The U(1) lattice gauge theory in three dimensions is a perfect laboratory to
study the properties of the confining string. On the one hand, thanks to the
mapping to a Coulomb gas of monopoles, the confining properties of the model
can be studied semi-classically. On the other hand, high-precision numerical
estimates of Polyakov loop correlators can be obtained via a duality map to a
spin model. This allowed us to perform high-precision tests of the universal
behavior of the effective string and to find macroscopic deviations with
respect to the expected Nambu-Goto predictions. These corrections could be
fitted with very good precision including a contribution (which is consistent
with Lorentz symmetry) proportional to the square of the extrinsic curvature in
the effective string action, as originally suggested by Polyakov. Performing
our analysis at different values of $\beta$ we were able to show that this term
scales as expected by Polyakov's solution and dominates in the continuum. We
also discuss the interplay between the extrinsic curvature contribution and the
boundary correction induced by the Polyakov loops.Comment: 7 pages, 2 pdf figures, contribution to the 32nd International
  Symposium on Lattice Field Theory "Lattice 2014" (23-28 June 2014, Columbia
  University, New York, NY, USA

Caselle, Michele

Panero, Marco

Pellegrini, Roberto

Vadacchino, Davide

English

arXiv

Institutional Research Information System University of Turin

PoS(LATTICE2014)349Effective string description of the interquarkpotential in the 3D U(1) lattice gauge theoryDavide Vadacchino∗, Michele CaselleDipartimento di Fisica Teorica, Università di Torinoand Istituto Nazionale di Fisica Nucleare, sezione di Torino,Via Pietro Giuria 1, I-10125 Torino, ItalyE-mail: vadacchi@to.infn.it, caselle@to.infn.itMarco PaneroInstituto de Física Téorica UAM/CSIC, Universidad Autónoma de MadridCalle Nicolás Cabrera 13-15, Cantoblanco E-28049 Madrid, SpainE-mail: marco.panero@inv.uam.esRoberto PellegriniPhysics Department, Swansea University,Singleton Park, Swansea SA2 8PP, UKE-mail: ropelleg@to.infn.itThe U(1) lattice gauge theory in three dimensions is a perfect laboratory to study the propertiesof the confining string. On the one hand, thanks to the mapping to a Coulomb gas of monopoles,the confining properties of the model can be studied semi-classically. On the other hand, high-precision numerical estimates of Polyakov loop correlators can be obtained via a duality mapto a spin model. This allowed us to perform high-precision tests of the universal behavior ofthe effective string and to find macroscopic deviations with respect to the expected Nambu-Gotopredictions. These corrections could be fitted with very good precision including a contribution(which is consistent with Lorentz symmetry) proportional to the square of the extrinsic curvaturein the effective string action, as originally suggested by Polyakov. Performing our analysis atdifferent values of β we were able to show that this term scales as expected by Polyakov’s solutionand dominates in the continuum. We also discuss the interplay between the extrinsic curvaturecontribution and the boundary correction induced by the Polyakov loops.IFT-UAM/CSIC-14-109The 32nd International Symposium on Lattice Field Theory,23-28 June, 2014Columbia University New York, NY∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(LATTICE2014)349Effective string in the 3D U(1) gauge theory Davide Vadacchino1. U(1) gauge theory in three spacetime dimensionsIn our recent work reported in ref. [1], we studied numerically the U(1) gauge theory in threespacetime dimensions. The Wilson discretization for the action of this theory isβ ∑x∈Λ∑1≤µ<ν≤3[1−ReUµν(x)], with β =1ae2, (1.1)where Λ is a cubic lattice of spacing a,Uµν(x) is the plaquette, andUµ(x) = exp[iaAµ (x+aµˆ/2)].A semi-classical analysis shows that this theory is confining for all values of β , and that it reducesto a theory of free massive scalars for β  1 [2, 3]. In this limit the mass of the lightest glueball(m0) and the string tension (σ ) are given bym0a= c0√8pi2βe−pi2βv(0), σa2 ≥ cσ√2pi2βe−pi2βv(0), (1.2)with c0 = 1 and cσ = 8. In agreement with previous numerical studies [4], we find that the stringtension saturates this bound and that both c0 and c1 are affected by the semi-classical approxima-tion, changing their values in the