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SELF-AVOIDING EFFECTIVE STRINGS

arXiv:hep-th/9111060v1 28 Nov 1991DFTT 47/91November 1991SELF-AVOIDING EFFECTIVE STRINGSIN LATTICE GAUGE THEORIES1M. Caselle2 and F. GliozziDipartimento di Fisica Teorica dell’Universit`a di TorinoIstituto Nazionale di Fisica Nucleare,Sezione di Torinovia P.Giuria 1, I-10125 Turin,ItalyAbstractIt is shown that the effective string recently introduced to describe thelong distance dynamics of 3D gauge systems in the confining phase has anintriguing description in terms of models of 2D self-avoiding walks in thedense phase.

The deconfinement point, where the effective string becomesN = 2 supersymmetric, may then be interpreted as the tricritical Θ pointwhere the polymer chain undergoes a collapse transition. As a consequence,a universal value of the deconfinement temperature is predicted.1Work supported in part by Ministero dell’Universit`a e della Ricerca Scientifica e Tecnologica2email address: Decnet=(39163::CASELLE) Bitnet=(CASELLE@TORINO.

INFN.IT)1

In the string picture of the long-distance dynamics of a gauge theory in theconfining phase, the quark pairs are linked together by a thin, fluctuating colourflux tube [1]. This effective string can be described as a conformal field theory(CFT) on a surface with quark lines as boundaries.Starting from general properties of lattice gauge theories in D space-time di-mensions, a specific CFT has been recently proposed [2] as a model for the effectivestring at zero temperature and it is identified by the following properties:i) it is composed, at large distances, by D−2 free bosons, describing the transversedisplacements of the string;ii) each bosonic field is compactified on a circle of radius Rt related to the finitethickness of the flux tube.

This allows fermionizing the model, which in turnguarantees that the colour flux tube does not self-overlap;iii) the boundary conditions on quark lines are dictated by the lattice gauge theory(LGT) and fulfil a sort of orthogonality constraint such that, whenever thequark and antiquark lines coincide, the effective string vanishes . This forcesthe adimensional radius ν = √πσRt to be exactly ν = 1/4.This model is in good agreement with results of Montecarlo simulations both in3 [3] and in 4 [4] dimensions.

In the following we will concentrate in particular onthree dimensional LGT. In this case one can also argue some interesting informationon the effective string picture at the deconfinement point, which turns out to berelated with a twisted N = 2 supersymmetric CFT’s with c = 1 [5].In D = 3 the model of ref [2] is simply a particular case of the gaussian modelwith actionS = g4πZdτdς ∂φ∂φ.

(1)Since the field φ is compactified on a circle, topologically non trivial configurationsare allowed. The partition function on the torus is obtained by taking into accountthe quantum fluctuations around each one of the classical solutions.

A standardcalculation gives (in the notation of ref [6, 7]):Zc[g, 1] =1η(q)η(¯q)Xn,m∈Zq( n√g +m√g)2/4¯q( n√g −m√g)2/4. (2)More general partition functions can be written if, for any reason, the solitonicconfigurations are somehow constrained:Zc[g, f] =1η(q)η(¯q)Xn∈Z/f,m∈Zfq( n√g +m√g)2/4¯q( n√g −m√g)2/4.

(3)It is easy to show thatZc[g, f] = Zc[gf 2, 1].1

It is well known that the ordinary Dirac fermion is exactly described byZc[g =1/2, 1] . Other Dirac fermions with non trivial spin structures are described byf ̸= 1.

The model of ref [2] is given by g = 1/2 and f = 1/2. The spectrum ofconformal weights is given byhn = n232 ,n = 0, 1, .

. .

(4)If we replace the torus with an infinite strip of width R, which is the appropriateworld sheet of the conformal theory describing the effective string, the spectrum ofphysical states propagating along the strip is a subset of those listed in eq . (4),depending on the boundary conditions on either side of the strip.

According to thepoint iii) , the gauge theory fixes uniquely these boundary conditions [2] in such away that the only propagating, physical states belong to the twisted sector of a freefermion with boundary phase 1/4, corresponding to a ground state of conformalweight h = 1/32. As a consequence, the static potential between the two quarks isgiven by [2]V (R) = σR + k + ˜c π24R + O(R) ;˜c = c −24h = 1/4 ,(5)where σ is the string tension , k is a non- universal constant, ˜c is the effectivecentral charge [8] controlling the universal Casimir energy generated by the finitesize of the strip [9].The constraint f = 1/2 , which is a direct consequence of the point iii) , has alsoa nice interpretation in the context of the critical O(n) models (or, more precisely,in terms of the corresponding SOS model [10]).

