Flux tubes and string breaking in three dimensional SU(2) Yang-Mills theory
aa r X i v : . [ h e p - l a t ] M a y Flux tubes and string breaking in three dimensional SU(2) Yang-Mills theory
Claudio Bonati ∗ and Silvia Morlacchi † Dipartimento di Fisica dell’Universit`a di Pisa and INFN - Sezione di PisaLargo Pontecorvo 3, I-56127 Pisa, Italy (Dated: May 21, 2020)We consider the three dimensional SU(2) Yang-Mills theory with adjoint static color sources,studying by lattice simulations how the shape of the flux tube changes when increasing the distancebetween them. The disappearance of the flux tube at string breaking is quite abrupt, but precursorsof this phenomenon are present already when the separation between the sources is smaller than itscritical value, a fact that influences also some details of the static potential.
I. INTRODUCTION
Color confinement is one of the main nonperturbativefeatures of nonabelian gauge theories. A first principleproof of this phenomenon is still lacking and constitutespart of the first Millennium Problem of the Clay Math-ematical Institute [1]. However a huge amount of in-formation about color confinement, both qualitative andquantitative, has been obtained by numerical simulationsof lattice discretized gauge theories.Starting from the seminal work in Ref. [2], the study offlux tubes between static color charges gained a promi-nent role in the investigation of color confinement [3–7].The flux tube between two sources has been investigated,for example, to test the predictions of effective string the-ory [8–13] and the dual superconductor picture of colorconfinement [14–19], but also the case of more than twocharges has been studied [20–23].So far the vast majority of flux tube investigations con-centrated on the Yang-Mills pure glue case, with sourcestransforming in the fundamental representation of thegauge group. In this setup a nonvanishing asymptoticstring tension is present , and the static potential risesindefinitely with the distance between the sources; thissignal that the flux tube always connects them, indepen-dently of their distance.Only recently investigations carried out in full QCDwith physical quark masses appeared [25, 26], howeverthe computational burden of simulations with dynami-cal light flavours makes impossible to obtain in this caseresults as accurate as those achieved for Yang-Mills the-ories. In particular, the QCD results obtained so far donot indicate any significant qualitative difference with re-spect to the pure glue case.Such a qualitative difference is however to be expected,since the asymptotic string tension vanishes in theories ∗ [email protected] † [email protected] This is what happens for most of the gauge groups used in theliterature and, in particular, for the SU(N) gauge groups. How-ever gauge groups exist for which gluons do screen fundamentalcharges, and when this happens the asymptotic string tensionvanishes, see e.g. [24] for the case of G . with dynamical matter fields in the fundamental repre-sentation of the gauge group. A striking consequenceof this fact is the peculiar behavior of the static poten-tial: for small distances between the sources the poten-tial looks like the one of the pure glue case, but whenthe separation increases beyond a critical value R c (thestring breaking length) the potential flattens, and doesnot grow anymore linearly with the distance between thecharges [24, 27–32].It is natural to expect the flux tube to disappear, orat least to be strongly suppressed, when the distance be-tween the sources approaches R c , but there are severalways in which this could happen: the flux tube could forexample behave as in the pure glue case for small dis-tances and then disappear abruptly at R c , or it couldstart to delocalize already when the sources are close toeach other. Which of these possibilities is the correct onecan only be established by numerical simulations, how-ever to perform such a study in QCD would be very de-manding from the computational point of view. We cannevertheless hope to gain at least some insight on whathappens in QCD by studying simplified models display-ing string breaking.In this work we use for this purpose the three dimen-sional SU(2) Yang-Mills theory with static sources trans-forming in the adjoint representation. It is indeed simpleto show that an adjoint charge can be screened by gluons,and this model has been already used in the past to nu-merically investigate string breaking and string decay inthe static potential [33–36] (see also [37] for the four di-mensional case and e.g. [38] for a non-lattice