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Register a new userConfinement studies in QCD with Dyson-Schwinger equations
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Authors
Marco Frasca
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We provide a study of quantum chromodynamics with the technique of Dyson-Schwinger equations in differential form. In this way, we are able to approach the non-perturbative limit and recover, with some approximations, the t Hooft limit of the theory. Quark mass in the propagator term goes off-shell at low-energies signaling confinement. A condition for such occurrence in the theory is provided.
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In this talk, we review some of the current efforts to understand the phenomenon of chiral symmetry breaking and the generation of a dynamical quark mass. To do that, we will use the standard framework of the Schwinger-Dyson equations. The key ingredient in this analysis is the quark-gluon vertex, whose non-transverse part may be determined exactly from the nonlinear Slavnov-Taylor identity that it satisfies. The resulting expressions for the form factors of this vertex involve not only the quark propagator, but also the ghost dressing function and the quark-ghost kernel. Solving the coupled system of integral equations formed by the quark propagator and the four form factors of the scattering kernel, we carry out a detailed study of the impact of the quark gluon vertex on the gap equation and the quark masses generated from it, putting particular emphasis on the contributions directly related with the ghost sector of the theory, and especially the quark-ghost kernel. Particular attention is dedicated on the way that the correct renormalization group behavior of the dynamical quark mass is recovered, and in the extraction of the phenomenological parameters such as the pion decay constant.
Using a technique devised by Bender, Milton and Savage, we derive the Dyson-Schwinger equations for quantum chromodynamics in differential form. We stop our analysis to the two-point functions. The t~Hooft limit of color number going to infinity is derived showing how these equations can be cast into a treatable even if approximate form. It is seen how this limit gives a sound description of the low-energy behavior of quantum chromodynamics by discussing the dynamical breaking of chiral symmetry and confinement, providing a condition for the latter. This approach exploits a background field technique in quantum field theory.
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.
We present a calculation of the three-quark core contribution to nucleon and Delta-baryon masses and Delta electromagnetic form factors in a Poincare-covariant Faddeev approach. A consistent setup for the dressed-quark propagator, the quark-quark, quark-diquark and quark-photon interactions is employed, where all ingredients are solutions of their respective Dyson-Schwinger or Bethe-Salpeter equations in a rainbow-ladder truncation. The resulting Delta electromagnetic form factors concur with present experimental and lattice data.
${cal N}=4$ Super Yang-Mills theory is a highly constrained theory, and therefore a valuable tool to test the understanding of less constrained Yang-Mills theories. Our aim is to use it to test our understanding of both the Landau gauge beyond perturbation theory as well as truncations of Dyson-Schwinger equations in ordinary Yang-Mills theories. We derive the corresponding equations within the usual one-loop truncation for the propagators after imposing the Landau gauge. We find a conformal solution in this approximation, which surprisingly resembles many aspects of ordinary Yang-Mills theories. We furthermore identify which role the Gribov-Singer ambiguity in this context could play, should it exist in this theory.
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