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Register a new userDyson-Schwinger equation constraints on the gluon propagator in BRST quantised QCD
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Authors
Peter Lowdon
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The gluon propagator plays a central role in determining the dynamics of QCD. In this work we demonstrate for BRST quantised QCD that the Dyson-Schwinger equation imposes significant analytic constraints on the structure of this propagator. In particular, we find that these constraints control the appearance of massless components in the gluon spectral density.
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We suggest a framework based on the rainbow approximation with effective parameters adjusted to lattice data. The analytic structure of the gluon and ghost propagators of QCD in Landau gauge is analyzed by means of numerical solutions of the coupled system of truncated Dyson-Schwinger equations. We find that the gluon and ghost dressing functions are singular in complex Euclidean space with singularities as isolated pairwise conjugated poles. These poles hamper solving numerically the Bethe-Salpeter equation for glueballs as bound states of two interacting dressed gluons. Nevertheless, we argue that, by knowing the position of the poles and their residues, a reliable algorithm for numerical solving the Bethe-Salpeter equation can be established.
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Douglas W. McKay
, Herman J. Munczek (Department of Physics andn Astronomy
, The University of Kansas
1996
We solve the Dyson--Schwinger equation for the quark propagator in a model with singular infrared behavior for the gluon propagator. We require that the solutions, easily found in configuration space, be tempered distributions and thus have Fourier transforms. This severely limits the boundary conditions that the solutions may satisify. The sign of the dimensionful parameter that characterizes the model gluon propagator can be either positive or negative. If the sign is negative, we find a unique solution. It is singular at the origin in momentum space, falls off like $1/p^2$ as $p^2rightarrow +/-infty$, and it is truly nonperturbative in that it is singular in the limit that the gluon--quark interaction approaches zero. If the sign of the gluon propagator coefficient is positive, we find solutions that are, in a sense that we exhibit, unconstrained linear combinations of advanced and retarded propagators. These solutions are singular at the origin in momentum space, fall off like $1/p^2$ asympotically, exhibit ``resonant--like behavior at the position of the bare mass of the quark when the mass is large compared to the dimensionful interaction parameter in the gluon propagator model, and smoothly approach a linear combination of free--quark, advanced and retarded two--point functions in the limit that the interaction approaches zero. In this sense, these solutions behave in an increasingly ``particle--like manner as the quark becomes heavy. The Feynman propagator and the Wightman function are not tempered distributions and therefore are not acceptable solutions to the Schwinger--Dyson equation in our model. On this basis we advance several arguments to show that the Fourier--transformable solutions we find are consistent with quark confinement, even though they have singularities on the
With the introduction of a spectral representation, the Schwinger-Dyson equation (SDE) for the fermion propagator is formulated in Minkowski space in QED. After imposing the on-shell renormalization conditions, analytic solutions for the fermion propagator spectral functions are obtained in four dimensions with a renormalizable version of the Gauge Technique anzatz for the fermion-photon vertex in the quenched approximation in the Landau gauge. Despite the limitations of this model, having an explicit solution provides a guiding example of the fermion propagator with the correct analytic structure. The Pad{e} approximation for the spectral functions is also investigated.
We calculate the variation of the chiral condensate in medium with respect to the quark chemical potential and evaluate the pion-nucleon sigma term via the Hellmann-Feynman theorem. The variation of chiral condensate in medium are obtained by solving the truncated Dyson-Schwinger equation for quark propagator at finite chemical potential, with different models for the quark-gluon vertex and gluon propagator. We obtain the value of the sigma term $sigma_{pi N}$ = 62(1)(2) MeV, where the first represents the systematic error due to our different model for the quark-gluon vertex and gluon propagator and the second represents a statistical error in our linear fitting procedure.
Any practical application of the Schwinger-Dyson equations to the study of $n$-point Greens functions of a field theory requires truncations, the best known being finite order perturbation theory. Strong coupling studies require a different approach. In the case of QED, gauge covariance is a powerful constraint. By using a spectral representation for the massive fermion propagator in QED, we are able to show that the constraints imposed by the Landau-Khalatnikov-Fradkin transformations are linear operations on the spectral densities. Here we formally define these group operations and show with a couple of examples how in practice they provide a straightforward way to test the gauge covariance of any viable truncation of the Schwinger-Dyson equation for the fermion 2-point function.
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