Skaler Gürsey modelinden esinlenmiş modeller.

To write a field theoretical model which has nonzero values for the coupling constants at zeroes of the beta function of the renormalization group is an endeavor which is still continuing in particle physics. The 4  theory is a "laboratory" where different methods in quantum field theory are first applied. The perturbatively nontrivial 4  in four dimensions was shown to go to a free theory as the cut-off is lifted. During the last twenty years, many papers were written on making sense out of "trivial models", interpreting them as effective theories without taking the cutoff to infinity. One of these models is the Nambu Jona-Lasinio model, hereafter NJL. Although this model is shown to be a trivial one in four dimensions, since the coupling constant goes to zero with a negative power of the logarithm of the ultraviolet cut-off, as an effective model in low energies it gives us important insight to several processes. There were also attempts, by Bardeen et al., to couple the NJL model to a gauge field, the so called gauged NJL model, to be able to get a non-trivial field theory. It was shown that if one has sufficient number of fermion flavors, such a construction is indeed possible. There are other models, made out of only spinors, which were constructed as alternatives of the original Heisenberg model, the first model given as "a theory of everything", using only spinors. The Gürsey model was proposed, before the NJL model, as a substitute for the Heisenberg model. The Classically the Gürsey model had the conformal symmetry. It had classical solutions, given by Kortel, which were interpreted as instantons and merons by Akdeniz, much like the solutions of the Yang-Mills (YM) theories. It had one important defect, though. Its non-polynomial Lagrangian made the use of standard methods in its quantization not feasible. Akdeniz et al. tried to make quantum sense of this model a while ago. They defined an equivalent Lagrangian, inspired by Gross-Nevue, which is polynomial. They quantized the equivalent model by this way the Gürsey model. They concluded that the model was resulted as a "trivial model". In other words the processes involving the constituent spinors resulted freely. Here we want to give a new interpretation of that work. We go to higher orders in our calculation in the new version, beyond the one loop for the scattering processes. It is shown that by using the Dyson- Schwinger and Bethe-Salpeter equations some of the fundamental processes can be better understood. We see that while the non-trivial scattering of the fundamental fields is not allowed, bound states can scatter from each other with non-trivial amplitudes. This phenomena can be understood as an example of treating the bound states, instead of the principal fields, as physical entities, that go through physical processes such as scattering. In our model we need an infinite renormalization in one of the diagrams. Further renormalization is necessary at each higher loop, like any other renormalizable model. The difference between our model and other renormalizable models lies in the fact that, although our model is a renormalizable one using naive dimensional counting arguments, we have only one set of diagrams which is divergent. We need to renormalize only one of the coupling constants by an infinite amount. This set of diagrams, corresponding to the scattering of two bound states to two bound states, have the same type of divergence in the dimensional regularization scheme for all odd number of loops. The contributions from even number of diagrams are finite, hence require no infinite renormalization. Using a new interpretation of the model and taking hints from the work of Bardeen et al., we studied a model, which classically simulates the Gürsey model, by coupling constituent U(1) gauge field to the spinors. We investigated whether this new coupling makes this new model a truly interacting one. We found that we are mimicking a gauge Higgs Yukawa (gHY) system, which had the known problems of the Landau pole, with all of its connotations of triviality. Then we studied our original model, coupled to a SU(N) gauge field, instead. We derived the renormalization group (RG) equations in one loop, and tried to derive the criteria for obtaining nontrivial fixed points for the coupling constants. Finally we showed that the renormalization group equations give indications of a nontrivial field theory when it is gauged with a SU(N) field. [ABSTRACT FROM AUTHOR]