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The energy density (39) as a function of x=aL/(2π). L−1 was varied from 260 to 280 MeV (from bottom to top).
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The effective potential of the order parameter for confinement is calculated for SU(N) Yang-Mills theory in the Hamiltonian approach. Compactifying one spatial dimension and using a background gauge fixing, this potential is obtained within a variational approach by minimizing the energy density for given background field. In this formulation the i...
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Citations
... More details on Polyakov gauge can be found in [6,127,128]. Besides the trivial simplification of the Polyakov loop, when imposing the Polyakov gauge, it turns out that the quantity hA 0 i becomes a good alternative choice for the order parameter instead of P, see Ref. [127] for an argument using Jensen's inequality for convex functions, see also [129][130][131]. For other arguments based on the use of Weyl chambers and within other gauges (see below), see Refs. ...
In earlier work, we set up an effective potential approach at zero temperature for the Gribov-Zwanziger model that takes into account not only the restriction to the first Gribov region as a way to deal with the gauge fixing ambiguity, but also the effect of dynamical dimension-two vacuum condensates. Here, we investigate the model at finite temperature in presence of a background gauge field that allows access to the Polyakov loop expectation value and the Yang-Mills (de)confinement phase structure. This necessitates paying attention to Becchi-Rouet-Stora-Tyutin and background gauge invariance of the whole construct. We employ two such methods as proposed elsewhere in literature: one based on using an appropriate dressed, Becchi-Rouet-Stora-Tyutin invariant, gluon field by the authors and one based on a Wilson-loop dressed Gribov-Zwanziger auxiliary field sector by Kroff and Reinosa. The latter approach outperforms the former in estimating the critical temperature for N=2, 3 as well as correctly predicting the order of the transition for both cases.
... Finite temperature calculations then involve the study of the ground state properties on the semicompactified spatial manifold R 2 × S 1 ðβÞ, and the Polyakov loop winds around the compactified spatial direction instead of the Euclidean time. This setting has been used successfully to compute the deconfinement phase transition in pure Yang-Mills theory [21][22][23]. In the present study, we extend these calculations to full QCD including dynamical quarks. ...
... The expectation value hÁ Á Ái 0 on the rhs of Eq. (23) is with the same wave functionals Eqs. (19) and (21) as in Coulomb gauge (hence the subscript "0"), but with the fields A being background transversaldA ¼ 0, and with the kernel ω in Eq. (19) promoted to a matrix in adjoint color space. (This will be discussed in the next subsection.) ...
... The reason for this shortcoming is that our trial Ansatz Eq. (18) is no longer sufficient at finite temperature: we should instead work with thermal states that involve arbitrary excitations above the ground state within a grand canonical ensemble. Such an approach has been attempted [19], but there is a simpler formulation which allows us to work with a trial vacuum wave functional and the usual minimization of the ground state energy [21,22,26]. The finite temperature T ¼ β −1 is here introduced by a compactification of the x 3 direction via the boundary conditions ...
We investigate the effective potential of the Polyakov loop, which is the order parameter for the deconfinement phase transition in finite temperature QCD. Our work is based on the Hamiltonian approach in Coulomb gauge where finite temperature T is introduced by compactifying one space direction. We briefly review this approach and extend earlier work in the Yang-Mills sector by including dynamical quarks. In a first approximation, we follow the usual functional approach and include only 1-loop contributions to the energy, with the finite temperature propagators replaced by their T=0 counterparts. It is found that this gives a poor description of the phase transition, in particular for the case of full QCD with Nf=3 light flavors. The physical reasons for this unexpected result are discussed, and pinned down to a relative weakness of gluon confinement compared to the deconfining tendency of the quarks. We attempt to overcome this issue by including the relevant gluon contributions from the 2-loop terms to the energy. We find that the 2-loop corrections have indeed a tendency to strengthen the gluon confinement and weaken the unphysical effects in the confining phase, while slightly increasing the (pseudo)critical temperature T* at the same time. To fully suppress artifacts in the confining phase, we must tune the parameters to rather large values, increasing the critical temperature to T*≈340 MeV for G=SU(2).
... Besides the trivial simplification of the Polyakov loop, when imposing the Polyakov gauge, it turns out that the quantity hA 0 i becomes a good alternative choice for the order parameter instead of P; see Ref. [71] for an argument using Jensen's inequality for convex functions, and see also Refs. [73][74][75]. For other arguments based on the use of Weyl chambers and within other gauges (see below), see Refs. ...
