Some Universal Features of the Effective String Picture

arXiv:hep-lat/9210031v1 26 Oct 1992DFTT-61/92October, 1992Some Universal Features of the Effective String Pictureof Pure Gauge Theories 1M.Caselle, F. Gliozzi and S.VintiDipartimento di Fisica Teorica dell’Universit`a di Torinoand INFN, Sezione di TorinoVia P. Giuria 1, I-10125 Torino, ItalyAbstractThe effective string describing the large distance behaviour of the quark sources ofgauge field theories in the confining phase in D=3 or D=4 space-time dimensions canbe formulated as a suitable 2D conformal field theory on surfaces with quark loops asboundaries.Some universal relations among the string tension , the thickness of thecolour flux tube , the deconfinement temperature and a lower bound of the glueball massspectrum are discussed.1Talk given by F.Gliozzi at LATTICE’92

1IntroductionThe colour flux joining a pair of quarks in the confining phase is concentrated inside atube of small but finite thickness. It is generally believed [1] that this thin tube behaveslike a vibrating string when these quarks are pulled very far apart.By combining these two simple properties with the L¨uscher [2] description of the fluxtube beyond the roughening, we derive, using a simple argument on the behaviour ofhorizontal Wilson loops at high temperature, a general relation between the deconfinementpoint Tc and the string tension σ which is universal i.e.

it does not depend on the specificchoice of the gauge group, being only a function of the dimension D of the space-time.This relation fits nicely the existing data of numerical simulations with pure lattice gaugetheories in D = 3 dimensions as well as the new data for SU(2) in D = 4, while the SU(3)value seems at present a bit different.Assuming further that the effective string cannot self-overlap, as suggested by thestrong coupling expansion of 3D Ising gauge model, we argue, using some recent re-sults [3, 4] on self-avoiding random surfaces, that this string is described, in the infraredlimit and at zero temperature, by a 2D conformal field theory (CFT) with D −2 freebosons compactified on a circle with a specific value of the radius. We argue that such acompactification radius approximately measures the thickness of the colour flux tube.The effective string picture arising in this way coincides with that proposed some timeago [5, 6] in order to fit accurately the numerical data on the expectation value of theWilson loops for various gauge systems in three and four space-time dimensions and basedon a different argument [7].The property of non-overlapping of the colour flux tube can also be used to obtaina universal lower bound on the mass spectrum of the glueball which is numerically wellverified by the mass of the 0+ glueball state both in SU(2) and SU(3) 4D gauge systems .1

2Transition TemperatureLarge space-like or horizontal Wilson loops (i.e. loops defined on a constant time slice)in a gauge system at a nonzero temperature cannot be used as order parameter of decon-finement.

In particular, at high temperature, they may show area law behaviour withoutstatic quarks being confined [8]. Nevertheless they provide us with some important infor-mation on the transition to the the gluon quark plasma.

Consider indeed the horizontalWilson loop in a 3D gauge system at a temperature T = 1/L as drawn in fig.1. Owingto the periodic boundary conditions along the imaginary time, the field x⊥describing thedisplacements of the effective string joining the q ¯q sources is obviously compactified on acircle of length L = 1/T, i.e.x⊥≡x⊥+ L.(1)x⊥may be considered as the Goldstone mode related to the spontaneous symmetry break-ing of the translational invariance along the time direction [2].1THorizontal Wilson loop at finite T bordering a surface of minimal area.Denoting by Wh(R, R′) an horizontal rectangular Wilson loop of size R × R′, the freeenergy FT = −log ⟨W⟩of the associated q ¯q pair has the following asymptotic expansionlimR′→∞FT (R, R′)R′= σhR −˜cπ24R + .

. .

(2)where σh is the horizontal string tension (which is different from the force experienced bya q ¯q pair in the time direction) and the 1/R term is the L¨uscher universal term, i.e. thezero-point energy contribution as it arises in the CFT of a strip (the world-sheet of the2

effective string) with the quark pair as boundaries: ˜c = c −24h(T) is the effective centralcharge and h(T) is the conformal weight of the minimal string state propagating along thestrip. In the CFT obeyed by x⊥one has c = 1 (c = D −2 for D space-time dimensions )and the spectrum of the allowed conformal weights is a known function of T = 1/L.Increasing the temperature of the system reduces the phase space of the colour fluxtube until it fills the whole space.At this point the flux tube begins to be squeezedbetween the two opposite sides of the temporal box; there will be a critical value Tc, lateridentified with the deconfinement temperature, such that for T ≥Tc the distribution of thecolour flux along the temporal axis becomes uniform.

As a consequence, the translationalinvariance in the time direction is restored, hence the Goldstone field x⊥describing thestring fluctuations disappears.It follows that the zero-point energy of the effective string must vanish i.e. ˜c = 0,which it is possible only if there is in the conformal spectrum a state of weighth(Tc) = c/24.

(3)Note that ˜c, being proportional to zero-point energy, measures the number of local degreesof freedom of the CFT. Its vanishing tells us that at the deconfining point the effectivestring theory has at most a discrete set of degrees of freedom, i.e.

it behaves like a topologi-cal conformal field theory (TCFT). Actually most TCFT’s may be formulated as (twisted)N = 2 superconformal theories (SCFT).

It turns out that in any N = 2 SCFT there isa physical state of conformal weight h = c/24 . Conversely one is led to conjecture thatany CFT with a weight h = c/24 is promoted to a N = 2 SCFT.

This is almost triviallytrue for c = 1 and c = 2, i.e. D = 3 and D = 4, which are the cases we are interested in.In particular, in the whole set of c = 1 CFT’s that can be written in terms of the fieldx⊥compactified on a circle of arbitrary radius, the only theory with a state of conformalweight h =124 is precisely the one where the conformal symmetry is promoted to a N=2extended supersymmetry.