This model can be defined throughanalytic continuation in n. The original O(n) model is defined initially for n ∈N,n ≥2 byZO(n) =Z Yid⃗SiY(1 + 1T⃗Sj ⃗Sk),(6)where ⃗S is n-component vector such that |⃗S|2 = n. By high temperature expansionone gets on the hexagonal lattice:ZO(n) =Xgraphs( 1T )NbnNl. (7)The sum is over all configurations of non intersecting, self- avoiding graphs, Nl isthe number of loops and Nb is the total number of bonds in the graph.

One cananalytically continue (7) to n ∈R. The model can then be explicitly transformedinto a SOS model (see ref.

[11] for details) and in this way can be solved. It can beshown that the model is critical for n ∈[−2, 2], and in the continuum critical limit,renormalizes onto the gaussian model defined by the action (1) and with a couplingconstant related to the parameter n byn = −2 cos πg.

(8)2

The two branches of the arc cosine with g ≤2 have well defined different meanings.Indeed it is known that the phase diagram of the model has a rich structure: it iscritical for Tc =q2 + √2 −n ; this is the so called dilute phase and it is related tothe branch g ∈[1, 2] . But this model is also critical in the whole region T < Tc (thedense phase), corresponding to the other branch g ∈[0, 1] .

The global propertiesof the system can be investigated by looking for instance at the theory formulatedon the cylinder or on the torus [6, 7]. Due to the particular mapping into an SOSmodel also defects at the boundaries of the type φ = π (and not only φ = 2π) areallowed.

This is coded in eq. (3) by f = 1/2.Among all the possible values of n in (7), a special role is played by the limitn →0 , which is related to the self avoiding walks (SAW) problem.

Looking at (8)we see that this exactly corresponds to our choice g = 1/2. This seems to indicatethat the model proposed in ref [2] gives an effective description of the CFT in thesurface bordered by the quarks in terms of self-avoiding walks in the dense phase.The CFT associated to the SAW problem in the dense phase is non unitary, withcentral charge c = −2 .

The spectrum of conformal weights is (see for instance [12]):hn = n2 −432,n = 0, 1, . .

. (9)which looks different from that of the gaussian model given in eq.

(4). Actually thepartition function of this non unitary theory on the torus and on the cylinder maybe expanded also in terms of c = 1 characters [6]3 .

In particular, the interquarkpotential is a function of the effective central charge ˜c˜c = c −24hl = 1 −3l24 ,(10)where l labels the lowest physical state which can propagate along the strip. Ifl = 1, we get exactly ˜c = 1/4 as in eq.

(5); this may be understood as follows.Notice that the equivalence between SOS model and the standard SAW problemis valid only on a simply connected surface.On a cylinder with quark lines asboundaries we must allow an odd number of self-avoiding loops wrapping aroundthe cylinder. In fact, only in this way the two opposite boundaries of the cylinderhave a shift in the boson field φ(R) −φ(0) = π , so that the boundary conditionsfulfil the orthogonality constraint described in the point iii).

As a consequence, theground state ( corresponding to l = 0 ) is projected out , so we have l = 1 in eq. (10), in agreement with eq.(5).

In such a case the partition function can be expanded interms of fermionic determinants with boundary phases ν = m/4 with m = 0, 1, 2, 3as shown in ref. [2, 13].3The central charge c = −2 and the non unitary spectrum of eq.

(9) can be obtained from thatof the gaussian, c = 1 model of eq. (4) adding a suitable charge at infinity, such that topologicallynon trivial loops have the same weights of the trivial ones in the partition function3

We want to give an heuristic argument suggesting that there is a set surfaces,spanned by the colour flux tubes in 2+1 dimensions, which generate 2D SAW con-figurations. In particular the rough surfaces, which control the functional integral inthe region where the string picture works, will generate SAW in the dense phase.

