The lattice Landau gauge photon propagator for 4D compact QED
TThe lattice Landau gauge photon propagator for 4D compact QED
Lee C. Loveridge , , ∗ Orlando Oliveira , † and Paulo J. Silva ‡ CFisUC, Department of Physics,University of Coimbra, 3004-516 Coimbra, Portugal Los Angeles Pierce College, 6201 Winnetka Ave.,Woodland Hills CA 91371, USA
In this work we report on the Landau gauge photon propagator computed for 4D compact QED inthe confined and deconfined phases and for large lattices volumes: , and . In the confinedphase, compact QED develops mass scales that render the propagator finite at all momentumscales and no volume dependence is observed for the simulations performed. Furthermore, for theconfined phase the propagator is compatible with a Yukawa massive type functional form. For thedeconfined phase the photon propagator seems to approach a free field propagator as the latticevolume is increased. In both cases, we also investigate the static potential and the average valueof the number of Dirac strings in the gauge configurations m . In the confined phase the mass gaptranslates into a linearly growing static potential, while in the deconfined phase the static potentialapproaches a constant at large separations. Results shows that m is, at least, one order of magnitudelarger in the confined phase and confirm that the appearance of a confined phase is connected withthe topology of the gauge group. I. INTRODUCTION AND MOTIVATION
The quest to understand quark confinement has, longago, lead to the formulation of QED on a hypercubiclattice [1] by Wilson. He was able to show that the com-pact formulation of QED has two phases. In the strongcoupling limit, i.e. at low β = 1 /e where e is the barecoupling constant, the theory is confining in the sensethat the static potential between fermions grows linearlywith the distance. In the weak coupling regime, i.e. atlarge β values, it was argued that the results of pertur-bation theory for the continuum formulation should berecovered. These predictions for the phase diagram ofQED have been confirmed with numerical simulationsand by various theoretical analyses of the theory [2–13].In numerical calculations, they typically use a mappinginto its dual formulation associated with the Z n symme-try group [14, 15] instead of performing a sampling of thecompact formulation of QED directly.For compact QED the transition between the confiningand the non-confining phase occurs at β = β c ≈ , see [7,16] and references therein. For β < β c the static potential V ( R ) is compatible with a Cornell type of potential V ( R ) = V + aR + σ R , (1)with the dimensionless string tension σ being a decreas-ing function of β as its critical value is approached frombelow [12]. Numerical simulations for β > β c suggestthat the theory contains a massless photon and repro-duces the behaviour associated with a free field theory. ∗ [email protected] † [email protected] ‡ [email protected] However, in the numerical simulations the reproductionof a free field theory for β > β c is never perfect [9, 17, 18].Compact QED, having a confined and a deconfinedphase, can also be viewed as a laboratory to try to un-derstand the differences between the two phases. In 3D,the confining mechanism is associated with the presenceof monopole configurations [19, 20] at low β values; seealso [21, 22] for numerical simulations. However, in 4Dthese classical configurations or their equivalent are notable to generate a linearly rising potential at large sep-arations and, therefore, cannot be at the origin of theconfining mechanism. It has been argued that a Cornellpotential type for the static potential is related to thetopology of the gauge group, see e.g. [23] and referencestherein.The interest in compact QED is not limited to its phasediagram. For example, it is not yet clear if the contin-uum limit of compact QED is a sensible theory. If thefermion sector of the theory is ignored, its continuum the-ory is a free field theory that one expects to recover forsufficiently high β values. The simulations performed sofar suggest that this is the case, but full agreement withthe results of a free field theory has not been achieved.The transition of the confined phase to the deconfinedphase seems to be first order which, once more, raisesthe question of how to take the continuum limit for com-pact QED. Besides