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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 one-loop contributions to the energy, with the finite temperature propagators replaced by their $T=0$ counter parts. It is found that this gives a poor description of the phase transition, in particular for the case of full QCD with $N_f = 3$ light flavours. 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 two-loop terms to the energy. We find that the two-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 $T^\ast$ 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 $T^\ast \approx 340\,\mathrm{MeV}$ for $G=SU(2)$., Comment: 38 pages, 13 figures (26 sub-figures), uses REVTeX4

We investigate the equal-time (static) quark propagator in Coulomb gauge within the Hamiltonian approach to QCD in $d=2$ spatial dimensions. Although the underlying Clifford algebra is very different from its counterpart in $d=3$, the gap equation for the dynamical mass function has the same form. The additional vector kernel which was introduced in $d=3$ to cancel the linear divergence of the gap equation and to preserve multiplicative renormalizability of the quark propagator makes the gap equation free of divergences also in $d=2$.

We investigate the equal-time (static) quark propagator in Coulomb gauge within the Hamiltonian approach to QCD. We use a non-Gaussian vacuum wave functional which includes the coupling of the quarks to the spatial gluons. The expectation value of the QCD Hamiltonian is expressed by the variational kernels of the vacuum wave functional by using the canonical recursive Dyson--Schwinger equations (CRDSEs) derived previously. Assuming the Gribov formula for the gluon energy we solve the CRDSE for the quark propagator in the bare-vertex approximation together with the variational equations of the quark sector. Within our approximation the quark propagator is fairly insensitive to the coupling to the spatial gluons and its infrared behaviour is exclusively determined by the strongly infrared diverging instantaneous colour Coulomb potential.

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 $S^1 (L) \times \mathbb{R}^2$ where $L$ is the length of the compactified dimension which defines the inverse temperature. The approach which is then applied to the Hamiltonian formulation of QCD in Coulomb gauge to study the chiral phase transition at finite temperatures., Comment: Invited talk given by H. Reinhardt at ICNFP2018, Kolymbari, Greece

We analyze the branching of center vortices in $SU(3)$ Yang-Mills theory in maximal center gauge. When properly normalized, we can define a branching probability that turns out to be independent of the lattice spacing (in the limited scaling window studied here). The branching probability shows a rapid change at the deconfinement phase transition which is much more pronounced in space slices of the lattice as compared to time slices. Though not a strict order parameter (in the sense that it vanishes in one phase) the branching probability is thus found to be a reliable indicator for both the location of the critical temperature and the geometric re-arrangement of vortex matter across the deconfinement phase transition., Comment: 11 pages, revtex4-1, expanded result section and discussion, improved figure layout, added references for the introduction, added acknowledgment and funding information

The chiral phase transition of the quark sector of QCD is investigated within the Hamiltonian approach in Coulomb gauge. Finite temperature T is introduced by compactifying one spatial dimension, which makes all thermodynamical quantities accessible from the ground state on the spatial manifold $\small\mathbb{R}^2 \times S^1(1/T)$. Neglecting the coupling between quarks and transversal gluons, the equations of motion of the quark sector are solved numerically and the chiral quark condensate is evaluated and compared to the results of the usual canonical approach to finite-temperature Hamiltonian QCD based on the density operator of the grand canonical ensemble. For zero bare quark masses, we find a second-order chiral phase transition with a critical temperature of about 92 MeV. If the Coulomb string tension is adjusted to reproduce the phenomenological value of the quark condensate, the critical temperature increases to 118 MeV., Comment: 18 pages, 7 figures (13 sub-figures), uses REVTeX4

