Quark Confinement from Color Confinement

We relate quark confinement, as measured by the Polyakov-loop order parameter, to color confinement, as described by the Kugo-Ojima/Gribov-Zwanziger scenario. We identify a simple criterion for quark confinement based on the IR behaviour of ghost and gluon propagators, and compute the order-parameter potential from the knowledge of Landau-gauge correlation functions with the aid of the functional RG. Our approach predicts the deconfinement transition in quenched QCD to be of first order for SU(3) and second order for SU(2) -- in agreement with general expectations. As an estimate for the critical temperature, we obtain T_c=284MeV for SU(3).


INTRODUCTION
Aside of the confinement of quarks, the confinement of gluons is a challenging and unsolved problem. Various scenarios predict the confinement mechanism to be manifest in the infrared domain of gauge-dependent correlation functions. In the Kugo-Ojima [1] and Gribov-Zwanziger scenarios [2] (KOGZ) an infrared enhancement of the ghost and an infrared suppression of the gluon signal confinement. These scenarios have been investigated by a variety of non-perturbative field theoretical tools such as functional methods [3,4,5] and lattice gauge theory [6]. The results provide strong support for these scenarios even though the infrared enhancement of the ghost is a subject of ongoing debate, for a summary see e.g. [8,9]. This paves the way for a comprehensive understanding of the non-perturbative mechanisms of stronglycoupled gauge systems.
A pressing open question is the relation of color confinement to quark confinement.
Typical quarkconfinement criteria based on the Wilson-loop or Polyakov-loop expectation value [10] in quenched QCD have so far remained inaccessible from the pure knowledge of low-order correlation functions of the gauge sector, although evidence for a linearly rising potential between static quarks has been collected within certain approximation schemes, e.g. [11,12,13].
In this letter, we propose a method for computing the full Polyakov-loop potential from background-fielddependent Green functions. Our approach relates the order parameter of quark confinement, the expectation value of the Polyakov loop, to the momentum dependence of gauge-dependent Green functions. This leads to a simple confinement criterion in any gauge. The method is explicitly applied in the Landau gauge, where it relates the KOGZ scenario of gluon confinement to quark confinement. We evaluate the effective potential of a purely temporal background field configuration A 0 , being directly related to the Polyakov loop variable, where P denotes time ordering, and the group trace is taken in the fundamental representation. The negative logarithm of the Polyakov loop expectation value relates to the free energy of a static fundamental color source. Moreover, L measures whether center symmetry is realised by the ensemble under consideration, see e.g. [14]. A center-symmetric confining (disordered) ground state ensures L = 0, whereas deconfinement L = 0 signals the ordered phase and center-symmetry breaking.
The order parameter L[A 0 ] is conveniently parametrised in the Polyakov gauge: ∂ 0 A 0 = 0 with A 0 in the Cartan subalgebra. Then, A 0 is sensitive to topological defects related to confinement [15], and also serves as a deconfinement order parameter. More specifically, L[A 0 ] is bounded from above by L[ A 0 ] owing to the Jensen inequality is nonzero in the center-broken phase. In the centersymmetric phase where the order parameter L[A 0 ] vanishes, also the observable L[ A 0 ] can be shown to be strictly zero [16]. This establishes both A 0 as well as L[ A 0 ] as a deconfinement order parameter.
In the present work, we compute the effective potential for A 0 from Green functions in the background-field formalism [17] in the Landau-DeWitt gauge by means of the functional RG. These Green functions can be deduced from that in the Landau gauge, that is at vanishing background field. Our construction relates gluon confinement encoded in the IR behaviour of Green functions to the potential of the order parameter for quark confinement, and provides a simple confinement criterion.

BACKGROUND-FIELD FLOWS
The effective potential is given by V (L[A 0 ]) = Γ/Ω, where Γ is the effective action taken at the mean field A 0 , and Ω is the space-time volume. We evaluate the effective action Γ in the background field approach, where Γ on the one hand depends on the field variable A, being the expectation value of the fluctuating quantum field. On the other hand, a dependence on an auxiliary background fieldĀ is introduced by gauge-fixing the fluctuating field with respect to the background,  [17].
