Dual Superconductor Mechanism of Confinement

arXiv:hep-lat/9210030v1 26 Oct 1992BARI - TH 110/92September, 1992Dual Superconductor Mechanism of Confinementon the LatticeP. Ceaa,b and L. CosmaibaDipartimento di Fisica dell’Universit`a di Bari, 70126 Bari, ItalyandbIstituto Nazionale di Fisica Nucleare, Sezione di Bari,70126 Bari, Italycea, cosmai @bari.infn.itAbstractWe investigate the dual superconductor mechanism of confinement forpure SU(2) lattice gauge theory in the maximally abelian gauge.

We focuson the the dual Meissner effect. We find that the transverse distribution ofthe longitudinal chromoelectric field due to a static quark-antiquark pairsatisfies the dual London equation.

Moreover we show that the size of theflux tube scales according to asymptotic freedom.HEP-LAT Preprint: hep-lat/9210030

Long time ago G. ’t Hooft and S. Mandelstam[1] proposed that the confiningvacuum is a coherent state of color magnetic monopoles. This proposal offers apicture of confinement whose physics can be clearly extracted.

Indeed the dualMeissner effect causes the formation of chromoelectric flux tubes between chro-moelectric charges leading to a linear confining potential. Following Ref.

[2], onecan study the monopole condensation by means of the so called Abelian projec-tion. It turns out that [3] it is possible to implement the Abelian projection onthe lattice, where the Abelian projection amounts to fix the gauge by diagonaliz-ing an operator which transforms according to the adjoint representation of thegauge group.

In this paper we consider the SU(2) gauge group with standardWilson action on a 124 lattice in the maximally Abelian gauge [4]. In this gaugeone diagonalize the lattice operatorX(x) =XµnUµ(x)σ3U†µ(x) + U†µ(x −ˆµ)σ3Uµ(x −ˆµ)o.

(1)The gauge is fixed iteratively via overrelaxation like the Landau gauge [5]. Weadopted a convergence criterion which coincides with the one of Ref.

[6]The aim of this paper is to analyze the dual Meissner effect by studying thecolor fields distribution due to a static quark-antiquark pair. To measure the colorfields we follow the method of Ref.

[7]. These authors measure the correlation ofa plaquette UP with a Wilson loop W. The plaquette is connected to the Wilsonloop by a Schwinger line L:ρW =DtrWLUPL†E⟨tr (W)⟩−12⟨tr (UP) tr (W)⟩⟨tr (W)⟩.

(2)By moving the plaquette UP with respect to the Wilson loop one can scan thestructure of the color fields. Note that in the na¨ıve continuum limit, the operatorρW is sensitive to the fields rather than to the square of the fields [7].2

Following the Abelian dominance idea [8] we investigate the color field distri-bution by measuringρabW =DtrWAUAPE⟨tr (WA)⟩−12DtrUAPtrWAE⟨tr (WA)⟩. (3)where the superscript means that the Wilson loop and the plaquette are builtfrom the abelian projected links.We use Wilson loop of size 5.

We average over 500 configurations (each oneseparated by 50 upgrades, after 3000 sweeps to equilbrate the lattice) in the range2.4 ≤β ≤2.525.The authors of Ref. [7] found a sizeable signal for the chromoelectric fieldparallel to the flux tube (UP parallel to W).

So we focus on the longitudinalchromoelectric field. By moving the plaquette outside the plane of Wilson loop upto distance 5 in lattice units, we measure the transverse profile of the longitudinalchromoelectric field in the middle of the flux tube.

In Fig. 1 we show the resultfor two different value of β.The authors of Ref.

[7] found that the transverse shape of the flux tube canbe fitted in accordance withEl(x⊥) = AG exph−mGx⊥−µ2Gx2⊥i, x⊥≥0 . (4)Equation (4) describes the flux tube like a relativistic string with gaussian fluc-tuations [12].We find that also our data can be fitted by Eq.

(4). In Fig.

