Regimes of 3D Yang--Mills theory in the presence of a constant vector background
D. R. Granado, A. J. G. Carvalho, A. Yu. Petrov, D. Vercauteren
RRegimes of D Yang–Mills theory in the presence of aconstant vector background
D. R. Granado a,b ∗ , A. J. G. Carvalho c † A. Yu. Petrov c ‡ , David Vercauteren a,b § a Institute of Research and Development,Duy Tan University, Da Nang 550000, Vietnam b Faculty of Natural Sciences,Duy Tan University, Da Nang 550000, Vietnam c Departamento de F´ısica, Universidade Federal da Para´ıba,Caixa Postal 5008, 58051-970, Jo˜ao Pessoa, Para´ıba, Brazil
Abstract
In this paper, we take into account the Gribov copies present in 3D Yang–Mills–Higgs theorywith a constant vector background whose presence breaks the Lorentz symmetry. The constant vectorbackground is introduced within the non-Abelian aether term. Here, we show that this term arises as aone-loop correction. The influence of the aether coupling constant on the system is treated afterwards.As a result, we find that for some values of it the theory can be driven from a nonperturbative regimeto a perturbative one. In this paper, we work with the Higgs field in the fundamental representationand in the Landau gauge.
The possibility of Lorentz symmetry violation has recently attracted a lot of attention from a varietyof viewpoints. It was proposed for the first time in the context of QED by Carroll, Field and Jackiw(CFJ) in the 90s: in [1] they suggested a consistent Lorentz-violating (LV) extension of QED involvinga constant axial vector b µ . This b µ vector induces a privileged space-time direction and therefore breaksthe Lorentz symmetry. Soon after that, a variety of LV extensions of the standard model were putforward by others [2], and many nontrivial issues have come under investigation. Among these, onecan mention birefringence and rotation of the polarization of an electromagnetic wave in vacuum (forexample [3]) shown to take place in various LV extensions of QED (for example [4, 5]), ambiguities inthe quantum corrections (for example [6]), and perturbative generation of new LV terms (see for example[1]). Plenty of experimental measurements hunting for Lorentz symmetry breaking have been carried out(see for example [7] and references therein). The renormalizability of minimal LV QED was furthermorediscussed in [8].Given the importance of the Higgs mechanism in the Standard Model, Lorentz breaking in the U (1)gauge-Higgs system has also enjoyed considerable interest in the last twenty years, with many differentscenarios and aspects having come under scrutiny [9, 10, 11, 12, 13, 14]. ∗ [email protected] † [email protected] ‡ petrov@fisica.ufpb.br § [email protected] a r X i v : . [ h e p - t h ] F e b he works listed above considered Lorentz symmetry breaking in the context of QED. One would thennaturally ask for a non-Abelian version of LV terms (a list of possible LV terms that can be considered,including non-Abelian ones, is given in [15].) The non-Abelian Carroll–Field–Jackiw (CFJ) term can begenerated perturbatively (see [16] for more details) and some consequences of adding this term have beendiscussed [17, 18, 19, 20]. The renormalizability of some non-Abelian systems involving additive termshas been explored as well [17, 18, 19]. Recently, the authors of [21] studied the path integral quantizationof the YM+CFJ system.When dealing with non-Abelian gauge theories, however, correct treating the nonpertubative regimeremains one of the greatest challenges in quantum field theory. It is well known that the perturbativeformalism fails for non-Abelian gauge theories at low energy and in the absence of the Higgs mechanism(or with only a small Higgs vacuum expectation value), since the coupling constant becomes strong. Toget reliable results in the infrared (IR) limit in the continuum formulation, non-perturbative methods arerequired. The papers [22, 23, 24, 25, 26, 27, 28] give a small selection of such methods and the resultsobtained with them. A number of studies over the past decade have focused on the gluon, quark and ghostpropagators in the infrared region, where color degrees of freedom are confined. Although these objects areunphysical by themselves (they are gauge dependent), they are nevertheless the basic building blocks,next to the interaction vertices, entering gauge invariant objects directly linked to