A study on four-fermionic tensorial interactions

Abstract

The study of four-fermionic interactions is a eld in particle physics which drew a lot of attention of the physicists for the past few decades. But due to the limitations of these theories, the models that are based on four-fermionic theories are generally called toy models. The existing toy models in late twentieth century like Nambu- Jona-Lasinio model and the famous Gross-Neveu model gave some start up to these kind of interactions. Although there were many set backs of the 1st model, but the 2nd one was well founded and gave nice results and conclusions. However the Gross- Neveu model was shown to be integrable only in two dimensions and the theory was essentially massless, it gave some enthusiasm to the researchers to formulate similar theories. After the success of Gross-Neveu, people started looking at it seriously. In literature, it is already seen that, theories involving four-fermion interactions give condensations of the composite elds and this type of eld theories or models serve the purpose of explaining color superconductivity in quantum chromodynamics, phase condensations, formation of cooper pairs and chiral condensations. So, by an intelligent guess, such models can give such interesting results.In our work, the construction of a four-fermion quadratic interacting eld is done by promoting the Gross-Neveu model into an interacting tensor eld. The chiral symmetry of the theory is veri ed. The model is rewritten in terms of auxiliary elds as it simpli es the calculations. The possible condensate is guessed and an auxiliary eld proportional to it is introduced. In order to study it, the guessing of auxiliary eldis done. Both symmetric and anti-symmetric forms of the auxiliary eld are tested for the validity under condensation conditions. The symmetric form of the auxiliary eld gave known result of scalar condensation of the eld which is like Gross-Neveu condensate in higher dimensions. The anti-symmetric form is also tested. To reduce the number of free parameters in the chosen form of the auxiliary eld, self-dual condition is employed. This is done to make the calculations simpler and to formulate a simpler model. From anti-symmetric case, under the self-duality condition, it is shown that if the condensate occurs then it will be imaginary. To check the condensation formation, generally search of other vacua states is performed and study of their stability is required.