The dual formulation of the compact U(1) lattice gauge theory in three
spacetime dimensions allows to finely study the squared width and the profile
of the confining flux tube on a wide range of physical interquark distances.
The results obtained in Monte Carlo simulations are compared with the
predictions of the effective bosonic-string model and with the dual
superconductor model. While the former fails at describing the data from a
quantitative point of view, the latter is in good agreement with it. An
interpretation of these results is proposed in light of the particular features
of the U(1) lattice gauge theory in 3D and a comparison with non-Abelian gauge
theories in four spacetime dimensions is discussed.Comment: 7 pages, 2 figures,Talk presented at the 34th International Symposium
  on Lattice Field Theory (Lattice 2016), 24-30 July 2016, Southampton, U

Caselle, Michele

Panero, Marco

Vadacchino, Davide

English

arXiv

Institutional Research Information System University of Turin

PoS(LATTICE2016)347Squared width and profile of the confining flux tubein the U(1) LGT in 3DDavide Vadacchino∗a,b, Michele Caselle,b and Marco PanerobaPhysics Department, College of Science, Swansea UniversitySingleton Park, Swansea SA2 8PP, United KingdombDepartment of Physics, University of Turin & INFN, TurinVia Pietro Giuria 1, I-10125 Turin, ItalyE-mail: vadacchi@to.infn.it, caselle@to.infn.it, panero@to.infn.itThe dual formulation of the compact U(1) lattice gauge theory in three spacetime dimensionsallows to finely study the squared width and the profile of the confining flux tube on a wide rangeof physical interquark distances. The results obtained in Monte Carlo simulations are comparedwith the predictions of the effective bosonic-string model and with the dual superconductor model.While the former fails at describing the data from a quantitative point of view, the latter is in goodagreement with it. An interpretation of these results is proposed in light of the particular featuresof the U(1) lattice gauge theory in 3D and a comparison with non-Abelian gauge theories in fourspacetime dimensions is discussed.34th annual International Symposium on Lattice Field Theory24-30 July 2016University of Southampton, UK∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/PoS(LATTICE2016)347Squared width and profile of the confining flux tube in the U(1) LGT in 3D Davide Vadacchino1. IntroductionAs is well known, the low-energy regime of confining theories is dominated by configurationsin which flux lines connecting a charge-anticharge pair are squeezed into a thin flux tube [1, 2, 3, 4].Since the study of its shape is one of the ways to gain insight into the confining mechanism, in thepast a large numerical effort has been devoted to its characterization, focusing in particular on thetransverse profile and on the squared string width, both for a “mesonic” and for a “baryonic” setup.There are two competing ways to describe those two observables. The first is effective stringtheory (EST), in which the flux tube is described by a one dimensional Nambu-Goto (NG) stringthat acquires a width as a result of quantum fluctuations. At leading order (LO) in an expansionaround the long-string limit, the model reduces to a free Gaussian string: its width is predicted tobe [5]w2(R) =12pislogRR0, (1.1)where s should coincide with the string tension σ characterizing the model, and R0 is a low-energyparameter. The latter takes into account that this description is only valid down to some distance atthe scale of 1/√σ . In this framework, a prediction has been computed up to the next-to-next-to-leading order [6, 7] both at zero and finite temperature, and, until very recently, all the numericalstudies have confirmed this prediction for both cases, both in the Abelian [8, 9, 10, 11, 12, 13]and non-Abelian case [14]; as shown in our recent work [15], however, the U(1) model in D = 3is a notable exception. The second way is related to the proposal of describing the vacuum of aconfining theory as a dual superconductor [16, 17]. According to this picture, the classical widthof the flux tube should not depend on the separation of the charges and its transverse profile shoulddecay exponentially, with a decay constant equal to the London penetration depth of the medium.The U(1) lattice gauge theory (LGT) in D = 3 is a perfect framework to compare these twoperspectives: its analytical treatment in the semiclassical approximation provides us with a deepunderstanding of how confinement works in this particular case, while its exact mapping to a dualspin model makes it very easy and fast to simulate.2. The U(1) LGT in D = 3The U(1) LGT in D = 3 is defined on a threedimensional square lattice Λ of spacing a withthe standard Wilson action,SW =1ae2 ∑x∈Λ ∑1≤µ<ν≤3[1−ReUµν (x)], (2.1)where Uµν(x) is the standard plaquette and e is the coupling. Its partition function readsZ =∫∏x∈Λ∏1≤µ<ν≤3dUµ(x)e−SW (2.2)where dUµ(x) is the Haar measure on the group manifold.The