physically relevantquantities such as the spectrum, decay constants, and critical exponents and temperatures.Notice that the continuum formulation requires gauge fixing, which, in non-Abelian theories, is muchless trivial than in QED [29, 30]. At the end of the 70’s, Gribov showed that the Faddeed–Popov gaugefixing procedure was not enough to fix all the gauge copies in Yang–Mills theory [29]. These extragauge copies are called “Gribov copies”, proved by Gribov to influence the system only in the low-energyregime of the theory. The new gauge fixing procedure proposed by Gribov imposes a restriction in thefunctional integral and leads to a direct modification of the gluon propagator: the gluon propagatorexhibits complex poles instead of real ones. The presence of these complex poles suggests that thedegrees of freedom have become unphysical or are confined [31, 32]. Since Gribov’s original paper [29],the subject has been developed even more [33, 34, 35, 36, 37, 38, 39, 40, 41] (see also the reviews [31, 32]).In [42, 43] the transition between a confining and a Higgs regime using the Gribov restriction was studied.Similar analyses were made for N = 1 super-Yang–Mills [44], Yang–Mills–Chern–Simons in 3 spacetimedimensions [45], and the Yang–Mills-aether system [46]. All these results enable us to state that the issueof the Gribov copies captures nontrivial aspects of the non-perturbative dynamics of Yang–Mills theories.One setting that has not yet been investigated is that of a Lorentz-breaking extension of a non-AbelianYang–Mills–Higgs theory. In order to rectify this, in this paper we consider SU ( N ) Yang–Mills theorywith a spontaneous symmetry breaking due to a fundamental Higgs field and with the presence of theso-called “aether term” [47], which, unlike the CFJ term, does not break the CPT symmetry. In thispaper, we consider the case of three spacetime dimensions.First we give a review of the Gribov formalism in section 2, which is followed by section 3 withour analysis of the Yang–Mills–Higgs–aether theory using the Gribov formalism. We end with someconcluding remarks in section 4. In this section, we present a short review of the work by Gribov presented in [29]. In this paper, newway to treat the nonperturbative regime of non-Abelian gauge theories was proposed. This approachis based on the fact that the Faddeev–Popov procedure is not sufficient to remove all the gauge copies2resent in the Yang–Mills path integral. This means that an extra restriction on the gauge field in theintegral is mandatory. As a consequence, a non-local term called Gribov mass arises in the system.The presence of such a term modifies the gauge propagator by removing the propagation of physicalexcitations. The Gribov mass vanishes in the high-energy regime of the theory and is highly relevantin the IR regime. This is Gribov’s interpretation of confinement, i.e. the excitations of the perturbativetheory are no longer present in the IR regime.
The Euclidean Yang–Mills path integral reads Z YM = (cid:90) DA e − (cid:82) d d x F aµν F aµν , (1)where F aµν is the field strength tensor: F aµν = ∂ µ A aν − ∂ ν A aµ + gf abc A bµ A cν . (2)Due to the gauge redundancy present in the partition function (1), the Faddeev–Popov (FP) procedureis necessary. In the Landau gauge, the gauge-fixed partition function reads Z FP = (cid:90) DADcD ¯ c e − S FP , (3)with S FP = 14 (cid:90) d d x F aµν F aµν + (cid:90) d d x (cid:16) b a ∂ µ A aµ + ¯ c a ∂ µ D abµ c b (cid:17) . (4)The fields (¯ c a , c a ) are the Faddeev–Popov ghosts, b a is the Lagrange multiplier implementing the Landaugauge condition: ∂ µ A µ = 0 , (5)and D abµ = ( δ ab ∂ µ + gf acb A cµ ) is the covariant derivative in the adjoint representation of SU ( N ). In [29],it is shown that even after the FP procedure, the partition function (3) is still plagued by the presenceof some physically equivalent gauge field configurations. The proposed solution is to restrict the gaugefields to a region, namely the first Gribov region, where the FP operator M ab is positive definite. Thisregion is defined as Ω = { A aµ ; ∂ µ A aµ = 0 ; M ab = − ∂ µ ( ∂ µ δ ab − gf abc A cµ ) > } . (6)As the FP operator is related to the inverse of the ghost field propagator, the extra restriction islinked to the ghost two-point function. This function can be computed as a functional of the gauge fieldup