continuum. Nevertheless, both m0 and σ are positive for all β , sothe model is confining. At finite lattice spacing, the m0/√σ ratio is given bym0√σ=2c0√cσ(2pi2β )3/4e−pi2v(0)β/2, (1.3)so it can be set to any value by tuning β .An exact duality transformation of this theory maps its partition function toZ = ∑{?s∈Z}∏linksI|d?s|(β ), (1.4)which defines a globally Z-invariant model, with integer-valued ?s variables, residing on dual sites.In eq. (1.4), Iα(z) is a modified Bessel function of the first kind, the product is taken over theelementary bonds of the dual lattice, and d?s denotes the difference of ?s variables at the ends of abond. In four dimensions, this duality yields a model with local Z symmetry [5].This duality maps the original gauge theory to a spin model, making it is faster and easier tosimulate. Furthermore, it is easy to add two opposite probe charges at distance R: this results intoa modified partition functionZR = ∑{?s∈Z}∏linksI|d?s+?n|(β ), (1.5)where ?n is a 1-form with values in Z, which is non-vanishing on a set of links dual to a surfacebounded by the worldlines of the original charges. Thus, the two-point Polyakov loop correlatorbecomes〈P?(R)P(0)〉= ZRZ. (1.6)Using the “snake algorithm” [6] and a hierarchical update scheme [7], these correlators can beevaluated to high precision, even at large R: the duality transformation allows one to bypass the2PoS(LATTICE2014)349Effective string in the 3D U(1) gauge theory Davide Vadacchinoexponential decay of the signal-to-noise ratio. Denoting the length of (the separation between) thePolyakov loops in units of the lattice spacing as nt (nR), one gets〈P?(R)P(0)〉=nRnt−1∏i=0Z(i+1)Z(i), (1.7)where the partition function in the numerator differs from that in the denominator by just one ?n ona bond of the dual lattice: thus the computation of the correlator is reduced to the computation ofnt local quantities in independent simulations.The dual formulation also offers insight into the confinement mechanism at work. As shown inref. [2], confinement is driven by monopole condensation, as advocated in the dual superconductorpicture of QCD. For this theory, however, Polyakov [8] even suggested an Ansatz for the effectiveaction which should describe the confining string, and for the dependence of its couplings on thelightest glueball mass and on the coupling of the original gauge theory. In the cited work it ispointed out that the action should beSPol = c1e2m0∫d2ξ√g + c2e2m0∫d2ξ√gK2 . (1.8)Identifying c1 and c2 with σ and α one finds√σ/α ∼ m0.This action was first proposed as a model for fluid membranes [9] and later as a way to stabilizethe Nambu-Goto action [10]. Its contribution to the interquark potential was first computed in thelimit of a large number of dimensions D [11], and then also for D generic [12]. The corrections itinduces onto higher-order terms of the spectrum have been recently calculated [13]. A discussionof the implications of this action for the static interquark potential in the 3D U(1) gauge theory ispresented in a companion contribution [14].2. Numerical evidence for a rigid-string contributionWe performed a set of simulations on the dually transformed U(1) lattice model with a local-update Metropolis algorithm. Here and in the following we will mostly use lattice units, settinga= 1. The computations were done on lattices of size L2×Nt , with L=Nt ranging from 64 to 128,at five values of β , from 1.7 to 2.4, see tab. 1: this setup enabled us to explore a large range of σand m0 values. Note that we always chose L > 10/√σ and L > 10/m0, to suppress finite-volumecorrections.We calculated the ratio of Polyakov loop correlators at distances differing by one lattice spac-ing, and constructed the quantityQ(R) =− 1NtlnG(R+1)G(R)=V (R+1)−V (R), (2.1)where G(R) = 〈P?