the question of the continuum limit, U (1) gauge theories are relevant to understand the Stan-dard Model, the comprehension of the Higgs sector callsfor simulations of Abelian Higgs models, and to describemany properties in Condensed Matter Physics. Further-more, recently the lattice QCD community has started tocompute QED corrections to QCD and, certainly, a goodunderstanding of lattice Abelian models is of paramountimportance. U (1) gauge theories are also a laboratory to-wards building simulations of gauge theories on quantum a r X i v : . [ h e p - l a t ] F e b computers.In the current work we are mainly interested in identi-fying the differences between the confined and deconfinedphase by looking at the Landau gauge photon propagatorin momentum space for the two phases of compact QED.The simulations described in this work do not take intoaccount the contribution of the fermionic degrees of free-dom to the dynamics. For the confined phase the theorydevelops a mass scale and the photon propagator is finiteand, at least for the simulations discussed, the presenceof the mass scale prevents or reduces the finite volumeeffects. On the other hand, in the deconfined phase thephoton propagator is compatible with a divergent /k or a higher power of k behaviour at low momenta k . Fi-nite volume effects are sizeable and the propagator ap-proaches that of a free field theory as the lattice volumeis increased. We hope that by studying the photon prop-agator one can arrive at a better understanding of theconfinement mechanism for QCD.The Landau gauge photon propagator for the confinedphase qualitatively follows the same type of behaviouras the Landau gauge gluon propagator in QCD for pureYang-Mills theories; see, for example, [24–30] for latticesimulations of the Landau gauge gluon propagator and[31–38] for continuum estimations of the same correlationfunction (see also the references therein). This suggeststhat the confinement mechanisms for the gluon and thephoton have some similarities and, in particular, thatthe confining theory is associated with dynamically gen-erated mass scales that make the propagator finite in thefull momentum range for compact QED and for QCD.For QED a photon mass has been related to a meaning-ful physical value of the expectation value of the vectorpotential squared, that is connected with the existenceof topological structures for the theory [13].Our work also aims to fill the gap in the literature[18, 39] and provide a large volume 4D lattice simulationof the Landau gauge photon propagator in momentumspace for β below and above β c . The two phases are dis-tinguished not only by the qualitative behaviour of thepropagators but also by their topological structures andthe static potential as computed from the Wilson loops.Our findings for the topological structures and static po-tential confirm previous results that can be found in theliterature.This work is organised as follows. In Sec. II we pro-vide the definitions used in our calculation ranging fromthe Wilson action, the gauge fixing procedure, the def-inition of the electromagnetic potential, the propagatorand number of Dirac strings crossing each plaquette. InSec. III the propagator and the static potential are dis-cussed for the confined phase, while in Sec. IV we discussthese properties for the deconfined phase. Finally, in Sec.V we summarise our results and discuss the differencesbetween the two phases. II. COMPACT QED: DEFINITIONS, LATTICESETUP AND DETAILS OF THE SIMULATION
In the current work we will consider the compact ver-sion of QED defined on an hypercubic lattice that is de-scribed by the Wilson action which, in Euclidean space,is given by S W ( U ) = β (cid:88) x (cid:88) (cid:54) µ,ν (cid:54) { − (cid:60) [ U µν ( x )] } , (2)where β = 1 /e , with e being the bare coupling constant,and the plaquette operator U µν ( x ) = U µ ( x ) U ν ( x + a ˆ e µ ) U † µ ( x + a ˆ e ν ) U † ν ( x ) (3)is written in terms of the link fields U µ ( x ) = exp (cid:110) i e a A µ (cid:16) x + a e µ (cid:17)(cid:111) , (4)with a being the lattice spacing, ˆ e µ the unit vector alongdirection µ and A µ is the bare photon field. In the contin-uum limit the plaquette operator (3) can also be written U µν ( x ) = exp (cid:26) i e (cid:73) C A µ ( z ) dz µ (cid:27) (5)where C is any closed curve around point x and infinitesi-mally close