Yang-Mills Theory in lambda-Gauges
aa r X i v : . [ h e p - l a t ] A ug Yang-Mills Theory in λ -Gauges Axel Maas, ∗ Tereza Mendes, † and ˇStefan Olejn´ık ‡ Theoretical-Physical Institute, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, D-07743 Jena, Germany Instituto de F´ısica de S˜ao Carlos, University of S˜ao Paulo,Caixa Postal 369, 13560-970 S˜ao Carlos, SP, Brazil Institute of Physics, Slovak Academy of Sciences, SK–845 11 Bratislava, Slovakia (Dated: July 3, 2018)The gauge-independent phenomenon of color confinement in Yang-Mills theory manifests itselfdifferently in different gauges. Therefore, the gauge dependence of quantities related to the infraredstructure of the theory becomes important for understanding the confinement mechanism. Par-ticularly useful are classes of gauges that are controlled by a single gauge parameter. We presentresults on propagators and the color-Coulomb potential for the so-called λ -gauges, which interpo-late between the (minimal) Landau gauge and the (minimal complete) Coulomb gauge. Results arereported for the SU(2) lattice gauge theory in three and four space-time dimensions. We investigateespecially intermediate and low momenta. We find a continuous evolution of all quantities with thegauge parameter, except at zero four-momentum. PACS numbers: 11.15.Ha 12.38.Aw 14.70.Dj
I. INTRODUCTION
The investigation of the confinement mechanism inQCD and Yang-Mills theory has progressed during thelast decade, but remains an unsolved and challengingproblem (see e.g. [1] for a brief review). A particularlyimportant question is that of gauge dependence.The confinement phenomenon has been investigated invarious gauges, with different degrees of sophistication,methods, and observables. Still, at the present stage it isnot yet clear to what extent the confinement mechanismitself is gauge-dependent. It is, however, sure that itsmanifestation in the structure of gauge-dependent quan-tities, like correlation functions, is different in differentgauges.For a full understanding of the confinement process,it is thus necessary to understand this mutability amongdifferent gauges. For such a task, classes of gauges areof special interest which offer parameters to deform thegauge condition smoothly. This allows one to track thechanges of correlation functions along the gauge orbit.One such class are the λ -gauges [2, 3], defined by thegauge condition ∂ ′ µ A µa ( x ) = 0 , (1) ∂ ′ µ ≡ ( λ ∂ , . . . , λ d − ∂ d − ) , where d is the number of space-time dimensions and the λ µ ’s are a set of gauge parameters. E. g., the Landau-gauge case will be given by λ µ = 1 for all µ . In princi-ple, it is necessary to add a prescription for dealing withGribov-Singer effects [4, 5]. We employ here the mini-mal extension of the perturbatively defined gauge (1) to ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] the non-perturbative domain by selecting a random Gri-bov copy among all Gribov copies belonging to the firstGribov region. This region is defined by those gaugecopies for which the associated Faddeev-Popov operator,defined below in equation (3), is non-negative.Clearly, if one of the λ µ ’s identically vanishes, thegauge condition is similar to a would-be Coulomb gauge,but has to be supplemented by further constraints toyield even a perturbatively complete gauge condition.In contrast, a complete Coulomb gauge condition is ob-tained by performing a limit λ µ → µ [6]. Itis not clear, whether the different lattice definitions of aperturbatively complete Coulomb gauge [6, 7] yield coin-ciding correlation functions. The results we present hereindicate that this may be the case, at least for a subclassof these definitions. However, in case of sufficiently non-smooth completions of the Coulomb gauge some effectscould be present [7].Here, we will concentrate on Landau-like λ -gauges, inwhich we choose only one of the λ µ ’s different from 1.As the lattice calculations presented here will be per-formed in Euclidean space-time at zero temperature, thedirection is irrelevant, but will conventionally be chosento be the time direction. Therefore we set λ = λ and λ = λ = · · · = λ d − = 1 henceforth.In these gauges the simplest correlation functions arethe gluon propagator and the Faddeev-Popov-ghost prop-agator. We will use here lattice gauge theory to deter-mine these propagators in the case of SU(2) Yang-Millstheory. Furthermore, in Coulomb gauge there exists anadditional color-Coulomb potential that bounds the con-ventional Wilson potential from above [8]. A confin-ing color-Coulomb potential is thus a necessary condi-tion for confinement, and we shall also investigate howthis potential develops from Landau to Coulomb gauge.There exists an associated order parameter, the so-calledresidual-gauge-symmetry order parameter [9], which sig-nals whether the residual gauge symmetry that exists inCoulomb gauge is unbroken. Also its evolution with thegauge parameter λ will be determined here.A word of caution is due here. The behavior of the cor-relation functions in the asymptotic infrared domain hasnot yet been completely explained even in Landau gauge.In fact, despite great effort invested in analytic [10–13]and numerical [14–17] studies, a consistent descriptionunifying the different approaches is still being elaborated.For recent reviews of analytic and lattice studies see re-spectively [18–20] and [20, 21]. Since the gauge-fixingprocess in the interpolating gauges is even more compu-tationally expensive than in ordinary Coulomb or Lan-dau gauge, it will not be possible here to reach volumespermitting to answer the question whether a scaling-typeor a decoupling-type behavior is observed for general λ -gauges. The main interest is therefore on the general de-formation of the relevant quantities with the gauge pa-rameter. Note that functional-equation studies predictthe existence of a scaling-type solution in these gaugesin four dimensions [22]. For three dimensions, this willbe shown in the appendix. Following the lines of the ar-gument given in [18], it is easily conceivable that also adecoupling type of solution exists in this class of gauges.In any case, we will not attempt to settle this questionhere.The lattice definition of the λ -gauges and the detailsof the production of the corresponding configurations areoutlined in Section II. The results for the propagators arepresented in Section III. Definitions of the propagatorsare also introduced there. Results for three dimensionsare given in Subsection III A, while those for four di-mensions are given in Subsection III B. Definitions andresults for the color-Coulomb potential and associatedquantities will be discussed in Section IV. A summary isgiven in Section V.The results presented in this paper extend earlier stud-ies [23]. In particular, we complement our previous in-vestigation [6], in which a qualitative difference of someparts of the gluon propagator was observed in d = 4.Some preliminary results were presented in Ref. [24]. Seealso [26] for investigations of other setups and quantitiesin this type of gauges. II. DEFINITIONS AND SET-UPA. Generation of gauge-fixed configurations