In this lecture I will explain the established pictures of the QCD vacuum and, in particular, the underlying confinement mechanism. These are: the magnetic monopole condensation (dual Mei\ss ner effect), the center vortex picture and the Gribov--Zwanziger picture. I will start by giving a survey of the common order and disorder parameters of confinement: the temporal and spatial Wilson loop, the Polyakov loop and the 't Hooft loop. Next the dual Mei\ss ner effect, which assumes a condensate of magnetic monopoles, will be explained as a picture of confinement. I will also show how magnetic monopoles arises in QCD after the so-called Abelian projection. The second lecture is devoted to the center vortex picture of confinement. Center vortices will be defined both on the lattice and in the continuum. Within the center vortex picture the emergence of the area law for the Wilson loop as well as the deconfinement phase transition at finite temperature will be explained. Furthermore, lattice evidence for the center vortex picture will be provided. Finally, I will discuss the topological properties of center vortices and their relation to magnetic monopoles. I will provide evidence from lattice calculations that center vortices are not only responsible for confinement but also for the spontaneous breaking of chiral symmetry. In the last lecture I will present the Hamiltonian approach to QCD in Coulomb gauge, which is then used to establish the Gribov--Zwanziger picture of confinement. Furthermore, I will also relate this scenario to the dual Mei\ss ner effect and the center vortex picture. Finally, I will study QCD at finite temperature within the Hamiltonian approach in a novel way by compactifying one spatial dimension. The Polyakov loop and the dual and chiral quark condensates will be evaluated as function of the temperature., Comment: Invited lecture given at the 53rd Karpacz Winter School of Theoretical Physics, 26 February to 4 March 2017

The variational Hamiltonian approach to Quantum Chromodynamics in Coulomb gauge is investigated within the framework of the canonical recursive Dyson--Schwinger equations. The dressing of the quark propagator arising from the variationally determined non-perturbative kernels is expanded and renormalized at one-loop order, yielding a chiral condensate compatible with the observations.

The chiral phase transition of the quark sector of QCD is investigated within the Hamiltonian approach in Coulomb gauge. Finite temperatures $T$ are introduced by compactifying one spatial dimension, which makes all thermodynamical quantities accessible from the ground state on the spatial manifold $\mathbb{R}^2 \times S^1(1/T)$. In the limit of a vanishing quark-gluon coupling, the equations of motion of the quark sector are solved numerically and the chiral quark condensate is evaluated and compared to the results of the usual canonical approach to finite-temperature Hamiltonian QCD based on the density operator of the grand canonical ensemble. For zero bare quark masses, we find a second-order chiral phase transition with a critical temperature of about $107 \, \mathrm{MeV}$., Comment: Manuscript has been superseded by a new submission with new techniques and a complete overhaul on a computational and formal level, including a change in the author list

The thermodynamics of $SU(2)$ Yang-Mills theory in the covariant variational approach is studied by relating the free action density in the background of a non-trivial Polyakov loop to the pressure of the gluon plasma. The correct subtraction of the vacuum contribution in the free action density is argued for. The Poisson resummed expression for the pressure can be evaluated analytically in limiting cases, and shows the correct Stefan-Boltzmann limit at $T \to \infty$, while the limit $T \to 0$ is afflicted by artifacts due to massless modes in the confined phase. Several remedies to remove these artifacts are discussed. Using the numerical $T=0$ solutions for the ghost and gluon propagators in the covariant variational approach as input, the pressure, energy density and interaction strength are calculated and compared to lattice data., Comment: 25 pages, 16 pdf figures, pdflatex; fixed a missing factor in the free energy; revised parts of section 4 and 5; added 4 references

Using Dirac's method for the quantization of constrained systems QED is canonically quantized in the front-form in a gauge which is the light-front analog of the Weyl gauge. From the obtained vacuum wave functional the spatial Wilson loop is calculated. The result known from the canonical instant-form quantization or from the covariant path integral quantization is found only if the static limit $x^+ \to 0$ is taken in a specific order. The same ambiguity is also found in the static photon propagator in coordinate space.