to one-loop order as G ( k, A ) = δ ab N − k (cid:16) δ ab + σ ab ( k, A ) (cid:17) , (7)where σ ( k, A ) is called the ghost form factor. If the gauge field A µ ( x ) has small amplitude, we have G ( k, A ) ≈ k − σ ( k, A )) . (8)Thus, the condition to stay within the first Gribov region can also be expressed as σ ( k, A ) < , (9)3his condition for the ghost form factor is known as the no-pole condition. After the FP procedureand constraining the path integral to a domain where the gauge field configuration satisfies the no-polecondition (8), the system has no more infinitesimal gauge copies. The no-pole restriction is implementedby means of the Heaviside step function θ : Z G = (cid:90) Ω DADcD ¯ c e − S FP = (cid:90) DADcD ¯ c θ (1 − σ ( k, A )) e − S FP . (10)The ghost two-point function in the presence of an external gauge field up to first order in the quantumfields reads G ( k, A ) = 1 k (cid:18) k µ k ν k N g V d ( N − (cid:90) d d p (2 π ) A aµ ( p ) A aν ( − p )( k − p ) (cid:19) . (11)It is known it suffices to take the limit k → in which case the ghost form factor reads σ (0 , A ) = N g dV ( N − (cid:90) d d p (2 π ) d A aµ ( p ) A aµ ( − p ) p . (12)Considering the integral representation of the Heaviside step function, the Gribov partition functionbecomes Z G = (cid:90) DADcD ¯ c (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) dβ πiβ e β (1 − σ (0 ,A )) e − S FP . (13)The integral over β can be performed in the saddle-point approximation elaborated on in the next section.Finally, we can write down the Gribov action, S G = S FP + βN g d ( N − (cid:90) d d x A aµ ( x ) (cid:2) ∂ (cid:3) − A aµ ( x ) − β . (14) As can be seen from the action (14), a mass parameter β is introduced into the theory. This β is nota free parameter of the theory. As required by the consistency of the model, it is determined by a gapequation.At the tree level in perturbation theory, the partition function (13) can be written as Z G = (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) dβ πiβ (cid:90) DADcD ¯ c exp (cid:26) − (cid:90) d d p (2 π ) d ) (cid:20) A aµ ( p ) Q abµν A bν ( − p ) + ¯ c a ( p ) P ab c b ( − p ) (cid:21) − β (cid:27) , (15)with Q abµν = δ ab (cid:20)(cid:18) N βg V d ( N −
1) 1 p + p (cid:19) δ µν + (cid:18) α − (cid:19) p µ p ν (cid:21) (16)and P ab = δ ab p . (17)Integrating out the fields, one ends up with Z G = (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) dβ iπ (cid:104) det Q abµν (cid:105) − / (cid:104) det P ab (cid:105) e β − ln β . (18) See [32] for a detailed computation of (11) and for further discussion. − / = e − Tr ln M for any matrix M, one rewrites the path integral ase − V E v = Z G = (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) dβ iπ e − f ( β ) , (19)with f ( β ) = β − ln β − d − d ( N − V (cid:90) d d p (2 π ) d ln (cid:18) p + βN g N − dV p (cid:19) . (20)As mentioned before, by means of the saddle-point approximation we have Z G ≈ e − f ( β ∗ ) , (21)where β ∗ is that value of β which satisfies the saddle-point equation ∂ E v ∂β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β = β ∗ = 0 , (22)leading us to the so-called gap equation d − d N g (cid:90) d d p (2 π ) d ) 1 p + γ ∗ . (23)In order to simplify the notation, we defined γ = g βNdV ( N − . As mentioned before, the gap equation (23)can be seen as a self-consistency condition of the model, i.e. the Gribov mass parameter is determinedby (23).From (16), it can be seen that the gluon propagator is influenced by the Gribov parameter γ . Thegluon propagator in the Landau limit ∆ → (cid:104) A aµ ( k ) A bν ( − k ) (cid:105) = δ ab k k + γ ∗ (cid:18) δ µν − k µ k ν k (cid:19) = δ ab (cid:18) k + iγ ∗ + 1 k − iγ ∗ (cid:19) (cid:18) δ µν − k µ k ν k (cid:19) . (24)Thus, from (24), it is clear that due to Gribov’s restriction, the gluon propagator displays complexconjugate poles. This prevents us from assigning an asymptotic single-particle interpretation to the gluonpropagator (its K¨all´en-Lehmann representation is not always positive [48]). As was mentioned before,Gribov interpreted this as confinement, i.e. the excitations of the perturbative theory are physicallyabsent in the IR regime. D Yang–Mills–aether theory with Higgs fields
