# 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

This work was partly supported by the FCT – Fun-dação para a Ciência e a Tecnologia, I.P., under ProjectsNos. UIDB/04564/2020 and UIDP/04564/2020. P. J. S.acknowledges ﬁnancial support from FCT (Portugal) un-der Contract No. CEECIND/00488/2017. The authorsacknowledge the Laboratory for Advanced Computing atthe University of Coimbra ( ) forproviding access to the HPC resource Navigator. [1] K. G. Wilson, doi:10.1103/PhysRevD.10.2445[2] T. Banks, R. Myerson and J. B. Kogut, Nucl. Phys. B , 493-510 (1977) doi:10.1016/0550-3213(77)90129-8[3] J. Glimm and A. M. Jaﬀe, Commun. Math. Phys. ,195 (1977) doi:10.1007/BF01614208[4] E. H. Fradkin and L. Susskind, Phys. Rev. D , 2637(1978) doi:10.1103/PhysRevD.17.2637[5] M. Creutz, L. Jacobs and C. Rebbi,doi:10.1103/PhysRevD.20.1915[6] A. H. Guth, Phys. Rev. D , 2291 (1980)doi:10.1103/PhysRevD.21.2291[7] B. E. Lautrup and M. Nauenberg, doi:10.1016/0370-2693(80)90400-1[8] J. Fröhlich and T. Spencer, Commun. Math. Phys. ,411-454 (1982) doi:10.1007/BF01213610[9] B. Berg and C. Panagiotakopoulos, Phys. Rev. Lett. ,94 (1984) doi:10.1103/PhysRevLett.52.94[10] J. B. Kogut, E. Dagotto and A. Kocic, Phys. Rev. Lett. , 772 (1988) doi:10.1103/PhysRevLett.60.772[11] J. Jersak, C. B. Lang and T. Neuhaus, Phys. Rev.D , 6909-6922 (1996) doi:10.1103/PhysRevD.54.6909[arXiv:hep-lat/9606013 [hep-lat]].[12] M. Panero, JHEP , 066 (2005) doi:10.1088/1126-6708/2005/05/066 [arXiv:hep-lat/0503024 [hep-lat]].[13] F. V. Gubarev, L. Stodolsky and V. I. Za-kharov, Phys. Rev. Lett. , 2220-2222 (2001)doi:10.1103/PhysRevLett.86.2220 [arXiv:hep-ph/0010057 [hep-ph]].[14] R. Balian, J. M. Drouﬀe and C. Itzykson,doi:10.1103/PhysRevD.10.3376[15] J. M. Drouﬀe, C. Itzykson and J. B. Zuber, Nucl. Phys. B , 132-134 (1979) doi:10.1016/0550-3213(79)90418-8[16] G. Arnold, B. Bunk, T. Lippert and K. Schilling,Nucl. Phys. B Proc. Suppl. , 864-866 (2003)doi:10.1016/S0920-5632(03)01704-3 [arXiv:hep-lat/0210010 [hep-lat]].[17] P. Coddington, A. Hey, J. Mandula and M. Ogilvie,Phys. Lett. B , 191-194 (1987) doi:10.1016/0370-2693(87)90366-2[18] A. Nakamura and M. Plewnia, Phys. Lett. B , 274-278 (1991) doi:10.1016/0370-2693(91)90247-N[19] A. M. Polyakov, Phys. Lett. B , 82-84 (1975)doi:10.1016/0370-2693(75)90162-8 [20] A. M. Polyakov, Nucl. Phys. B , 429-458 (1977)doi:10.1016/0550-3213(77)90086-4[21] M. N. Chernodub, E. M. Ilgenfritz andA. Schiller, Phys. Rev. D , 034502 (2003)doi:10.1103/PhysRevD.67.034502 [arXiv:hep-lat/0208013 [hep-lat]].[22] B. L. G. Bakker, M. N. Chernodub and A. I. Veselov,Phys. Lett. B , 338-344 (2001) doi:10.1016/S0370-2693(01)00180-0 [arXiv:hep-lat/0011062 [hep-lat]].[23] J. Greensite, Lect. Notes Phys. , 1-211 (2011)doi:10.1007/978-3-642-14382-3[24] D. Dudal, O. Oliveira and P. J. Silva, AnnalsPhys. , 351-364 (2018) doi:10.1016/j.aop.2018.08.019[arXiv:1803.02281 [hep-lat]].