More on the three-gluon vertex in SU(2) Yang-Mills theory in three and four dimensions
aa r X i v : . [ h e p - l a t ] J u l More on the three-gluon vertex in SU(2) Yang-Millstheory in three and four dimensions
Axel Maas, Milan Vujinovi´cInstitute of Physics, NAWI Graz, University of Graz,Universit¨atsplatz 5, A-8010 Graz, AustriaJuly 21, 2020
Abstract
The three-gluon vertex has been found to be a vital ingredient innon-perturbative functional approaches. We present an updated latticecalculation of it in various kinematical configurations for all tensorstructures and multiple lattice parameters in three dimensions, andin a subset of those in four dimensions, for SU(2) Yang-Mills theoryin minimal Landau gauge. In three dimensions an unambiguous zerocrossing for the tree-level form-factor is established, and consistencyfor all investigated form factors with a power-like divergence towardsthe infrared. The results in four dimensions are consistent with sucha behavior, but do not yet reach deep enough into the infrared toestablish it.
Vertices are the central quantities to encode interactions. Of particularimportance among them are the primitively divergent ones. In Yang-Millstheory these are the three-gluon vertex, the ghost-gluon vertex, and the four-gluon vertex. Besides the fact that they encode themselves information onthe interactions, they are also the building blocks for solutions of functionalequations, like Dyson-Schwinger equations and functional renormalizationgroup equations.Among the primitively divergent vertices the ghost-gluon vertex showedso far least modification from its tree-level behavior [1–10]. The three-gluonvertex, however, showed quite surprising features, especially at low momenta[2–6, 8, 9, 11–19]. Lattice results find that in two dimensions its tree-levelform factor shows unambiguously a zero crossing at about 450(50) MeVmomenta on the largest investigated volume together with a power-like di-vergence towards negative infinity [3]. In three dimensions, results suggested1he existence of such a zero-crossing at a few hundred MeV [4, 12]. However,only the lowest non-vanishing momentum point has been found to signal azero crossing, which is notoriously affected by systematic errors. In fourdimensions, it depends on the judgment of statistical and systematic uncer-tainties, whether a zero crossing has been observed [4, 11, 18, 19]. Functionalresults [5, 6, 8, 9, 13–17] strongly suggest that the zero crossing is alwayspresent, though it is not yet unambiguously decided whether in higher di-mension a divergence occurs. The situation for the other tensor structureshas been much less investigated, but depending on definitions they alsoshow non-trivial behavior [5, 6, 8, 9, 11, 13–15, 17]. The four-gluon vertexis substantially more involved, though results indicate a similar non-trivialbehavior [5, 6, 20].Here, we will investigate the three-gluon vertex of SU(2) Yang-Mills the-ory using lattice gauge theory in minimal Landau gauge in three dimensionsfor all tensor structures in a number of kinematic configurations at highstatistics. We establish its zero crossing in the thermodynamic limit ataround 340 −
580 MeV, depending on momentum configuration, and findsubstantial evidence in favor of a power-like divergence towards zero mo-mentum. A corresponding cross check in four dimensions, which accountsfor the same type of systematic uncertainties, but only for the tree-leveltensor structure, is compatible with the qualitative behavior in three di-mensions. We establish an upper limit for the zero crossing of 170 − Our aim is twofold. One is the behavior of the tree-level form factor at verysmall momenta, especially in three dimensions. As the ghost-gluon vertexexhibited unexpected large lattice artifacts in some, but not all, momentumconfigurations previously investigated [1], an important step is to constrainlattice artifacts. Thus, we used for this the relatively straight-forward ap-proach of [12] to determine the vertex function on a number of differentlattice settings. The list of lattice setups is listed in table 1. Especially,the results are based on the unimproved Wilson action in minimal Landaugauge. 