Weshall use this argument to study the behaviour of the effective string as a functionof the physical temperature of the gauge lattice system.Consider a pair of quark -antiquark lines parallel to the imaginary time axis ofa 3D cubic lattice of size ∞2 × L , with periodic boundary conditions in the timedirection, describing a gauge system at the temperature T = 1/L . The surfaceswept by the elementary colour flux tube joining this pair of quarks in the strongcoupling region is topologically a cylinder , with the two quark lines of length L asboundaries.

Describing this surface as the world sheet of a free bosonic string istoo drastic an approximation, which does not allow to take into account the correctboundary conditions [2].A better picture of the observed flux tube , which has a finite thickness, isobtained surrounding the quark sources by a set of flux rings, like in fig. 1.

In the3D lattice these rings sweep toroidal surfaces linked to the quark lines.r✛✚✘✙✞✝☎✆r✓✒✏✑fig. 1A colour flux tube joining a quark pair ( the two dots ), surrounded by flux ringslinked to the quark sources.The intersections of the cylinder with the tori are self-avoiding loops whichcan wrap the tori (there is an odd number of homotopically non trivial SAW foreach torus).When these surfaces are all smooth, it is immediate to see, usingstrong coupling arguments, that their contribution to the correlation function of thetwo quark lines is partly suppressed by the vacuum diagrams (in Z2 gauge theorythe cancellation is complete).

On the contrary, whenever one of these surfaces iscrumpled up , there is a strong contribution due to excluded volume effects .There is a sort of duality between the configurations of the cylinder and those ofthe tori: If the cylinder is very rough, the tori are forced to be smooth by excludedvolume effects, and vice versa. The former configurations are those that shoulddominate at very low temperature and large distances, where the configurationsavailable for the cylinder grow with the area between the two quark lines, while thetori configurations grow only with the length L = 1/T of the quark lines.

In such acase it is reasonable to assume that the intersections of the rough cylinder with the4

tori are self-avoiding loops in the dense phase. Probably the above considerationsmight be made more precise introducing, for instance, the concept of winding ofSAW around the (quark) sources , like in ref [14].

We use them only to guide ourintuition in the study of the confining phase.Let us push now a bit further our construction. As we increase the physicaltemperature T of the LGT, the tori become more and more rough ( indeed the toriconfigurations will dominate the liberated quark phase).

Then, the excluded volumeeffects become more and more important , and produce vacancies in the dense phase.These vacancies simulate the introduction of a solvent in a polymer chain . At acertain value of the concentration and of the temperature Θ, the polymer undergoesa collapse transition, which corresponds to a tricritical point [15].It is now tempting to argue that this transition coincides precisely with thequark deconfinement.

Actually the partition function for the model of SAW at thetricritical Θ point has been calculated in ref. [16, 13] and turns out to be describedby a gaussian model (3) with g = 2/3 and f = 1/2.

Remarkably enough , thisCFT coincides exactly with the model of the effective string at the deconfiningtemperature proposed in ref. [5] .Notice that this model was based on a completely different argument, namelythat at the deconfinement point , where the effective string disappears, also theCasimir effect must vanish, implying ˜c = 0.

This may be also understood by notingthat ˜c measures the number of physical degrees of freedom of the effective string,so it must go to zero when the string vanishes. The fact that the effective string atthe deconfining transition has no (local) physical degrees of freedom suggests thatit might be described by a topological conformal theory .

Indeed it has been shown[5] that there is only one CFT with c = 1 and vanishing effective central charge andit corresponds to the compactification radiir ≡sgf 22=n2√3 , (n = 1, 2, 3, 6) ,(11)where the c = 1 conformal symmetry is promoted to a N = 2 supersymmetry [17]which supports the topological nature of the model.On the other hand, these compactification radii are simply related to the physicaltemperature T of the 3D gauge lattice theory through r =q σπ/2T ; for this reasonthey have been used [5] to select four special values of T, one of which shouldcorrespond to the deconfining transition Tc.Now, the analogy with the tricritical point of the polymer chain we have estab-lished in the present paper allows us to identify the correct value of the transitionpoint.According to eq. (11), the value of g and f singled out by the Θ pointcorresponds to n = 1.

It turns out that the corresponding value of the criticaltemperature Tc/√σ =√3/√π is universal and fits well to the numerical data [5].5

We thank R. Fiore, I. Pesando, P. Provero and S. Vinti for many useful andinteresting discussions.6

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