Minimal λ -gauges on the lattice are a straightforwardgeneralization of the conventional minimal Landau gauge[23, 27]. In particular, to fix the gauge, a gauge trans-formation g ( x ) is determined, which minimizes the func-tional E ( g ( x )) = 1 − a d dV X µ λ µ X x tr (cid:2) U gµ ( x ) + U g + µ ( x ) (cid:3) . (2)Here a is the lattice spacing and V = N d the latticevolume. U µ ∈ SU ( N c ), with N c = 2 for the present SU(2) case, is a link variable in the lattice configurationof the gauge field, and its gauge transform U gµ is given by U gµ ( x ) = g ( x ) U µ ( x ) g + ( x + a~e µ ) . For a more detailed discussion of this gauge condition, inparticular for the limit of λ → M ab ( x, y ) ω b ( y ) = δ xy X µ λ µ { G abµ ( y )[ ω b ( y ) − ω b ( y + e µ )] − G abµ ( y − e µ )[ ω b ( y − e µ ) − ω b ( y )]+ X c f abc [ A bµ ( y ) ω c ( y + e µ ) − A bµ ( y − e µ ) ω c ( y − e µ )] } , (3) G abµ ( x ) = 18 tr (cid:0) { τ a , τ b } (cid:2) U µ ( x ) + U µ ( x ) + (cid:3)(cid:1) ,A aµ = √ β ia tr (cid:0) τ a ( U µ − U + µ ) (cid:1) + O ( a ) , with τ a the generators of SU(2). As said above, we selectalways a random copy from the set of Gribov copies satis-fying the constraint in Eq. (1) through the minimizationof the functional in Eq. (2), thereby implementing theso-called minimal version of all of these gauges.The functional (2) can be minimized using the samemethods as in Landau gauge [27]. In particular, astochastic overrelaxation algorithm is used, analogous tothe Landau-gauge case [28]. One should note that therequired number of gauge-fixing sweeps increases withdecreasing λ . Hence simulations at smaller and smaller λ become computationally more and more expensive. Al-though the number of gauge-fixing sweeps can be de-creased by multiple updates of each individual time-slicebefore changing the time-slice, this does not decrease thetotal computing time needed, due to the increase of thenumber of updates on single time-slices.Thus, we limit our systematic investigations here to λ ≥ − . Nevertheless, checks on small volumes downto λ = 10 − do not show any qualitatively new effects.We distinguish furthermore between three types ofCoulomb gauge fixing: • Simple Coulomb gauge, defined by setting λ to 0. • Complete Coulomb gauge, defined by fixing theresidual (perturbative) gauge freedom in the simpleCoulomb gauge in a Landau-like manner [6]. • Limiting Coulomb gauge, defined by the limit λ →
0. Note that while this limit is well-definedin the sense of a gauge condition, the individ-ual Green’s functions do not necessarily have apointwise-smooth limit .The last two versions of the Coulomb gauge should co-incide in the limit of λ → III. PROPAGATORS IN λ -GAUGES Setting λ to a value different from 1 yields a gaugecondition (1) that is no longer manifestly Lorentz (Eu-clidean) invariant. Thus, as in Coulomb gauge, the en-ergy p and the spatial momentum | ~p | have to be consid-ered as independent variables.Furthermore, the gluon is a vector particle. Hence,it is no longer possible to describe it by a single scalarfunction as in Landau gauge, but two independent onesare required. For practical purposes we choose these twoindependent functions to be D ( p , | ~p | ) = 1( N c − < A a A a > , (4) D tr ( p , | ~p | ) = 1( N c − d − × (cid:18) δ ij − p i p j p + ~p (cid:19) < A ai A aj >, (5)where i and j run only over spatial indices. D tr is eval-uated at zero d − d -momentum, by setting the second term in the aboveprojector to zero and multiplying by ( d − / ( d − d -momentum. Note that at λ = 0, thesecond term in (5) is not contributing, since in this case p i A i = 0, due to the Coulomb gauge condition. The re-lation to the conventional choice in DSE studies [22] isdetailed in Appendix A. However, our results suggest that the limit is smooth except at p = 0. Note that these are essentially the same definitions as were usedin [6], since there only the case p = 0 has been investigated.This definition has been chosen for numerical convenience. Seeappendix A for the relation to the definition of [22] and a generaldecomposition. Herein, the gluon field A aµ in momentum space is ob-tained from the position-space variables by A aµ ( p ) = e − iπPµN X X e πi P µ PµXµN A aµ ( x ) , where on a finite lattice the components P µ of P have theinteger values − N/ , . . . , N/ X µ of X are the (integer) coordinates in the lattice rang-ing from 0 to N −
1. The physical momenta p to whichthe integer lattice ones P correspond are p µ = 2 a sin P µ πN . The ghost propagator D G ( p , | ~p | ) is, as in Landaugauge, defined by the Fourier transform of the inverseFaddeev-Popov operator (3). As this operator is symmet-ric, the inversion can be done using a conjugate gradientmethod, as in ordinary Landau gauge [28]. To determinethe renormalization effects in four dimensions, it is fur-thermore useful to define the dressing function G via D G ( p , | ~p | ) = 1 λp + ~p G ( p , | ~p | ) . (6)Thereby the tree-level part 1 / ( λp + ~p ) of the propagatoris explicitly factored out. A. Propagators in 3 dimensions
In three space-time dimensions, Yang-Mills theory in λ -gauges is, as in Landau gauge, finite on the level ofcorrelation functions. Hence no renormalization effectsoccur. In particular, the gauge-parameter λ is not differ-ent from its tree-level value.It is worthwhile to study the three different propaga-tors, the one of the transverse gluon (5), the one of thetemporal gluon (4), and the one of the ghost (6), indi-vidually.The transverse gluon propagator is shown for var-ious kinematic configurations and values of λ in Figure 1.There are a number of interesting observations, outlinednext.With respect to the spatial momentum, the clearly dis-tinct maximum of the transverse gluon propagator inLandau gauge moves towards larger momenta with de-creasing λ , albeit rather slowly. As for the dependenceon the temporal momentum, the dependence on p be-comes flat at low momenta and the flat region becomesmore extended when decreasing λ . However, the p = 0point is always lower, and may be influenced differentlyby finite-size effects. Already at λ = 1 /
10, the maximumseems to be gone. The propagator at λ = 1 /
100 is evenmore similar to the one of a massive particle. This is,however, a finite-volume effect. A comparison of differentvolumes, presented in Figure 2, shows that the volumesin Figure 1 are too small to resolve the position of the [ G e V ] p | [ G e V ] p | ] - [ G e V t r D =1/100 λ at tr Spatial gluon propagator D [ G e V ] p | [ G e V ] p | ] - [ G e V t r D =0 λ at tr Spatial gluon propagator D [ G e V ] p | [ G e V ] p | ] - [ G e V t r D =1 λ at tr Spatial gluon propagator D [ G e V ] p | [ G e V ] p | ] - [ G e V t r D =1/10 λ at tr Spatial gluon propagator D | [GeV]p|0 0.5 1 1.5 2 ] - | ) [ G e V p ( , | t r D -1 |p as a function of | tr D [GeV] p0 0.5 1 1.5 2 ] - , ) [ G e V ( p t r D -1 as a function of p tr D FIG. 1: The transverse gluon propagator D tr for different values of λ and different kinematic configurations, in the 3 d case on a40 lattice. The top-left panel is the propagator as a function of pure spatial momenta ~p , the top-right panel of pure temporalmomenta p . The spatial momenta are measured along the x -axis. Results shown are for λ = 1, i.e. Landau gauge (circles), λ = 1 / λ = 1 /
10 (squares), λ = 1 /
20 (triangles), λ = 1 /
100 (stars) and λ = 0 (upside-down triangles). The lowerpanels show the full momentum dependence at λ = 1 (middle-left panel), λ = 1 /
10 (middle-right panel), λ = 1 /
100 (bottom-leftpanel), and λ = 0 (bottom-right panel). All results are at β = 4 .