to it. On an hypercubic lattice, the term thatappears in the exponential is the change of the photonfield around a plaquette centered at x + a (ˆ e µ + ˆ e ν ) / andwe write U µν ( x ) = exp (cid:110) i e a (cid:16) ∆ A µν ( x ) (cid:17)(cid:111) (6)It follows from the definitions that everywhere in the lat-tice − π (cid:54) e a A µ (cid:54) π and − π (cid:54) e a ∆ A µν ( x ) (cid:54) π , i.e.the quantities e a A µ and e a ∆ A µν take values on com-pact spaces. Note that, in general, e a ∆ A µν is not givenby the sum of e a A µ over each of the links that definethe plaquette but, instead, ∆ A µν ( x ) = A µ (cid:16) x + a e µ (cid:17) + A ν (cid:16) x + a ˆ e µ + a e ν (cid:17) − A µ (cid:16) x + a ˆ e ν + a e µ (cid:17) − A µ (cid:16) x + a e ν (cid:17) + 2 π m µν ( x ) e a (7)where m µν ( x ) is an integer number that measures thenumber of Dirac strings that cross the plaquette . Theinteger field m µν ( x ) can be measured by combining in-formation on the links and plaquettes [40]. We will notstudy m µν ( x ) in detail but will report on its mean valueover the lattice, i.e. m = 16 V (cid:88) x,µ<ν m µν ( x ) (8) In 3D, m µν ( x ) can be identified with the number of monopolesin the plaquette. where V is the number of lattice points. Indeed, as dis-cussed below m can be used to distinguish between theconfined and deconfined phases, with the configurationsin the confined phase having a much larger mean numberof Dirac strings crossing each plaquette.The generating functional of the compact QED Green’sfunctions is given by Z = (cid:90) D A exp {− S W ( U ) } , (9)where S W is defined in Eq. (2). For the importancesampling we rely on an implementation of the HybridMonte Carlo method [41] based on QDP++ and Chromalibraries [42].The rotation of the links obtained by importance sam-pling to the Landau gauge is formulated as an optimiza-tion problem, over the gauge orbits. Setting the opti-mization problem depends on the definition of the photonfield.The gauge field can be computed with a linear defini-tion that, for the U (1) theory, reads e a A µ (cid:16) x + a e µ (cid:17) = U µ ( x ) − U † µ ( x )2 i . (10)If one relies on this definition the gauge fixing is per-formed by maximizing the functional F [ U ; g ] = 1 V D (cid:88) x,µ (cid:60) (cid:2) g ( x ) U µ ( x ) g † ( x + a ˆ e µ ) (cid:3) , (11)where V is the total number of lattice points and D theEuclidean spacetime dimension. It can be shown, see e.g.[43], that in this way the continuum Landau gauge con-dition is reproduced up to corrections O ( a ) . For the U (1) gauge theory there is no clear way to set the latticespacing and, therefore, this procedure can introduce sig-nificant deviations of the continuum Landau gauge whenapplied to the confined ( β (cid:46) ) or to the deconfined( β (cid:38) ) phase. The setup just described, that uses a lin-ear definition of the gauge field, is used for non-Abeliangauge theories defined on a lattice from simulations thatare close to continuum physics. In a first step to definethe Landau gauge for the photon field, we used the aboveprocedure relying on a steepest descent algorithm withFourier acceleration, see [43] for details and references,and controlling the approach to the Landau gauge withthe quantity ∆( x ) = (cid:88) ν (cid:104) U ν ( x − a ˆ e ν ) − U ν ( x ) (cid:105) , (12)a lattice version of − ∂ · A ( x ) . The maximisation wasstopped when θ = 1 V (cid:88) x (cid:12)(cid:12)(cid:12) ∆( x ) (cid:12)(cid:12)(cid:12) < − . (13) On the other hand, the Euclidean photon field can becomputed using a logarithmic definition e a A µ (cid:16) x + a e µ (cid:17) = − i ln (cid:16) U µ ( x ) (cid:17) . (14)This is an exact definition, up to machine precision, thatdoes not call for the use of a small lattice spacing. Then,following [44] adapted to the Abelian theory, the Landaugauge condition is achieved by maximizing the functional (cid:101) F [ U ; g ] = 1 V D (cid:88) x,µ (cid:26) − a e (cid:104) A ( g ) µ (cid:16) x + a e µ (cid:17)(cid:105) (cid:27) (15)over the gauge orbits. In the Eq. (15) the field e a A ( g ) isthe photon field given by Eq. (14) after the links U µ ( x ) have been gauge transformed by g ( x ) . The approach to-wards the Landau gauge can be monitored using (cid:101) ∆( x ) = a e (cid:88) ν (cid:104) A ν ( x − a e ν ) − A ν ( x + a e ν ) (cid:105) , (16)once more a lattice version of − ∂ · A ( x ) . In our compu-tation of the Landau gauge propagator, after the maxi-mization problem associated with the functional given inEq. (11), a maximization of the functional (15) is alsoperformed. In this way one aims to reduce possible devi-ations of the continuum Landau gauge for both phases ofthe theory. In this second maximization we use again asteepest descent algorithm with Fourier acceleration andthe gauge fixing was stopped when (cid:101) θ = 1 V (cid:88) x (cid:12)(cid:12)(cid:12) (cid:101) ∆( x ) (cid:12)(cid:12)(cid:12) < − . (17)In both stages the maximisation of the gauge fixingfunctional is done with a Fourier accelerated steepest de-scent method that calls for the PFFT library [45] to dothe required fast Fourier transformations. The completenumerical simulation, i.e. the importance sampling, thegauge fixing and the computation of all quantities, wereperformed in the Navigator cluster [46] of the Universityof Coimbra.From the definition (14) for the Euclidean spacetimephoton field, the momentum space photon field is givenby A µ ( p ) = (cid:88) x e − ip · ( x + a ˆ e µ ) A µ (cid:16) x + a e µ (cid:17) (18)and the Landau gauge propagator reads (cid:104) A µ ( p ) A µ ( p ) (cid:105) = V δ ( p + p ) D µν ( p ) (19)where (cid:104)· · · (cid:105) stands for the vacuum expectation value. Ina lattice simulation, the vacuum expectation values areaccessed via the generation of a set of configurations sam-pled accordingly with the probability distribution (9) andtaking averages of the products of gauge fields, such asthose in Eq. (19), over the full set of gauge configura-tions. For the analysis of the propagator, it will be as-sumed that the propagator has the same tensor structureas the continuum theory, i.e. D µν ( p ) = (cid:18) δ µν − p µ p ν p (cid:19) D (ˆ p ) (20)where the function D (ˆ p ) , named propagator below, is afunction of the tree level improved momenta ˆ p = 2 a sin (cid:16) πL n µ (cid:17) ,n µ = − L , − L , . . . , , , . . . , L − (21)where L is the number of lattice points in each side ofthe hypercubic lattice. The rationale to use ˆ p instead ofthe naive lattice momenta p = 2 πa L n µ , (22)comes from lattice perturbation theory that requires ˆ p instead of p . In the lattice evaluation of the gluon prop-agator the improved momentum also helps to suppressfinite spacing effects in the propagator [47]. In order tofurther suppress the effects due to the use of finite latticespacing we perform the conical and cylindrical cuts intro-duced in [47] for momenta a ˆ p > Λ IR and, following theprocedure devised in [24], below this threshold we con-sider all the momenta available to get information on theinfrared region. The choice of the infrared threshold Λ IR is a compromise between taking into account extra data,allowing larger fluctuations, and resulting in a smoothcurve for D ( p ) . The choice of this threshold does notchange the overall behaviour of the lattice data and Λ IR will be chosen differently for each simulation. In the fol-lowing we use Λ IR = 0 . for the largest lattice volumeand Λ IR = 0 . for the two smallest lattices.The description of the lattice propagator with the con-tinuum tensor structure as given by Eq. (20) is question-able, especially concerning the confined phase. Similarstudies for the gluon propagator show that the latticepropagator has other tensor structures not considered inEq. (20) and, in principle, they should also be consideredhere. However, given that the definition of the latticeLandau gauge returns a transverse gauge field, one ex-pects a gauge propagator that should also be transverse.Furthermore, the studies performed for the gluon prop-agator suggest that the introduction of momentum cutsselects the set of momenta where the finite lattice effectsare minimised. This gives us confidence that the sameshould apply to the photon propagator.If one assumes a tensor structure as given by Eq. (20),then the type of momentum considered in the projectoris irrelevant as long as one measures the propagator formfactor using D (ˆ p ) = 13 (cid:88) µ =1 D µµ ( p ) . (23) Confined Phase ( b = 0.8) Deconfined Phase ( b = 1.2) Figure 1. Mean value, over the lattice, of the plaquette forall simulations. In the simulation we did not always keepthe values of the plaquette for all computed trajectories and,therefore, the plots have regions with no data.