In the present study, we identify the background field with the Polyakov loop field,Ā = A 0 . For evaluating the effective potential V (L[A 0 ]), it suffices to consider A 0 as constant, yielding We compute the effective potential non-perturbatively by means of the functional RG (FRG) for the effective action [18], for reviews see [19,20]. The flow equation for Γ[Φ,Ā] in the background-field approach reads where Γ (n,m) k [20,21,22]. The regulator function R k implements an IR regularisation at p 2 ≃ k 2 , and the trace Tr sums over momenta, internal indices and species of fields. The flow (4) interpolates between the classical action in the UV and the quantum effective action Γ = Γ k=0 in the IR. For Φ = 0, Eq. (4) entails the flow of Γ k [A] = Γ k [0,Ā = A], and as a specifically interesting case, that of Background-field flows have been applied successfully to non-perturbative analyses of chiral properties in full QCD [23], including quantitative estimates of the critical temperature of the chiral transition from first principles.
The flow (4) is solved utilising optimisation ideas [20,24] that minimise the truncation error. Here, we use a specific optimised regulator [20], supplemented by k-dependent fields Φ such that Γ  [20,22]: the right-hand side of (4) depends on Γ , the flow of which cannot be extracted from ∂ t Γ k [A]. This has been neglected in previous non-perturbative applications [25] but turns out to be crucial for confinement. Hence the key input, the twopoint function Γ (2,0) k [0, A] in the background field, has to be computed separately.

EFFECTIVE ACTION FROM LANDAU-GAUGE PROPAGATORS
First, we observe that in the Landau-DeWitt gauge the longitudinal components of Green functions decouple from the transversal dynamics, which further reduces the truncation error, for a detailed discussion see [8]. Moreover, Γ (2,0) k [0, 0](p 2 ) corresponds to the propagator in the Landau gauge, since the background field gauge with gauge condition (2) reduces to the Landau gauge for vanishing background field. The Landau-gauge propagator has been computed within functional methods, [3,8,26], as well as within lattice gauge theory [6]; for reviews and further literature, see [8,14,19,20,27].
Recalling the results for Landau-gauge propagators, the gluon propagator can be displayed as where Π L,µν (p) = p µ p ν /p 2 , P T = 1 − P L , 1l ab = δ ab , and ξ denotes the gauge parameter. For the ghost, we have The longitudinal dressing function obeys Z L = 1 + O(ξ) and hence drops out of all diagrams beyond one loop in the Landau gauge ξ = 0. The dressing functions Z A,C encode the nontrivial behavior of the full propagators.
In the deep infrared, they exhibit the leading momentum behaviour In the last years it has become clear that Landau gauge Yang-Mills admits a one-parameter family of infrared solutions consistent with renormalisation group invariance [8]. Despite some formal progress the full understanding of the underlying structure is a subject of current research. Technically, the parameter corresponds to an infrared boundary condition, the value of Z C (0), and is also relates to Z A (p 2 → 0) [8]. This fact is reflected in recent lattice solutions [29] and indications thereof have also been seen in the strong coupling limit [30]. For Z C (p 2 → 0) → 0 it can be shown that there is a unique scaling solution, [31,32]. Then the two exponents are related and obey the sum rule in d dimensional spacetime [4,28,31]. Possible solutions are bound to lie in the range κ C ∈ [1/2 , 1], see [28]. For the truncation used in most DSE and FRG computation, we are led to being the value for the optimised regulator [5]. The regulator dependence in FRG computations leads to a range of κ C ∈ [0.539 , 0.595], see [5]; for a specific flow, see [33]. These results entail the KOGZ confinement scenario: the gluon is infrared screened, whereas the ghost is infrared enhanced with κ C > 1/2. In turn it can be shown that for non-vanishing Z C (0) the gluon propagator tends to a constant in the infrared, p 2 Z A (p 2 ) → m 2 , for related work see e.g. [8,34,35,36,37,38,39]. Note that the gluon propagator then does not correspond to the propagator of a massive physical particle. Instead, we observe clear indications for positivity violation in the numerical solutions for the gluon propagator related to gluon confinement, [8,41]. Still the gluon decouples from the dynamics as does a massive particle, hence the name decoupling solution. The value of Z C seems to be bounded by its perturbative value from above, and the gluon mass parameter is bounded from below [8]. The qualitative infrared behaviour is then given by the infrared exponents We emphasise that even though the infrared exponents for the scaling solution (10) and the decoupling solution (11) are