2able 1: Number and parameters of the configurations used,ordered by dimension, lattice spacing, and physical volume.See [12, 21] for technical details. Auto-correlation times oflocal observables have been monitored to ensure decorrela-tion. See [22] on details of how the lattice spacing was de-termined. Config. is the number of configurations. Tensorindicates whether this setting has been used to determinetree-level (tl) or non-tree-level (ntl) form factors. d N β a [fm] a − [GeV] L [fm] config. Tensor3 40 3.18 0.246 0.802 9.84 77879 tl3 60 3.18 0.246 0.802 14.8 67388 tl3 80 3.18 0.246 0.802 19.7 57620 tl3 32 3.60 0.210 0.940 6.71 33696 ntl3 48 3.60 0.210 0.940 10.1 33696 ntl3 64 3.60 0.210 0.940 13.4 33696 ntl3 80 3.60 0.210 0.940 16.8 33696 ntl3 32 4.00 0.184 1.07 5.89 33696 ntl3 48 4.00 0.184 1.07 8.83 33696 ntl3 64 4.00 0.184 1.07 11.8 33696 ntl3 80 4.00 0.184 1.07 14.7 33696 ntl3 32 4.40 0.164 1.20 5.24 33696 ntl3 48 4.40 0.164 1.20 7.86 33696 ntl3 64 4.40 0.164 1.20 10.5 33696 ntl3 80 4.40 0.164 1.20 13.1 33696 ntl3 40 5.61 0.123 1.60 4.92 86451 tl3 60 5.61 0.123 1.60 7.38 62667 tl3 80 5.61 0.123 1.60 9.84 58793 tl3 40 10.5 0.0616 3.21 2.46 69093 tl3 60 10.5 0.0616 3.21 3.70 69006 tl3 80 10.5 0.0616 3.21 4.93 49202 tl4 16 2.1306 0.246 0.800 3.94 63772 tl4 24 2.1306 0.246 0.800 5.90 49600 tl4 32 2.1306 0.246 0.800 7.87 44671 tl4 16 2.3936 0.123 1.60 1.97 78668 tl4 24 2.3936 0.123 1.60 2.95 70703 tl4 32 2.3936 0.123 1.60 3.94 35256 tl4 16 2.5977 0.0616 3.20 0.986 50940 tl4 24 2.5977 0.0616 3.20 1.48 65792 tl4 32 2.5977 0.0616 3.20 1.96 37327 tlThe form factors Γ j are generally obtained [12] by projection and am-3utation as Γ j = Γ jµνρabc (cid:10) A aµ A bν A cν (cid:11) Γ jαβγdef D dgασ D ehβω D fiγδ Γ jσωδghi . Herein the D are the gluon propagators. They can be found in [1] forthe range of lattice settings used here. The Γ j are suitable base tensors,of which four transverse ones are sufficient to determine the three-gluonvertex in Landau gauge completely [23]. Thus, the form factors Γ j yield thedeviations from the base tensors Γ j , where any constant prefactors can beabsorbed judiciously into these base tensors. Here, the base tensor Γ willbe the lattice tree-level tensor from [24], as was used in [12].For the remaining three base tensors, we use the Bose-symmetric basisdeveloped in [17, 21]. For the non-tree-level vertices we are primarily inter-ested in an exploration of their low-momentum behavior. Due to asymptoticfreedom, they should approach zero at sufficiently large momenta in three di-mensions . Given the complexity in deriving the non-tree-level lattice formfactor at leading perturbative order [24], which to our knowledge has notyet been calculated in three dimensions, we employ their continuum versions[17] instead, and only determine them up to ap <
1. It turns out that thisis not a serious limitation, as above this momenta all of them are found forall considered momentum configurations to be consistent with zero withinerrors.We consider thus the following three additional tensor structures [17]Γ µνρ ( p , p , p ) = t µ t ν t ρ (1)Γ µνρ ( p , p , p ) = p t µ δ νρ + p t ν δ ρµ + p t ρ δ µν (2)Γ µνρ ( p , p , p ) = ω t µ δ νρ + ω t ν δ ρµ + ω t ρ δ µν (3) t = p − p t = p − p t = p − p ω = p − p ω = p − p ω = p − p using unimproved lattice momenta [21]. Note that for SU(2) the only colorstructure is the totally anti-symmetric Levi-Civita tensor, which is thereforesuppressed. In contrast to the tree-level vertex, all of them have a largermass dimension, due to the different momentum structure. This is takencare of when isolating the dimensionless form factors. Note that there may be potentially logarithmic deviations, due to the fact that non-perturbative resummation in three dimensions does yield non-trivial additional corrections[25].
