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Inspired by the center-vortex dominance in the infrared sector of $SU(N)$ Yang-Mills theory observed on the lattice, we propose a vacuum wave functional localized on an ensemble of correlated center vortices endowed with stiffness and magnetic monopoles that change the orientation of the vortex flux. In the electric-field representation, this wave functional becomes an effective partition function for N complex scalar fields. The inclusion of both oriented and non-oriented vortices as well as so-called N-vortex matchings leads to an effective potential that has only a center symmetry left. In the center-vortex condensed phase, this symmetry is spontaneously broken. In this case, the Wilson loop average can be approximated by a solitonic saddle-point localized around the minimal surface. The asymptotic string tension thus obtained displays Casimir scaling.

By means of lattice calculations, center vortices have been established as the infrared dominant gauge field configurations of Yang-Mills theory. In this work, we investigate an ensemble of center vortices in D = 3 Euclidean space-time dimension where they form closed flux loops. To account for the properties of center vortices detected on the lattice, they are equipped with tension, stiffness and a repulsive contact interaction. The ensemble of oriented center vortices is then mapped onto an effective theory of a complex scalar field with a U(1) symmetry. For a positive tension, small vortex loops are favoured and the Wilson loop displays a perimeter law while for a negative tension, large loops dominate the ensemble. In this case the U(1) symmetry of the effective scalar field theory is spontaneously broken and the Wilson loop shows an area law. To account for the large quantum fluctuations of the corresponding Goldstone modes, we use a lattice representation, which results in an XY model with frustration, for which we also study the Villain approximation., Comment: 23 pages

I report on recent results obtained within the Hamiltonian approach to QCD in Coulomb gauge. Furthermore this approach is compared to recent lattice data, which were obtained by an alternative gauge fixing method and which show an improved agreement with the continuum results. By relating the Gribov confinement scenario to the center vortex picture of confinement it is shown that the Coulomb string tension is tied to the spatial string tension. For the quark sector a vacuum wave functional is used which explicitly contains the coupling of the quarks to the transverse gluons and which results in variational equations which are free of ultraviolet divergences. The variational approach is extended to finite temperatures by compactifying a spatial dimension. The effective potential of the Polyakov loop is evaluated from the zero-temperature variational solution. For pure Yang--Mills theory, the deconfinement phase transition is found to be second order for SU(2) and first order for SU(3), in agreement with the lattice results. The corresponding critical temperatures are found to be $275 \, \mathrm{MeV}$ and $280 \, \mathrm{MeV}$, respectively. When quarks are included, the deconfinement transition turns into a cross-over. From the dual and chiral quark condensate one finds pseudo-critical temperatures of $198 \, \mathrm{MeV}$ and $170 \, \mathrm{MeV}$, respectively, for the deconfinement and chiral transition., Comment: Talk given by H. Reinhardt at "5th Winter Workshop on Non-Perturbative Quantum Field Theory", 22-24 March 2017, Sophia-Antipolis, France. arXiv admin note: text overlap with arXiv:1609.09370, arXiv:1510.03286, arXiv:1607.08143

I report on recent results obtained within the Hamiltonian approach to QCD in Coulomb gauge. By relating the Gribov confinement scenario to the center vortex picture of confinement it is shown that the Coulomb string tension is tied to the spatial string tension. For the quark sector a vacuum wave functional is used which results in variational equations which are free of ultraviolet divergences. The variational approach is extended to finite temperatures by compactifying a spatial dimension. For the chiral and deconfinement phase transition pseudo-critical temperatures of 170 MeV and 198 MeV, respectively, are obtained., Comment: Talk given by H. Reinhardt at "XIIth Quark Confinement and the Hadron Spectrum", 28 August-4 September 2016, Thessaloniki, Greece

I will review essential features of the Hamiltonian approach to QCD in Coulomb gauge showing that Gribov's confinement scenario is realized in this gauge. For this purpose I will discuss in detail the emergence of the horizon condition and the Coulomb string tension. I will show that both are induced by center vortex gauge field configurations, which establish the connection between Gribov's confinement scenario and the center vortex picture of confinement. I will then extend the Hamiltonian approach to QCD in Coulomb gauge to finite temperatures, first by the usual grand canonical ensemble and second by the compactification of a spatial dimension. I will present results for the pressure, energy density and interaction measure as well as for the Polyakov loop., Comment: 13 pages, 10 figures, talk given by H. Reinhardt at "5th International Conference on New Frontiers in Physics", 6-14 July 2016, Kolymbari, Greece