In this section, in order to describe a mechanism through which the desired non-Abelian aether termarises in 3 D , we describe its perturbative generation, and, by means of the approach described in theprevious section, investigate the different regimes of the 3 D Yang–Mills–Higgs–aether theory. In the thermodynamic limit, the term ln β can be disregarded, since β is proportional to the volume V . .1 Perturbative generation of the aether term in D Various Lorentz-breaking terms, including the aether term, either Abelian or non-Abelian ones, canbe generated perturbatively as one-loop corrections in some theory involving a (non-Abelian) gaugefield coupled to spinors with inclusion of Lorentz-breaking parameters. This methodology was proposedalready in [2] and applied to the Abelian aether term in various space-time dimensions from 3 to 5in [50, 51], and to its four-dimensional non-Abelian analogue in [49]. Now, let us discuss the three-dimensional non-Abelian aether term. We start with the extended spinor QED action which can betreated as a natural 3 D analogue of the action proposed in [51]: S = (cid:90) d x ¯ ψ i (cid:16) iγ µ ( ∂ µ δ ij − ieA aµ ( T a ) ij + b µ δ ij ) + g(cid:15) µνλ b µ F aνλ ( T a ) ij − mδ ij (cid:17) ψ j , (25)where ( T a ) ij are the generators of the corresponding Lie algebra, and both the gauge field A µ = A aµ T a and the non-Abelian stress tensor F µν = F aµν T a are Lie-algebra valued. The vertex involving A µ is furtherreferred to as the minimal one, and the one involving the F µν as the non-minimal one. We note that,unlike in the four-dimensional case, there is no chirality in three dimensions, and the analogue of the γ matrix given by γ γ γ is proportional to the unit matrix. It must be emphasized that the presence ofthe gauge covariant derivative D ijµ = ∂ µ δ ij − ieA aµ ( T a ) ij is necessary for the full-fledged gauge invarianceof the action while otherwise, in the absence of the minimal coupling, only restricted gauge invariance,with constant gauge parameters without dependence on the space-time coordinates, is possible.Completely analogously to [51], there could be three contributions to the aether term – the one formedby two minimal vertices, the mixed one, and the one formed by two non-minimal vertices. However,straightforward calculations show that the purely minimal contribution vanishes. This fact can be justifiedas follows: the integral over momenta in the corresponding Feynman diagram is the same for the Abelianand the non-Abelian case, and the Abelian minimal aether-like contribution (proportional to e ) is zerosince in this case the Lorentz-breaking vector b µ is ruled out by the simple gauge transformation A µ → A µ − b µ . Hence the minimal contribution of second order in b µ is zero independently on the gauge group.The mixed aether contribution (proportional to eg ) also can easily be shown to vanish in the Abeliancase (indeed, the only relevant term, after the above-mentioned gauge transformation, turns out to beof first order in b µ ), hence, by gauge symmetry reasons, the non-Abelian generalization of this term willalso absent (indeed, the Abelian contribution is the quadratic part of the non-Abelian one).Therefore, the only nontrivial aether-like contribution is the purely non-minimal one. In this casewe can straightforwardly apply the results obtained in [50], with the only difference a factor κ arisingfrom the definition of the trace tr( T a T b ) = κδ ab and coming from the product of two generators in twovertices, and write down the desired aether term S aether = 4 | m | g κ π b µ F aµν b λ F λνa . (26)So, we explicitly demonstrated how the 3 D non-Abelian aether term arises. It is clear that in fourdimensions, the non-Abelian aether contributions will be generated for all three cases, not only theminimal one studied in [49]. However, the contributions involving either one or two non-minimal verticeswill be ambiguous in full analogy with [51]. 