Solution Of Schwinger-Dyson Equations Research Articles

Graph polynomials associated with Dyson-Schwinger equations

Quantum motions are encoded by a particular family of recursive Hochschild equations in the renormalization Hopf algebra which represent Dyson-Schwinger equations, combinatorially. Feynman graphons, which topologically complete the space of Feynman diagrams of a gauge field theory, are considered to formulate some random graph representations for solutions of quantum motions. This framework leads us to explain the structures of Tutte and Kirchhoff-Symanzik polynomials associated with solutions of Dyson-Schwinger equations. These new graph polynomials are applied to formulate a new parametric representation for large Feynman diagrams and their corresponding Feynman rules.

Schwinger-Dyson renormalization group

We use the Schwinger-Dyson equations as a starting point to derive renormalization flow equations. We show that Katanin's scheme arises as a simple truncation of these equations. We then give the full renormalization group equations up to third order in the irreducible vertex. Furthermore, we show that to the fifth order, there exists a functional of the self-energy and the irreducible four-point vertex whose saddle point is the solution of Schwinger-Dyson equations.

Schwinger-Dyson equations and disorder

Using simple models in D=0+0 and D=0+1 dimensions we construct partition functions and compute two-point correlations. The exact result is compared with saddle-point approximation and solutions of Schwinger-Dyson equations. When integrals are dominated by more than one saddle-point we find Schwinger-Dyson equations do not reproduce the correct results unless the action is first transformed into dual variables.

QCD tests with an infrared finite gluon propagator and coupling constant

Recent progress in the solution of Schwinger-Dyson equations, as well as lattice simulation of pure glue QCD, indicate that the gluon propagator and coupling constant are infrared finite. Such non-perturbative information can be introduced in the QCD perturbative expansion in the scheme named Dynamical Perturbation Theory. We exemplify this procedure with the calculation of some two-body non-leptonic annihilation B meson decays, which show agreement with the experimental data in the case of a gluon propagator characterized by a dynamical gluon mass of 500MeV, compatible with the value found in several processes computed with this method. We give a preliminary account of the application of this procedure at the loop level in the case of the Bjorken sum rule.

Confinement property in SU(3) gauge theory

We study the confinement property of the pure SU(3) gauge theory, combining in this effort the nonperturbative gluon and ghost propagators obtained as solutions of Dyson-Schwinger equations with solutions of an integral ladder diagram summation type equation for the Wilson loop. We obtain the string potential and effective UV coupling.

Vacuum condensates of QCD**Supported by National Natural Science Foundation of China (10647002,10565001) and Natural Science Foundation of Guangxi (0575020, 0542042, 0481030)

We study the properties of QCD vacuum state in this paper. The values of various local quark vacuum condensates, quark-gluon mixed vacuum condensates, and the structure of non-local quark vacuum condensate are predicted by the solution of Dyson-Schwinger Equations in ``rainbow'' approximation with three sets of different parameters for effective gluon propagator. The light quark virtuality is also obtained in a consistent way. Our all theoretical results here are in good agreement with the empirical values used widely in literature and many other theoretical calculations.

Dynamically Running Mass of Light Quark and QCD Vacuum Condensates**The project supported in part by National Natural Science Foundation of China under Grant No. 10247004, and the Natural Science Foundation of Guangxi Province of China under Grant No. 0339008

Based on Dyson–Schwinger equations (DSEs) in “rainbow” approximation, the dynamically running mass of light quark and QCD vacuum condensate are investigated. The structure of non-local quark vacuum condensate, the values of local vacuum condensate of quarks and quark-gluon mixture, and dynamical transition of quark mass from current quark to constituent quark are illustrated. At the same time, according to the knowledge and experience learned from an extensive study of the solutions of DSEs, a parameterized form of confining quark propagator is suggested for a practical use. The new parameterized form of quark propagator is analytic everywhere in the finite complex p2-plane and has no Lehmann representation. The predictions for p2-dependence of effective quark masses, defined by the self-energy functions and both from the numerical solutions of DSEs and from its parameterized form, are shown dynamically. Our conclusion is that all numerical results are consistent with empirical values used in QCD sum rules and lattice QCD calculations. For a qualitative study, the parameterized form is a sufficiently good approximation to confining quark propagator

Freezing of the QCD coupling constant and solutions of Schwinger-Dyson equations

We compare phenomenological values of the frozen QCD running coupling constant $({\ensuremath{\alpha}}_{s})$ with two classes of infrared finite solutions obtained through nonperturbative Schwinger-Dyson equations. We use these same solutions with frozen coupling constants as well as their respective nonperturbative gluon propagators to compute the QCD prediction for the asymptotic pion form factor. Agreement between theory and experiment on ${\ensuremath{\alpha}}_{s}(0)$ and ${F}_{\ensuremath{\pi}}{(Q}^{2})$ is found only for one of the Schwinger-Dyson equation solutions.

Finite nonperturbative solutions of Dyson-Schwinger equations in QED in the infrared domain.

Assuming an ansatz for the vertex function and the electron propagator suggested by the Ward identity we solve the Dyson-Schwinger equations in quantum electrodynamics in the infrared domain. The nonperturbative results obtained in this way are in agreement with perturbation theory. In our approach the loop integrals are finite. Our procedure is self-consistent, no divergences arise during calculations, and only a finite renormalization has to be carried out.