2. Also, λ = 0 corresponds to the simple-Coulomb-gaugecase. Results at β = 6 . TABLE I: Information on the configurations considered in our numerical simulations. The value “0 fixed” for λ denotes thecase when the residual gauge freedom (after setting λ to 0) is fixed according to [6]. The value of the lattice spacing a has beentaken from [29] for the three-dimensional case and from [30, 31] for the four-dimensional case. Sweeps indicates the number ofsweeps [28] between two consecutive gauge-fixed measurements. In all cases 200 thermalization sweeps have been performed.More details on the generation of configurations, error-determination, etc. can be found in [28]. d = 3 dimensions d = 4 dimensions λ N β a − [GeV] Conf. Sweeps V /d [fm] N β a − [GeV] Conf. Sweeps V /d [fm]1 40 4.2 1.136 586 50 6.9 22 2.2 0.938 425 50 4.61 40 6.0 1.733 689 50 4.5 22 2.5 2.309 532 50 1.91 22 2.8 5.930 342 50 0.731 40 2.2 0.938 50 8.41/2 40 4.2 1.136 466 50 6.9 22 2.2 0.938 380 50 4.61/2 40 6.0 1.733 542 50 4.5 22 2.5 2.309 372 50 1.91/2 22 2.8 5.930 362 50 0.731/10 20 4.2 1.136 481 30 3.51/10 20 6.0 1.733 555 30 2.31/10 40 4.2 1.136 424 50 6.9 22 2.2 0.938 311 50 4.61/10 40 6.0 1.733 461 50 4.5 22 2.5 2.309 370 50 1.91/10 22 2.8 5.930 399 50 0.731/10 60 4.2 1.136 314 70 10 40 2.2 0.938 50 8.41/10 60 6.0 1.733 322 70 6.8 70 2.2 0.938 310 14.71/20 40 4.2 1.136 447 50 6.9 22 2.2 0.938 364 50 4.61/20 40 6.0 1.733 463 50 4.5 22 2.5 2.309 307 50 1.91/20 22 2.8 5.930 409 50 0.731/20 40 2.2 0.938 50 8.41/20 70 2.2 0.938 336 14.71/100 14 2.2 0.938 418 30 2.91/100 14 2.5 2.309 420 30 1.21/100 40 4.2 1.136 356 50 6.9 22 2.2 0.938 262 50 4.61/100 40 6.0 1.733 478 50 4.5 22 2.5 2.309 367 50 1.91/100 22 2.8 5.930 332 50 0.731/100 32 2.2 0.938 50 6.71/100 40 2.2 0.938 50 8.41/100 70 2.2 0.938 116 14.70 40 4.2 1.136 375 50 6.9 22 2.2 0.938 327 50 4.60 40 6.0 1.733 352 50 4.5 22 2.5 2.309 340 50 1.90 22 2.8 5.930 319 50 0.730 fixed 40 4.2 1.136 377 50 6.9 22 2.2 0.938 434 50 4.60 fixed 40 6.0 1.733 314 50 4.5 22 2.5 2.309 309 50 1.90 fixed 22 2.8 5.930 401 50 0.73 maximum at λ < ∼ /
10. It is visible that the maximumreappears at smaller momenta on a larger volume. Cor-responding calculations to track the maximum at evenlower values of λ become, however, exceedingly costly.For all three types of Coulomb gauge the transversegluon propagators as functions of spatial momentum co-incide. This is not surprising. Quantities that are onlydefined on a given time-slice, such as the spatial gluonpropagator, should be independent of how the inter-time-slice gauge degree of freedom is fixed. Therefore, in thefigures only one case, that of the simple Coulomb gauge,is shown.We thus see that, in 3 d Coulomb gauge, the transversepropagator for spatial momenta is similar to the one ob-tained in four dimensions in simple Coulomb gauge [32].It has a maximum and is infrared-suppressed.On the other hand, the transverse propagator as a function of energy in Coulomb gauge is approximatelyconstant. In the simple Coulomb gauge this is a con-sequence of the unfixed residual gauge degree of free-dom. That this persists even in the case of fully fixedCoulomb gauges is not trivial [6], but has been deducedalso in functional studies [33]. In any case, in the alterna-tive ways to fix the remaining gauge freedom, the gaugecondition cannot impose a strict constraint on the gluonfields on any given configuration . Hence, the fields stillwash out when averaged over configurations, leaving onlyan instantaneous propagator. For this reason, the prop- Consider, e.g., ∂ A = 0. Such a constraint is only ful-filled on average, < ∂ A > = 0. The integrated constraint ∂ R d d − x i A ( x , x i ) = 0 is, however, true for any individualconfiguration. TABLE II: Configurations used for the determination of thecolor-Coulomb potential and the residual-gauge-symmetry or-der parameter. For details, see Table I. All calculations havebeen performed in four dimensions. The columns with various β ’s indicate the number of configurations used. The numberof sweeps was always 50. V /d [fm] λ N β = 2 . β = 2 . β = 2 . β = 2 .
21 4 188 188 188 0.841 6 190 190 190 1.31 8 191 191 191 1.71 10 192 192 192 2.11 12 193 193 193 2.51 16 195 195 195 3.41 20 196 196 196 4.21 24 104 197 5.01/10 4 188 188 188 0.841/10 6 190 190 190 1.31/10 8 191 191 191 1.71/10 10 192 192 192 2.11/10 12 193 193 193 2.51/10 16 195 195 195 3.41/10 20 180 196 196 4.21/10 24 94 298 197 5.01/100 4 188 188 188 0.841/100 6 190 190 190 1.31/100 8 191 191 191 1.71/100 10 192 192 192 2.11/100 12 193 193 193 2.51/100 16 100 195 195 3.41/100 20 40 192 138 4.21/100 24 29 197 197 5.00 4 188 188 188 0.840 6 190 190 190 1.30 8 191 191 191 1.70 10 192 192 192 2.10 12 193 193 193 2.50 16 195 195 195 3.40 20 115 196 196 4.20 24 70 197 197 5.00 fixed 4 188 188 188 0.840 fixed 6 190 190 190 1.30 fixed 8 191 191 191 1.70 fixed 10 192 192 192 2.10 fixed 12 193 193 193 2.50 fixed 16 195 195 195 3.40 fixed 20 82 196 196 4.20 fixed 24 90 197 197 5.0 agator is constant as a function of the energy. This isobserved in all three types of Coulomb gauge and shownin Figure 1 for the simple-Coulomb case. Note that thisargument does not apply to the zero-energy mode. Thismode coincides with the zero mode depending on spatialmomenta, leading to the same value at zero d -momentumfor the propagator, irrespective of the dependence on spa-tial momentum or on energy. If the propagator as a func-tion of spatial momentum thus vanishes in the infinite-volume limit, it may have a discontinuity at zero as a [GeV] p0 0.5 1 1.5 2 ] - , ) [ G e V ( p t r D -1 × as a function of p tr Volume dependence of D
FIG. 2: The transverse gluon propagator D tr as a function of p at λ = 1 /
10 for different volumes, in the 3 d case. Openand full symbols correspond to β = 6 . β = 4 .