The chiral and deconfinement phase transitions are investigated within the variational Hamiltonian approach to QCD in Coulomb gauge. The temperature $\beta^{- 1}$ is introduced by compactifying a spatial dimension. Thereby the whole temperature dependence is encoded in the vacuum state on the spatial manifold $\mathbb{R}^2 \times S^1 (\beta)$. The chiral quark condensate and the dual quark condensate (dressed Polyakov loop) are calculated as function of the temperature. From their inflection points the pseudo-critical temperatures for the chiral and deconfinement crossover transitions are determined. Using the zero-temperature quark and gluon propagators obtained within the variational approach as input, we find $168 \, \mathrm{MeV}$ and $196 \, \mathrm{MeV}$, respectively, for the chiral and deconfinement transition.

The variational approach to QCD in Coulomb gauge developed previously by the T\"ubingen group is improved by enlarging the space of quark trial vacuum wave functionals through a new Dirac structure in the quark-gluon coupling. Our ansatz for the quark vacuum wave functional ensures that all linear divergences cancel in the quark gap equation resulting from the minimization of the energy calculated to two-loop order. The logarithmic divergences are absorbed in a renormalized coupling which is adjusted to reproduce the phenomenological value of the quark condensate. We also unquench the gluon propagator and show that the unquenching effects are generally small and amount to a small reduction in the mid-momentum regime.

The variational Hamiltonian approach to QCD in Coulomb gauge is reviewed and the essential results obtained in recent years are summarized. First the results for the vacuum sector are discussed, with a special emphasis on the mechansim of confinement and chiral symmetry breaking. Then the deconfinement phase transition is described by introducing temperature in the Hamiltonian approach via compactification of one spatial dimension. The effective action for the Polyakov loop is calculated and the order of the phase transition as well as the critical temperatures are obtained for the color group SU(2) and SU(3). In both cases, our predictions are in good agreement with lattice calculations., Comment: Talk given by H. Reinhardt at the 4th International Conference on New Frontiers in Physics (ICNFP 2015)

From continuum studies it is known that the Coulomb string tension $\sigma_C$ gives an upper bound for the physical (Wilson) string tension $\sigma_W$ [D. Zwanziger, Phys. Rev. Lett. 90, 102001 (2003)]. How does however such relationship translate to the lattice? In this paper we give evidence that there, while the two string tensions are related at zero temperature, they decouple at finite temperature. More precisely, we show that on the lattice the Coulomb gauge confinement scenario is always tied to the spatial string tension, which is known to survive the deconfinement phase transition and to cause screening effects in the quark-gluon plasma. Our analysis is based on the identification and elimination of center vortices which allows to control the physical string tension and study its effect on the Coulomb gauge observables. We also show how alternative definitions of the Coulomb potential may sense the deconfinement transition; however a true static Coulomb gauge order parameter for the phase transition is still elusive on the lattice., Comment: 12 pages, 14 figures. Figures and changes in text added to match published version

Yang-Mills theory is studied at finite temperature within the Hamiltonian approach in Coulomb gauge by means of the variational principle using a Gaussian type ansatz for the vacuum wave functional. Temperature is introduced by compactifying one spatial dimension. As a consequence the finite temperature behavior is encoded in the vacuum wave functional calculated on the spatial manifold $\mathbb{R}^2 \times \mathrm {S}^1 (L)$ where $L^{-1}$ is the temperature. The finite-temperature equations of motion are obtained by minimizing the vacuum energy density to two-loop order. We show analytically that these equations yield the correct zero-temperature limit while at infinite temperature they reduce to the equations of the $2$+$1$-dimensional theory in accordance with dimensional reduction. The resulting propagators are compared to those obtained from the grand canonical ensemble where an additional ansatz for the density matrix is required., Comment: 10 pages, 7 figures

We review our lattice results concerning the Gribov-Zwanziger confinement mechanism in Coulomb gauge. In particular, we verify the validity of Gribov's IR divergence condition for the Coulomb ghost form factor. We also show how the quark self-energy is, like that of the transverse gluon, IR divergent, thus effectively extending the Gribov-Zwanziger scenario to full QCD., Comment: 7 pages, 4 figures. References updated. Presented at the 31st International Symposium on Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, Germany