(x)P(x+R)〉, P(x) is the Polyakov loop through the site x and V (R) denotes thepotential between two static sources at distance R. Q(R) relates the effective string action to thestatic interquark potential, removing constant and perimeter terms. Using the algorithm describedabove, we evaluated Q(R) to high precision for several values of β and for 1/√σ < R< L/2.3PoS(LATTICE2014)349Effective string in the 3D U(1) gauge theory Davide VadacchinoThe data was first fitted with the standard Nambu-Goto effective string expectation1QNG(R) = σ[√(R+1)2− pi12σ−√R2− pi12σ]. (2.2)Q(R) was fitted to the data (using the string tension σ as the fitted parameter), for R from Rminto R = L/2. Rmin was increased, starting from 1/√σ , until a reduced χ2 of order 1 could beobtained, yielding the results reported in the second column of tab. 1. For β < 2 a good fit couldbe obtained for small Rmin, but for β ≥ 2, the minimal R had to be increased to larger and largervalues. The deviation from the Nambu-Goto behavior is manifest in fig. 2, where the differences[Q(R)−QNG(R)] (using the asymptotic values of σ from tab. 1 for QNG(R)) are shown.It is well-known that Lorentz invariance sets strong constraints onto the effective string ac-tion [15]. For the 3D U(1) theory, Polyakov [8] presented an argument for the existence of amapping between gauge and string degrees of freedom, and suggested the form of the action de-scribing the string dynamics, including a term proportional to the square of the extrinsic curvatureof the worldsheet. The resulting string is often referred to as the “rigid string”. Here we follow theanalysis performed in ref. [14], from which we can infer that the shape of Q(r) allowed by Lorentzinvariance, computed to the leading order in the extrinsic curvature term, isQ(R) = QNG(R)+Qr(R)+Qb(R) (2.3)withQb(R) =−b2pi360[1(R+1)4− 1R4], Qr(R) =− m2pi∞∑n=1K1 (2nm(R+1))−K1(2nmR)n, (2.4)for m =√σ/(2α). Qb(R) is a boundary contribution and Qr(R) is the contribution from theextrinsic curvature term in the effective string action. Including the next-to-leading-order (NLO)contribution in the extrinsic curvature term would correspond to replacing Qr(R) withQ′r(R) = Qr(R)+2120mσ( pi24)2[ 1(R+1)4− 1R4]. (2.5)The [Q(R)−QNG(R)] differences were at first fitted to the boundary correction Qb(R) alone.However, reasonable values of χ2red could only be reached for very large Rmin, and the resultingvalues for b2 were inconsistent with the expected scaling behavior. b2 should scale as σ−3/2, butwe found that b2σ3/2 increases from 0.033(3) (for β = 1.7) to 0.62(6) (for β = 2.4). To ruleout the possibility that the deviations could be due solely to a boundary term, we performed atwo-parameter fit of the [Q(R)−QNG(R)] differences to a correction term with a free exponent b,Q′b(R) = k[1(R+1)b− 1Rb]. (2.6)In this case, at every value of the coupling that we investigated, the results of the fits for b rangedbetween 2 and 3 and reasonable χ2red values could only be obtained for very large Rmin, resulting in1We performed our fits using the NG expression to all orders in the 1/R expansion, as in eq. (2.2), which is equiva-lent, within errors, to its truncation at order O(R−3).4PoS(LATTICE2014)349Effective string in the 3D U(1) gauge theory Davide Vadacchinoβ σ m0 L,Nt1.7 0.122764(2) 0.88(1) 641.9 0.066824(6) 0.56(1) 642.0 0.049364(2) 0.44(1) 642.2 0.027322(2) 0.27(1) 642.4 0.015456(7) 0.197(10) 128Table 1: Information on the setup of our simulations.β m m0 m/m01.7 0.28(9) 0.88(1) 0.32(10)1.9 0.25(4) 0.56(1) 0.45(7)2.0 0.17(2) 0.44(1) 0.39(4)2.2 0.11(1) 0.27(1) 0.41(4)2.4 0.06(2) 0.20(1) 0.30(10)Table 2: Best-fit results for m obtained using a three-parameter fit to our data, as explained in the text.k values essentially compatible with zero, within their uncertainties. A boundary-type correctionwas therefore rejected as the explanation for the observed deviations.We then tried to fit the [Q(R)−QNG(R)] differences with the rigid-string prediction Qr(R).2 Inthis case, fits with m as the only free parameter successfully describe the data, even for small valuesof Rmin, and the resulting m values show the expected scaling behavior, m ∝ m0 (see tab. 2). Forexample, for β = 2.2, a χ2red of order 1 is obtained for Rmin√σ = 2.15: a clear improvement overthe one-parameter fit to the pure Nambu-Goto model. Our results for Q(R) at β = 2.2 are shownin fig. 1, with the fitted curves.We also tested if the NLO correction eq. (2.5) from the rigidity term could be seen in our latticedata, by fitting the [Q(R)−QNG(R)] differences, with m as the only fitted parameter. This leads tobetter fits, especially for 1 < Rmin√σ < 2, but the χ2red values are still larger than one. Moreover, thebest-fit values for m exhibit changes larger than their uncertainties. Therefore, this term cannot beneglected, within the precision of our lattice data. Note that this correction is of the same order asthe boundary correction previously discussed. Hence