2, respec-tively. The lattice sizes are 20 (circles), 40 (squares), and60 (triangles). function of energy.Concluding, the spatial propagator D tr ( p ) is instan-taneous and infrared-suppressed in all versions of theCoulomb gauge considered here.Although not shown, a further observation is that forthe spatial propagator as a function of ~p the dependenceon β at fixed lattice size weakens with decreasing λ . How-ever, at the same time the β -dependence increases formomenta along the time direction. This can indicateeither a change in the volume dependence of the propa-gators with a change in λ , or a different β -dependence.The former is more likely, as can be seen in Figure 2: Thevolume dependence is stronger than in corresponding cal-culations in Landau gauge in three dimensions [28, 29].It is furthermore visible that the β -dependence dimin-ishes with increasing volume, making it more probablethat this is a finite-volume effect, rather than a finite- β effect.The case of the temporal propagator is at the firstglance a bit simpler, as D ( p ,
0) vanishes identically for p = 0 due to the gauge condition. The remaining resultsare shown in Figure 3. As a function of spatial momenta,the maximum moves towards smaller momenta with de-creasing λ . Again, already at λ = 1 /
10, resolving themaximum is no longer possible for these volumes. Atthe smallest values of λ , the propagator looks more likea divergent function, rather than an infrared-suppressedfunction, except at the zero-momentum point for finite λ . However, this is again only a finite-volume effect,as is visible in Figure 4. This changes drastically forthe Coulomb gauges, where the propagator diverges for ~p → ~
0, independently of p , as it does already in pertur- | [GeV]p|0 0.5 1 1.5 2 ] - | ) [ G e V p ( , | D |p as a function of | D FIG. 3: The temporal gluon propagator D for different valuesof λ and different kinematic configurations, in the 3 d case on a40 lattice. The top-left panel is the propagator as a function ofpure spatial momenta ~p . The momenta are measured along the x -axis. Results shown are for λ = 1, i.e. Landau gauge (circles), λ = 1 / λ = 1 /
10 (squares), λ = 1 /
20 (triangles), λ =1 /
100 (stars), and λ = 0 (upside-down triangles). The panelsbelow show the full momentum dependence at λ = 1 (middle-left panel), λ = 1 /
10 (middle-right panel), λ = 1 /
100 (bottom-left panel), and λ = 0 (bottom-right panel). All results are at β = 4 .
2. Also, λ = 0 corresponds to the simple-Coulomb-gaugecase. Results at β = 6 . [ G e V ] p | [ G e V ] p | ] - [ G e V D =1/100 λ at Temporal gluon propagator D [ G e V ] p | [ G e V ] p | ] - [ G e V D =0 λ at Temporal gluon propagator D [ G e V ] p | [ G e V ] p | ] - [ G e V D =1 λ at Temporal gluon propagator D [ G e V ] p | [ G e V ] p | ] - [ G e V D =1/10 λ at Temporal gluon propagator D bation theory. Thus, for all three definitions of Coulombgauge the infrared-divergent gluon propagator is recov-ered. Note in particular that also the value at p = 0 isnon-vanishing in the Coulomb limit. Furthermore, thelimit λ → ghost propagator is shown in Figure 5.The dependence on λ is clearly seen to increase witha decreasing ratio of p / | ~p | . There is essentially no λ - | [GeV]p|0 0.2 0.4 0.6 0.8 ] - | ) [ G e V p ( , | D |p as a function of | D FIG. 4: The temporal gluon propagator D as a function of | ~p | at λ = 1 /
10 for different volumes, in the 3 d case. Openand full symbols correspond to β = 6 . β = 4 .
2, respec-tively. The lattice sizes are 20 (circles), 40 (squares), and60 (triangles). dependence at p = 0. Furthermore, as in Landau gauge,always a strong infrared enhancement is visible. Hencethe most interesting case is the one of pure temporal mo-menta. In this case the propagator seems (to leadingorder) to be scaled by a constant factor, which is both λ - and β -dependent. This is tested explicitly in Figure6. Multiplying the dressing function by λ α , with α cho-sen to be 0.15 for β = 4 . β = 6 .