The variational approach to Yang-Mills theory in Coulomb gauge is extended to full QCD. For the quark sector we use a trial wave functional, which goes beyond the previously used BCS-type state and which explicitly contains the coupling of the quarks to transverse gluons. This quark wave functional contains two variational kernels: One is related to the quark condensate and occurs already in the BCS-type states. The other represents the form factor of the coupling of the quarks to the transverse gluons. Minimization of the energy density with respect to these kernels results in two coupled integral (gap) equations. These equations are solved numerically using the confining part of the non-Abelian color Coulomb potential and the lattice static gluon propagator as input. With the additional coupling of quarks to transverse gluons included the low energy chiral properties increase substantially towards their phenomenological values. We obtain a reasonable description of the chiral condensate, which for a vanishing current quark mass is obtained in the range of 190-235 MeV. The coupling of the quarks to the transverse gluons enhances the constituent quark mass by about 60% in comparison to the pure BCS ansatz.

We show that in the lattice Hamiltonian limit all Coulomb gauge propagators are consistent with the Gribov-Zwanziger confinement mechanism, with an IR enhanced effective energy for quarks and gluons and a diverging ghost form factor compatible with a dual-superconducting vacuum. Multiplicative renormalizability is ensured for all static correlators, while for non-static ones their energy dependence plays a crucial role in this respect. Moreover, from the Coulomb potential we can extract the Coulomb string tension \sigma_C ~ 2 \sigma., Comment: 8 pages, 5 figures. Talk given at the conference "Confinement X, Quark Confinement and the Hadron Spectrum", 8-12 October 2012, TUM Campus Garching, Munich, Germany

We calculate the Coulomb ghost propagator G(|p|) and the static Coulomb potential V_C(|r|) for SU(2) Yang-Mills theory on the lattice. In view of possible scaling violations related to deviations from the Hamiltonian limit we use anisotropic lattices to improve the temporal resolution. We find that the ghost propagator is infrared enhanced with an exponent kappa_gh ~ 0.5 while the Coulomb potential exhibits a string tension larger than the Wilson string tension, sigma_C ~ 2 sigma. This agrees with the Coulomb "scaling" scenario derived from the Gribov-Zwanziger confinement mechanism., Comment: 23 pages, 5 figures. Some issues clarified and extended, references added. To appear in PRD

We calculate the lattice quark propagator in Coulomb gauge both from dynamical and quenched configurations. We show that in the continuum limit both the static and full quark propagator are multiplicatively renormalizable. From the propagator we extract the quark renormalization function Z(|p|) and the running mass M(|p|) and extrapolate the latter to the chiral limit. We find that M(|p|) practically coincides with the corresponding Landau gauge function for small momenta. The computation of M(|p|) can be however made more efficient in Coulomb gauge; this can lead to a better determination of the chiral mass and the quark anomalous dimension. Moreover from the structure of the full propagator we can read off an expression for the dispersion relation of quarks, compatible with an IR divergent effective energy. If confirmed on larger volumes this finding would allow to extend the Gribov-Zwanziger confinement mechanism to the fermionic sector of QCD., Comment: 14 pages, 7 figures, 1 table. References added and corrected, some issues clarified. To appear in PRD

Recently it was claimed that QCD condensates were associated with the internal dynamics of the hadrons. We challenge this "in-hadron" picture of the QCD condensates and show that it conflicts well established concepts and experimental facts for the example of the quark condensate. We explicitly demonstrate the vacuum nature of the quark condensate by a model calculation., Comment: Five pages, two figures, similar to version to be published in PRD

We present the current status of ongoing efforts to use functional methods, Dyson-Schwinger equations and functional renormalization group equations, for the description of the infrared regime of nonabelian (pure) gauge theories in the Coulomb gauge. In particular, we present a new determination of the color-Coulomb potential with the help of the functional renormalization group that results in an almost linearly rising potential between static color charges at large spatial distances., Comment: 11 pages, 3 figures. Slightly improved version of J. Phys.: Conf. Ser. 287 (2011) 012023

The Dyson-Schwinger equation for the 4-point quark Green's functions is studied. In the limit of the heavy quark mass and with the truncation to include only the dressed two point functions for the Yang-Mills sector, we provide an exact solution for the 4-point quark Green's functions, in both quark-antiquark and diquark channels, and show that the corresponding poles relate to the bound state energy of the heavy quark systems. Moreover, a natural separation between physical and unphysical poles in the Green's functions emerges., Comment: 21 pages, 10 figures

