Yang-Mills Theory In Coulomb Gauge Research Articles

Dyson-Schwinger approach to Hamiltonian quantum chromodynamics

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 terms 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.

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.

Deconfinement phase transition in the Hamiltonian approach to Yang-Mills theory in Coulomb gauge

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.

The ghost–gluon vertex in Hamiltonian Yang–Mills theory in Coulomb gauge

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.

Hamiltonian flow in Coulomb gauge Yang-Mills theory

We derive a new functional renormalization group equation for Hamiltonian Yang-Mills theory in Coulomb gauge. The flow equations for the static gluon and ghost propagators are solved under the assumption of ghost dominance within different diagrammatic approximations. The results are compared to those obtained in the variational approach and the reliability of the approximations is discussed.

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 Schrödinger 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.

On the connection between Hamilton and Lagrange formalism in quantum field theory

The connection between the Hamilton and the standard Lagrange formalism is established for a generic quantum field theory with vanishing vacuum expectation values of the fundamental fields. The effective actions in both formalisms are the same if and only if the fundamental fields and the momentum fields are related by the stationarity condition. These momentum fields in general differ from the canonical fields as defined via the effective action. By means of functional methods a systematic procedure is presented to identify the full correlation functions, which depend on the momentum fields, as functionals of those usually appearing in the standard Lagrange formalism. Whereas Lagrange correlation functions can be decomposed into tree diagrams, the decomposition of Hamilton correlation functions involves loop corrections similar to those arising in n-particle effective actions. To demonstrate the method we derive for theories with linearized interactions the propagators of composite auxiliary fields and the ones of the fundamental degrees of freedom. The formalism is then utilized in the case of Coulomb gauge Yang–Mills theory for which the relations between the two-point correlation functions of the transversal and longitudinal components of the conjugate momentum to the ones of the gauge field are given.

Boost operators in Coulomb gauge QCD: The pion form factor and Fock expansions in $ \phi$ radiative decays

In this article we re-derive the Boost operators in Coulomb Gauge Yang-Mills theory employing the path integral formalism and write down the complete operators for QCD. We immediately apply them to note that what are usually called the pion square, quartic ... charge radii, defined from derivatives of the pion form factor at zero squared momentum transfer, are completely blurred out by relativistic and interaction corrections, so that it is not clear at all how to interpret these quantities in terms of the pion charge distribution. The form factor therefore measures matrix elements of powers of the QCD boost and Mœller operators, weighted by the charge density in the target’s rest frame. In addition we remark that the decomposition of the \( \eta{^\prime}\) wave function in quarkonium, gluonium, ..., components attempted by the KLOE Collaboration combining data from \( \phi\) radiative decays, requires corrections due to the velocity of the final-state meson recoiling against a photon. This will be especially important if such decompositions are to be attempted with data from J/\( \psi\) decays.

Slavnov–Taylor identities in Coulomb gauge Yang–Mills theory

The Slavnov–Taylor identities of Coulomb gauge Yang–Mills theory are derived from the (standard, second order) functional formalism. It is shown how these identities form closed sets from which one can in principle fully determine the Green’s functions involving the temporal component of the gauge field without approximation, given appropriate input.

Truncating First-Order Dyson–Schwinger Equations in Coulomb Gauge Yang–Mills Theory

The non-perturbative domain of QCD contains confinement, chiral symmetry breaking, and the bound state spectrum. For the calculation of the latter, the Coulomb gauge is particularly well-suited. Access to these non-perturbative properties should be possible by means of the Green's functions. However, Coulomb gauge is also very involved, and thus hard to tackle. We introduce a novel BRST-type operator r, and show that the left-hand side of Gauss' law is r-exact. We investigate a possible truncation scheme of the Dyson-Schwinger equations in first-order formalism for the propagators based on an instantaneous approximation. We demonstrate that this is insufficient to obtain solutions with the expected property of a linear-rising Coulomb potential. We also show systematically that a class of possible vertex dressings does not change this result.

Perturbation theory in the Hamiltonian approach to Yang-Mills theory in Coulomb gauge

We study the Hamiltonian approach to Yang-Mills theory in Coulomb gauge in Rayleigh-Schroedinger perturbation theory. The static gluon and ghost propagator as well as the potential between static colour sources are calculated to one-loop order. Furthermore, the one-loop $\beta$-function is calculated from both the ghost-gluon vertex and the static potential and found to agree with the result of covariant perturbation theory.