approach).Our principal aim is the study of the flux tube behavioras a function of the distance between the adjoint sources,and in particular for distances close to the critical value R c . However, to better appreciate the similarities anddifferences with respect to the case without string break-ing, we will also perform a precision study of the staticpotential in the unbroken string phase.The paper is organized as follows: in Sec. II we sum-marize the numerical setup adopted and we describe theobservables used to study the flux tube. Numerical re-sults are reported in Sec. III A and III B for the flux tubeand the static potential respectively. Finally in Sec. IVwe summarize the results obtained and we draw our con-clusions. II. NUMERICAL SETUP
As anticipated in the introduction, in this workwe use the three-dimensional SU(2) Yang-Mills theorywith static sources in the adjoint representation as atestbed to investigate the behavior of flux tubes closeto string breaking. The usual Wilson discretization [39]is adopted, which for the case of the gauge group SU(2)can be written in the form S = X x , µ>ν β (cid:18) −
12 Π µν ( x ) (cid:19) . (1)In this expression x is a point of a three dimen-sional isotropic lattice with periodic boundary condi-tions, µ, ν ∈ { , , } denote two lattice directions, andΠ µν ( x ) = Tr (cid:2) U µ ( x ) U ν ( x + ˆ µ ) U † µ ( x + ˆ ν ) U † ν ( x ) (cid:3) (2)is the trace of the product of the link variables aroundthe plaquette in position x laying in the plane ( µ, ν ). Theupdate is performed by using standard heatbath [40, 41]and microcanonical [42] moves, in the ratio of 1 to 5.On the contrary of what happens in four-dimensionalgauge theories, the gauge coupling is not dimensionlessin three space-time dimensions, and the bare continuumcoupling g is related to the β value entering Eq. (1) by therelation aβ = 4 /g , where a denotes the lattice spacing.As a consequence there is no dimensional transmutationin the three-dimensional case, and dimensionless physicalobservables can be expanded in inverse powers of β in theweak coupling limit. In particular we will sometimes usethe following approximate expression for the square rootof the string tension [43] a √ σ = 1 . β + 1 . β + O ( β − ) , (3)which is valid for β ≥ . d can be computed by using F adj ( d ) = − aN t log h Tr P adj ( )Tr P adj ( d ˆ1 ) i , (4)where N t is the temporal extent of the lattice, latticetranslation and rotation invariances have been used and P adj ( x ) denotes the adjoint Polyakov loop in position x .The trace of P adj ( x ) can be immediately related to thetrace of the Polyakov loop in the fundamental represen-tation P fund ( x ) = N t − Y k =0 U ( x + k ˆ0 ) (5)(where periodic boundary conditions are implied and 0denotes the temporal direction) by the relationTr P adj ( x ) = | Tr P fund ( x ) | − . (6) To investigate the flux tube between two static adjointcharges separated by a distance d along ˆ1 (this choice ofthe direction is purely conventional and irrelevant for thefinal result) we use the observable ρ adj µν ( d, x t ) = h Tr P adj ( )Tr P adj ( d ˆ1 )Π µν ih Tr P adj ( )Tr P adj ( d ˆ1 ) i − h Π µν i , (7)where Π µν stands forΠ µν ( d ˆ1 / x t ˆ2 ) , (8)i.e. for the plaquette oriented in the ( µ, ν ) plane, posi-tioned midway between the static sources at a transversedistance x t . On the lattice the d/ ⌊ d/ ⌋ .This “midpoint” flux tube is the one that has been mostinvestigated in the literature, mainly because in this waywe minimize the effect of the static charges. Of courseit would be interesting to extend the study to have acomplete picture of the whole flux tube also closer to thestatic sources, but this would require a decomposition of ρ adj µν in near and far-field components (see [18]). F adj ( d ) and ρ adj µν ( d, x t ) are the natural generalizationsto the adjoint case of the usual expressions for funda-mental static sources, and it is simple to show that inthe naive continuum limit ρ adj µν reduces to the variationof h F µν i (no sum intended) induced by the presence ofthe adjoint static sources. Moreover ρ adj µν is multiplica-tively renomalizable, and its renormalization constant isthe same of Π µν , which also coincides with that of ρ fund µν .In order to avoid computing this renormalization con-stant we will use in the following the ratio R µν ( d, x t ) = ρ adj µν ( d, x t ) ρ fund10 ( d, , (9)which has a well defined continuum limit if numerator andenominator are computed at the same lattice spacing.To obtain accurate estimates of F adj ( d ) and ρ adj µν ( d, x t )we use both multihit [44] and multilevel [45] noise reduc-tion algorithms. The application of these algorithms isstraightforward, once the components of P adj ( x ) are ex-plicitly written in term of P fund ( x ) by using the relation P adj ab ( x ) = 12 Tr( σ a P fund ( x ) σ b [ P fund ( x )] † ) , (10)where σ a denotes a Pauli matrix. The optimal valuesfor the number of levels, the size of the slices and thenumber of updates to be used in the multilevel algorithmhas been determined by minimizing the fluctuations ofTr P adj ( )Tr P adj ( d ˆ1 ) at fixed simulation time.The optimal number of hits to be used in the multi-hit turned out to be quite insensitive to the distance d between the sources, while the optimal setup for the mul-tilevel algorithm typically consisted of a single level forsmall distances between the sources, and of two levels forlarger values of d . Let us consider for example the case x t / a ρ a d j µ ν ( d = a , x t ) L=16, (1,0)L=16, (2,0)L=32, (1,0)L=32, (2,0) x t / a -2 × -4 -1 × -4 × -4 × -4 ρ a d j µ ν ( ) - ρ a d j µ ν ( ) (1,0)(2,0) FIG. 1. Comparison of the estimates obtained for the quantity ρ adj µν ( d = 4 a, x t ) at β = 6 . L = 16 and L = 32). Results refer to the longitudinal (1 , ,
0) components of the chromoelectricfield. of the lattice 64 at β = 11 . d = 4 a consisted of a single level algorithm with slices ofthickness 4 a and 600 updates for slice, while for d = 15 a we used two slices of thickness 4 a and 8 a , with 10000 and10 updates for slice respectively.In all the cases data corresponding to different valuesof d and/or x t came from different simulations, and theyare thus statistically independent of each other. Statis-tical errors have been estimated by means of standardblocking, jackknife and bootstrap procedures. III. NUMERICAL RESULTSA. Flux tube
In this section we report our results concerning the be-havior of the flux tube close to string breaking, obtainedby studying the dependence of ρ adj µν ( d, x t ) (as a functionof the transverse distance x t ) on the separation d be-tween the adjoint static charges. The majority of oursimulations have been performed on a 32 lattice, but weresorted also to different lattice sizes to investigate finitevolume and finite lattice spacing effects.We mainly focus on the longitudinal component of thechromoelectric field (corresponding to ρ adj10 with the con-ventions of the previous section), which turns out to bethe dominant component of the flux tube also in the ad-joint case. However the study of the two other com-ponents of the field strength is important to identify thedisappearance of the flux tube: since string breaking hap-pens when the two charges are at a finite distance fromeach other, we cannot expect the longitudinal chromo-electric field to vanish at string breaking, because thenear-field of the charges is always present (see [18]). Thenatural expectation is that the longitudinal componentof the chromoelectric field became of the same size as theother components at string breaking.As a first step we investigate which lattice sizes are needed in order not to have significant finite volume ef-fects. For this purpose we estimated ρ adj µν ( d, x t ) for d = 4 a at coupling β = 6 .
0, using two different lattice sizes, i.e. L = 16 and L = 32. As can be seen from the numeri-cal results reported in Fig. 1, finite size effects are wellunder control in this setup, and the longitudinal compo-nent of the chromoelectric field is indeed the dominantcomponent of the flux tube.To study the dependence of the adjoint flux tube on thedistance d between the static sources, we thus start byusing a fixed scale approach on a 32 lattice at β = 6 . R c ≈ a (see [35] and Sec. III B), andresults for ρ adj µν ( d, x t ) obtained in this setup are shown inFig. 2, both for the longitudinal component ρ adj10 and forthe transverse ones ρ adj20 and ρ adj12 .From data in Fig. 2 we can already draw several inter-esting observations: first of all it is evident that the lon-gitudinal component of the adjoint flux tube decreasesby increasing the distance between the sources. Whilethe huge decrease from d = 4 a to d = 8 a can be ascribedto the closeness of the sources (and thus to the presenceof the Coulomb component at d = 4 a ), the differencesbetween d = 8 a and d = 9 a can not be interpreted inthis way. Indeed the transverse components of the fieldstrength do not change significantly, and the same hap- x t / a ρ a d j ( d , x t ) d=4ad=8ad=9ad=10a x t / a ρ a d j µ ν ( d , x t ) d=8a, (2,0)d=9a, (2,0)d=8a, (1,2)d=9a, (1,2) FIG. 2. Results for ρ adj µν ( d, x t ) obtained on a 32 lattice at cou-pling β = 6 .