6 .2 Gauge propagator The Yang–Mills–Higgs–aether Euclidean action before gauge fixing and implementing the Gribovformalism reads S = (cid:90) d x (cid:18) (cid:0) F aµν (cid:1) + α a µ F aµν a δ F aδν (cid:19) + ( D ijµ Φ j ) † ( D ikµ Φ k ) + λ (cid:16) Φ † Φ − ν (cid:17) . (27)To this action, we add a Landau gauge fixing term and the contributions coming from the Gribovformalism.We can write the quadratic terms as S quad = (cid:90) d k (2 π ) (cid:18) A aµ ( k ) Q abµν A bν ( − k ) (cid:19) , (28)where we introduced the inverse propagator operator Q abµν = δ ab (cid:20)(cid:18) k + γ k + g ν (cid:19) δ µν + (cid:18) − (cid:19) k µ k ν + α (cid:0) ( a.k ) δ µν − ( a · k ) a ν k µ − a µ k ν ( a · k ) + k a µ a ν (cid:1)(cid:21) , (29)where we still have γ = βNg V ( N − the Gribov parameter and ∆ the gauge fixing parameter, which mustbe put to zero for the Landau gauge.In this ∆ → (cid:104) A aµ ( k ) A bν ( k ) (cid:105) = δ ab F ( k ) (cid:20)(cid:18) δ µν − k µ k ν k (cid:19) − F ( k ) (cid:0) ( a · k ) k µ − k a µ (cid:1) (cid:0) ( a · k ) k ν − k a ν (cid:1)(cid:21) , (30)where F ( k ) = k k + γ + g ν k + α ( a.k ) k , (31a) F ( k ) = α (1 + αa ) k + γ + g ν k . (31b)The poles of F ( k ) are found at k equal to minus m ± = − g ν ξ ( θ ) ± ξ ( θ ) (cid:112) ( g ν − ξ ( θ ) γ ) (32)where we defined θ as the angle between k µ and a µ , and ξ ( θ ) = 1 + αa cos θ . The poles of F ( k ) can befound by putting θ = 0. Repeating the steps that led to (23), one finds that the gap equation in our case is given by d = N g V ( N −
1) Tr( Q abµν ) − , (33)where Q abµν was defined in (29). 7o compute the trace, we use a basis in which Q abµν is diagonal. For any vector v µ orthogonal both to k µ and a µ , we have Q abµν v ν = δ ab (cid:18) k + γ k + g ν α ( a · k ) (cid:19) v µ , (34)which gives us the first d − k µ we have: Q abµν k ν = δ ab (cid:18) γ k + g ν k (cid:19) k µ . (35)This yields one more eigenvalue. In order to find the last eigenvalue, we consider a vector in the ( a µ , k µ )plane but orthogonal to k µ : Q abµν (cid:18) a ν − a · kk k ν (cid:19) = δ ab (cid:18) k + γ k + g ν αk a (cid:19) (cid:18) a µ − a · kk k µ (cid:19) . (36)To conclude, we have:Tr( Q abµν ) − = V ( N − (cid:34) ( d − (cid:90) d d k (2 π ) d k + γ k + g ν + α ( a · k ) + (cid:90) d d k (2 π ) d γ k + g ν + k + (cid:90) d d k (2 π ) d k + γ k + g ν + αa k (cid:35) . (37)The second term is zero in the limit ∆ →
0. For d = 3, the integrals are furthermore finite as is usualin odd-dimensional space-times, making the dimensional regularization unnecessary. This leaves us withthe following gap equation:3 = N g (cid:32)(cid:90) d k (2 π ) ξ ( θ ) k + g ν k + γ + (cid:90) d k (2 π ) ζk + g ν k + γ (cid:33) , (38)where we put ζ = 1 + αa and we still have ξ ( θ ) = 1 + αa cos θ .The integrals on the r.h.s. are finite, such that we can argue using normal integration rules: • Taking the derivative of the r.h.s. with respect to γ yields integrals of minus a square, which isnegative. This means that the r.h.s. of the gap equation decreases with γ . • After changing variables k µ → γq µ , it is possible to expand the integrals with respect to large γ ,yielding results behaving as 1 /γ . This means that the r.h.s. goes to zero for large γ .As a result, the r.h.s. is a monotonously decreasing function of γ , with limiting value zero. The gapequation can therefore only have a solution for real (positive) γ if and only if the r.h.s. of the gap equationevaluated at γ = 0 is more than 3:3 < N g (cid:32)(cid:90) d k (2 π ) ξ ( θ ) k + g ν k + (cid:90) d k (2 π ) ζk + g ν k (cid:33) . (39)If this inequality is not satisfied, there can be no physical solutions, but one checks easily that the no-polecondition (9) is automatically satisfied in that case, and the Gribov formalism is therefore not needed.The integrand of the second integral can be rewritten using the Feynman parametrization:1 ζ k ( k + g ν ζ ) = 1 ζ (cid:90) dx ( k + x g ν ζ ) (40)8ntegrating over k µ and then over x immediately yields12 πgν √ ζ . (41)For the first integral, use spherical coordinates and that (cid:90) dxαx + β = 1 √ αβ arctan( x (cid:113) αβ ) + C , (42a) (cid:90) dx √ αx + 1 = 1 √ α arsinh( x √ α ) + C , (42b)to find 12 πgνa √ α arsinh( a √ α ) . (43)The condition for a solution to the gap equation is therefore3 < N g πν √ (cid:18) arsinh( a √ α ) a √ α + 1 √ αa (cid:19) , (44)or, for a small Lorentz-breaking term, which is the physical case, g > πν √ N (cid:18) αa · · · (cid:19) . (45)This is the blue line separating regimes I and II in Figure 2: there is no nontrivial Gribov parameter inthe weak-coupling strong-Higgs regime I, while the Gribov parameter is nonzero (regimes II to IV) forstrong coupling or weak Higgs term. Some numerically determined values of the Gribov parameter areplotted in Figure 1. One notices that a positive value for αa pulls the Gribov parameter down, while anegative aether term pushes it up.