Topological susceptibility in SU(2) Yang-Mills theory in the Hamiltonian approach in Coulomb gauge

The topological susceptibility is calculated within the Hamiltonian approach to Yang-Mills theory in Coulomb gauge, using the vacuum wave functional previously determined by a variational solution of the Yang-Mills Schroedinger equation. The numerical result agrees qualitatively with the predictions of lattice simulations.

Dielectric Function of the QCD Vacuum

It is shown that the inverse of the ghost form factor in the Hamilton approach to Yang-Mills theory in Coulomb gauge can be interpreted as the color dielectric function of the QCD vacuum. Furthermore, the horizon condition to the ghost form factor implies that in the infrared the QCD vacuum is a perfect color diaelectric medium and therefore a dual superconductor. The dielectric function is explicitly calculated within a previously developed variational approach, using a specific ansatz for the vacuum wave functional.

Yang-Mills vacuum in Coulomb gauge inD=2+1dimensions

The variational approach to the Hamilton formulation of Yang-Mills theory in Coulomb gauge developed by the present authors previously is applied to Yang-Mills theory in $2+1$ dimensions and is confronted with the existing lattice data. We show that the resulting Dyson-Schwinger equations (DSE) yield consistent solutions in $2+1$ dimensions only for infrared divergent ghost form factor and gluon energy. The obtained numerical solutions of the DSE reproduce the analytic infrared results and are in satisfactory agreement with the existing lattice date in the whole momentum range.

Subcritical solution of the Yang-Mills Schrödinger equation in the Coulomb gauge

In the Hamiltonian approach to Coulomb gauge Yang-Mills theory, the functional Schroedinger equation is solved variationally resulting in a set of coupled Dyson-Schwinger equations. These equations are solved self-consistently in the subcritical regime defined by infrared-finite form factors. It is shown that the Dyson-Schwinger equation for the Coulomb form factor fails to have a solution in the critical regime where all form factors have infrared divergent power laws.

Two-point functions of Coulomb gauge Yang-Mills theory

The functional approach to Coulomb gauge Yang-Mills theory is considered within the standard, second order, formalism. The Dyson-Schwinger equations and Slavnov-Taylor identities concerning the two-point functions are derived explicitly and one-loop perturbative results are presented.

Perturbation theory of Coulomb gauge Yang-Mills theory within the first order formalism

Perturbative Coulomb gauge Yang-Mills theory within the first order formalism is considered. Using a differential equation technique and dimensional regularization, analytic results for both the ultraviolet divergent and finite parts of the two-point functions at one-loop order are derived. It is shown how the nonultraviolet divergent parts of the results are finite at spacelike momenta with kinematical singularities on the light-cone and subsequent branch cuts extending into the timelike region.

The ’t Hooft loop in the Hamiltonian approach to Yang-Mills theory in Coulomb gauge

The spatial 't Hooft loop, which is a disorder parameter dual to the temporal Wilson loop, is calculated using the nonperturbative Yang-Mills vacuum wave functional determined previously by a variational solution of the Yang-Mills Schr\"odinger equation in Coulomb gauge. It is shown that this wave functional yields indeed a perimeter law for large spatial 't Hooft loops signaling confinement.

Propagator Dyson-Schwinger equations of Coulomb gauge Yang-Mills theory within the first order formalism

Coulomb gauge Yang-Mills theory within the first order formalism is considered with a view of deriving the propagator Dyson-Schwinger equations. The first order formalism is studied with special emphasis on the Becchi-Rouet-Stora (BRS) invariance and it is found that there exists two forms of invariance---invariance under the standard BRS transform and under a second, nonstandard transform. The field equations of motion and symmetries are derived explicitly and certain exact relations that simplify the formalism are presented. It is shown that the Ward-Takahashi identity arising from invariance under the nonstandard part of the BRS transform is guaranteed by the functional equations of motion. The Feynman rules and the general decomposition of the two-point Green's functions are derived. The propagator Dyson-Schwinger equations are derived and certain aspects (energy independence of ghost Green's functions and the cancellation of energy divergences) are discussed.

Confining solution of the Dyson-Schwinger equations in Coulomb gauge

The Dyson-Schwinger equations arising from minimizing the vacuum energy density in the Hamiltonian approach to Yang-Mills theory in Coulomb gauge are solved numerically. A new solution is presented which gives rise to a strictly linearly rising static quark potential and whose existence was previously observed in the infrared analysis of the Dyson-Schwinger equations. For the new solution we also present the static quark potential and calculate the running coupling constant from the ghost-gluon vertex.