We remind the reader that for zero momentum the prop-agator is given by δ µν D (0) and, therefore, the computa-tion of D (0) requires a different normalisation factor.In the current work, we aim to see how the pho-ton propagator behaves in the confined and deconfinedphases. To achieve such a goal we perform Monte Carlosimulations of the theory at β = 0 . (confined phase) andat β = 1 . (deconfined phase). In order to check for finitevolume effects in both cases we perform simulations on , and hypercubic lattices. For each β valueand lattice volume, the propagators were computed usingthe last (in the Markov chain) 200 gauge configurations.The configurations used in the calculation of propagatorhave a separation of 10 trajectories for the smaller latticevolume, for both β values considered herein, and also forthe simulation in the confined phase ( β = 0 . ). Inthe remaining simulations we used a separation of 100trajectories in the corresponding Markov chain. ^ a ² e ² D ( p ^ ² ) Confined Phase ( b = 0.8) Figure 2. The Landau photon propagator in the confinedphase for all the lattice volumes. The solid black line refersto the fit to the lattice data discussed in the text. Seetext for further details. In Fig. 1 the mean values of the plaquette over thelattice are shown for each of the Markov chains. In thesimulations the value of the plaquette at the end of eachtrajectory was not always kept and, therefore, in the re-construction of the plaquette history we lost some of thedata points. As the Fig. shows the mean value of theplaquette seems to be independent of the lattice volumefor each β and in the deconfined region, i.e. for the simu-lation with β = 1 . , the plaquette is significantly larger.This result suggests that the U (1) links approach unityas β is increased.For the computation of statistical errors for all thequantities reported here, i.e. propagators, Wilson loopsand monopole densities, we rely on the bootstrap methodwith a 67.5% confidence level. The quoted errors associ-ated with the fits assume Gaussian error propagation. III. PHOTON IN THE CONFINED PHASE
The Landau gauge photon propagator for compactQED in the confined phase with β = 0 . and for thevarious lattice volumes can be seen in Fig. 2. The datadoes not follow the behaviour of a free particle propaga-tor and deviations from a /p functional form are clearlyseen. Indeed, the various data sets seem to be closer tothe qualitatively behaviour of the QCD gluon propaga-tor [24, 25, 27]. Moreover, the propagator being finiteover the full range of momenta suggests that 4D com-pact QED generates a mass gap dynamically, as is alsoobserved in 3D simulations [21, 22]. The data for the var-ious volumes is compatible within one standard deviationand, therefore, shows no volume dependence.It seems that the presence of the mass gap is sufficientto reduce the volume dependence of D ( p ) . This con- ( R , T ) R = 1R = 2R = 3R = 4R = 5R = 6R = 7R = 8R = 9r = 10R = 11R = 12
Confined Phase (96 - b = 0.8) Figure 3. Wilson loop W ( R, T ) at β = 0 . and for L = 96 . trasts with what is observed for the propagator in thedeconfined phase; more on this topic later.A possible way to identify the mass gap is by fittingthe lattice data to a given functional form. We foundthat, for all volumes, the lattice data is well described bya Yukawa type propagator a e D ( p ) = z p + m , (24)where z , p and m are dimensionless quantities.The fits using the full range of momenta result in a χ /d.o.f. = lattice, 1.31 for the lattice dataand 1.08 for the lattice data. The corresponding fit-ting parameters are z = 21 . , m = 2 . , z = 21 . , m = 2 . , z = 21 . , m = 2 . , respectively, and are all compatiblewithin one standard deviation. We have observed thatincreasing Λ IR results in smaller values for the χ /d.o.f. in all cases. In Fig. 2 the solid black line represents thefunctional form given in Eq. (24) with z and m givenby the estimation of the fit to the lattice data from thelargest volume. Similar curves using the other two setsof parameters could be drawn