0, yieldsroughly agreeing dressing functions. This is a genuine β -dependent effect, as the same exponents appear on a 20 lattice at the same values of β . This may indicate thatfor this correlation function the scaling regime is movedas a function of β . The reason for this movement of thescaling region could be that the theory in Coulomb gaugeis no longer finite, and renormalization has to be takeninto account, even in three dimensions. If this should oc-cur smoothly, the scaling region without renormalizationshould move to larger β with decreasing λ . An alternativeexplanation of this may be a combination of finite-size ef-fects, leading seemingly to a β -dependence, and a genuinenon-trivial dependence of the dressing function G on λ .Such a non-trivial dependence is in fact found in func-tional calculations in the far infrared (see Appendix Aand [22]). However, if this is the correct interpretation,than this non-trivial λ -dependence seems to pertain torather large momenta.If the exponent α should be non-zero in the infinite-volume and continuum limit, then this would imply thatthe ghost propagator diverges for all p , independently ofthe spatial momentum, in the limit λ →
0. This would bethe behavior expected in the Gribov-Zwanziger scenario[4, 34]. It is not possible with the method used here to de-termine the ghost propagator in the case of λ = 0, asthe corresponding Faddeev-Popov operator is differentfrom (3). In this case, it would be necessary to invertthe Faddeev-Popov operator in the sub-space orthogonalnot only to constant gauge transformations, as in Landaugauge [28], but also to all purely temporal gauge transfor-mations. If, however, the limiting Coulomb gauge λ → λ = 0coincide also for the ghost propagator (as in the caseof the gluon propagator), then an infrared-divergent, in-stantaneous ghost propagator would be obtained. Thisis the limiting behavior that is suggested from Figure 5.Concluding, the results show how the propagatorsevolve smoothly with λ from a Landau-like behavior toa Coulomb-like behavior. However, below a certain mo-mentum, which decreases with decreasing λ , the prop-agators always show qualitatively a Landau-like behav-ior. In particular, the temporal propagator as a func-tion of spatial momenta shows a more and more diverg-ing behavior, as in Coulomb gauge, at intermediate mo-menta, before becoming infrared-suppressed, as in Lan-dau gauge. On the other hand, the transverse propaga-tor as a function of transverse momenta becomes moreand more infrared-suppressed, as it is in Coulomb gauge.Finally, the ghost propagator as a function of pure tem-poral momenta shows a rather pronounced dependenceon λ and β .All in all, the propagators show a smooth developmenttowards a Coulomb-like behavior in an intermediate mo-mentum regime. Only in the far infrared does the originalLandau-like behavior persist. The limit λ → A com-pletely unfixed, the limiting and the complete Coulombgauge fix this component. However, in both cases thegluon propagators coincide within statistical errors. Thatis a consequence of the additional constraint not being alocal condition on the fields. It only ensures ∂ A = 0on average, just as in the case of an unfixed temporal di-rection. To completely understand the properties of thelimit of the interpolating gauge, this has to be investi-gated further, but is already in line with arguments fromfunctional calculations [33]. B. Propagators in 4 dimensions
The situation in four space-time dimensions turns outto be qualitatively the same as in three dimensions. Still, | [GeV]p|0.5 1 1.5 2 ] - | ) [ G e V p G ( , | |pGhost dressing function G as a function of | [GeV] p0.5 1 1.5 2 ] - , ) [ G e V G ( p Ghost dressing function G as a function of p [ G e V ] p | [ G e V ] p | | ) p , | G ( p =1 λ Ghost dressing function G at [ G e V ] p | [ G e V ] p | | p , | G ( p =1/10 λ Ghost dressing function G at [ G e V ] p | [ G e V ] p | | ) p , | G ( p =1/100 λ Ghost dressing function G at
FIG. 5: The ghost dressing function G for different values of λ and different kinematic configurations, in the 3 d case on a 40 lattice. The top-left panel is the propagator as a function ofpure spatial momenta ~p , the top-right panel of pure temporalmomenta p . The spatial momenta are measured along the x -axis. Results shown are for λ = 1, i.e. Landau gauge (circles), λ = 1 / λ = 1 /
10 (squares), λ = 1 /
20 (triangles), and λ = 1 /
100 (stars). The lower panels show the full momentumdependence at λ = 1 (middle-left panel), λ = 1 /
10 (middle-rightpanel), and λ = 1 /
100 (bottom-left panel). All results are at β = 4 .
2. Also, λ = 0 corresponds to the simple-Coulomb-gaugecase. Results at β = 6 . quantitatively, differences appear. One is due to thefact that finite-volume effects are harder to control, sincecalculations are in general more costly. The second isthe need for renormalization even outside the Coulombgauge: The theory is no longer finite in four dimensions.This point makes the situation a bit more intricate, as thegauge parameter λ also gets renormalized by a renormal-ization constant Z λ . This influences of course the defini-tion of the ghost dressing function (6). In addition, theghost dressing function itself gets renormalized by thewave-function renormalization constant e Z . The corre- sponding relative renormalization constants between two β -values will be determined by e Z ( β ) e Z ( β ) = D G (0 , | ~p | , β ) D G (0 , | ~p | , β ) (7) Z λ ( β ) Z λ ( β ) ≈ D G ( p , , β ) D G ( p , , β ) D G (0 , | ~p | , β ) D G (0 , | ~p | , β ) . (8)The absolute normalization can be chosen at will, and isof no importance here. The second relation is based on(6), and would be exact only if the dressing function did0 [GeV] p0.5 1 1.5 2 ] - , ) [ G e V G ( p α λ Ghost dressing function G as a function of p
FIG. 6: The rescaled ghost dressing function λ α G for differentvalues of λ for temporal momenta p , in the 3 d case. Symbolsused are the same as in the top row of Figure 5, with opensymbols corresponding to β = 6 . β = 4 . not depend on β . Within the domain of perturbationtheory, this dependence exists, but is only logarithmic.For the quite similar values of β employed here, relation(8) is hence an acceptable approximation at sufficientlylarge momenta. TABLE III: Ratios (7–9) of the renormalization constants.The matching condition was a continuous matching at thelargest edge-momenta of the smaller β -value. Note that at λ = 1 the gauge parameter is not renormalized because Lan-dau gauge is a fixed-point of the renormalization group. At λ = 0 no results on the ghost propagator are available, seetext. At λ = 1 the two gluon renormalization constants coin-cide trivially. Note that, at least in the simple Coulomb gauge,it is strictly speaking not possible to renormalize purely mul-tiplicatively [36]. λ β β Z ( β )˜ Z ( β ) Z λ ( β ) Z λ ( β ) Z tr3 ( β ) Z tr3 ( β ) ≈ Z ( β ) Z ( β ) In case of the gluon propagator, the corresponding ra- tios of renormalization constants will be defined by Z tr3 ( β ) Z tr3 ( β ) = D tr (0 , | ~p | , β ) D tr (0 , | ~p | , β ) , (9) Z ( β ) Z ( β ) = D (0 , | ~p | , β ) D (0 , | ~p | , β ) . (10)Note that due to the lack of Euclidean invariance bothpropagators have to be renormalized separately [2, 3],and therefore independent wave-function renormalizationconstants Z tr3 for the transverse gluon propagator and Z for the temporal gluon propagator have to be introduced.This is in contrast to Landau gauge. Also, note thatwhen reaching Coulomb gauge the combination g D isa renormalization group invariant, and the correspondinginverse renormalization constant of the propagator aloneyields the renormalization of the coupling constant [2].The approximate renormalization constants are givenin Table III. The ratios for the transverse and the tem-poral part of the gluon propagator are found to be essen-tially equal, and thus only one is listed. Of course, theseratios cannot be determined precisely, as the same mo-menta are not available for different lattices. The renor-malization has been performed to obtain a smooth con-nection at the largest edge-momentum of the lattice withthe larger physical volume at the same lattice volume .In addition, statistical fluctuations and systematic errorsalso prevent a precise measurement within the presentscope, so these results are more qualitative than quan-titative. However, the error on the ratio is of the samesize as the statistical and systematic errors of the propa-gators, and thus a few, up to ten, percent for statisticalerrors and similarly for the systematic errors. The abso-lute normalization is chosen not to change the results at β = 2 . λ . How-ever, the renormalization constant of λ itself dependson λ . This was somewhat anticipated. When movingfurther away from a fixed point of the renormalizationgroup, renormalization effects should become strongerand stronger with “distance”. That Landau gauge is afixed point is explicitly demonstrated by the decrease in Z λ with increasing β . From here on, only renormalizedresults will be shown.Aside from these renormalization effects, the resultsin four dimensions show the same qualitative behavioras in three dimensions. However, this is only true whencomparing roughly the same volumes, or, in fact, slightlysmaller volumes in the three-dimensional case, to the vol-umes available in four dimensions. In particular, nearlynone of the infrared effects visible in three dimensions can The ultraviolet properties of the gluon propagator are onlyweakly affected by finite-volume effects. Hence the difference ofphysical volume can be ignored for the purpose of determiningthe renormalization constants at ultraviolet momenta. λ . This has already been discussed previ-ously in [6]. Hence, here only a more compact presenta-tion of the results will be given compared to the one inthree dimensions in the previous section. In particular,as the results for non-vanishing energy and non-vanishingspatial momenta only interpolate smoothly between dif-ferent behaviors if one of them is zero, here only thesecases will be shown. Of course, in these kinematic con-figurations the results also look qualitatively similar tothe three-dimensional case.The results for the transverse gluon propagator areshown in Figure 7. As a function of spatial momen-tum | ~p | , the propagator becomes more and more infrared-suppressed, as already observed for three dimensions. Noclear maximum is visible at large λ , owing to the smallvolumes in four dimensions. However, for the large vol-umes at values of λ ≤ .