rather different, the propagators do only differ in the deep infrared. It has been suggested in [8] that the infrared boundary condition is directly related to the global part of the gauge fixing, and hence to different resolutions of the Gribov problem. Indeed in [29] the infrared boundary condition has been implemented directly as a global completion of the gauge fixing. Note also, that for Landau gauge Yang-Mills with standard local BRST invariance the requirement of global BRST singles out the scaling solution. The existence of such a formulation on the lattice has been shown recently in [42]. In summary the results are affirmative for the above interpretation and are supported by results in the strongcoupling limit [30] for different implementations of lattice Landau gauge. In turn, it has been also shown in a series of works that an infrared condition also is present in Landau gauge Yang-Mills with the horizon function, e.g. [35,36,37,38]. The latter introduces an explicit (or soft) breaking of BRST invariance as it restricts the functional integral to the first Gribov region. Still this does not fix global gauge degrees of freedom as also the first Gribov region contains infinite many gauge copies. The possibility of a scaling solution in this framework hints at the validity of Zwanziger proposal: full BRST invariance is recovered in the thermodynamic limit if the path integral is restricted to the fundamental modular domain with only one gauge copy.
In summary a consistent picture has emerged with nicely relates all current results. The confirmation of this picture certainly would provide further insight to the confinement mechanism. For the present work, we simply note that the scaling solution is singled out by global BRST invariance which allows the construction of a physical Hilbert space from gauge fixed correlation functions. Nonetheless, the whole one-parameter family provides consistent gauge-fixed correlation functions of Yang-Mills theory and physical observables should be insensitive to the parameter choice. In the present work, we can test this statement.
We proceed by extending the Landau-gauge propa-gator to that in a given backgroundĀ. The Landaugauge two-point function Γ (2,0) k [0, 0](p 2 ) is, apart from its Lorentz structure provided by the projection operators P T/L (p), a function of only the momentum squared p 2 , cf. Eq. (6). At vanishing temperature, the background field propagator Γ (2,0) k [0, A] can be related to the Landau-gauge propagator in a unique fashion owing to gauge covariance, with non-singular f (0) in order to ensure the proper limit of a vanishing background. The projection operators P T/L implicitly contained in Γ At finite temperature, the Polyakov loop L is a further invariant, and the 00 component of the gluon two-point function (12) receives further contributions proportional to derivatives of L. For constant fields A 0 , we arrive at as the f term in (12) vanishes: F (A 0 ) = 0. In this letter, we take only the explicit T dependence due to Matsubara frequencies into account and drop any implicit T dependence: first, this amounts to dropping the L contribution in (13). This term is related to the second derivative of the effective potential V (2) k via Nielsen identities [20,22], and can indeed be estimated by V (2) k . Its influence on the confinement-deconfinement phase transition temperature is parametrically suppressed, and can be neglected for a first estimate of the critical temperature T c . Second, this amounts to using the zero-temperature propagators. First results indeed indicate that transversal and longitudinal gluon and ghost propagators are little modified [43,44,45] for higher Matsubara frequencies 2πT n for n > 2, 3. The biggest change appears in the gluon propagator longitudinal with respect to the heat bath that develops some enhancement compared to the transversal counterpart. The inclusion of the full temperature dependence is necessary for an accurate determination of, e.g., the critical exponents or the equation of state (see, e.g., [46]). This will be subject of a forthcoming paper.

A SIMPLE ORDER-DISORDER CONFINEMENT CRITERION
The preceding analysis gives rise to a simple confinement criterion which relates the IR behaviour of gluon and ghost 2-point functions to the deconfinement order parameter. Integrating the flow (4), we obtain where the counterterms (c.t.) denote the appropriate UV initial conditions of the flow, and the O(∂ t Γ (2,0) k ) terms correspond to integrated RG improvement terms. The first term is explicitly regulator-independent, and so is the improvement term. This can be used to show within the specific choice (5) that the improvement term is subdominant for the following analytic argument, which is confirmed by the full numerical solution.