periodicity of SW with respect to the plaquette complex phase is the most fertile featureof this model. Much like what happens in the XY model in D = 2, at weak coupling (β  1),1PoS(LATTICE2016)347Squared width and profile of the confining flux tube in the U(1) LGT in 3D Davide Vadacchinothe partition function splits into a part describing spin waves and another describing topologicalexcitations. The latter can be shown to behave as magnetic monopoles in the presence of an electriccurrent loop, whose condensation is responsible for a non-vanishing mass gap m0 and for the area-law decay of expectation values of large loops,m0a = c0√8pi2βe−pi2v(0)β , σa2 ≥ cσ√2pi2βe−pi2v(0)β , (2.3)where v(0)' 0.2527 . . . is the zero-distance Coulomb potential in lattice units, and semiclassicallyc0 = cσ/8 = 1; for further details, see ref. [18].From the numerical point of view, we define an exact duality mapping to a spin model definedon the dual lattice with integer elementary degrees of freedom ?s and partition functionZ = ∑{?s∈Z}∏bondsI|d?s|(β ), (2.4)where d denotes a difference at neighboring sites, and Iν(x) is the modified Bessel function of thesecond kind of order ν .In the dual formulation it is easy to incorporate sources of the gauge field straight into thepartition function. Representing a charge-anticharge pair at distance R by two Polyakov loopswinding around the system in opposite directions, the corresponding two-point correlation function〈P(0)P†(x)〉 reads〈P(0)P†(x)〉= ZRZ=1Z ∑{?s∈Z}∏bondsI|d?s+?n|(β ) (2.5)where the non-dynamical variable ?n is non-vanishing on the links dual to a surface bounded bythe loops.3. The squared width and the flux-tube transverse profileIn this work we studied two related quantities: the squared width of the flux tube and thetransverse profile of the flux tube. The latter can be obtained from direct lattice calculations viael(xt) =〈P?(R)P(0)El(xt)〉〈P?(R)P(0)〉 −〈P?(R)P(0)El(xt)〉= 〈d?l +? n〉√β, (3.1)where xt is the transverse direction in the symmetry plane of the charges. The squared string widthw2 is then defined asw2 =∫dxt x2t el(xt)∫dxt el(xt). (3.2)A summary of the numerical setup can be found in the first four columns of tab. 2. For eachvalue of β we explored a wide range of charge separations, going from R∼ a/√σ to several timesthat quantity. For each value of R/a, we measured el(xt) for xt going from the axis connecting thecharges to values of xt such that el ∼ 0. Once the transverse profile was obtained, the computationof w2(R) was performed with a naive discretization of eq. 3.2. The farthest points in the transverseprofile, for which el ∼ 0, just contribute to the systematic error of w2; we define our estimate of thestring width by truncating the sums at values of |xt | where w2 reaches a plateau. The values thusobtained are reported in tab. 1.2PoS(LATTICE2016)347Squared width and profile of the confining flux tube in the U(1) LGT in 3D Davide Vadacchinow2/a2R/a β = 1.7 β = 2.0 β = 2.2 β = 2.44 – 8.59(21) 14.26(2) –6 4.00(10) 10.46(28) 19.1(5) –8 4.32(6) 12.20(24) 21.4(5) –10 4.74(6) 13.75(27) 26.2(5) 47.2(1.4)12 4.98(6) 14.72(31) 27.0(5) 59.2(3.3)14 5.23(6) 15.49(27) 30.4(6) 55.4(2.8)16 5.43(6) 16.52(31) 32.1(6) 67.2(3.2)18 5.63(8) 16.70(31) 34.5(6) 65.1(3.3)20 5.78(8) 16.92(31) 36.2(7) 64.6(3.3)22 5.81(6) 18.0(4) 38.5(7) 71.5(3.2)24 – – – 76.6(3.5)26 – – – 75.3(3.3)28 – – – 76.7(3.4)Table 1: Squared width w2 of the flux tube at different values of β = 1/(ae2), as a function of the distanceR between the static sources.4. Results and conclusionsThe first conclusion we can draw is that the string width is certainly not a constant with thecharge separation. We must thus reject the dual superconductor approximation, at least at tree leveland for finite lattice spacing. It is now natural to try and fit eq. (1.1) to the data using σ and R0 asfitting parameters. We started by fitting the whole available R/a range and then progressively dis-carded the smallest distances until an acceptable reduced χ2 (that we denote as χ2r in the following)was obtained. The results are reported in tab. 2 and the fits are shown in fig. 1.β σa2 m0a L, Nt Rmin/a sa2 R0/a χ2r d.o.f.1.7 0.122764(2) 0.88(1) 64 6 0.108(3) 0.41(42) 0.53 72.0 0.049364(2) 0.44(1) 64 6 0.0271(8) 0.996(74) 0.58 82.2 0.027322(2) 0.27(1) 96 8 0.0109(4) 1.72(13) 0.75 92.4 0.015456(7) 0.165(10) 96 12 0.0044(2) 3.01(25) 0.48 6Table 2: Results of the fits of w2 to eq. (1.1), at the different values of β and for the lattice sizes L, Nt that westudied (for which the values of the string tension and lightest glueball mass are displayed in the second andthird column). The fifth column shows the minimal value of R/a included in the fit, while the sixth and theseventh column show the fitted parameters. The χ2r of the fit and the number of degrees of freedom (d.o.f.)are reported in the last two columns. Note that, although the dependence of the data on R is logarithmic, aspredicted by eq. (1.1), the fitted values of sa2 are never