1 the dashed lineis the result of the fit (4) (AG is fixed by El(x⊥= 0)). Moreover in Fig.

2 wecheck the scaling of µG. We find that µG scales andµGΛL= 83 ± 2.

(5)3

Such nice scaling property is not shared by AG and mG. The value (5) is quiteclose to the one µGΛL = 75±2 obtained in Ref.

[7] by using non abelian quantities ona 164 cooled lattice. Thus, we feel that this result support the Abelian dominanceidea.On the other hand, we find that the data are compatible with another func-tional form, namelyEl(x⊥) = AMK0 (µMx⊥) , x⊥> 0 ,(6)where K0 is the modified Bessel function of order zero.

Equation (6) is a straigth-forward consequence of the dual superconductor hypothesis (see also the infraredeffective theory of the monopole condensation proposed in Ref. [9]).

Indeed, letus consider a second kind superconductor in an external static magnetic field.If we have an isolated vortex line, then in the London limit the magnetic fieldsatisfies the London equation [10]h −λ2 ∇2 h = ϕ0 δ(2)(x⊥),(7)where h is the magnetic field parallel to the vortex line, x⊥the transverse distancefrom the vortex line, and ϕ0 the magnetic flux. The penetration depth λ is relatedto the photon mass by the well known relationλ = 1mγ.

(8)The solution of Eq. (7) ish(x⊥) = ϕ02π1λ2 K0x⊥λx⊥> 0 .

(9)Interchanging magnetic with electric, we are lead to consider the fit Eq. (6) withµM = 1λ and AM ∼µ2M.4

Recently, it has been shoved that [11] that the longitudinal electric field of theU(1) flux tube satisfies the dual London equation. We will compare our methodwith the one of Ref.

[11] in a separate paper.The results of the fit Eq. (6) are plotted as solid line in Fig.

1. Both fitsEqs.

(4) and (6) give a comparable reduced χ2. Moreover µM scales (see Fig.

3):µMΛL= 132 ± 2 . (10)It turns out that (see Fig.

4):AM = (0.23705 ± 0.00841) µ2M . (11)Following Ref.

[12] we can define the width of the flux tube byD =R d2x⊥x2⊥El(x⊥)R d2x⊥El(x⊥). (12)From Equations (12) and (6)we obtainD =2µM.

(13)Using [7] ΛL = 6.8 ± 0.2MeV and Eq. (10) we getD = 0.44 ± 0.02 fm(14)which is close to the value estimated in Ref.

[7].In a previous paper [13] we propose a method to measure the abelian photonmass by means of the connected correlation function of an operator with thequantum number of the photon. In order to check Eq.

(8) we recorded also theabelian photon mass. In Figure (5) we display µM/mγ versus β.

Even thoughthere are large statistical fluctuations, mainly due to the abelian photon mass,we see that the relation Eq. (8) is consistent with Monte Carlo data.5

In conclusion we have showed that the transverse distribution of the longitu-dinal chromoelectric field due to a static quark-antiquark pair satisfies the dualLondon equation. However, care must be taken of self-energy effects from theWilson line.

So our results should be checked on larger lattices. We showed thatthe London penetration length scales according to asymptotic freedom.

This sug-gests that the penetration length is a gauge invariant physical quantity. Somepreliminary results on a larger lattice [14] support this conclusion.

Needless tosay, this matter will be deepen in future studies.AknowledgementsWe thank A. Di Giacomo, R.W.

Haymaker and T. Suzuki for useful discus-sions.6

FIGURE CAPTIONSFig. 1Transverse distribution of the longitudinal chromoelectric field at a)β = 2.4 and b) β = 2.5.

Dashed and solid lines refer to Eqs. (4) and(6) respectively.Fig.

2Asymptotic scaling of µG. The dashed line is the fitted value Eq.

(5).Fig. 3Asymptotic scaling of µM.

The dashed line is the fitted value Eq. (10).Fig.

4The ratio AM/m2M versus β.Fig. 5The ratio µM/mγ versus β.

The solid line corresponds to µM/mγ = 1.7

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