Dyson--Schwinger equations are an established, powerful non-perturbative tool for QCD. In the Hamiltonian formulation of a quantum field theory they can be used to perform variational calculations with non-Gaussian wave functionals. By means of the DSEs the various $n$-point functions, needed in expectation values of observables like the Hamilton operator, can be thus expressed in terms of the variational kernels of our trial ansatz. Equations of motion for these variational kernels are derived by minimizing the energy density and solved numerically., Comment: Talk given by D. Campagnari at XIIth Quark Confinement and the Hadron Spectrum, August 28-September 04, 2016, Thessaloniki, Greece, submitted to the proceedings

The variational approach to QCD in Coulomb gauge is revisited. By assuming the non-Abelian Coulomb potential to be given by the sum of its infrared and ultraviolet parts, i.e.~by a linearly rising potential and an ordinary Coulomb potential, and by using a Slater determinant ansatz for the quark wave functional, which contains the coupling of the quarks and the gluons with two different Dirac structures, we obtain variational equations for the kernels of the fermionic vacuum wave functional, which are free of ultraviolet divergences. Thereby, a Gaussian type wave functional is assumed for the gluonic part of the vacuum. By using the results of the pure Yang--Mills sector for the gluon propagator as input, we solve the equations for the fermionic kernels numerically and calculate the quark condensate and the effective quark mass in leading order. Assuming a value of $\sigma_{\mathrm{C}} = 2.5 \sigma$ for the Coulomb string tension (where $\sigma$ is the usual Wilsonian string tension) the phenomenological value of the quark condensate $\langle \bar{\psi} \psi \rangle \simeq (-235 \, \mathrm{MeV})^3$ is reproduced with a value of $g \simeq 2.1$ for the strong coupling constant of the quark-gluon vertex.

Following a recent proposal by Cooper and Zwanziger we investigate via $SU(2)$ lattice simulations the effect on the Coulomb gauge propagators and on the Gribov-Zwanziger confinement mechanism of selecting the Gribov copy with the smallest non-trivial eigenvalue of the Faddeev-Popov operator, i.e.~the one closest to the Gribov horizon. Although such choice of gauge drives the ghost propagator towards the prediction of continuum calculations, we find that it actually overshoots the goal. With increasing computer time, we observe that Gribov copies with arbitrarily small eigenvalues can be found. For such a method to work one would therefore need further restrictions on the gauge condition to isolate the physically relevant copies, since e.g.~the Coulomb potential $V_C$ defined through the Faddeev-Popov operator becomes otherwise physically meaningless. Interestingly, the Coulomb potential alternatively defined through temporal link correlators is only marginally affected by the smallness of the eigenvalues., Comment: 11 pages, 13 figures. Discussion and conclusions extended, some plots changed. To appear in PRD

The partition function of a relativistic invariant quantum field theory is expressed by its vacuum energy calculated on a spatial manifold with one dimension compactified to a 1-sphere $S^1 (\beta)$, whose circumference $\beta$ represents the inverse temperature. Explicit expressions for the usual energy density and pressure in terms of the energy density on the partially compactified spatial manifold $\mathbb{R}^2 \times S^1 (\beta)$ are derived. To make the resulting expressions mathematically well-defined a Poisson resummation of the Matsubara sums as well as an analytic continuation in the chemical potential are required. The new approach to finite-temperature quantum field theories is advantageous in a Hamilton formulation since it does not require the usual thermal averages with the density operator. Instead, the whole finite-temperature behaviour is encoded in the vacuum wave functional on the spatial manifold $\mathbb{R}^2 \times S^1 (\beta)$. We illustrate this approach by calculating the pressure of a relativistic Bose and Fermi gas and reproduce the known results obtained from the usual grand canonical ensemble. As a first non-trivial application we calculate the pressure of Yang-Mills theory as function of the temperature in a quasi-particle approximation motivated by variational calculations in Coulomb gauge.

