Resolution of Gauss' law in the maximal Abelian gauge
We resolve Gauss' law in the maximal Abelian gauge supplemented by the Coulomb gauge for the Abelian gauge field and derive the gauge-fixed Hamiltonian of QCD.
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arXiv subjects
H. Reinhardt
Publications and source records attributed to H. Reinhardt.
The 't Hooft loop from a center-vortex wave functional
Previously, we proposed an infrared vacuum wave functional for $SU(N)$ Yang-Mills theory peaked at thin center vortices and showed that it yields an area law for the Wilson loop. In this work, we use this wave functional to calculate the spatial 't Hooft loop, for which we find a perimeter law, in accordance with 't Hooft's criterion for confinement.
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The infrared Yang-Mills wave functional due to percolating center vortices
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.
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Effective theory of the D = 3 center vortex ensemble
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.
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Hamiltonian approach to QCD in Coulomb gauge - a survey of recent results
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.
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Hamiltonian approach to QCD in Coulomb gauge at zero and finite temperature
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.
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Chiral and deconfinement phase transition in the Hamiltonian approach to QCD in Coulomb gauge
The chiral and deconfinement phase transitions are investigated within the variational Hamiltonian approach to QCD in Coulomb gauge. The temperature $β^{- 1}$ is introduced by compactifying a spatial dimension. Thereby the whole temperature dependence is encoded in the vacuum state on the spatial manifold $\mathbb{R}^2 \times S^1 (β)$. 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.
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Hamiltonian approach to QCD in Coulomb gauge: Gribov's confinement scenario at work
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.
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Improved variational approach to QCD in Coulomb gauge
The variational approach to QCD in Coulomb gauge developed previously by the Tübingen 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.
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Hamiltonian approach to QCD in Coulomb gauge: From the vacuum to finite temperatures
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.
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Coulomb vs. physical string tension on the lattice
From continuum studies it is known that the Coulomb string tension $σ_C$ gives an upper bound for the physical (Wilson) string tension $σ_W$ [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.
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Finite-temperature Yang-Mills theory in the Hamiltonian approach in Coulomb gauge from a compactified spatial dimension
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.
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Confinement in Coulomb gauge
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.
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Quark Sector of the QCD Groundstate in Coulomb Gauge
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.
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Lattice Coulomb propagators, effective energy and confinement
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 σ_C ~ 2 σ.
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Ghost propagator and the Coulomb form factor from the lattice
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.
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Running mass, effective energy and confinement: the lattice quark propagator in Coulomb gauge
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.
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Equal-time two-point correlation functions in Coulomb gauge Yang-Mills theory
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.
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