but the curves are indis-tinguishable to the naked eye from the curve shown.The low β phase of compact QED was investigatedby Wilson in [1], where he computed the static potentialfrom Wilson loops. Indeed, it was shown that, at low β ,the Wilson loop follows an area law and, therefore, the as-sociated static potential grows linearly with the distancebetween sources. It is in this sense that compact QEDat low β values is a confining theory. This observationmotivated us to compute Wilson loops and we took onlythose loops whose spatial part is along one of the latticeaxis to measure the static potential V ( R ) . The Wilsonloop can be seen in Fig. 3. Note that we use no trick toimprove the signal to noise ratio, the noise for W ( R, T ) is large for some cases and increases with R . This is an V ( R ) from T = 3 - 96 from T = 2 - 96 from T = 3 - 48 from T = 2 - 48 Confined Phase ( b = 0.8)
Figure 4. The static potential V ( R ) for β = 0 . and for L = 96 and L = 48 . indication that the static potential grows with R giventhat W ( R, T ) = e − V ( R ) T . (25)Further, it is clear from Fig. 3 that exponential be-haviour sets in for quite small T . Then, from the datafor W ( R, T ) one can measure V ( R ) from V ( R ) = log (cid:18) W ( R, T ) W ( R, T + 1) (cid:19) (26)and in Fig. 4 we show V ( R ) computed from taking T = 2 and T = 3 . The data in Fig. 4 should be regarded as anupper bound on V ( R ) . The results summarised in Figs. 3and 4 confirms that V ( R ) grows with R and suggest thatthe data is compatible with linear behaviour at large R .In this sense the simulation confirms that compact QEDis a confining theory at low β values.The static potential for 4D compact QED was com-puted in [12] from Polyakov loops, exploring dualitytransformations, and it was found that in the confinedphase V ( R ) grows linearly with the distance for suffi-ciently large R as also found in our simulations. IV. PHOTON IN THE DECONFINED PHASE
The nature of the photon propagator at large β is ex-pected to be rather different than that observed in Fig. 2.Indeed, as can be seen in Fig. 5, for the deconfined phasewith β = 1 . the photon propagator seems to diverge atzero momentum. Furthermore, if at low β the propagatoris blind to the finite volume effects, the data for the vari-ous volumes in Fig. 5 are not compatible with each otherwithin one standard deviation. The propagator data forthe smallest volume is above the other two sets ofpropagator data in the mid range momenta, while the ^ a ² e ² D ( p ^ ² ) Deconfined Phase ( b = 1.2) ^ a ² e ² D ( p ^ ² ) Figure 5. The Landau photon propagator in the deconfinedphase for all the lattice volumes. Note that the in the innerplot, the vertical scale is linear. data associated with the lattice is between the datacomputed with the smallest and the largest lattice vol-umes. However, at zero momenta the largest a D (0) isassociated with the largest volume, followed by the data and by the data in decreasing order of values.For momenta such that a p (cid:38) all the data sets seemsto be compatible within one standard deviation, see theinner plot in Fig. 5.The data in Fig. 5 suggest that the photon propagatordiverges as momentum approaches zero. If the data is tobe associated with a free field theory, it should reproducethe behaviour of a free field propagator. However, in afinite volume Monte Carlo simulation deviations from thecontinuum free field theory are expected as the simulationis performed on a finite lattice. The approach to thecontinuum behaviour can be tested by fitting the latticedata to the functional form a D ( a ˆ p ) = Z ( a ˆ p ) + Z ( a ˆ p ) . (27)If the theory reproduces a free field theory a Z (cid:54) = 0 isa manifestation of finite volume effects and one expects Z to become smaller as the lattice volume is increased.The direct fit using the full range of momenta and tak-ing into account the statistical errors of the lattice datareturns values of the χ /d.o.f. (cid:38) . For the smallestvolume, for a ˆ p (cid:62) . the fit has a χ /d.o.f. = 1 . with Z = 