We compute the effective action of the Polyakov loop in SU(2) and SU(3) Yang-Mills theory using a previously developed covariant variational approach. The formalism is extended to background gauge and it is shown how to relate the low order Green's functions to the ones in Landau gauge studied earlier. The renormalization procedure is discussed. The self-consistent effective action is derived and evaluated using the numerical solution of the gap equation. We find a clear signal for a deconfinement phase transition at finite temperatures, which is second order for SU(2) and first order for SU(3). The critical temperatures obtained are in reasonable agreement with high precision lattice data., Comment: pdflatex, 30 pages, 10 pdf figures

The general method for treating non-Gaussian wave functionals in the Hamiltonian formulation of a quantum field theory, which was previously proposed and developed for Yang--Mills theory in Coulomb gauge, is generalized to full QCD. For this purpose the quark part of the QCD vacuum wave functional is expressed in the basis of coherent fermion states, which are defined in term of Grassmann variables. Our variational ansatz for the QCD vacuum wave functional is assumed to be given by exponentials of polynomials in the occurring fields and, furthermore, contains an explicit coupling of the quarks to the gluons. Exploiting Dyson--Schwinger equation techniques, we express the various $n$-point functions, which are required for the expectation values of observables like the Hamiltonian, in terms of the variational kernels of our trial ansatz. Finally the equations of motion for these variational kernels are derived by minimizing the energy density.

We extend the covariant variational approach for SU(N) Yang-Mills theory in Landau gauge to non-zero temperatures. The renormalization of the zero-temperature case is revisited and it is shown that the same counterterms are sufficient to render the low-order Green's function finite at non-zero temperature. We compute the ghost and gluon propagator numerically and show that it agrees in all qualitative respects with the results of high-precision lattice calculations., Comment: pdflatex, 36 pages, 10 pdf figures

We study, for $SU(2)$ Yang-Mills theories discretized on a lattice, a non-local topological order parameter, the center flux ${{z}}$. We show that: i) well defined topological sectors classified by $\pi_1(SO(3))=\mathbb{Z}_2$ can only exist in the ordered phase of ${{z}}$; ii) depending on the dimension $2 \leq d\leq 4$ and action chosen, the center flux exhibits a critical behaviour sharing striking features with the Kosterlitz-Thouless type of transitions, although belonging to a novel universality class; iii) such critical behaviour does not depend on the temperature $T$. Yang-Mills theories can thus exist in two different continuum phases, characterized by an either topologically ordered or disordered vacuum; this reminds of a quantum phase transition, albeit controlled by the choice of symmetries and not by a physical parameter., Comment: 32 pages, 15 figures. Typos corrected; to appear on PRD

The canonical recursive Dyson--Schwinger equations for the three-gluon and ghost-gluon vertices are solved numerically. The employed truncation includes several previously neglected diagrams and includes back-coupling effects. We find an infrared finite ghost-gluon vertex and an infrared diverging three-gluon vertex. We also compare our results with those obtained in previous calculations, where bare vertices were used in the loop diagrams., Comment: Version accepted by Phys. Rev. D

We study the static gluon and quark propagator of the Hamiltonian approach to Quantum Chromodynamics in Coulomb gauge in one-loop Rayleigh--Schr\"odinger perturbation theory. We show that the results agree with the equal-time limit of the four-dimensional propagators evaluated in the functional integral (Lagrangian) approach.

Recent results obtained for the deconfinement phase transition within the Hamiltonian approach to Yang-Mills theory are reviewed. Assuming a quasiparticle picture for the grand canonical gluon ensemble the thermal equilibrium state is found by minimizing the free energy with respect to the quasi-gluon energy. The deconfinement phase transition is accompanied by a drastic change of the infrared exponents of the ghost and gluon propagators. Above the phase transition the ghost form factor remains infrared divergent but its infrared exponent is approximately halved. The gluon energy being infrared divergent in the confined phase becomes infrared finite in the deconfined phase. Furthermore, the effective potential of the order parameter for confinement is calculated for SU$(N)$ Yang-Mills theory in the Hamiltonian approach by compactifying one spatial dimension and using a background gauge fixing. In the simplest truncation, neglecting the ghost and using the ultraviolet form of the gluon energy, we recover the Weiss potential. From the full non-perturbative potential (with the ghost included) we extract a critical temperature of the deconfinement phase transition of $269$ MeV for the gauge group SU$(2)$ and $283$ MeV for SU$(3)$., Comment: 10 pages, 17 figures, talk given by Hugo Reinhardt at "2nd International Conference on New Frontiers in Physics" in Kolymbari, Crete