0. In the upper panel the longitudinal componentis reported, while in the lower panel the transverse directionsare shown (data have been slightly shifted to improve thereadability). Notice the different scales on the vertical axis ofthe two panels. x t σ R ( d = a , x t ) β =6.0 β =5.8 β =5.6 β =5.5 x t σ R ( d = a , x t ) β =6.0 β =5.8 β =5.5 FIG. 3. Numerical results for the ratio R µν ( d = 9 a, x t ) de-fined in Eq. (9) obtained using a 32 lattice. pens for the flux tube in the fundamental representation:for comparison ρ fund10 ( d, x t = 0) changes by less than 4%when going from d = 8 a to d = 9 a , to be compared withthe 21% change of ρ adj10 ( d, x t = 0).Another important thing to notice is that the longi-tudinal component of the adjoint flux tube is about afactor three larger than the transverse components at d = 8 a and 9 a , however at string breaking (i.e. d = 10 a ) ρ adj10 ( d, x t ) suddenly drops and become compatible withthe transverse components. As previously discussed thisis the smoking gun signal of the flux tube disappearance,since for finite R c we can not expect the longitudinal (orany other) component to vanish. A hint that at d = 10 a the physics of the system is changing comes also from thescaling of error-bars: from Fig. 2 we see that ρ adj10 ( d, x t )data at d = 10 a have errors which are approximatelythree times those at d = 9 a , despite the fact that thestatistics accumulated for d = 10 a is about six timeslarger than the one used for the other distances. A pos-sible interpretation of this fact is that for d < a the fluxtube is present and the main sources of statistical errorin ρ adj10 are the fluctuations of Polyakov loops, which arehowever kept well under control by using the multilevelalgorithm. For d = 10 a the string is broken and fluctua-tions in the plane containing the plaquette increase (the“broken ends” of the string moves freely), thus reducingthe effectiveness of the error reduction of our implemen-tation of the multilevel algorithm.To have a finer control of the separation between the σ d π σ w k w ln(d/d ) d=8a β =6.0 d=9a β =6.0 d=9a β =5.8d=9a β =5.6 d=9a β =5.5 FIG. 4. Dependence of the width of the flux tube (as definedin Eq. (12)) on the distance between the adjoint charges. sources and better resolve the distances close to stringbreaking, we now abandon the fixed scale approach andchange the distance between the sources by varying thelattice spacing. More in detail we keep d = 9 a on a 32 lattice, and we increase the lattice spacing by decreas-ing the value of the coupling constant β in the range[5 . , . β = 6 . . a p σ ( β )needed for 5 . ≤ β ≤ . ρ adj µν , due to the presenceof the lattice dependent renormalization, so we use theratio R µν defined in Eq. (9). In Fig. 3 we present ourresults for the longitudinal component R ( d = 9 a, x t )at four different values of the coupling β in the range[5 . , . , R is steeplydecreasing when increasing the distance between theadjoint Polyakov loops. In particular, its peak valueat x t = 0 reduces approximately by a factor of threewhen increasing the separation between the sources from ≈ .
02 fm (at β = 6 .
0) to ≈ .
13 fm (at β = 5 . R also decreases when increasingthe lattice spacing, but in a less dramatic way than thelongitudinal component.From Fig. 3 it is not completely clear if the longitudi- β a √ σ a lattice. Conversion tophysical units is performed by using 1 / √ σ = 0 .
45 fm. x t σ R ( d , x t ) β =6.0, d=8a, L=32 β =8.6392, d=12a, L=48 FIG. 5. Continuum scaling of R ( d, x t ) for d ≈ .
91 fm. nal flux tube just gets rescaled when approaching stringbreaking, or it is also slightly distorted (i.e. the rescalingfactor is different for different values of x t ). To betterinvestigate this point we tried computing the flux tubewidth defined by w ( d ) = R ∞ x t ρ adj10 ( d, x t )d x t R ∞ ρ adj10 ( d, x t )d x t . (11)This quantity does not need any renormalization and itwas computed by using a spline interpolation of the datafor ρ ( d, x t ).Numerical results for w ( d ) are shown in Fig. 4, and aslight increase of the flux tube width with the distancebetween the sources seems to be present. While there isno reason for Effective String Theory (EST) to providerobust results for theories with string breaking, it is nev-ertheless interesting to compare the observed behaviorwith the one predicted by EST. In particular in Fig. 4we also report the result of a best fit of the form2 πσw ( d ) = k w log( d/d ) , (12)which for k w = 1 is the form expected on the basis of EST(see e.g. [8]). The functional form in Eq. (12) well de-scribes numerical data for w ( d ) but with k w = 0 . w ( d ) on the distance d ismild enough that also a lineal function correctly repro-duces data. From this fact we can conclude that the fluxtube is not simply rescaled as d approaches R c , it getsslightly broader but the numerical accuracy is not enoughto reliably fix the functional form of w ( d ).To close this section we verify that lattice discretiza-tion artifacts do not significantly affect the results pre-sented so far. For this purpose we compare data obtainedby using two different lattice spacings, which have beendetermined by using Eq. (3) to keep the value of d con-stant in physical units. We used a 32 lattice at coupling β = 6 . lattice with β = 8 . a ( β = 6) ≈ .