We consider a BRST-invariant generalization of the “massive background Landau gauge,” resembling the original Curci-Ferrari model that saw a revived interest due to its phenomenological success in modeling infrared Yang-Mills dynamics, including that of the phase transition. Unlike the Curci-Ferrari model, however, the mass parameter is no longer a phenomenological input, but it enters as a result of dimensional transmutation via a BRST-invariant dimension-2 gluon condensate. The associated renormalization constant is dealt with using Zimmermann’s reduction of constants program, which fixes the value of the mass parameter to values close to those obtained within the Curci-Ferrari approach. Using a self-consistent background field, we can include the Polyakov loop and probe the deconfinement transition, including its interplay with the condensate and its electric–magnetic asymmetry. We report a continuous phase transition at Tc≈0.230 GeV in the SU(2) case and a first-order one at Tc≈0.164 GeV in the SU(3) case, values which are again rather close to those obtained within the Curci-Ferrari model at one-loop order.
... These equations can be for instance the set of quantum equations of motion, known as Dyson-Schwinger equations [40][41][42][43][44], or the hierarchy of renormalization group equations that one derives from the Wetterich equation [45][46][47]. Related approaches include the use of n-particle-irreducible effective actions [48,49], the Hamiltonian formalism and its variational principle [50,51], or Dyson-Schwinger equations modified through the pinch technique [52]. ...
... As the temperature is increased, the minimum of the potential moves continuously from the confining, center-symmetric state at r = π to a pair of degenerate states (r, 2π − r) connected by center symmetry. The transition is then of the second order type, in agreement with the results of lattice simulations [145][146][147][148][149] or continuum non-perturbative approaches [48,49,51,73,74,76,[156][157][158][159][160][161]. ...
In the case of non-abelian gauge theories, the standard Faddeev-Popov (FP) gauge-fixing procedure in the Landau gauge is known to be incomplete due to the presence of gauge-equivalent field configurations. A widespread belief is that the proper analysis of the low energy properties in this gauge requires the extension of the gauge-fixing procedure beyond the FP recipe. This manuscript reviews various applications of the Curci-Ferrari (CF) model, a phenomenological proposal for such an extension, based on the decoupling properties of Landau gauge correlators as computed on the lattice. We investigate the predictions of the model concerning the deconfinement transition of strongly interacting matter at finite temperature, first in the case of pure Yang-Mills theory and then in the case of heavy-quark Quantum Chromodynamics. We show that most qualitative aspects and also many quantitative features of the deconfinement transition can be accounted for within the CF model, with only one additional parameter, adjusted from comparison to lattice simulations. Moreover, these features emerge in a systematic and controlled perturbative expansion, as opposed to the ill-defined perturbative expansion within the FP model in the infrared. Applications at finite temperature and/or density require one to consider a background extension of the Landau gauge, the so-called Landau-deWitt gauge. The manuscript is also intended as a thorough but pedagogical introduction to these techniques, including the rationale for introducing a background, the role of the Weyl chambers, and the complications that emerge due to the sign problem in the case of a real quark chemical potential. It also investigates the fate of the correlation functions as computed in the CF model and conjectures a specific behavior for the corresponding functions evaluated on the lattice, in the case of the SU(2) gauge group.
... In recent years, much valuable progress has been made toward the understanding of non-Abelian gauge theories at finite temperature using background field gauge (BFG) methods [1,2] in the Landau-DeWitt gauge, in combination with several functional methods [3][4][5][6][7][8][9][10][11][12][13][14]. On the one hand, BFG methods provide an efficient way to describe the confinement/deconfinement order parameter (the Polyakov loop or any of its proxies [3]) because the related center symmetry is explicit at the quantum level and is easily maintained in approximation schemes [15][16][17]. ...
We evaluate the finite temperature scalar sunset diagram with imaginary square masses, that appears in the Gribov-Zwanziger approach to Yang-Mills (YM) theory beyond one-loop order. Since YM theory at finite temperature is governed by center-symmetry and the Polyakov loop, we also include the possibility of a constant temporal background gauge field in the form of color-dependent imaginary chemical potentials.
... Related computations are available using different techniques to cope with nonperturbative propagators at finite temperature, see e.g. [14,15,16,17,18,19,20,21,22,23,24]. In [25,26,27], it was already pointed out that the Gribov-Zwanziger quantization offers an interesting way to illuminate some of the typical infrared problems for finite temperature gauge theories. ...
... Besides the trivial simplification of the PL, when imposing the Polyakov gauge it turns out that the quantity A 0 becomes a good alternative choice for the order parameter instead of P. This extra benefit can be proven by means of Jensen's inequality for convex functions and is carefully explained in [16], see also [15,17,18,19,20]. As mentioned before, with the Polyakov gauge imposed to the background fieldĀ µ , the time-component becomes diagonal and time-independent. ...