1, a shallow maximum is ob-served. This maximum moves to larger momenta withdecreasing λ , as in the three-dimensional case. This canalso be seen in a direct comparison of different volumes,shown in Figure 8. This confirms the results obtainedin [6]. The observed infrared-suppressed gluon propaga-tor in Coulomb gauge is in accordance with the resultsof [7, 32, 35], though here the momentum-dependent as-pects of renormalization in Coulomb gauge still have tobe taken into account [36].The behavior as a function of temporal momentum p is also reminiscent of the results in three dimensions. Butdue to the restriction in volume the effects are even lesspronounced. Still, it is visible how the propagator be-comes flatter with decreasing λ and is constant at λ = 0,except at zero momentum.Hence, up to the effects of the smaller volumes andrenormalization, the transverse gluon propagator be-haves qualitatively as in three dimensions. In particu-lar, as a function of spatial momentum, it becomes moreinfrared-suppressed with decreasing λ .The results for the temporal gluon propagator areshown in Figure 9. The situation is essentially the sameas in three dimensions. The propagator as a function ofspatial momenta becomes more and more enhanced inthe infrared. Of course, for the volumes accessible here,it is not possible to check whether it then bends overagain in the far infrared, as it was observed in three di-mensions in sufficiently large volumes. When adding adependence also on the temporal momenta, it is foundthat the propagator increases with decreasing λ as well.Note that this implies a much stronger falloff towardszero temporal momenta, which always occurs due to thegauge condition for vanishing spatial momenta.The dependence on volume is less pronounced (andless spectacular so far) than in case of the spatial gluonpropagator, and hence will not be shown explicitly.Finally, the ghost dressing function is shown in Fig-ure 10. It shows a clearly infrared enhanced behavior at [GeV] p0 0.5 1 1.5 ] - , ) [ G e V ( p t r D as a function of p tr Spatial gluon propagator D | [GeV]p|0 0.5 1 1.5 ] - | ) [ G e V p ( , | t r D |p as a function of | tr Spatial gluon propagator D
FIG. 7: The transverse gluon propagator D tr for differentvalues of λ . The top panel is the propagator as a function ofpure temporal momenta p , the bottom panel of pure spatialmomenta ~p . The spatial momenta are measured along the x -axis. Results shown are for λ = 1, i.e. Landau gauge (circles), λ = 1 / λ = 1 /
10 (squares), λ = 1 /
20 (triangles), λ = 1 /
100 (stars), and λ = 0 (upside-down triangles). Allresults are at β = 2 .
2. Also, λ = 0 corresponds to the simple-Coulomb-gauge case. Results at larger β look similar, butare more strongly affected by finite-size effects. In case of thespatial momenta, the results for λ = 1 come from a lattice ofsize 40 , while λ = 1 / λ = 1 /
20, and λ = 1 /
100 are from alattice of size 70 . In all other cases, the lattice size was 22 . small momenta, essentially unaltered compared to Lan-dau gauge. Only in the pure temporal case is an addi-tional increase with λ for all momenta observed which isnot a pure renormalization effect, but instead similar tothe situation in three dimensions.As in three dimensions, it is possible to let all curvesat ~p = ~ λ α . In contrast2 | [GeV]p|0 0.5 1 1.5 ] - | ) [ G e V p ( , | t r D -1 × | and volumep as a function of | tr Spatial gluon propagator D
FIG. 8: The transverse gluon propagator D tr as a functionof | ~p | at λ = 1 /
100 for different volumes, in the 4 d case.Circles, squares, triangles, stars, and open circles correspondto systems of size 14 , 22 , 32 (from [6]), 40 , and 70 , all at β = 2 . | [GeV]p|0 0.5 1 1.5 ] - | ) [ G e V p ( , | D |p as a function of | Temporal gluon propagator D
FIG. 9: The temporal gluon propagator D for different val-ues of λ as a function of pure spatial momenta ~p , in the 4 d case. The momentum is measured along the x -axis. Symbolshave the same meaning as in Figure 7. to three dimensions, the exponent α is essentially β -independent, as a consequence of renormalization. It in-creases slowly with λ , more or less leveling off at thesmallest achieved λ value at around α ≈ .
08. Thisrescaling leads to an apparent weaker divergence of theghost dressing function. However, with such a small α ,the effect is rather weak.Thus the results in four dimensions are, up to finite- | [GeV]p|0.5 1 1.5 ] - | ) [ G e V p G ( , | |pGhost dressing function G as a function of | [GeV] p0.5 1 1.5 ] - , ) [ G e V G ( p Ghost dressing function G as a function of p
FIG. 10: The ghost dressing function G for different values of λ as a function of pure spatial momenta ~p , in the 4 d case, inthe top panel. The dependence on pure temporal momenta p is given in the bottom panel. Results shown are for λ = 1, i.e.Landau gauge (circles), λ = 1 / λ = 1 /
10 (squares), λ = 1 /
20 (triangles), and λ = 1 /
100 (stars), all at β = 2 . lattice. Results at larger β are essentially identical. volume effects, qualitatively the same as in three dimen-sions. When decreasing λ then at least at mid-momentaa Coulomb-like behavior is observed. In particular thetransverse gluon propagator as a function of spatial mo-mentum becomes more infrared-suppressed, the tempo-ral propagator as function of spatial momentum becomesmore mid-momentum enhanced. The ghost seems to bemore or less inert, but as it is the quantity usually leastaffected by changes which are effectively volume changes,this is not surprising.For Coulomb gauge the same applies, as was alreadysaid in the three-dimensional case: On the level of the3 | [GeV]p|0 0.5 1 1.5 | ) p ( , | c | ) / D p ( , | t r D |p as a function of | C /D tr D FIG. 11: The ratio of the transverse gluon propagator (5) tothe instantaneous would-be Coulomb-gauge propagator (11)as a function of spatial momentum for λ = 1 (circles), λ =1 /
10 (squares), λ = 1 /
20 (triangles), and λ = 1 /
100 (stars).The lattice was of size 40 at β = 2 . gluon propagators all definitions of the Coulomb gaugeare indistinguishable. This approach can be made evenquantitative, e. g. by comparing the gluon propagator asa function of spatial momentum defined according to (5)with the would-be Coulomb-gauge propagator, i.e. thetime-average of (5) D C ( | ~p | ) = X t (cid:18) δ ij − p i p j ~p (cid:19) < A ai ( t, ~p ) A aj ( t, − ~p ) > ( N c − d − V . (11)This comparison is made in Figure 11. It is explicitlyvisible how the transverse gluon propagator tends to theinstantaneous one, in particular in the infrared. In fact,at the smallest non-vanishing momenta the rate of ap-proach is roughly proportional to the unrenormalized λ beyond Landau gauge. IV. COULOMB POTENTIAL
One remarkable property of Coulomb gauge is the ex-istence of an upper bound to the conventional Wilsonpotential in terms of a gauge-dependent quantity, theso-called color-Coulomb potential [8]. This conditionis often called informally the “no-confinement-without-Coulomb-confinement” one, and has been repeatedlystudied [9, 25, 26].The Coulomb potential can be straightforwardly de-fined by use of link variables as V C ( R, − E = − ln (cid:28)
12 tr (cid:0) U ( ~x, t ) U +0 ( ~y, t ) (cid:1)(cid:29)(cid:12)(cid:12)(cid:12)(cid:12) R ≡| ~x − ~y | , (12) R [fm]0 0.5 1 1.5 2 ( R ) [ a . u . ] C V Coulomb potential
FIG. 12: The color-Coulomb potential (12) as a function ofdistance on a 24 lattice at β = 2 .