4s in [12] introduced, we will consider three different momentum con-figurations. One is the symmetric one, in which all momenta are of equalmagnitude. The second is the back-to-back configuration, in which one mo-mentum vanishes. The third has two momenta at 90 degrees, but otherwiseunconstrained. This last one is only considered for the tree-level form factor.Note that the form factors are all symmetric in all arguments, like the vertexitself. The choice of basis (1-3) has the consequence that for the symmetricconfiguration all but the Γ form factor vanish for kinematical reasons [17]. The results for the tree-level form factor in three dimensions are shownin figure 1. It is visible that at large momenta the form factor behavesas expected, and approaches one quickly above roughly 2 GeV. It is alsovisible that the form factor drops below zero around 300-500 MeV, relativelyindependent of the lattice parameters. However, lattice artifacts play arelevant role at low momenta. It is visible that an increase in volume atfixed discretization tends to make the form factor more negative, while a finerdiscretization at fixed volume tends to do the opposite. As a consequence,the results on the coarser lattices are systematically below the ones on thefiner lattices in the infrared. This effect is statistically significant, and nota small effect.We fitted each lattice setup by two ans¨atzeΓ ( p ) = 1 − a ( p ) − b (4)Γ ( p ) = 1 − a p + m , (5)i. e. by a power-law ansatz and a dominant-pole ansatz, allowing explicitlyfor zero mass. The latter would be expected in a perturbative setting. Noneof the lattice setups show explicitly a flattening at small momenta, and thuswe did not include a fit with a crossing of the zero momentum axis with flatslope.We did the fits explicitly for the back-to-back momentum configurationand the symmetric momentum configuration. We included all points below1 GeV momentum as well as all other points with less than 25% relativeerror in the fits. The fits become harder and harder on smaller volumes, asthere is less and less distinction from the asymptotic constant behavior. Wedid not obtain reasonable fit results for the dominant-pole ansatz (5), butachieved very good fit-quality, with χ values between 1 and 3 in most cases,with the power-law ansatz (4). This is strikingly different from the situationin presence of a Brout-Englert-Higgs effect [26], where (5) works excellent5 [GeV] -1
10 1 / ) π ( p , p , Γ -16-14-12-10-8-6-4-202 Three-gluon vertex, symmetric p [GeV] -1
10 1 / ) π ( p , p , Γ -16-14-12-10-8-6-4-202 Lattice parameters, a=0.25 fm (20 fm) , a=0.12 fm (9.8 fm) , a=0.062 fm (4.9 fm), a=0.25 fm (15 fm) , a=0.12 fm (7.4 fm) , a=0.062 fm (3.7 fm) , a=0.25 fm (9.8 fm) , a=0.12 fm (4.9 fm) , a=0.062 fm (2.5 fm)Power-law fit Three-gluon vertex, orthogonal momenta with two equal p [GeV] -1
10 1 / ) π ( , p , Γ -16-14-12-10-8-6-4-202 Three-gluon vertex, back-to-back q [ G e V ] k [ G e V ] / ) π ( p , q , Γ -16-14-12-10-8-6-4-202 Three-gluon vertex, orthogonal momenta
Figure 1: The three-gluon vertex tree-level form factor in three dimensions.The top-left panel shows a cut along the diagonal of the lower-right plot.The latter shows the situation with two momenta being orthogonal to eachother, with interpolation of the data points for the largest volume at thecoarsest discretization. The top-right panel shows the back-to-back mo-mentum configuration and the bottom-left panel the symmetric momentumconfiguration. For momenta larger than 1 GeV only points with a relativeerror of less than 50% are shown. The fit (4) shown is to the largest volumein the symmetric configuration (bottom-left panel) and in the back-to-backconfiguration (top panels).for the three-gluon vertex. We concentrate therefore in the following on thepower-law ansatz (4). The fit is also shown in figure 1. It is also visible, bycomparing the fit to a slightly different momentum configuration, that theparameters are angle-dependent.This behavior is emphasized by plottingln | − Γ | , (6)rather than the dressing function itself, against ln p . Because of (4), this6 [GeV] -1