We review a method, suggested many years ago, to numerically measure the relative amplitudes of the true Yang-Mills vacuum wavefunctional in a finite set of lattice-regulated field configurations. The technique is applied in 2+1 dimensions to sets of abelian plane wave configurations of varying amplitude and wavelength, and sets of non-abelian constant configurations. The results are compared to the predictions of several proposed versions of the Yang-Mills vacuum wavefunctional that have appeared in the literature. These include (i) a suggestion in temporal gauge due to Greensite and Olejnik; (ii) the "new variables" wavefunction put forward by Karabali, Kim, and Nair; (iii) a hybrid proposal combining features of the temporal gauge and new variables wavefunctionals; and (iv) Coulomb gauge wavefunctionals developed by Reinhardt and co-workers, and by Szczepaniak and co-workers. We find that wavefunctionals which simplify to a "dimensional reduction" form at large scales, i.e. which have the form of a probability distribution for two-dimensional lattice gauge theory, when evaluated on long-wavelength configurations, have the optimal agreement with the data., Comment: 19 pages, 11 figures

I briefly review results obtained within the variational Hamiltonian approach to Yang-Mills theory in Coulomb gauge and confront them with recent lattice data. The variational approach is extended to non-Gaussian wave functionals including three- and four-gluon kernels in the exponential of the vacuum wave functional and used to calculate the three-gluon vertex. A new functional renormalization group flow equation for Hamiltonian Yang--Mills theory in Coulomb gauge is solved for the gluon and ghost propagator under the assumption of ghost dominance. The results are compared to those obtained in the variational approach., Comment: 6 pages, 9 figures. Invited talk given by H. Reinhardt at "T(r)opical QCD 2010", September 26--October 1, 2010, Cairns, Australia

The frustrated Ising model on a two-dimensional lattice with open boundary conditions is revisited. A hidden Z2 gauge symmetry relates models with different frustrations which, however, share the same partition function. By means of a duality transformation, it is shown that the partition function only depends on the distribution of gauge invariant vortices on the lattice. We finally show that the exact ground state energy can be calculated in polynomial time using Edmonds' algorithm., Comment: 4 pages, 2 figures

The participants in this discussion session of the QCHS 9 meeting were each asked the following question: "What would be the most useful piece of information that you could obtain, by whatever means, that would advance your own program, and/or our general understanding of confinement?" This proceedings contains a brief summary of each panel member's contribution to the discussion, provided by the panel members themselves., Comment: 8 pages. Roundtable discussion at the Quark Confinement and Hadron Spectrum 9 conference (Madrid, Aug. 29 - Sep. 3, 2010). To appear in the conference proceedings, published by the AIP. v2: References added. v3: Reference corrected

We present results for a numerical study of the ghost propagator in Coulomb gauge whereby lattice results for the spatial gluon propagator are used as input to solving the ghost Dyson-Schwinger equation. We show that in order to solve completely, the ghost equation must be supplemented by a boundary condition (the value of the inverse ghost propagator dressing function at zero momentum) which determines if the solution is critical (zero value for the boundary condition) or subcritical (finite value). The various solutions exhibit a characteristic behavior where all curves follow the same (critical) solution when going from high to low momenta until `forced' to freeze out in the infrared to the value of the boundary condition. The boundary condition can be interpreted in terms of the Gribov gauge-fixing ambiguity; we also demonstrate that this is not connected to the renormalization. Further, the connection to the temporal gluon propagator and the infrared slavery picture of confinement is discussed., Comment: 3 pages, 2 figures, talk presented at "Quark Confinement and the Hadron Spectrum IX", Madrid, August 30-September 3, 2010, to appear in the proceedings

We discuss the gluon propagator in 3- and 4-dimensional Yang-Mills theories in Coulomb gauge and compare it with the corresponding Landau gauge propagator, showing that for both the relevant IR mass scale coincides. We also report preliminary results on Coulomb gauge ghost form factor and quark propagators and give a comment on the gluon propagator's strong coupling limit., Comment: 7 pages, 6 Figure. Talk given at Lattice 2010. Typos corrected

In the heavy quark limit of Coulomb gauge QCD and by truncating the Yang-Mills sector to include only dressed two-point functions, an analytic nonperturbative solution to the Faddeev equation for three-quark bound states in the case of equal quark separations is presented. A direct connection between the temporal gluon propagator and the three-quark confinement potential is provided and it is shown that only color singlet qqq (baryon) states are physically allowed., Comment: 9 pages, 1 figure