91 fm ≈ a ( β = 8 . . (13) r min / a k fit of V(d)fit of V’(d) FIG. 6. Values of the coefficient k defined in Eq. (15) ob-tained by fitting on the interval [ r min , a ] data for V adj (cor-responding to integer values of r min /a ) or its first derivative(corresponding to half-integer values of r min /a ). Data havebeen estimated by using a 64 lattice at β = 11 . The results obtained with this setup for R ( d, x t ) arepresented in Fig. 5, and it is clear that lattice artifactsare well under control, being at most of the same size ofstatistical errors. B. Static potential
In this section we describe the results of our study ofthe static potential between adjoint color charges, per-formed for distances between the sources small enough tobe in the unbroken string regime. The aim of this studyis to understand if the breaking of the string, associatedto the dependence of the flux tube on d discussed in theprevious section, has some precursor in the behavior ofthe static potential.One of the most typical properties of the static poten-tial between fundamental charges is the presence of theso called Luscher term [46]. This is just the first termof the EST expansion of the static potential in powers of σr (see e.g. [47] for a recent review), and it is character-ized by the fact of having an universal coefficient, whichdepends only on the space-time dimensionality but noton the gauge group nor on other high-energy propertiesof the theory (as far as an asymptotic string tension ex-ists). In our three-dimensional setup the large distancebehavior of the fundamental static potential is thus V fund ( d ) = σd + π d + O ( d − ) . (14)Does something analogous to the Luscher term ex-ist also for the static potential V adj ( d ) between adjointsources? While V adj has been previously investigatedseveral times [33–36], to the best of our knowledge an ac-curate investigation of the presence of the Luscher termin V adj has not been carried out so far . We thus try to This issue was mentioned in [35] but the Authors report that no σ d V ( d ) / σ / β =11.3138 β =8.34688 β =6.0 β =6.0 (fund) FIG. 7. Continuum scaling of the adjoint static potential:data have been obtained on lattices with L = 32 ( β = 6 . L = 48 ( β = 8 . L = 64 ( β = 11 . β = 6 . fit data for V adj according to the ansatz V adj ( d ) = σd + k π d , (15)where k is a free parameter. Such an ansatz is reasonableonly for d < R c however, just like in standard EST, valuesof d which are too small have to be excluded from the fit,since they are contaminated by the Coulomb interactionbetween the sources (that in our case is logarithmic).In Fig. 6 we show our estimates for the parameter k en-tering Eq. (15), obtained by fitting data for V adj ( d ) com-puted on a 64 lattice at coupling β = 11 . β = 6 .
0, sowe expect R c ≈ a , and indeed up to d = 18 a we foundno signal of string breaking. In Fig. 6 we also report esti-mates obtained by fitting the two-point finite differenceapproximation of the derivative of V adj d V adj d r ( r + a/ ≃ V adj ( r + a ) − V adj ( r ) a (16)instead of the static potential itself, which give consis-tent results. From Fig. 6 we see that k is definitely notconsistent with 1, and this fact can be interpreted as asignal for d < R c that the string will break by increasingthe distance between the sources.Finally, in Fig. 7 we show the continuum scaling of V adj for three different values of the lattice spacing (whichgoes from a ≈ .
11 fm at β = 6 . a ≈ .