compatible with σa2.As we can see, the LO order prediction from NG correctly describes the logarithmic depen-dence of w2 on R/a, but the fitted value of the parameter s is not the one predicted by EST. Notethat for every value of β , the smallest value in the range of the fitted R/a is always greater than1/√σ . Moreover, as we found by simple inspection, the best-fit values of the parameter s actually3PoS(LATTICE2016)347Squared width and profile of the confining flux tube in the U(1) LGT in 3D Davide Vadacchino101R/a020406080100w2 a2 =1.7, sa2 = 0.1064(37), R0/a = 0.430(52)2r = 0.57, Rmin/a = 6 =2.0, sa2 = 0.02708(81), R0/a = 0.996(74)2r = 0.47, Rmin/a = 4 =2.2, sa2 = 0.01127(31), R0/a = 1.593(92)2r = 0.79, Rmin/a = 4 =2.4, sa2 = 0.00498(12), R0/a = 2.28(11)2r = 1.58, Rmin/a = 4Figure 1: Results of the fits of the squared flux-tube width to the data, for the whole range of interquarkdistances.scale with m0 rather than with σ .The NLO correction to the logarithmic behavior, that we expect to give a detectable contribu-tion to the fits in the short- to intermediate-distance range, does not improve the agreement betweenthe data and the EST prediction. As fig. 1 shows, the manifest logarithmic behavior persists downto distances comparable with 1/√σ .One can obtain further insight from the study of the transverse profile of the flux tube. ESTpredicts a Gaussian transverse profile p(xt) = C0 exp(−x2t /δ), while the solution of the equationsof motion of the dual superconductor model yield an exponentially decaying profile, characterizedby the London penetration depth λ = 1/mv,el(xt) =Φ m2v e−mv|xt |. (4.1)Trying to fit any range of the profile with a Gaussian yields unacceptable χ2r values, while the tails,defined as the values of el(xt) with xt > xct , for some xct which is to be determined from the data,are correctly described by an exponential. For an example of fit of eq. (4.1 to the data see fig. 2and tab. 3. From the fitted values we see that, the quantity mv does not dependent on the separationof the charges and is compatible with m0, as predicted by the dual-superconductor model. Thequantities xct and Φ are weakly growing with R for every value of the coupling constant.The dual superconductor model is thus a good description for the tails of the transverse profile,but fails to describe a bell-shaped inner core of radius xct .As we already remarked, our study in ref. [15] was motivated by the existence of two compet-ing descriptions for the flux tube of a confining theory, and by the peculiar behavior of the U(1)model in D = 3 in its flow towards the continuum limit. In this model it is possible to obtain veryprecise data for two observables related to the shape of the flux tube, that can be used to comparethe agreement of both theories to the data. From the discussion above, we conclude that the pre-diction of EST up to NLO cannot correctly describe the data. Perhaps this is signaling us that, as4PoS(LATTICE2016)347Squared width and profile of the confining flux tube in the U(1) LGT in 3D Davide Vadacchino15 10 5 0 5 10 15xt/a10−310−210−1log(a2El)Figure 2: The transverse string profile, for β = 2.0 and R = 16a. Note that the vertical axis is displayedin a logarithmic scale. In this plot, the exponential decay of the tails of el(xt) manifests itself in the linearbehavior observed for |xt |> 8a: the green dash-dotted line is the result of the fit to eq. (4.1) in this range.xct /a Φ amv χ2r1 7.4(2) 0.323(8) 93.52 6.9(1) 0.368(6) 21.53 6.86(4) 0.405(4) 3.864 6.99(3) 0.428(4) 1.485 7.21(6) 0.444(5) 0.966 7.25(13) 0.447(7) 0.967 7.11(28) 0.44(1) 1.038 7.3(6) 0.44(2) 1.069 6.9(1.0) 0.44(3) 1.1210 6.5(1.9) 0.43(5) 1.18Table 3: Results of the fits of eq. (4.1) for the Monte Carlo data obtained at β = 2.0 and R = 10a.was done for the static potential, the prediction should be supplemented by the contribution from arigidity term in the effective string action [18]. We also showed that the flux-tube transverse profileis not Gaussian, and has exponentially decaying tails that can be characterized by m0, as predictedby the dual-superconductor model.Note that in non-Abelian gauge theories, in which typically the m0/√σ ratio is roughly inde-pendent of a (up to small discretization effects), the question whether an Abrikosov-like behaviorof the flux tube is observed or not, cannot be answered by an analysis relying on the variation of theseparation between the charges. It is then natural to focus on the transverse profile. Unfortunately,in the cases that we studied this observable gives no conclusive results, since it appears to includea mixture of Gaussian and exponential contributions.AcknowledgmentsThe work of D. 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Squared width and profile of the confining flux tube in the U(1) LGT in 3D

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Squared width and profile of the confining flux tube in the U(1) LGT in 3D

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