10 1 / ) | ) π ( p , p , Γ l n ( | - -1-0.500.511.522.53 Three-gluon vertex, symmetric p [GeV] -1
10 1 / ) | ) π ( p , p , Γ l n ( | - -1-0.500.511.522.53 Lattice parameters, a=0.25 fm (20 fm) , a=0.12 fm (9.8 fm) , a=0.062 fm (4.9 fm), a=0.25 fm (15 fm) , a=0.12 fm (7.4 fm) , a=0.062 fm (3.7 fm) , a=0.25 fm (9.8 fm) , a=0.12 fm (4.9 fm) , a=0.062 fm (2.5 fm)Power-law fit Three-gluon vertex, orthogonal momenta with two equal p [GeV] -1
10 1 / ) | ) π ( , p , Γ l n ( | - -1-0.500.511.522.53 Three-gluon vertex, back-to-back q [ G e V ] -1 k [ G e V ] -1
10 1 / ) | ) π ( p , q , Γ l n ( | - -1-0.500.511.522.53 Three-gluon vertex, orthogonal momenta
Figure 2: The three-gluon vertex tree-level form factor in three dimensionsplotted as ln | − Γ | The top-left panel shows a cut along the diagonal of thelower-right plot. The latter shows the situation with two momenta beingorthogonal to each other, with interpolation of the data points for the largestvolume at the coarsest discretization. The top-right panel shows the back-to-back momentum configuration and the bottom-left panel the symmetricmomentum configuration. Above 1 GeV only points with a relative error ofless than 50% are shown. The fit is the same as in figure 1.should yield a linear behavior in ln p at small momenta. This is indeed thecase, as is visible in figure 2.To allow for a continuum extrapolation, the fits have been done for alllattice settings. The results are shown in figure 3. The prefactor shows littledependence on the lattice parameters within uncertainty, but depends on themomentum configuration. This emphasizes a slight angular dependency ofthe form factor, as seen already in figures 1 and 2 where the fit from the back-to-back configuration does not too well describe the equal-momentum config-uration at 90 degrees. The exponent is substantially volume-dependent, andin different ways for the different kinematic configurations. It has, within7 -1 a MomentumBack-to-back a=0.25 fmSymmetric a=0.25 fmBack-to-back a=0.12 fmSymmetric a=0.12 fmBack-to-back a=0.062 fmSymmetric a=0.062 fm
Prefactor of the fit ] -1 b Exponent of the fit
Figure 3: The fit parameters of (4) in 3 dimensions, together with a fit forthe volume dependence. For the prefactor the fit is linear in the inverselattice extent, and quadratic for the exponent.errors, also not reached its infinite-volume behavior on the current latticesettings. The infinite-volume-extrapolated results for the back-to-back con-figuration and the symmetric configuration areΓ b = 1 − . p − . Γ s = 1 − . p − . +2 − , respectively. These extrapolated values also confirm the slight angular de-pendence. Note that the value of the exponent is well within errors belowtwo, and thus too small for a massless pole behavior. It is even furtherbelow the prediction of a scaling behavior [27], which is larger than two. Inaddition, this yields a zero crossing at 485 +32 − MeV in the symmetric con-figuration and 460 +122 − MeV in the back-to-back configuration in the ther-modynamic limit, slightly above the one on the largest volumes employedhere.
