedusingtheno- of these representations converges in the weak coupling tations of external calculus on the lattice [12]. p-forms limit, while the other is more suitable for the strong- on the lattice are associatedwith p-simplices, e.g. scalar couplingexpansion[10]. Configurationsofcentervortices functions areassociatedwith lattice sites,1-forms - with (Z magnetic strings) can be directly constructed from 2 lattice links, 2-forms - with lattice plaquettes etc. Cor- m ’s. Namely, the worldsheets of center vortices are the l respondingly, scalar functions will be denoted by a sub- surfaces on the dual lattice which consist of plaquettes script s, 1-forms – by a subscript l and 2-forms – by a p∗ for which ∗dm = 1 [6, 7]. It is straightforward to l subscript p. External and co-external differentials are show that such surfaces are indeed closed if one notes denoted as d and δ, respectively. Scalar product of two that for Z - valued 2-formsco-externalderivative yields 2 p-formsf andg is denotedas(f,g)andthe Hodgeoper- the 1-form which is equal to 1 only if the link belongs ator is denoted as ∗. Hodge operator on the lattice acts to the boundary of the surface made of plaquettes for between p-forms defined on the simplices of the original which this 2-form is nonzero. Since δ∗dm = ∗d∗∗dm = l l lattice and D−p-forms defined on the simplices of the ∗ddm =0,thesurfacesconstructedfromdualplaquettes l dual lattice. It can be shown that the operators d, δ, ∗ with nonzero ∗dm are always closed. Such one-to-one l andthe scalarproductof p-formsonthe lattice havethe correspondencebetweenco-closedZ -valued2-formsand 2 same properties as in the continuum theory [12]. In all closedsurfacesallowsonetorewritethesumoverm ’sas l operations on Z -valued forms summation is understood asumoverallclosednon-intersectingvortexworldsheets 2 as summation modulo 2. Σ on the dual lattice: m 3 P ml+m[C] Z(β)W[C]= dg˜lTr g˜l (−1)l∈C ! XmlSOZ(3) Yl lY∈C exp β/2 (−1)dml+mpTrg˜ = dg˜Tr g˜ p l l ! Yp (cid:16) (cid:17) mp∗;Xδmp∗=0SOZ(3) Yl lY∈C (−1)m[C]+p∈PΣC∗mp∗ exp β/2 (−1)mp∗+mpTrg˜p = Yp (cid:16) (cid:17) = (−1)L[Σm,C] dg˜Tr g˜ l l ! Σm;X∂Σm=0 SOZ(3) Yl lY∈C (−1)m[C] exp(β/2 (−1)mpTrg˜ ) exp(−β (−1)mpTrg˜ ) (7) p p Yp ∗pY∈Σm where mp∗ = ∗dml, ΣC is an arbitrary surface spanned on the Wilson loop and L[Σm,C] = ∗mp∗ is the p∈ΣC topological winding number of the surface Σ and the loop C. The factor L[Σ ,C] in (7) implies Pthat the Wilson m m loop changessign eachtime it is crossedby center vortex. Note that the representation(7) is gauge-dependent,since the action associated center vortices depends on the non-invariant functions m . The factor (−1)m[C] is also gauge- p dependent. Although the Wilson loop remains gauge-independent in any case, one can try to fix the gauge in such a way that the contribution of center vortices to the expectation value (7) becomes dominant and the contribution of the terms (−1)m[C] and Tr g˜ to the string tension can be neglected. It turns out that such gauge-fixing l (cid:18)l∈C (cid:19) procedure is exactly the maximalQcenter projection [6, 7, 14], which rotates all link variables as close as possible to some element of the group center. It is also interesting to note that the presence of topological monopoles changes the action of center vortices, which indicates that monopoles can induce some nontrivial dynamics on the vortex worldsheet. Localization of Abelian monopoles on center vortices and the associated two-dimensionaldynamics were indeed observed in lattice simulations [7, 8]. The expression (6) can be also represented as a sum over worldsheets of Z electric strings – closed self-avoiding 2 surfaces Σ which open on the loop C [9]. This is achieved by expanding the statistical weight of each plaquette in e powers of (−1)dml using the identity exp((−1)mx) = chx (1+(−1)mthx). The product of all the weights is then expandedinto the sumof products of (−1)dml overdifferent sets oflattice plaquettes. After summationoverm each l such product contributes to the sum only if each m enters the product an even number of times. In this case the l corresponding set of plaquettes forms a surface Σ bounded by the loop C [9], i.e. the sum over all sets of lattice e plaquettes reduces to a sum over all such Σ : e m[C]+ P mp Z(β)W [C]= dg˜lTr g˜l (−1) Σemin ! Σe;X∂Σe=CSOZ(3) Yl lY∈C P jl∗ ch (β/2Trg˜p) th (β/2Trg˜p)(−1)l∗∈V (8) Yp pY∈Σe where Σ is the surface of the minimal area spanned of SU(2) gauge group (or, more generally, in all repre- emin on the loop C and the Stokes theorem was used to rep- sentations with half-integer spin) can be represented as resent m as a sum over all links dual to a sum over all surfaces of Z electric strings Σ which p 2 e p∈Σe,Σemin openonit. Itisnotdifficulttoderivesimilarrepresenta- cubes whichPbelong to the volume V boundedby Σ and e tions for the partition function of the theory or another Σ . Note that both combinations of Z variables in emin 2 observablessuchast’HooftlooporthecorrelatorsofWil- (8), m[C]+ mp and jl∗, are gauge-invariant, son loops. Σemin l∗∈V therefore the rPepresentationP(8) is gauge-independent. In order to see whether the representations (8) and Thus the Wilson loop in the fundamental representation (7) can be used in the continuum limit of the theory, 4 one should study the scaling of the total area, or, equiv- Thispartitionfunctioninterpolatesbetweenthepartition alently, of the density of center vortices or Z electric functions of SU(2) and SO(3) lattice gauge theories at 2 strings. Consider first the density ofcenter vortices. Ac- µ =0 and µ →∞ respectively. e e cording to (7), some lattice plaquette ∗p belongs to cen- Now the average total area of Σ in lattice units can e ter vortex if for the dual plaquette p signTrg =−1, l be found by differentiating (10) over µ : l∈p e correspondingly, the probability thatQa given plaquette belongs to center vortex is: ∂ h|Σ |i=− lnZ(β,µ )| = e ∂µ e µe=0 1− signTrg 1−h signTrg i e l l P =h lQ∈p 2 i= lQ∈p 2 (9) = 1−hexp(2−βTrgp)i (11) p X P is nothing but the density of vortices in lattice units, which is by definition gauge-dependent. Consider first The expectation value hexp(−βTrg )i tends to zero as thetheorywithoutgauge-fixingandintegrate signTrg p l β →∞andthecontinuumlimitisapproached,therefore l∈p over gauge orbits g → h g h−1. A simple Qcalculation in the continuum limit of the theory Z2 electric strings l s l s′ which involves character decomposition of signTrg and occupy half of all lattice plaquettes and their physical density diverges as a−4. Nevertheless, the sum (8) re- usingthe”gluing”formulafortheintegralsofgroupchar- acters gives the result dh signTr h g h−1 = mains well-defined at a → 0 and in fact sums up to s s l s′ s∈p l∈p exp(−σ|Σemin|),whichcanonlybeexplainedbytheex- α χ (g ), wRheQre theQcoefficient(cid:0)s α beh(cid:1)ave act cancellation of contributions from different surfaces k k p k k=1/2,3/2,... P jl∗ with opposite signs of (−1)l∗∈V , i.e. due to the pres- as (−P1)k+1/2k−6. Thus integrating (9) over gauge or- ence of Z topological monopoles. Indeed, if the term 2 bits yields some gauge-invariant function of the plaque- with jl∗ is omitted, the expression (8) can be consid- tte variable g . It can be shown that this function p ered as the partition function of Z lattice gauge theory takes values in the range −1/3+2/π2,1/3−2/π2 . As 2 with fluctuating, but always positive coupling. It can 1/3−2/π2≈0.130691<1,thedensityofcentervortices be shown that in the weak coupling limit electric strings (cid:2) (cid:3) inlatticeunitsremainsfiniteatweakcoupling,andtheir in Z gauge theory also occupy half of all lattice pla- 2 physical density diverges as a−4. On the other hand, quettes. The sum over such creased surfaces with pos- any perturbative calculation around the vacuum gl = 1 itive weights can only lead to perimeter dependence of yields exactly zero density of center vortices, which thus the Wilson loop, which is indeed the case for Z lat- N appear as truly nonperturbative objects. The situation tice gauge theories in the weak coupling limit [10]. The changes dramatically after the maximal center gauge is m[C]+ P mp imposed – in this case the total area of center vortices is terms Tr g˜l and (−1) Σemin are also not minimized and their density in lattice units goes to zero (cid:18)l∈C (cid:19) likelyto contQributeto the full stringtension, since it can as a2, thus their physical density remains finite in the be shown that the expression (8) yields physical string continuum limit [6, 7]. tension even when these terms are omitted [16]. Thus Since statistical weights of Z electric strings in the 2 it is reasonable to conjecture that in the weak coupling sum (8) are not positive, strictly speaking they can not limitZ electricstringsareconfiningduetothepresence be interpreted as random surfaces [15]. Nevertheless, 2 one can define the vacuum expectation value of their of topological monopoles with currents jl∗. It could be interesting to study numerically the properties of such density by introducing the ”chemical potential” µ for e topological monopoles. electric strings and by differentiating the partition function Z(β,µe) over it. This amounts to multiplying the To conclude, it was shown that unlike Z2 center vor- statistical weight of each surface by an additional factor tices,whichremainphysicalinthecontinuumlimit[6,7], exp(−µe|Σe|), where |Σe| is the total area of Σe. The their duals – Z2 electric strings – can not be consis- partition function of the theory can be calculated from tently described as random surfaces in the continuum (8)byshrinkingthe loopC tozero. Aftersuchmodifica- theory. Instead, electric strings condense in a creased tion of (8) one can reverse all the transformations which phase with infinite Hausdorf dimension, but neverthe- led to this representation and write the partition func- less due to some cancelations betweensurfaces with pos- tion of the theory at nonzero µ in terms of originallink itiveandnegativestatisticalweightsthe minimalsurface e variables as: Σemin dominates in the Wilson loop. In fact the for- mation of some creased structures is typical for subcritical strings [15]. For instance, subcritical Nambu-Goto Z(β,µ )= dg e l strings exist only as branchedpolymers [15]. 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