The effective action in (14) involves the Lapacian −D 2 for vanishing field strength. In the constant A 0 background, we use the parametrisation gA a 0 = 2πT φ a , where φ a is a vector in the Cartan subalgebra. The spectrum of the Laplacian then reads where the ν ℓ denote the N 2 c − 1 eigenvalues of the hermitian color matrix T a φ a /|φ|, (T a ) bc = −if abc being the generators of the adjoint representation. From Eq. (15), it is clear that φ is a compact variable.
At high temperature, 2πT ≫ Λ QCD , the effective potential is dominated by the perturbative regime, and the background-covariant inverse propagators of both gluons and ghosts are approximately given by their tree-level values Γ (2),tree (−D 2 ) = −D 2 . The perturbative limit of the effective potential V in d > 2 is given by the wellknown Weiss potential [47], where the terms in curly brackets in the first row denote the contributions from transversal gluons, longitudinal gluons and ghosts, respectively. In the second row, we have dropped a T -and field-independent constant. The Weiss potential exhibits maxima at the center-symmetric points where L[ A 0 ] = 0, implying that the perturbative ground state is not confining, L = 0. Now, we perform the same analysis at low temperature 2πT ≪ Λ QCD . The series in (16) converges rather rapidly due to the 1/n d suppression of higher terms. Hence, the effective potential V (φ a ) is dominantly induced by fluctuations with momenta near the temperature scale p 2 ∼ (2πT ) 2 . This does not change qualitatively in the presence of a non-trivial momentum dependence of the propagators. We conclude that only the first 10-20 Matsubara frequencies play a rôle. Moreover, changing the propagator for the first two or three Matsubara frequencies, even though their weight is higher, only gives rise to minimal changes in the potential. This fully justifies the zero-temperature estimate on the propagators.
With the parametrisation (6), (7), the dressing functions Z A (p 2 ), Z C (p 2 ) in the KOGZ scenario are characterised by the power-law behaviour (8) in the deep IR, p 2 ≪ Λ 2 QCD . For low enough temperature, the spectral window −D 2 [A 0 ] ≃ (2πT ) 2 is in this asymptotic regime, and thus the effective potential arises dominantly from fluctuations in the deep IR, If the anomalous dimensions are such that the expression in curly brackets becomes negative, the effective potential is reversed and the confining center-symmetric points become order-parameter minima. We conclude that the effective action (17) predicts a center-symmetric quark-confining ground state if Provided that the O(∂ t Γ k (2, 0)) terms in Eq.
which is satisfied for the numerical values for the scaling exponents κ d in d = 2, 3, 4, see [4,28]. Specifically in d = 4, we have Eq. (10), and hence For the decoupling solution (11), we are led to Both values imply confinement, and hence the whole one parameter family of solutions is confining. Note that this is to be expected as corresponding propagators can be obtained within lattice simulations with different gauge fixings.
The above confinement criterion has to be compared to the Kugo-Ojima criterion for color confinement κ > 0 and the Zwanziger horizon condition for the ghost κ > 0 and for the gluon κ > 1/2 in d = 4. The Kugo-Ojima criterion and the Zwanziger horizon condition are necessary but not sufficient for confinement. Indeed for 0 <κ < 1/4 in four dimensions, we observe that the Kugo-Ojima criterion is satisfied but does not lead to confinement according to the present confinement criterion (19). We would also like to emphasise that, in effective theories for QCD, Eq. (18) only serves as a necessary condition. It only restricts the propagators, and other Green functions in effective theories might violate related constraints.

RESULTS FOR THE PHASE TRANSITION
In contradistinction to the simple confinement criterion put forward above, the physics of the confinementdeconfinement phase transition, e.g., the transition temperature and the order of the phase transition, is determined by the dynamics of the system and not by its IR asymptotics. Indeed, we find that fluctuations in the nonperturbative mid-momentum regime induce the centersymmetric minimum of the A 0 potential long before the propagators acquire their deep IR scaling form (8). As only the deep infrared is sensitive to the infrared boundary condition the critical temperature is insensitive to this choice which is confirmed in the explicit computation.