The effective potential of the order parameter for confinement is calculated within the variational approach to the Hamilton formulation of Yang-Mills theory. Compactifying one spatial dimension and using a background gauge fixing this potential is obtained by minimizing the energy density for a given constant and color diagonal background field directed along the compactified dimension. Using Gaussian type trial wave functionals I establish an analytic relation between the propagators in the background gauge at finite temperature and the corresponding zero temperature propagators in Coulomb gauge. In the simplest truncation, neglecting the ghost and using the ultraviolet form of the gluon energy one recovers the Weiss potential. On the other hand from the infrared form of the gluon energy one finds an effective potential which yields a vanishing Polyakov loop indicating the confined phase. From the full non-perturbative potential (with the ghost included) one extracts a critical temperature of the deconfinement phase transition of 269 MeV for the gauge group SU(2) and 283 MeV for SU(3)., Comment: 7 pages, 9 figures, talk given by Hugo Reinhardt at "QCD-TNT-III: From quarks and gluons to hadronic matter: A bridge too far?", 2-6 September 2013, Trento, Italy

The quark gap equation under the rainbow truncation, with two versions of a phenomenological one-gluon exchange interaction and in the presence of a uniform magnetic field is considered. It is argued that in order to describe the quark condensate in the limit of vanishing magnetic fields, one must sum over the Landau levels. The resulting chiral quark condensate rises quadratically for small magnetic fields and linearly for large fields, in qualitative agreement with various recent lattice results. It is observed that when discussing quarks, the magnitude of the magnetic field must be considered relative to the scale of the strong interaction., Comment: 10 pages, 3 figures, talk given at "QCD-TNT-III: From quarks and gluons to hadronic matter: A bridge too far?", 2-6 September 2013, Trento, Italy

Our previous studies of Coulomb gauge Yang-Mills theory are extended to finite temperature. We investigate the SU(2) static gluon and ghost propagators and show results for the Coulomb potential, with a focus on the Gribov ambiguity. To compute these quantities at high temperatures and to solve scaling violations we use the anisotropic Wilson gauge action., Comment: 7 pages, 4 figures. Presented at the 31st International Symposium on Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, Germany

The nonperturbative quark gap equation under the rainbow truncation and with two versions of a phenomenological one-gluon exchange interaction is studied in the presence of a uniform external magnetic field, with emphasis on the small field limit. The chiral quark condensate, magnetic moment and susceptibility are calculated and compared to recent lattice data., Comment: 19 pages, 7 figures

We investigate the low-order Green's functions of SU(N) Yang-Mills theory in Landau gauge, using a covariant variational principle based on the effective action formalism. Employing an approximation to the Faddeev-Popov determinant established previously in the Hamiltonian approach in Coulomb gauge leads to a closed set of integral equations for the ghost and gluon propagator. We carry out the renormalization and the infrared analysis of this system of equations. Finally, we solve the renormalized system numerically and compare with lattice results and other functional approaches., Comment: 33 pages, LaTeX, 6 eps figures, 1 table