This allows us to select a compactification radius and hence a3

specific value for Tc [9]:Tc√σ =√3p(D −2)π. (4)The D dependence has been inserted to take into account also the other interesting caseof D = 4, which can be treated in the same way [10].Remarkably enough, our determination of Tc coincides with the Hagedorn temperatureand with the value predicted for the Nambu-Goto string.

Our argument suggests that thistemperature is universal and does not depend on the gauge group.3Self-avoiding StringThe strong coupling expansions of any gauge theory show that the colour flux tube cannotself-overlap freely as Nambu-Goto string would imply, but must obey some constraintwhich depends on the gauge group. The simplest case is the ZZ2 gauge group where thisconstraint tells us that the flux tube is self-avoiding.Actually it has been shown [3] the exact equivalence between a model of random self-avoiding surfaces embedded in a three-dimensional lattice and a O(N) lattice gauge theoryfor any N. Moreover this model has a phase transition belonging to the same universalityclass of the ZZ2 gauge model.This suggests assuming that the effective string is described at least approximately,for any gauge group, by a self-avoiding string.It is well known that the theory of random surfaces can be formulated in terms of a 2Dquantum gravity coupled to some matter fields describing the embedding of the surfacesin a target space.

In the case of a 3D euclidean space, these matter fields are of coursethe three coordinates xi(ς, τ), (i = 1, 2, 3), describing the embedding of the surface as afunction of the world-sheet parameters ς and τ. Their contribution to the central chargeis cx = 3.

The self-avoiding property induces further conformal matter which controlsthe self-intersections of the surface.There are two kinds of such matter fields for 3Dembedding [4]: the lines of self- intersection may be associated to a free fermion ψ with4

cψ = 1/2 while the end-points of these lines, which are topological defects of the embeddedsurface, are described by an anyon χ with cχ = −11/2; then cmatter = cx + cψ + cχ = −2.As an aside remark, note that this is exactly the central charge of the conformal matterdescribing a self-avoiding polymer in the dense phase; the deconfinement point, where theeffective string becomes N=2 supersymmetric, may then be interpreted as the tricriticalΘ point of the polymer [11].For very large interquark distances there are reasons to believe that the effective stringis described only in terms of matter conformal fields. Actually this is what happens in thetransverse gauge [12] where, as we already anticipated, the asymptotic string is described,for D=3, by a free bosonic field x⊥with c = 1.

Consistency with the above-mentionedtheory of random surfaces implies that there should be a c = 1 CFT equivalent to theone with c = −2. Surprising as it may be, this equivalence exists and is unique [13]:with suitable boundary conditions [11] the partition function for the c = −2 CFT canbe expressed in terms of a gaussian model where the free field x⊥is compactified like ineq.

(1), but with a compactification length L′ = 2Rf slightly smaller then that associatedto the (inverse) of critical temperature, indeed one finds [11, 10]√σRf =qπ(D −2)/4 . (5)In analogy with the string picture at the deconfinement point, we think that Rf measuresthe transverse radius of the colour flux tube at zero temperature.

Notice that this gaussianmodel with precisely such a compactification length has been selected among variouspossible descriptions of the asymptotic effective string as the one which fits better to thedata of numerical simulations in many different gauge systems [5, 6, 7].The idea that the asymptotic effective string is described by a compactified bosonallows us also to get a lower bound for the glueball mass. This spectrum can be evaluatedby studying the exponential decay of the correlation function of small quark loops at largedistance.

If γ is a small circular loop with center on a point x and Wx(γ) is the associated5

Table 1: Comparing string values with data from LGT simulations3D Gauge Systems4D Gauge SystemsstringZZ2SU(2)stringSU(2)SU(3)Tc/√σ0.9771.17(10) [9]0.94(3) [14]0.6910.69(2) [16]0.56(3) [16]mG/√σ2.5964.77(5) [18]3.6713.7(2) [17]3.5(2) [17]Tc/mG0.3760.1880.180(16) [16]0.176(20) [16]√σRf0.4430.6270.4 ÷ 0.6 [15]Wilson loop operator, we have, asymptotically⟨Wx(γ)Wx+L(γ)⟩∼e−mGL,(6)where mG is the mass of the lowest glueball. In a string picture this expectation value canbe written as the partition function of a CFT on a surface with the topology of a cylinderof length L. The minimal radius Rm of this cylinder is not determined by the geometryof the system, rather it should be generated dynamically.

If we assume that Rm is largeenough to apply CFT formulas, we get an asymptotic expansion similar to eq. (2)⟨WxWx+L⟩∼e(−σ2πRmL+˜cL/12Rm) ,(7)where the first term at the exponent is the usual area term and the second one is theuniversal quantum correction.

Comparing eq. ( 6) with eq.

(7) yieldsmG ≃2πσRm −D −212Rm. (8)In the bosonic string picture there is no natural lower bound for Rm: the minimal areais obtained for Rm →0 , where the quantum contribution diverges.

On the contrary, theself- avoiding string picture we are describing gives obviously Rm ≃Rf, otherwise thereis an overlapping of the colour flux tube. From eqs.

(5) and (8) we getmG√σ ≃√D −23π2 −26√π. (9)In Table I the values of the observables we have determined in eqs.

(4),(5) and (9)through our asymptotic effective string scheme, are compared with the corresponding6

data from numerical simulations on lattice gauge theories with various gauge groups. Theagreement to the numerical simulations on LGT is in general rather good.

The fact thatthe lowest observed glueball mass for 3D SU(2) LGT is much larger than the lower boundfixed by the string might indicate that in this case the ground string state is decoupled. Itwould be interesting to do new numerical simulations in order to complete the table andto test the universality of these string formulas with other gauge groups.References[1] H.B.

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