it is possible that both boundary and rigid-string corrections are present. To test this hypothesis, we set m and σ to the previously obtainedbest-fit values, and carried out a one-parameter fit of [Q(R)−QNG−Q′r(R)] with the boundarycorrection Vb(R), using b2 as the free parameter. This leads to much better fits, including at smallRmin√σ : χ2red values around one are obtained already for Rmin√σ ∼ 1.65, with a small but non-negligible value of b2σ3/2 = 0.005(1). This shows that also this term is non-vanishing, and thatit should be included in the analysis. However, disentangling the contributions from the boundaryand the rigidity terms would require more precise data.The Rmin dependence of the fits reveals that the rigid-string correction could affect the value ofσ . Carrying out two-parameter fits of Q(R) to the function QNG +Qr(R) (or QNG +Q′r(R)) using σand m as free parameters, we find that the best-fit value of σ has a non-negligible impact on m andon its uncertainty, and that including the NLO rigid-string correction changes the result for m.It is clear that both the best-fit values of σ and b2 affect our estimate of m. In order to accountfor this interplay, we chose the result of a three-parameter fit to the data (using σ , m and b2 as freeparameters) as our estimate for m. As σ and b2 are not fixed, this leads to a slightly larger error onm. We report these estimates for m in tab. 2. The last column shows the m/m0 ratio, which revealsgood scaling properties. Our final estimate for the rigid-string parameter, including both statistical2The sum over Bessel functions was truncated at n= 100; the corresponding error was much smaller than the otheruncertainties of our data.5PoS(LATTICE2014)349Effective string in the 3D U(1) gauge theory Davide Vadacchino1 2 3 4 5R√σ0.0280.0290.0300.031Q(R)aNG, χ 2r =0.91, Rmin√σ=4.3Lu¨scherNG+VE , χ2r =1.04, Rmin√σ=2.15NG+VE +V′2 , χ2r =1.2, Rmin√σ=1.82β=2.2Figure 1: Data and best-fit curves for β = 2.2 with QNG, QNG+ Lüscher,Vext, and Vext +V ′2. The last two aretwo-parameter (σ and m) fits.1 2 3 4 5R√σ0.00000.00010.00020.00030.00040.00050.00060.0007(Q(R)−QNG(R))ama=0.112(1), χ 2r =1.08, Rmin√σ =2.31ma=0.101(1), χ 2r =1.31, Rmin√σ =2.31ma=0.099(2), b2σ3/2 =0.005(1), χ 2r =1.13, Rmin√σ =1.65Figure 2: Fits of the [Q−QNG(R)] differences at β = 2.2, with Vext, Vext +V ′2 and Vext +V ′2+ boundary term.and systematic uncertainties into the error budget, ismm0= 0.35(10). (2.7)3. Concluding remarksOur results show that in the 3D U(1) model, large deviations from the Nambu-Goto modelappear for β ≥ 2. These deviations grow as β is increased towards the continuum limit, and are6PoS(LATTICE2014)349Effective string in the 3D U(1) gauge theory Davide Vadacchinoaccounted for by an extrinsic-curvature term in the effective string action. The rigidity parameterm scales with the mass m0 of the lightest glueball, and becomes dominant in the continuum limit,making this string very different from the Nambu-Goto one.A necessary future step in our analysis is to disentangle the NLO contribution of the rigidityterm from the one due to the boundary term, and to study the fate of the rigid string at finitetemperature. Another interesting extension of this work would be to study the behavior of thestring width at large β , which should differ from the Nambu-Goto prediction.Acknowledgments This work is supported by the Spanish MINECO (grant FPA2012-31686and “Centro de Excelencia Severo Ochoa” programme grant SEV-2012-0249).References[1] M. Caselle, M. Panero, R. Pellegrini, and D. Vadacchino, (2014), 1406.5127.[2] A. M. Polyakov, Nucl.Phys. B120, 429 (1977).[3] M. Göpfert and G. Mack, Commun.Math.Phys. 82, 545 (1981).[4] M. Loan, M. Brunner, C. Sloggett, and C. Hamer, Phys.Rev. D68, 034504 (2003), hep-lat/0209159.[5] M. Zach, M. Faber, and P. Skala, Phys.Rev. D57, 123 (1998), hep-lat/9705019. M. Panero, JHEP0505, 066 (2005), hep-lat/0503024. M. Panero, Nucl.Phys.Proc.Suppl. 140, 665 (2005),hep-lat/0408002. E. Cobanera, G. Ortiz, and Z. Nussinov, Adv.Phys. 60, 679 (2011), 1103.2776.Y. D. Mercado, C. Gattringer, and A. Schmidt, Phys.Rev.Lett. 111, 141601 (2013), 1307.6120.[6] P. de Forcrand, M. 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Effective string description of the interquark potential in the 3D U(1) lattice gauge theory

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