We investigate the temporal Wilson loop using the Hamiltonian approach to Yang-Mills theory. In simple cases such as the Abelian theory or the non-Abelian theory in (1+1) dimensions, the known results can be derived using unitary transformations to take care of time evolution. Alternatively, the exact solution can also be found in Coulomb gauge using the exact ground state wave functional which is known explicitly in these simple cases. The Coulomb gauge technique can also be applied to the more realistic case of Yang-Mills theory in (3+1) dimensions, where one has to rely on the approximate vacuum wave functional obtained e.g. in recent variational approaches. We use this formulation to compute the temporal Wilson loop in (3+1) dimensional Yang-Mills theory, and find that the Wilson and Coulomb string tension agree within this approximation scheme. Possible improvements of these findings are briefly discussed., Comment: 8 pages; uses PoS.cls; talk given by M. Quandt at "Quark Confinement and the Hadron Spectrum X", Munich, Germany, Oct. 8-12, 2012

The deconfinement phase transition of SU(2) Yang-Mills theory is investigated in the Hamiltonian approach in Coulomb gauge assuming a quasi-particle picture for the grand canonical gluon ensemble. The thermal equilibrium state is found by minimizing the free energy with respect to the quasi-gluon energy. At the deconfinement phase transition the gluon energy, being infrared divergent in the confined phase, becomes infrared finite in the deconfined phase, while the ghost form factor remains infrared divergent in the deconfined phase but its infrared exponent is approximately halved. Using the lattice results for the gluon propagator to fix the scale the deconfinement transition temperature is obtained in the range of 275 to 290 MeV., Comment: 8 pages, 7 eps figures. Talk given by Hugo Reinhardt at Xth Quark Confinement and the Hadron Spectrum, October 8--12, 2012 TUM Campus Garching, Munich, Germany

The effective potential of the order parameter for confinement is calculated within the Hamiltonian approach to Yang--Mills theory. Compactifying one spatial dimension and using a background gauge fixing this potential is obtained by minimizing the energy density for a given background field. Using Gaussian type trial wave functionals I establish an analytic relation between the propagators in the background gauge at finite temperature and the corresponding zero temperature propagators in Coulomb gauge. In the simplest truncation, neglecting the ghost and using the ultraviolet form of the gluon energy one recovers the Weiss potential. From the fully non-perturbative potential (with the ghost included) one extracts a critical temperature of the deconfinement phase transition of 270 MeV for the gauge group SU(2)., Comment: 8 pages, 6 eps figures. Talk given by Hugo Reinhardt at Xth Quark Confinement and the Hadron Spectrum, October 8--12, 2012 TUM Campus Garching, Munich, Germany

The general method for treating non-Gaussian wave functionals in the Hamiltonian formulation of a quantum field theory, which was previously developed and applied to Yang--Mills theory in Coulomb gauge, is generalized to full QCD. The Hamiltonian Dyson-Schwinger equations as well as the quark and gluon gap equations are derived and analysed in the IR and UV momentum regime. The back-reaction of the quarks on the gluon sector is investigated., Comment: 7 pages, 3 eps figures. Talk given by D. Campagnari at Xth Quark Confinement and the Hadron Spectrum, October 8--12, 2012 TUM Campus Garching, Munich, Germany

I will discuss the gap and Bethe-Salpeter equations for quantum chromodynamics in Coulomb gauge under a leading order truncation scheme. Within this scheme, the role of charge conservation and the cancellation of infrared divergences becomes particularly clear. The quark gap equation exhibits not only chiral symmetry breaking, but explicitly reproduces the heavy quark limit. The corresponding gluon equation has a massive solution with interesting nonperturbative renormalization properties. I will further discuss various aspects of the Bethe-Salpeter equation for pseudoscalar and vector mesons with arbitrary quark masses and present numerical results for the meson masses and leptonic decay constants., Comment: 8 pages, 5 figures, talk presented at "Quark Confinement and the Hadron Spectrum X", Munich, October 8-12, 2012

The Bethe-Salpeter equation and leptonic decay constants for pseudoscalar and vector quark-antiquark mesons with arbitrary quark masses are studied in Coulomb gauge, under a leading order truncation. As input, we use a pure linear rising potential, supplemented by a contact term arising from the conservation of total color charge. It is shown how the equations can be written in terms of manifestly finite functions, despite the infrared singular interaction. The resulting equations are solved numerically. Both the pattern of dynamical chiral symmetry breaking and the leading order heavy quark limit are visible., Comment: 23 pages, 8 figures