2. Other volumes anddiscretizations follow the expected pattern. The gauges are λ = 1 (circles), λ = 1 /
10 (squares), λ = 1 /
100 (stars), sim-ple Coulomb gauge (upside-down-triangles) and fully fixedCoulomb gauge (triangles). where E is a constant energy offset, which can be re-moved by renormalization of the potential. For conve-nience, it is here always chosen such that the Coulombpotential is zero at zero distance. Of course, this identi-fication is only possible up to O ( a ), but studies at vari-ous β indicated that the effects are small, but discernible.However, they affected essentially only the value of thestring tension, but neither the functional shape of V C ,nor its dependence on λ , and an extensive analysis istherefore skipped.The result is shown in Figure 12. It is clearly vis-ible that the color-Coulomb potential for all gauges,except Coulomb gauge, is asymptotically flat. How-ever, with decreasing λ , a linear rise is seen initially atsmall distances. Finally, all three possible definitions ofthe Coulomb gauge, simple Coulomb gauge, fully-fixedCoulomb gauge, and the limit λ → / ( λ Λ YM ), though, of course,the functional shape could be different. This is only anestimate, but in line with the interpretation given in [6].The fact that once more all three implementations ofCoulomb gauge are the same is at this point not surpris-ing. However, there exists an order parameter Q for the4 ] -1 -1 × -1 × -1 × Q -1 Residual symmetry
FIG. 13: The remnant-residual-symmetry order parameter Q (13) as a function of L = aN x at β = 2 .
2. Though the pre-factors appearing are strongly β -dependent, as investigationsat β = 2 . β = 2 . λ = 1 (circles), λ = 1 / λ = 1 /
100 (stars), simple Coulomb gauge (upside-down-triangles) and fully fixed Coulomb gauge (triangles). residual gauge freedom of Coulomb gauge, defined as [9] Q = 1 N t N t X n =1 *r
12 tr ˜ U ( t ) ˜ U ( t ) + (13)˜ U ( t ) = 1 N x N y N z X x,y,z U ( x, t ) . If for N x N y N z N t → ∞ Q is vanishing, the residual gaugesymmetry is unbroken. Since it has been shown that thisis equivalent to a non-zero Coulomb string-tension [9], itis possible to anticipate from Figure 12 already the re-sult that Q will vanish for all definitions of the Coulombgauge. In fact, this is seen in the results presented inFigure 13. In all definitions of the Coulomb gauge Q vanishes with volume. This also implies that althoughthe residual gauge symmetry has been fixed in the fully-fixed version of Coulomb gauge and by taking the limit ofthe interpolating gauge, the residual gauge symmetry isnot broken on average. On the other hand, for all gaugesexcept Coulomb gauge it is once more visible how for de-creasing λ at small volumes first a Coulomb-like patternemerges before the original Landau-gauge pattern, withbroken residual gauge symmetry, becomes manifest.It should be noted that Q = 0 implies ˜ U ( t ) = 0, i. e.,a zero matrix. The second type of Coulomb gauge, inwhich the temporal gauge freedom of Coulomb gauge is We are grateful to Daniel Zwanziger for pointing this out. fixed in a Landau-like manner, is actually the conditionto minimize [6] X x,y,z,t tr U ( x, t ) ∝ X t tr ˜ U ( t ) . In the confining case, this sum is thus always exactly zeroin the infinite-volume limit, and therefore the correspond-ing gauge-fixing step is trivial. Thus, in the infinite-volume limit the fully-fixed Coulomb gauge and the sim-ple Coulomb gauge coincide. That this appears to bealso the case at finite volume, as found here, appearsnon-trivial.
V. SUMMARY
Summarizing, a systematic investigation of correlationfunctions of Yang-Mills theory in the class of λ -gauges in-terpolating between the Coulomb and the Landau gaugein three and four dimensions has been performed.The main result is threefold. On the one hand,the gauge parameter interpolating between both gaugesparametrizes a certain length scale. For distances shorterthan this scale, all the determined correlation functionsshow an essentially Coulomb-like behavior, at least upto momenta where perturbation theory becomes thedominant contribution. For distances longer than thisscale, the behavior is essentially that as in the Landaugauge. There is no indication that this separation ofCoulomb-like and Landau-like behavior will change forlarger volumes and/or finer discretizations, even thoughthe asymptotic infrared behavior in Landau gauge is stillsensitive to such changes.Furthermore, the investigation of different realizationsof the Coulomb gauge, with and without fixing the resid-ual gauge degree of freedom, did not show any effect forany of the correlation functions. This is in accordancewith the reasoning of [6] and of the continuum investiga-tions in [33].Finally, the evolution of correlation functions with thisgauge parameter from Landau gauge to Coulomb gaugeis smooth, with the only possible exception of the pointat zero four-momentum for the temporal gluon propa-gator: In the interpolating gauge it is forced to vanishthere, while in Coulomb gauge it is expected to diverge.However, for any non-zero four-momenta, its behavior issmooth.Thus the interpolating gauge indeed turns into an in-terpolation of scales. The results interpolate in the farinfrared from a Landau-like behavior over a Coulomb-likebehavior to the perturbative λ -like behavior. Hence thename of interpolating gauges is truly justified.Concluding, it is remarkable that all energy-dependence is actually vanishing in all definitions of theCoulomb gauge. Thus, all investigated correlation func-tions become time-independent, and therefore do not de-scribe propagating degrees of freedom, in contrast to5physical (bound) states. In this sense, Coulomb gaugecan be regarded as a physical gauge, for any of the inves-tigated versions. Acknowledgments
We are grateful to Attilio Cucchieri for helpful dis-cussions and to Daniel Zwanziger for remarks on themanuscript. A. M. was supported by the DFG un-der grant numbers MA 3935/1-1, MA 3935/1-2, andMA 3935/5-1, and by the FWF under grant numbersP20330 and M1099-N16. T. M. was partially supportedby FAPESP and by CNPq. ˇS. O. was supported bythe Slovak Grant Agency for Science, project VEGA No.2/0070/09, by ERDF OP R&D, project CE meta-QUTEITMS 26240120022, and via Center of Excellence SASQUTE. The ROOT framework [37] has been used in thisproject. Part of our simulations were done on the IBMsupercomputer at S˜ao Paulo University (FAPESP grant04/08928-3).