2 . and Z = 0 . . For the dataand for a ˆ p (cid:62) . it follows that χ /d.o.f. = 1 . with Z = 2 . and Z = 0 . . On the otherhand for the largest lattice volume, due to large fluc-tuations that are observed at larger momenta, one cannever achieve a reasonable χ /d.o.f. However, by dou-bling the statistical errors on the definition of the χ , thedata becomes compatible with (27) for a ˆ p (cid:62) . . In thiscase the fit has χ /d.o.f. = 1 . with Z = 2 . and ^ a ² e ² D ( p ^ ² ) Deconfined Phase ( b = 1.2) Figure 6. The Landau photon propagator in the deconfinedphase and the fits to Eq. (27). Z = 0 . . Note that in all cases one has Z ≈ . ,while Z decreases with the lattice volume.The lattice data together with the fits can be seen inFig. 6. In general and for the corresponding fittingranges, the curves overlap the Monte Carlo data. Fur-ther, the coefficient Z ≈ . , we are quoting thevalue of the fit to the largest lattice volume, seems to benearly independent of the volume. The data in Fig. 6suggest that Z is independent of L , while Z is sensitiveto L . Indeed this coefficient goes from Z = 0 . for the smallest volume to Z = 0 . for the largestvolume, which is about / of the value for the smallestvolume; note that the inverse of the ratio of the latticesizes is 1/3. This result for Z suggests that the data forthe propagator seems to converge to the propagator of afree field theory in the infinite volume limit. This state-ment has to be read with care due to the use of σ in thedefinition of the minimising χ for the largest volume.That the fitting range does not start at the smallest non ( R , T ) R = 1R = 2R = 3R = 4R = 5R = 6R = 7R = 8R = 9R = 10R = 11R = 12R = 13R = 14R = 15R = 16R = 17R = 18R = 19R = 20
Deconfined Phase (96 - b = 1.2) V ( R ) Deconfined Phase ( b = 1.2) Figure 7. The Wilson loop (top) for L = 96 and the statticpotential (bottom) for all the lattice volumes in the deconfinedphase. vanishing momentum for each volume is not surprising,as finite volume effects, that should appear at the small-est momenta, are to be expected. We have also triedfitting the data with an almost free field propagator, i.e.assuming D ( p ) = Z / ( p ) α and leaving Z and α as freeparameters. The fits to this last functional form have thesame problems as those mentioned before but it turns outthat α ≈ , i.e. the lattice data for the propagator followsclosely the behaviour of a free field theory . In order to quantify the typical values of α let us report on itsvalues given by fitting to the propagator data replacing σ by σ in the definition of the minimising χ . Demanding that the χ /d.o.f. (cid:54) , it follows that for the smallest lattice volume thefitting range starts at a ˆ p = 0 . and has α = 1 . / , the fittingrange for the data starts at a ˆ p = 0 . and has α = 1 . ,while for the largest volume the fitting range starts at a ˆ p = 1 and has α = 0 . . σ a ² Figure 8. The string tension as a function of /L . The linesare connect the origin where σ a = 0 with the value found forthe largest lattice volume. The shaded region represents theone standard deviation on the result for σ a from the largestvolume. The above analysis suggests that the Monte Carlopropagator data almost reproduce a free field theorypropagator. Let us check the results for the static po-tential, computed from Wilson loops as was done for theconfined phase. The Wilson loop for various values of R is given in Fig. 7 for the largest lattice volume and itlooks rather different from the Wilson loop for the con-fined phase reported in Fig. 3. If for the confined phasethe slope increases with R , for the deconfined phase theslope of the log W ( R, T ) seems to be the same for all R .Indeed, measuring V ( R ) from the effective mass and tak-ing its value for T = 9, one gets the bottom plot of Fig. 7.The large distance behaviour of V ( R ) is sensitive to finitevolume effects and the slope of V ( R ) for large R becomessmaller as L is increased. In Fig. 8 we show the stringtension measured by fitting V ( R ) in the range R = 9 − for