The effective potential of the order parameter for confinement is calculated within the Hamiltonian approach by compactifying one spatial dimension and using a background gauge fixing. Neglecting the ghost and using the perturbative gluon energy one recovers the Weiss potential. From the full non-perturbative potential calculated within a variational approach a critical temperature of the deconfinement phase transition of 269 MeV is found for the gauge group SU(2)., Comment: 5 pages, 3 eps figures; revised version; accepted for publication in Physics Letters B

We consider a variational approach to the finite temperature Yang-Mills theory in the Coulomb gauge. The partition function is computed in the ensemble of glueballs and quasi-gluons which emerge as eigenstates of the Coulomb gauge Hamiltonian. We compute the energy density and pressure and compare with results of lattice computations for both SU(2) and SU(3). The emergence of a phase transition is discussed., Comment: 10 pages, 7 eps figures

The deconfinement phase transition of SU(2) Yang--Mills theory is investigated in the Hamiltonian approach in Coulomb gauge assuming a quasi-particle picture for the grand canonical gluon ensemble. The thermal equilibrium state is found by minimizing the free energy with respect to the quasi-gluon energy. Above the deconfinement phase transition the ghost form factor remains infrared divergent but its infrared exponent is approximately halved, while the gluon energy, being infrared divergent in the confined phase, becomes infrared finite in the deconfined phase. For the effective gluon mass we find a critical exponent of 0.37. Using the lattice results for the gluon propagator to fix the scale, the deconfinement transition temperature is obtained in the range of 275 to 290 MeV., Comment: 20 pages, 13 figures, accepted for publication by Phys. Rev. D

The heavy quark sector of Coulomb gauge QCD is investigated, by making a heavy quark mass expansion of the QCD action and restricting to the leading order. With the truncation of the Yang-Mills sector to include only dressed two-point functions, we study the Dyson-Schwinger equations for the four-point quark Green's functions (proper and amputated). In this limit, we provide an exact solution for the four-point quark Green's functions and show that the corresponding poles relate to the bound state energy of the heavy quark system. Moreover, a natural separation between the physical and unphysical poles in the Green's functions emerges., Comment: 12 pages, 5 figures, talk given at the International Workshop on QCD Green's Functions, Confinement and Phenomenology, 5-9 September 2011, Trento, Italy

Coulomb gauge QCD in the first order formalism can be written in terms of a ghost-free, nonlocal action that ensures total color charge conservation via Gauss' law. Making an Ansatz whereby the nonlocal term (the Coulomb kernel) is replaced by its expectation value, the resulting Dyson-Schwinger equations can be derived. With a leading order truncation, these equations reduce to the gap equations for the static gluon and quark propagators obtained from a quasi-particle approximation to the canonical Hamiltonian approach. Moreover a connection to the heavy quark limit can be established, allowing an intuitive explanation for the charge constraint and infrared divergences., Comment: 12 pages, 2 figures, talk given at the "II International Workshop on QCD Green's Functions, Confinement and Phenomenology", Trento, Italy, September 2011

I will review results obtained recently within the Hamilton approach to QCD in Coulomb gauge. The focus will be on finite-temperature Yang--Mills theory and chiral symmetry breaking in QCD., Comment: 12 pages, 14 figures, talk given by H. Reinhardt at the "II International Workshop on QCD Green's Functions, Confinement and Phenomenology", Trento, Italy, September 2011. Submitted to the proceedings

A truncation scheme for the Dyson-Schwinger equations of quantum chromodynamics in Coulomb gauge within the first order formalism is presented. The truncation is based on an Ansatz for the Coulomb kernel occurring in the action. Results at leading loop order and in the infrared are discussed for both the Yang-Mills and quark sectors. It is found that the resulting equations for the static gluon and quark propagators agree with those derived in a quasi-particle approximation to the canonical Hamiltonian approach. Moreover, a connection to the heavy quark limit is established. The equations are analyzed numerically and it is seen that in both the gluonic and quark sectors, a nontrivial dynamical infrared mass scale emerges., Comment: 27 pages, 11 figures