The relationship between the nonperturbative Green's functions of Yang-Mills theory and the confinement potential is investigated. By rewriting the generating functional of quantum chromodynamics in terms of a heavy quark mass expansion in Coulomb gauge, restricting to leading order in this expansion and considering only the two-point functions of the Yang-Mills sector, the rainbow-ladder approximation to the gap and Bethe-Salpeter equations is shown to be exact in this case and an analytic, nonperturbative solution is presented. It is found that there is a direct connection between the string tension and the temporal gluon propagator. Further, it is shown that for the 4-point quark correlation functions, only confined bound states of color-singlet quark-antiquark (meson) and quark-quark (baryon) pairs exist., Comment: 22 pages, 6 figures

The connection of Gribov's confinement scenario in Coulomb gauge with the center vortex picture of confinement is investigated. For this purpose we assume a vacuum wave functional which models the infrared properties of the theory and in particular shows strict confinement, i.e. an area law of the Wilson loop. We isolate the center vortex content of this wave functional by standard lattice methods and investigate their contributions to various static propagators of the Hamilton approach to Yang-Mills theory in Coulomb gauge. We find that the infrared properties of these quantities, in particular the infrared divergence of the ghost form factor, are dominated by center vortices., Comment: 18 pages, 5 figures

We show that in SU(2) Yang-Mills theories a simple relation exists between lattice gluon propagators in Coulomb and Landau gauge and discuss the physical implications of such result. In particular, the realization of the Gribov-Zwanziger confinement mechanism in Coulomb gauge, linked to dual-superconductivity, would imply that the standard BRST charge must be ill defined non perturbatively. As a consequence, the Kugo-Ojima confinement criterion, which relies on BRST charge conservation beyond perturbation theory, would not be fulfilled., Comment: 18 pages, 5 figures. Some sections have been extended and references have been added. To appear in PRD.

I review results recently obtained within the Hamiltonian approach to Yang-Mills theory in Coulomb gauge. In particular, I will present results for the ghost and gluon propagators and compare these with recent lattice data. Furthermore, I will give an interpretation of the inverse of the ghost form factor as the dielectric function of the Yang-Mills vacuum. Our ansatz for the vacuum wave functional will be checked by means of functional renormalization group flow equations, which are solved for the gluon energy and the ghost form factor. Finally, we calculate the Wilson loop for the vacuum wave functional obtained from the variational approach, using a Dyson equation., Comment: 12 pages, 20 figures. Invited talk given by H. Reinhardt at the Workshop on QCD Green's Functions, Confinement and Phenomenology, September 2009, Trento

We apply a functional perturbative approach to the calculation of the equal-time two-point correlation functions and the potential between static color charges to one-loop order in Coulomb gauge Yang-Mills theory. The functional approach proceeds through a solution of the Schroedinger equation for the vacuum wave functional to order g^2 and derives the equal-time correlation functions from a functional integral representation via new diagrammatic rules. We show that the results coincide with those obtained from the usual Lagrangian functional integral approach, extract the beta function, and determine the anomalous dimensions of the equal-time gluon and ghost two-point functions and the static potential under the assumption of multiplicative renormalizability to all orders., Comment: 33 pages, 7 figures with Feyman diagrams generated with pstricks; revised version with additional references and comments on possible applications added in the conclusions; accepted for publication in Nucl. Phys. B

The Dyson equation proposed for planar temporal Wilson loops in the context of supersymmetric gauge theories is critically analysed thereby exhibiting its ingredients and approximations involved. We reveal its limitations and identify its range of applicability in non-supersymmetric gauge theories. In particular, we show that this equation is applicable only to strongly asymmetric planar Wilson loops (consisting of a long and a short pair of loop segments) and as a consequence the Wilsonian potential can be extracted only up to intermediate distances. By this equation the Wilson loop is exclusively determined by the gluon propagator. We solve the Dyson equation in Coulomb gauge for the temporal Wilson loop with the instantaneous part of the gluon propagator and for the spatial Wilson loop with the static gluon propagator obtained in the Hamiltonian approach to continuum Yang-Mills theory and on the lattice. In both cases we find a linearly rising color potential., Comment: 12 pages, 7 figures