The results presented below are achieved by numerically integrating the flow equation (4) in order to obtain the potential for an A 0 background. The present truncation is optimised by using Landau-gauge propagators and RG improvement terms at zero temperature computed from the FRG for different infrared boundary conditions. It is also compared to results obtained by using fits to Landau-gauge propagators as measured by lattice gauge theory [7] and the RG improvement computed in [8]. For our numerical study of the order-parameter potential we have suitably amended the lattice propagators by the perturbative behaviour in the UV and the corresponding power laws (8) in the IR. In Fig. 1 we show the gluon and ghost propagators as obtained from FRG computations [8] and lattice simulations [7]. There is an impressive agreement of the results for the ghost and gluon propagators for momenta larger than about p 700 MeV which holds for the whole one parameter family of solutions including the scaling one. The results for the ghost dressing from scaling solution of the FRG and lattice simulations start deviating for p 700 MeV whereas the scaling solution for the gluon starts deviating for even lower momenta. Since the lowest non-vanishing Matsubara mode is associated with momenta at about |p| ∼ 2πT c ∼ 1700 MeV, the differences in the IR are hardly probed in the present study of the deconfinement phase transition. This is confirmed by the explicit computation. In the vacuum limit, T → 0, the picture arising from the preceding simple confinement criterion is confirmed: a sufficient amount of gluon screening with or without an IR enhancement of the ghost creates a centerdisordered ground state with quark confinement.
The confinement-deconfinement transition is taking place in the mid-momentum regime that interpolates between the perturbative regime and the IR asymptotics. The effective potentials for SU(2) and SU(3) for various temperature values near the phase transition are displayed in Fig. 2. For SU(3) (right panel), the slice of the potential in A 8 0 direction is depicted where the relevant minima for the phase transition occur. Reading off A 0 from the minimum of the potential at a given temperature, we can determine L[ A 0 ] which is plotted in Fig. 3. For SU(2) (blue/dashed line), the phase transition is of second order. For SU(3) (black/solid line), we clearly observe a first-order phase transition at a critical temperature of T c ≃ 284 ± 10MeV with a lattice string tension √ σ = 440MeV, that is T c / √ σ = 0.646 ± 0.023. The error relates to the uncertainties of the fits for the lattice propagators which exceed the estimate on the systematic error in the FRG computation. The result compares favourably both qualitatively and quantitatively with lat- tice simulations, see e.g. [7,48]. Also, our result for L[ A 0 ] in the deconfined phase is higher than the lattice measurement of the Polyakov-loop expectation value L in agreement with the Jensen inequality L[ A 0 ] > L . Note however that this statement has to be taken with care as the lattice result involves a non-trivial renormalisation factor which is absent in the definition of L[ A 0 ]. Indeed, L[ A 0 ] ≤ 1 whereas the renormalised Polyakov loop L ren necessarily exceeds unity for some temperature range as can be deduced from perturbation theory. As discussed above, corrections to our estimate arise from finite-T modifications of the propagators as well as from order-parameter fluctuations; the latter are more pronounced for SU (2) owing to the second-order nature of the transition. As expected, the critical temperature is not sensitive to the one-parameter family of solutions, it is only sensitive to the mid-momentum regime at about 1 GeV. Indeed, this also explains the fact that the gluon mass parameter is restricted from below: small gluon mass parameters would also trigger changes in the midmomentum regime and almost certainly change physical quantities such as the critical temperature.
In summary, we have established a simple confinement criterion that relates quark confinement to the infrared behaviour of ghost and gluon Green functions. This confinement criterion is applicable in arbitrary gauges. Our full numerical analysis of the IR dynamics predicts a second-order phase transition for SU(2) and a first-order phase transition for SU(3), the critical temperature of which is in quantitative agreement with lattice results. The related Polyakov loop potential also plays an important rôle for full QCD computations with dynamical quarks within functional methods, for first results on the QCD phase diagram see [49]. lattice data and useful discussions. HG acknowledges DFG support under Gi 328/1-4. JB acknowledges support by the Natural Sciences and Engineering Research Council of Canada (NSERC). TRIUMF receives federal funding via a contribution agreement through the National Research Council of Canada.