15 20 25 30 Ng / ν γ / g ν α a = α a = α a =- Figure 1:
Values of the dimensionless Gribov parameter γ/gν as function of
N g/ν from a numerical solution of(38) for select values of the Lorentz breaking αa . From the above, we conclude there are at least two regimes: one with and one without Gribovparameter. The situation becomes more complex when looking at the behavior of the gluon propagator.In the absence of a nonzero Gribov parameter, the gluon simply behaves like a massive but Lorentz-brokenHiggs type gluon. 9 - α a Ng / ν I IIIIIa IIIbIV
Figure 2:
Boundaries between different behaviors of the poles of the gluon propagator (32). See section 3.4 foran in-depth discussion.
Once the Gribov parameter is nonzero, this begins to change. At not very high values of the Gribovparameter, however, the gluon propagator still has real, massive poles like in the pure Higgs–aethercase. This is what happens in regime II of Figure 2. At some point, however, the discriminant of thedenominators of (31) changes sign, at least for some values of θ . The green line in Figure 2 indicatesa change in sign for θ = 0 (and thus also for the denominator of F ( k )), while the red line indicates achange in sign for θ = π . The sign changes for other values of θ happen between these two lines.As a result we have the regimes labeled IIIa and IIIb, where the gluon has real or complex polesdepending on the direction of propagation relative to the aether field. In the strong-coupling weak-Higgsregime IV, all poles have nonzero imaginary parts, and no physical gluons can propagate.One sees that all the transition lines in Figure 2 increase with αa . As a result, introducing apositive aether term has a qualitative effect comparable to lowering the coupling strength or increasingthe strength of the Higgs background, while a negative aether term is similar to stronger coupling orweaker Higgs. In this work we presented a first study of non-Abelian Yang–Mills–Higgs with an additive Lorentzbreaking aether term in 3 spacetime dimensions. To justify the presence of the non-Abelian aether term,we performed its perturbative generation, demonstrating that it arises as a one-loop correction. As a by-product, we argued that the non-Abelian aether-like term in 4 D receives contribution from non-minimalcouplings, too. We used the Gribov–Zwanziger formalism to fix the gauge in Landau gauge withoutinfinitesimal gauge copies, which gives insight in the nonperturbative dynamics of the theory.We found that a positive aether parameter αa reduces the value of the Gribov parameter and (ifsufficiently large) can turn the theory from showing a nonpertubative behavior to a perturbative one. Anegative value of αa has the opposite effect.Compared to the 3D SU ( N ) Yang–Mills–Higgs case, we find one extra intermediate regime: in betweenthe regime with real poles and the one with complex poles in the gluon propagator, there is an additionalregime where the reality of the gluon propagator poles depends on the direction of propagation. Thedifferent regimes are depicted in Figure 2. 10 next step in this line of research would be to consider the theory in an appropriate temporalbackground which allows to access the vacuum expectation value of the Polyakov loop [52, 53, 54, 55],which would tell us whether the theory is effectively confined in the regime where the gluon propagatorhas complex poles and deconfined otherwise. Acknowledgments
The work by A. Yu. P. has been partially supported by CNPq, project 301562/2019-9.
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