057 fm at β =11 . V adj (2 / √ σ ) = 7 √ σ ,and an almost perfect scaling is observed, which impliesalso in this case the absence of significant cut-off effects. IV. CONCLUSIONS
In this paper we have studied color flux tubes in atheory which displays string breaking, and in particulartheir behavior when the separation between the staticsources approaches the string breaking distance R c . Forthis purpose we used as testbed the three-dimensionalSU(2) Yang-Mills theory with charges transforming inthe adjoint representation of the gauge group.We have shown that the adjoint flux tube, like thefundamental one, consists mainly of the longitudinalchromoelectric field for distances d between the sourcesthat are smaller than R c . As the critical distance R c is approached, the longitudinal chromoelectric field getsstrongly suppressed, becoming of the same size of thetransverse fields at R c . The disappearance of the fluxtube is quite abrupt, and the value of R ( d, x t = 0)(which is related to square of longitudinal chromoelec-tric field inside the flux tube) decreases approximatelyby a factor of 3 when the relative difference between d and R c reduces below 10%.This rapid disappearance is the one that could havebeen naively guessed from the behavior of the adjointstatic potential V adj ( d ), which suddenly switches froman approximately linear grow to a constant plateau at d ≃ R c . We have however seen that precursors of stringbreaking are present for d smaller than R c , which are ba-sically related to the failure of standard effective stringtheory. The scaling of the square width w ( d ) of theflux tube with the distance d follows (at least within thepresent accuracy) the expected logarithmic behaviour,but the value of the coefficient differs from the universaleffective string prediction. Similarly, an analogous of theLuscher term is present also in V adj ( d ), but again numer-ical data are not compatible with the expected universalcoefficient.Future studies should be aimed at extending this anal-ysis to other models, to understand to which amountthe phenomenology at string breaking observed in thethree-dimensional SU(2) Yang-Mills case is generic and,in particular, is relevant for QCD. For the same reasonit would be very interesting to investigate if there is arelation between the values of the coefficients k w and k in Eqs. (12),(15) (or better, their deviations from theEST predictions) and some nonuniversal property of thetheory, like its spectrum. Acknowledgements
Numerical simulations have beenperformed on the CSN4 cluster of the Scientific Com-puting Center at INFN-PISA. It is a pleasure to thankMichele Caselle for useful comments and discussions. stable fit parameter was found. , 374 (1983).[3] A. Di Giacomo, M. Maggiore and S. Olejnik, Phys. Lett.B , 199 (1990).[4] A. Di Giacomo, M. Maggiore and S. Olejnik, Nucl. Phys.B , 441 (1990).[5] G. S. Bali, K. Schilling and C. Schlichter, Phys. Rev. D , 5165 (1995) [hep-lat/9409005].[6] R. W. Haymaker, V. Singh, Y. C. Peng and J. Wosiek,Phys. Rev. D , 389 (1996) [hep-lat/9406021].[7] P. Cea and L. Cosmai, Phys. Rev. D , 5152 (1995)[hep-lat/9504008].[8] A. Allais and M. Caselle, JHEP , 073 (2009)[arXiv:0812.0284 [hep-lat]].[9] F. Gliozzi, M. Pepe and U.-J. Wiese, Phys. Rev. Lett. , 232001 (2010) [arXiv:1002.4888 [hep-lat]].[10] F. Gliozzi, M. Pepe and U.-J. Wiese, JHEP , 057(2011) [arXiv:1010.1373 [hep-lat]].