We consider finite-temperature $SU(2)$ gauge theory in the continuum formulation. Choosing the Landau gauge, the existing gauge copies are taken into account by means of the Gribov-Zwanziger quantization scheme, which entails the introduction of a dynamical mass scale (Gribov mass) directly influencing the Green functions of the theory. Here, we determine simultaneously the Polyakov loop (vacuum expectation value) and Gribov mass in terms of temperature, by minimizing the vacuum energy with respect to the Polyakov-loop parameter and solving the Gribov gap equation. The main result is that the Gribov mass directly feels the deconfinement transition, visible from a cusp occurring at the same temperature where the Polyakov loop becomes nonzero. Finally, problems for the pressure at low temperatures are reported.
... Related computations are available using different techniques to cope with nonperturbative propagators at finite temperature, see e.g. [14,15,16,17,18,19,20,21,22,23,24]. In [25,26,27], it was already pointed out that the Gribov-Zwanziger quantization offers an interesting way to illuminate some of the typical infrared problems for finite temperature gauge theories. ...
... Besides the trivial simplification of the PL, when imposing the Polyakov gauge it turns out that the quantity A 0 becomes a good alternative choice for the order parameter instead of P. This extra benefit can be proven by means of Jensen's inequality for convex functions and is carefully explained in [16], see also [15,17,18,19,20]. As mentioned before, with the Polyakov gauge imposed to the background fieldĀ µ , the time-component becomes diagonal and time-independent. ...
We consider finite-temperature SU(2)gauge theory in the continuum formulation. Choosing theLandau gauge, the existing gauge copies are taken into account by means of the Gribov-Zwanzigerquantization scheme, which entails the introduction of a dynamical mass scale (Gribov mass) directly influencing the Green functions of the theory. Here, we determine simultaneously the Polyakov loop (vacuum expectation value) and Gribov mass in terms of temperature, by minimizing the vacuum energy with respect to the Polyakov-loop parameter and solving the Gribovgap equation. The main result is that the Gribov mass directly feels the deconfinement transition, visible from a cusp occurring at the same temperature where the Polyakov loop becomes nonzero. Finally, problems for the pressure at low temperatures are reported.
... As such, we propose a different approach. The ultimate goal of this research program is to investigate what happens with the Gribov-Zwanziger theory at finite temperature, to investigate the response of the Green functions and their feedback on the deconfinement transition, if any, which can be investigated by including an appropriate temporal background [79][80][81][82], which allows to access the vacuum expectation value of the Polyakov loop. An important first step in this direction is to pinpoint a desirable T = 0 vacuum state to start from. ...
Abstract We revisit the effective action of the Gribov–Zwanziger theory, taking into due account the BRST symmetry and renormalization (group invariance) of the construction. We compute at one loop the effective potential, showing the emergence of BRST-invariant dimension 2 condensates stabilizing the vacuum. This paper sets the stage at zero temperature, and clears the way to studying the Gribov–Zwanziger gap equations, and particularly the horizon condition, at finite temperature in future work.
... As such, we propose a different approach. The ultimate goal of this research program is to investigate what happens with the Gribov-Zwanziger theory at finite temperature, to investigate the response of the Green functions and their feedback on the deconfinement transition, if any, which can be investigated by including an appropriate temporal background [114][115][116][117], which allows to access the vacuum expectation value of the Polyakov loop. An important first step in this direction is to pinpoint a desirable T = 0 vacuum state to start from. ...
We revisit the effective action of the Gribov-Zwanziger theory, taking into due account the BRST symmetry and renormalization (group invariance) of the construction. We compute at one loop the effective potential, showing the emergence of BRST-invariant dimension 2 condensates stabilizing the vacuum. This paper sets the stage at zero temperature, and clears the way to studying the Gribov-Zwanziger gap equations, and particularly the horizon condition, at finite temperature in future work.
... The circumference of the circle represents the inverse temperature. In Ref. [37], this novel approach has been used to calculate the effective potential of the Polyakov loop in pure Yang-Mills theory using, however, the zero-temperature gluon and ghost propagator. The correct order of the deconfinement phase transition (second order for SU (2) and first order for SU(3)) were obtained with critical temperatures in the range between 270 MeV and 290 MeV. ...
A novel approach to the Hamiltonian formulation of quantum field theory at finite temperature is presented. The temperature is introduced by compactification of a spatial dimension. The whole finite-temperature theory is encoded in the ground state on the spatial manifold S 1 ( L ) × R 2 where L is the length of the compactified dimension which defines the inverse temperature. The approach is then applied to the Hamiltonian formulation of QCD in Coulomb gauge to study the chiral phase transition at finite temperatures.