The Dyson-Schwinger equation for the ghost-gluon vertex of the Hamiltonian approach to Yang-Mills theory in Coulomb gauge is solved at one-loop level using as input the non-perturbative ghost and gluon propagators previously determined within the variational approach. The obtained ghost-gluon vertex is IR finite but IR enhanced compared to the bare one by 15% to 25%, depending on the kinematical momentum regime., Comment: 7 pages, 5 eps figures

We investigate the temporal Wilson loop using the Hamiltonian approach to Yang-Mills theory. In simple cases such as the Abelian theory or the non-Abelian theory in (1+1) dimensions, the known results can be reproduced using unitary transformations to take care of time evolution. We show how Coulomb gauge can be used for an alternative solution if the exact ground state wave functional is known. In the most interesting case of Yang-Mills theory in (3+1) dimensions, the vacuum wave functional is not known, but recent variational approaches in Coulomb gauge give a decent approximation. We use this formulation to compute the temporal Wilson loop and find that the Wilson and Coulomb string tension agree within our approximation scheme. Possible improvements of these findings are briefly discussed., Comment: 24 pages, 4 eps-figures; new version matches published one

Spontaneous breaking of chiral symmetry is investigated in the Hamiltonian approach to QCD in Coulomb gauge. The quark wave functional is determined by the variational principle using an ansatz which goes beyond the commonly used BCS-type of wave functionals and includes the coupling of the quarks to the transversal spatial gluons. Using the lattice gluon propagator as input it is shown that the low energy chiral properties of the quarks, like the quark condensate and the constituent quark mass, are substantially increased by the coupling of the quarks to the spatial gluons. Our results compare favourably with the phenomenological values., Comment: 4 pages, 2 figures

We study the finite-temperature phase of a gluon ensemble in a variational approximation to QCD in the Coulomb gauge. We derive and numerically solve the underlying Dyson-Schwinger equations up to one-loop order. Assuming the subcritical solution at T=0, we find a sharp transition in the infrared value of the gluon energy at a critical temperature., Comment: 11 pages, 6 eps figures

Using simple models in D=0+0 and D=0+1 dimensions we construct partition functions and compute two-point correlations. The exact result is compared with saddle-point approximation and solutions of Schwinger-Dyson equations. When integrals are dominated by more than one saddle-point we find Schwinger-Dyson equations do not reproduce the correct results unless the action is first transformed into dual variables., Comment: 7 pages, 6 figures

We consider the Hamiltonian formulation of Yang-Mills theory in the Coulomb gauge and apply the recently developed technique of Hamiltonian flows. We formulate a flow equation for the color Coulomb potential which allows for a scaling solution that results in an almost linearly rising confining potential., Comment: 6 pages, 4 figures

A general method for treating non-Gaussian wave functionals in quantum field theory is presented and applied to the Hamiltonian approach to Yang-Mills theory in Coulomb gauge in order to include a three-gluon kernel in the exponential of the vacuum wave functional. The three-gluon vertex is calculated using the propagators found in the variational approach with a Gaussian trial wave functional as input., Comment: 3 pages, 4 figures, talk presented at "Quark Confinement and the Hadron Spectrum IX", Madrid, August 30-September 3, 2010, to appear in the proceedings

A new functional renormalization group equation for Hamiltonian Yang-Mills theory in Coulomb gauge is presented and solved for the static gluon and ghost propagators under the assumption of ghost dominance. The results are compared to those obtained in the variational approach., Comment: 3 pages, talk presented at the conference "Quark Confinement and the Hadron Spectrum IX", August 30 - September 3, 2010, Madrid, to appear in the AIP proceedings