[25] A. G. Duarte, O. Oliveira and P. J. Silva,Phys. Rev. D , no.1, 014502 (2016)doi:10.1103/PhysRevD.94.014502 [arXiv:1605.00594[hep-lat]].[26] A. Cucchieri, D. Dudal, T. Mendes and N. Van-dersickel, Phys. Rev. D , no.9, 094513 (2016)doi:10.1103/PhysRevD.93.094513 [arXiv:1602.01646[hep-lat]].[27] O. Oliveira and P. J. Silva, Phys. Rev. D ,114513 (2012) doi:10.1103/PhysRevD.86.114513[arXiv:1207.3029 [hep-lat]].[28] D. Dudal, O. Oliveira and N. Vandersickel, Phys. Rev.D , 074505 (2010) doi:10.1103/PhysRevD.81.074505[arXiv:1002.2374 [hep-lat]].[29] I. L. Bogolubsky, E. M. Ilgenfritz, M. Muller-Preusskerand A. Sternbeck, Phys. Lett. B , 69-73 (2009)doi:10.1016/j.physletb.2009.04.076 [arXiv:0901.0736[hep-lat]].[30] A. Cucchieri and T. Mendes, PoS LATTICE2007 , 297(2007) doi:10.22323/1.042.0297 [arXiv:0710.0412 [hep-lat]].[31] M. Q. Huber, Phys. Rev. D , 114009 (2020)doi:10.1103/PhysRevD.101.114009 [arXiv:2003.13703[hep-ph]].[32] M. Q. Huber, Phys. Rept. , 1-92 (2020)doi:10.1016/j.physrep.2020.04.004 [arXiv:1808.05227[hep-ph]].[33] A. K. Cyrol, L. Fister, M. Mitter, J. M. Pawlowski andN. Strodthoﬀ, Phys. Rev. D , no.5, 054005 (2016) doi:10.1103/PhysRevD.94.054005 [arXiv:1605.01856[hep-ph]].[34] C. S. Fischer, A. Maas and J. M. Pawlowski,Annals Phys. , 2408-2437 (2009)doi:10.1016/j.aop.2009.07.009 [arXiv:0810.1987 [hep-ph]].[35] A. C. Aguilar, D. Binosi and J. Papavassiliou, Phys. Rev.D , 025010 (2008) doi:10.1103/PhysRevD.78.025010[arXiv:0802.1870 [hep-ph]].[36] P. Boucaud, J. P. Leroy, A. Le Yaouanc, J. Micheli,O. Pene and J. Rodriguez-Quintero, JHEP ,099 (2008) doi:10.1088/1126-6708/2008/06/099[arXiv:0803.2161 [hep-ph]].[37] D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickeland H. Verschelde, Phys. Rev. D , 065047 (2008)doi:10.1103/PhysRevD.78.065047 [arXiv:0806.4348 [hep-th]].[38] R. Alkofer, W. Detmold, C. S. Fischer andP. Maris, Phys. Rev. D , 014014 (2004)doi:10.1103/PhysRevD.70.014014 [arXiv:hep-ph/0309077 [hep-ph]].[39] A. Nakamura and R. Sinclair, Phys. Lett. B , 396-402(1990) doi:10.1016/0370-2693(90)91403-X[40] T. A. DeGrand and D. Toussaint,doi:10.1103/PhysRevD.22.2478 [41] S. Duane, A. D. Kennedy, B. J. Pendleton andD. Roweth, Phys. Lett. B , 216-222 (1987)doi:10.1016/0370-2693(87)91197-X[42] R. G. Edwards et al. [SciDAC, LHPC and UKQCD],Nucl. Phys. B Proc. Suppl. , 832 (2005)doi:10.1016/j.nuclphysbps.2004.11.254 [arXiv:hep-lat/0409003 [hep-lat]].[43] P. J. Silva and O. Oliveira, Nucl. Phys. B , 177-198(2004) doi:10.1016/j.nuclphysb.2004.04.020 [arXiv:hep-lat/0403026 [hep-lat]].[44] E. M. Ilgenfritz, C. Menz, M. Muller-Preussker,A. Schiller and A. Sternbeck, Phys. Rev. D et al. [UKQCD], Phys. Rev.D , 094507 (1999) [erratum: Phys. Rev. D61