The results for the non-tree-level form factors are shown in figure 4. Onlyin the symmetric case, where only one of them is potentially non-vanishing,any statistically substantial deviations from zero is visible. In the orthogonalcase all non-tree-level form factors are within errors consistent with zero.However, this needs to be interpreted as upper limits only. Still, even if theyare non-zero, they are substantially smaller than the tree-level form factor.This is in good agreement with results from functional studies [5, 6, 8, 9, 13–17].The result in the symmetric case is still marginally compatible with anon-zero and negative value below roughly 500 MeV. It is still much smaller8 [GeV]0 0.2 0.4 0.6 0.8 1 / ) π ( p , p , Γ -0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.1 =0.94 GeV -1 L=6.7 fm, a =0.94 GeV -1 L=10 fm, a=0.94 GeV -1 L=13 fm, a =0.94 GeV -1 L=17 fm, a=1.1 GeV -1 L=5.9 fm, a =1.1 GeV -1 L=8.8 fm, a=1.1 GeV -1 L=12 fm, a =1.1 GeV -1 L=15 fm, a=1.2 GeV -1 L=5.2 fm, a =1.2 GeV -1 L=7.9 fm, a=1.2 GeV -1 L=11 fm, a =1.2 GeV -1 L=13 fm, a symmetric Γ Form factor p [GeV]0 0.2 0.4 0.6 0.8 1 / ) π ( , p , Γ -1-0.8-0.6-0.4-0.200.20.40.6 back-to-back Γ Form factor p [GeV]0 0.2 0.4 0.6 0.8 1 / ) π ( , p , Γ -2.5-2-1.5-1-0.500.511.5 back-to-back Γ Form factor p [GeV]0 0.2 0.4 0.6 0.8 1 / ) π ( , p , Γ -1-0.500.511.522.5 back-to-back Γ Form factor
Figure 4: The non-tree-level form-factor of three-gluon vertex in three di-mensions for the investigated momentum configurations. The symmetriccase with only one non-trivially non-vanishing form factor is shown in thelower-left panel. The remainder are the three non-tree-level form factorsin the back-to-back momentum configuration. The fits are discussed in thetext.than the tree-level form factor. Above this momentum, it is again too smallto be resolved by the current statistics. The same form-factor shows in theback-to-back configuration an upward fluctuation below 500 MeV momen-tum.Given the size of the fluctuations, these results strongly suggest thatsubstantially more than an order of magnitude of statistics is necessary tohave a chance of seeing a signal for any statistical sound non-zero behaviorin the back-to-back case. This is also necessary in the symmetric case toobtain a statistically significant sound signal for a deviation from zero inthe infrared. Thus, this needs to be postponed until such an amount ofcomputing time becomes available for this purpose.However, the behavior seen is indeed consistent with the same power-law9 [GeV] -1
10 1 / ) π ( p , p , Γ -1-0.500.511.52 Three-gluon vertex, symmetric p [GeV] -1
10 1 / ) π ( p , p , Γ -1-0.500.511.52 Lattice parameters, a=0.25 fm (7.9 fm) , a=0.12 fm (3.9 fm) , a=0.062 fm (2.0 fm) , a=0.25 fm (5.9 fm) , a=0.12 fm (3.0 fm) , a=0.062 fm (1.5 fm) , a=0.25 fm (3.9 fm) , a=0.12 fm (2.0 fm) , a=0.062 fm (0.99 fm)Power-law fit Three-gluon vertex, orthogonal momenta with two equal p [GeV] -1
10 1 / ) π ( , p , Γ -1-0.500.511.52 Three-gluon vertex, back-to-back q [ G e V ] k [ G e V ] / ) π ( p , q , Γ -1012 Three-gluon vertex, orthogonal momenta
Figure 5: Same as in figure 1, but in four dimensions.behavior as for the tree-level. This is indicated by the fits using the sameexponents as in figure 1, and only adjusting the prefactor to -0.025 in thesymmetric case and to 0.015, -0.09, and 0.025 in the back-to-back case forthe three form factors. The only other change was to drop the tree-levelone. This suggests that the qualitative behavior of all form factors are thesame within the present statistical uncertainty, when the tree-level behavioris taken into account.