[11] N. Cardoso, M. Cardoso and P. Bicudo, Phys. Rev. D , 054504 (2013) [arXiv:1302.3633 [hep-lat]].[12] A. Amado, N. Cardoso and P. Bicudo, arXiv:1309.3859[hep-lat].[13] M. Caselle, M. Panero and D. Vadacchino, JHEP ,180 (2016) [arXiv:1601.07455 [hep-lat]].[14] M. S. Cardaci, P. Cea, L. Cosmai, R. Falconeand A. Papa, Phys. Rev. D , 014502 (2011)[arXiv:1011.5803 [hep-lat]].[15] P. Cea, L. Cosmai and A. Papa, Phys. Rev. D , 054501(2012) [arXiv:1208.1362 [hep-lat]].[16] P. Cea, L. Cosmai, F. Cuteri and A. Papa, Phys. Rev. D , 094505 (2014) [arXiv:1404.1172 [hep-lat]].[17] P. Cea, L. Cosmai, F. Cuteri and A. Papa, JHEP ,033 (2016) [arXiv:1511.01783 [hep-lat]].[18] M. Baker, P. Cea, V. Chelnokov, L. Cosmai, F. Cu-teri and A. Papa, Eur. Phys. J. C , 478 (2019)[arXiv:1810.07133 [hep-lat]].[19] N. Battelli and C. Bonati, Phys. Rev. D , 114501(2019) [arXiv:1903.10463 [hep-lat]].[20] F. Okiharu and R. M. Woloshyn, Nucl. Phys. Proc.Suppl. , 745 (2004) [hep-lat/0310007].[21] F. Bissey, F. G. Cao, A. R. Kitson, A. I. Signal,D. B. Leinweber, B. G. Lasscock and A. G. Williams,Phys. Rev. D , 114512 (2007) [hep-lat/0606016].[22] P. Bicudo, N. Cardoso and M. Cardoso, Prog. Part. Nucl.Phys. , 440 (2012) [arXiv:1111.0334 [hep-lat]].[23] A. S. Bakry, X. Chen and P. M. Zhang, Phys. Rev. D ,114506 (2015) [arXiv:1412.3568 [hep-lat]].[24] B. H. Wellegehausen, A. Wipf and C. Wozar, Phys. Rev. D , 016001 (2011) [arXiv:1006.2305 [hep-lat]].[25] P. Cea, L. Cosmai, F. Cuteri and A. Papa, Phys. Rev. D , 114511 (2017) [arXiv:1702.06437 [hep-lat]].[26] C. Bonati, S. Cal`ı, M. D’Elia, M. Mesiti, F. Negro,A. Rucci and F. Sanfilippo, Phys. Rev. D , 054501(2018) [arXiv:1807.01673 [hep-lat]].[27] O. Philipsen and H. Wittig, Phys. Rev. Lett. , 4056(1998) Erratum: [Phys. Rev. Lett. , 2684 (1999)][hep-lat/9807020].[28] C. E. Detar, O. Kaczmarek, F. Karsch and E. Laermann,Phys. Rev. D , 031501 (1998) [arXiv:hep-lat/9808028[hep-lat]].[29] C. W. Bernard, T. A. DeGrand, C. E. Detar, P. La-cock, S. A. Gottlieb, U. M. Heller, J. Hetrick, K. Orginos,D. Toussaint and R. L. Sugar, Phys. Rev. D , 074509(2001) [arXiv:hep-lat/0103012 [hep-lat]].[30] F. Gliozzi and A. Rago, Nucl. Phys. B , 91 (2005)[hep-lat/0411004].[31] G. S. Bali et al. [SESAM Collaboration], Phys. Rev. D , 114513 (2005) [hep-lat/0505012].[32] J. Bulava, B. Hrz, F. Knechtli, V. Koch, G. Moir,C. Morningstar and M. Peardon, Phys. Lett. B , 493(2019) [arXiv:1902.04006 [hep-lat]].[33] P. W. Stephenson, Nucl. Phys. B , 427 (1999)[hep-lat/9902002].[34] O. Philipsen and H. Wittig, Phys. Lett. B , 146(1999) [hep-lat/9902003].[35] S. Kratochvila and P. de Forcrand, Nucl. Phys. B ,103 (2003) [hep-lat/0306011].[36] M. Pepe and U.-J. Wiese, Phys. Rev. Lett. , 191601(2009) [arXiv:0901.2510 [hep-lat]].[37] K. Kallio and H. D. Trottier, Phys. Rev. D , 034503(2002) [arXiv:hep-lat/0001020 [hep-lat]].[38] A. Agarwal, D. Karabali and V. P. Nair, Nucl. Phys. B , 216 (2008) [arXiv:0705.0394 [hep-th]].[39] K. G. Wilson, Phys. Rev. D , 2445 (1974).[40] M. Creutz, Phys. Rev. D , 2308 (1980).[41] A. D. Kennedy and B. J. Pendleton, Phys. Lett. ,393 (1985).[42] M. Creutz, Phys. Rev. D , 515 (1987).[43] M. J. Teper, Phys. Rev. D , 014512 (1998)[hep-lat/9804008].[44] G. Parisi, R. Petronzio and F. Rapuano, Phys. Lett. , 418 (1983).[45] M. Luscher and P. Weisz, JHEP , 010 (2001)[hep-lat/0108014].[46] M. Luscher, Nucl. Phys. B , 317 (1981).[47] B. B. Brandt and M. Meineri, Int. J. Mod. Phys. A31