the various volumes. The corresponding χ /d.o.f. forthe various fits are always below 0.5. The dashed blueline connects the origin with the result for the largestvolume, while the shaded region takes into account theone standard deviation on σa for L = 96 . Our resultsseems to be compatible with a vanishing string tensionin the infinite volume limit.The short distance behaviour of V ( R ) is difficult to un-derstand from the computed Wilson loop directly. TheMonte Carlo data for the photon propagator is compat-ible with free field propagator behaviour at high p andapproaching /p as the volume is increased and, there-fore, one expects to have, in the infinite volume limit, V ( R ) ∝ /R at short distances. We have tried to disen-tangle the short distance behaviour from the V ( R ) MonteCarlo data but the results were inconclusive.Our simulations for the deconfined phase of compactQED suggest that for the β considered here, the finitevolume effects are still not negligible even for a latticevolume as large as . m b = 0.848 b = 0.896 b = 0.8 m b = 1.248 b = 1.296 b = 1.2 Figure 9. Average Dirac string density over the lattice asgiven by Eq. (8) for the confined (top) and deconfined (bot-tom) phases for the thermalised gauge configurations. Thehorizontal axis refers to the configuration number.
V. SUMMARY AND CONCLUSION
In the current work the Landau gauge photon propa-gator is investigated for compact QED in the strong cou-pling (confining) and weak coupling (free field theory)regimes and for various lattice volumes. By computingthe static potential, our simulation confirms that at low β the theory is confining and the behaviour of the photonpropagator in momentum space follows closely a Yukawatype of propagator, i.e. 4D compact QED has a massgap. Moreover, in the confining phase the theory devel-ops a mass scale that makes the photon propagator finitein the full momentum range.For the deconfined phase, the photon propagator seemsto approach a free field type of propagator as the infinitevolume is approached. We have observed that the match-ing with a free field theory is not perfect with both thephoton propagator and the static potential showing somedeviations from the expected behaviour, that we inter-preted as being due to finite volume effects. Indeed, thedeviations from a free field theory are reduced, in all thecomputed quantities, as the lattice volume is increased.Given that at low momenta the propagator of a free fieldtheory diverges and the lattice regularizes both the UVand IR divergences, in a sense the deviations from thefree field theory results are not unexpected.Comparing the confined and deconfined phase results,it seems that it is the generation of a mass gap thatoccurs for the confined phase that turns the propagatoressentially independent of the lattice volume. This is asituation that is also seen in the simulations for QCD.If the phase diagram for compact QED as a functionof β has two different phases, one can ask how can onedistinguish them. According to [19] the appearance ofthe confining phase for the 3D theory is due to the pres-ence of monopole configurations. The monopoles are con-nected with the topology of the gauge group and theybecomes irrelevant for the dynamics at large β values.The 4D equivalent to the monopole configurations areDirac strings that should be seen on a finer analysis ofthe gauge configurations. In Fig. 9 we report on the av-erage number of Dirac strings over the lattice, computedwith the definitions (7) and (8), for all the lattices. Theplots show that m is independent of L in both phases, al-though the fluctuations for the smaller lattices are muchlarger, and that m is about a factor of fifty larger in theconfined phase when compared to its value in the de-confined phase. This result suggests that, indeed, theDirac strings are responsible for the confined phase in4D compact QED in agreement with the suggestion of [19]. However, further studies are required to draw firmconclusions. ACKNOWLEDGMENTS
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