The four-dimensional case is much harder to assess, as the higher computa-tional costs make it much harder to obtain the corresponding high statisticson large volumes. Still, it is possible to see whether the deviation from onetowards small momenta can be as well described by the power-law fit (4) asin three dimension.The results for the tree-level form factor are shown in figure 5. Thereare two distinct behaviors. 10 [GeV] -1
10 1 / ) | ) π ( p , p , Γ l n ( | - -2-1.5-1-0.500.51 Three-gluon vertex, symmetric p [GeV] -1
10 1 / ) | ) π ( p , p , Γ l n ( | - -2-1.5-1-0.500.51 Lattice parameters, a=0.25 fm (7.9 fm) , a=0.12 fm (3.9 fm) , a=0.062 fm (2.0 fm) , a=0.25 fm (5.9 fm) , a=0.12 fm (3.0 fm) , a=0.062 fm (1.5 fm) , a=0.25 fm (3.9 fm) , a=0.12 fm (2.0 fm) , a=0.062 fm (0.99 fm)Power-law fit Three-gluon vertex, orthogonal momenta with two equal p [GeV] -1
10 1 / ) | ) π ( , p , Γ l n ( | - -2-1.5-1-0.500.51 Three-gluon vertex, back-to-back q [ G e V ] -1 k [ G e V ] -1
10 1 / ) | ) π ( p , q , Γ l n ( | - -2-101 Three-gluon vertex, orthogonal momenta
Figure 6: Same as in figure 2, but in four dimensions.One is in the ultraviolet. There, a distinct dependency on the latticespacing is observable in the back-to-back configuration, which is not presentin the other momentum configurations. Though not visible on the scaleof figure (1) it is also present in three dimensions. A similar observationof pronounced discretization-dependency was also made for the ghost-gluonvertex [1] and (quenched) scalar-gluon vertices [28]. It therefore appears tobe a rather general feature of the back-to-back configuration. Therefore, thismomentum configuration is not as suitable to extract ultraviolet propertiesthan as, say, the symmetric momentum configuration.As is visible in figure 5 no statistically significant zero-crossing is ob-served. The infrared behavior is again better emphasized by the transformedform (6). The behavior towards small momenta, as is visible in figure 6, isindeed consistent with a power-law-like deviation from one, as described by(4). Again, the results exhibit a slight angular dependency.Performing the fits in the same way as in three dimensions yields thefit parameters shown in figure 7. Again no pronounced dependency on thediscretization is observed. The infinite-volume limits of the fits are also11 -1 a MomentumBack-to-back a=0.25 fmSymmetric a=0.25 fmBack-to-back a=0.12 fmSymmetric a=0.12 fmBack-to-back a=0.062 fmSymmetric a=0.062 fm
Prefactor of the fit ] -1 b Exponent of the fit
Figure 7: The fit parameters of (4) in 4 dimensions, together with a fit forthe volume dependence. For the prefactor the fit is quadratic in the inverselattice extent, and linear for the exponent.again different for different angles, and are found to beΓ b = 1 − . p − . Γ s = 1 − . +14 − p − . +4 − , which correspond to an expected zero crossing at 151 +20 − MeV and 201 +106 − MeV for the back-to-back momentum configuration and the symmetric mo-mentum configuration, respectively. In these momenta ranges the resultsin figure 5 are indeed compatible with zero within statistical uncertainty.Hence, the four dimensional form factor shows qualitatively the same be-havior as in three dimensions, though with a substantially smaller exponent.This is in as far interesting, as this implies a decrease of the exponent fromabout 2.2 to roughly 1.5 and 0.6 from two to four dimensions, crossingsomewhere between two and three dimensions the expected behavior for amassless pole. These findings are consistent with those for SU(3) Yang-Millstheory and QCD [11, 29] and other results [5, 6, 8, 9, 13–17, 19].
In summary, we presented the most comprehensive study of the three-gluonvertex in three dimensions to date. We established an unambiguous zerocrossing of the tree-level form factor and find substantial evidence in favorof an infrared divergence with a strength slightly less than that expectedfrom a massless pole. We also presented the first investigation of non-tree-level form factors in three dimensions, and find them to be of negligiblesize in comparison to the tree-level form factor above roughly 500 MeV.12owards the infrared, we find hints of a similar qualitative behavior as forthe tree-level form factor, though with an order of magnitude suppression.In addition we find evidence that in four dimensions a similar qualitativedependency prevails, though with a much weaker infrared divergence. Con-sequently, we do not yet reach deep enough into the infrared to establish azero crossing unambiguously, but could establish the momentum range inwhich this should occur. As the infrared turns out to be little affected bydiscretization this suggests that relatively coarse lattices of order 48 withhigh statistics should be able to establish a zero crossing reliably, if it isindeed there. This is, however, beyond our current capacity.We find therefore evidence that the critical infrared exponent decreasesquickly with dimensionality, crossing the massless-pole value somewhere be-tween two and three dimensions. The actual strength, i. e. prefactor, ishowever remarkably similar in all dimensions. We also find evidence for aslight quantitative angle-dependency of the form factors. Also, we find thatthe back-to-back configuration suffers, as with all other investigated verticesso far, from substantial discretization artifacts in the ultraviolet. This sug-gests to use other momentum configurations exclusively to investigate thehigh-momentum features, like anomalous dimensions. Acknowledgments
M. V. was supported by the FWF under grant number J3854. Thecomputations have been performed on the HPC clusters at the Universityof Graz and we are grateful for its fine operation.