Appendix A: Infrared analysis usingDyson-Schwinger equations
In this appendix, connection to the commonly usedframework for interpolating gauges in functional meth-ods will be made. Furthermore, it will be shown thatthe persistence of the Landau-like behavior in the far in-frared (at least for non-perturbative λ -gauges) allows fora scaling solution not only in four dimensions, as shownin [22], but also in three dimensions.In functional calculations, it is convenient to use in-stead of the quantities D and D tr , defined in equations(4–5) above, different ones. Writing the gluon propagatoras (no summation implied) D µν = (cid:18) δ µ δ ν − λδ µ ( δ ν − p p ν ~p − λ ( δ µ − δ ν p p µ ~p (A1)+ λ ( δ µ − δ ν − p µ p ν p ~p (cid:19) d +( δ µ − δ ν − (cid:18) δ µν − p µ p ν ~p (cid:19) d tr , defines two scalar functions, d and d tr .Due to the appearance of | ~p | in the denominator, thispossibility to parametrize the gluon propagator is nottoo useful in lattice calculations. The scalar functions d and d tr are related to the ones defined in (4–5) by d = D d tr = D tr − λ p ~p ( p + ~p ) D D tr = d tr + λ p ~p ( p + ~p ) d . The main difference in the results between both defini-tions is that the maximum in spatial (temporal) direc-tions is more pronounced for d tr ( D tr ) than for D tr ( d tr ),when comparing both results with the lattice data. Forthe results presented in the main text, this is only a weak,quantitative effect.The DSEs and their solutions can then be obtainedalong the same lines as in four dimensions [22]. Theequations are given by D G ( p ) − = ¯ pp (A2) − C A g (2 π ) Z d k ¯ p µ D µν ( k )¯ p ν D G ( p + k ) D µν ( p ) = D µρ ( p ) D tl ρσ ( p ) D σν ( p )+ C A g (2 π ) Z d kD µρ ( p )¯ k ρ ¯ k σ D σν ( p ) × D G ( k ) D G ( p + k ) , ¯ p µ = Hp µ = ( p λ, ~p ) µ , using the metric H = diag( λ, , λd p = d ¯ p ¯ D µν (¯ p ) = 1 λ D µν ( p )¯ D G (¯ p ) = D G ( p ) . Note that this implicitly defines in general rather compli-cated functions ¯ D µν and ¯ D G . Rewriting equations (A3– Spatial directions implies still a non-zero temporal component,as otherwise both definitions coincide. D G (¯ p ) − = ¯ pp (A3) − C A g (2 π ) Z d ¯ k ¯ p µ ¯ D µν (¯ k )¯ p ν ¯ D G (¯ p + ¯ k )¯ D µν (¯ p ) = ¯ D µρ (¯ p ) D tl ρσ ( p ) ¯ D σν ( p ) (A4)+ C A g (2 π ) Z d ¯ k ¯ D µρ (¯ p )¯ k ρ ¯ k σ ¯ D σν (¯ p ) × ¯ D G (¯ k ) ¯ D G (¯ p + ¯ k ) . Except for the tree-level terms, this system of equationslooks now formally as in Landau gauge. In the gluonequation, (A4), the tree-level term will turn out to besub-leading, and can thus be neglected. In case of theghost equation, this is in general more subtle. However,when making an infrared ansatz of type¯ D G (¯ p ) ∼ ¯ p κ − (cid:18) δ µν − ¯ p µ ¯ p ν ¯ p (cid:19) ¯ D µν (¯ p ) ∼ ¯ p t − , the system needs to be renormalized to possess a solution.It is then possible to argue as in four dimensions [22] thatthe tree-level term can be removed in the renormalizationprocess. Thus the system becomes completely equivalentto the Landau gauge case, and thus has the same twosolutions [38] 2 κ + t = 12 κ = − κ = − . . . . (A5) In particular, this leads to the infrared behavior for theoriginal functions of D G ( p ) ∼ ~p + λ p ) − κ (A6) d tr ( p ) ∼ λ ( ~p + λ p ) − t d ( p ) ∼ λ ~p ( ~p + λ p ) − t . The ghost diverges at zero ¯ p . Its anomalous scaling with λ for ~p = ~ λ − κ in (A6). The transverse gluon propa-gator is consistently suppressed from all directions. Thetemporal gluon propagator vanishes identically for ~p = ~ t ,but due to the explicit factor of ~p in the numerator. Infact, 2 − t is not larger than zero. [1] R. Alkofer and J. Greensite, J. Phys. G , S3 (2007)[arXiv:hep-ph/0610365].[2] L. Baulieu and D. Zwanziger, Nucl. Phys. B , 527(1999) [arXiv:hep-th/9807024].[3] A. Burnel, (ed.), Lect. Notes Phys. , 1-235 (2008).[4] V. N. Gribov, Nucl. Phys. B , 1 (1978).[5] I. M. Singer, Commun. Math. Phys. , 7 (1978).[6] A. Cucchieri, A. Maas, and T. Mendes, Mod. Phys. Lett.A , 2429 (2007) [arXiv:hep-lat/0701011].[7] G. Burgio, M. Quandt, and H. Reinhardt, Phys. Rev.Lett. , 032002 (2009) [arXiv:0807.3291 [hep-lat]];M. Quandt, G. Burgio, S. Chimchinda, and H. Reinhardt,PoS CONFINEMENT8 , 066 (2008) [arXiv:0812.3842[hep-th]].[8] D. Zwanziger, Phys. Rev. Lett. , 102001 (2003)[arXiv:hep-lat/0209105].[9] J. Greensite, ˇS. Olejn´ık, and D. Zwanziger, Phys. Rev. D , 074506 (2004) [arXiv:hep-lat/0401003].[10] J. Braun, H. Gies, and J. M. Pawlowski, Phys. Lett. B684 , 262-267 (2010) [arXiv:0708.2413 [hep-th]]. [11] S. P. Sorella, Phys. Rev.
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