References [1] A. Maas, (2019), 1907.10435.[2] A. Maas, Phys. Rep. , 203 (2013), 1106.3942.[3] A. Maas, Phys. Rev.
D75 , 116004 (2007), 0704.0722.[4] A. Cucchieri, A. Maas, and T. Mendes, Phys. Rev.
D77 , 094510 (2008),0803.1798.[5] M. Q. Huber, (2020), 2003.13703.[6] L. Corell, A. K. Cyrol, M. Mitter, J. M. Pawlowski, and N. Strodthoff,SciPost Phys. , 066 (2018), 1803.10092.[7] B. Mintz, L. Palhares, S. Sorella, and A. Pereira, Phys. Rev. D ,034020 (2018), 1712.09633. 138] A. K. Cyrol, L. Fister, M. Mitter, J. M. Pawlowski, and N. Strodthoff,Phys. Rev. D94 , 054005 (2016), 1605.01856.[9] M. Q. Huber, Phys. Rev.
D93 , 085033 (2016), 1602.02038.[10] A. Sternbeck and M. M¨uller-Preussker, Phys. Lett.
B726 , 396 (2012),1211.3057.[11] A. Sternbeck et al. , PoS
LATTICE2016 , 349 (2017), 1702.00612.[12] A. Cucchieri, A. Maas, and T. Mendes, Phys. Rev.
D74 , 014503 (2006),hep-lat/0605011.[13] A. Aguilar et al. , Eur. Phys. J. C , 154 (2020), 1912.12086.[14] A. Aguilar, M. Ferreira, C. Figueiredo, and J. Papavassiliou, Phys.Rev. D , 094010 (2019), 1903.01184.[15] A. L. Blum, R. Alkofer, M. Q. Huber, and A. Windisch, EPJ WebConf. , 03001 (2017), 1611.04827.[16] A. Blum, M. Q. Huber, M. Mitter, and L. von Smekal, Phys. Rev. D89 , 061703 (2014), 1401.0713.[17] G. Eichmann, R. Williams, R. Alkofer, and M. Vujinovic, Phys. Rev.
D89 , 105014 (2014), 1402.1365.[18] A. Athenodorou et al. , Phys. Lett.
B761 , 444 (2016), 1607.01278.[19] A. G. Duarte, O. Oliveira, and P. J. Silva, Phys. Rev.
D94 , 074502(2016), 1607.03831.[20] A. K. Cyrol, M. Q. Huber, and L. von Smekal, Eur. Phys. J.
C75 , 102(2015), 1408.5409.[21] M. Vujinovic and T. Mendes, Phys. Rev. D , 034501 (2019),1807.03673.[22] A. Maas, Phys. Rev. D91 , 034502 (2015), 1402.5050.[23] J. S. Ball and T.-W. Chiu, Phys. Rev.
D22 , 2550 (1980).[24] H. J. Rothe,
Lattice gauge theories: An Introduction (World Sci. Lect.Notes Phys., 2005).[25] R. Jackiw and S. Templeton, Phys.Rev.
D23 , 2291 (1981).[26] A. Maas, S. Raubitzek, and P. T¨orek, Phys. Rev.
D99 , 074509 (2019),1811.03395. 1427] M. Q. Huber, R. Alkofer, C. S. Fischer, and K. Schwenzer, Phys. Lett.
B659 , 434 (2008), 0705.3809.[28] A. Maas, Phys. Rev. D , 114503 (2019), 1902.10568.[29] A. Sternbeck, EPJ Web Conf.137