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Cosmology and Quantum Field Theory II: Study of an extended Nambu-Jona-Lasinio model with a Dynamical Coupling

We study the cosmological implications of the Nambu-Jona-Lasinio (NJL model) when the coupling constant is field dependent. The NJL model has a four-fermion interaction describing two different phases due to quantum interaction effects and determined by the strength of the coupling constant g. It describes massless fermions for weak coupling and a massive fermions and strong coupling, where a fermion condensate is formed. In the original NJL model the coupling constant g is indeed constant, and in this work we consider a modified version of the NJL model by introducing a dynamical field dependent coupling motivated by string theory. The effective potential as a function of the varying coupling (aimed to implement a natural phase transition) is seen to develop a negative divergence, i.e. becomes a "bottomless well" in certain limit region. Although we explain how an lower unbounded potential is not necessarily unacceptable in a cosmological context, the divergence can be removed if we consider a mass term for the coupling-like field. We found that for a proper set of parameters, the total potential obtained has two minima, one located at the origin (the trivial solution, in which the fluid associated with the fields behave like matter); and the other related to the non-trivial solution. This last solution has three possibilities: 1) if the minimum is positive V_{min}>0, the system behave as a cosmological constant, thus leading eventually to an accelerated universe; 2) if the minimized potential vanishes V_{min}=0, then we have matter with no acceleration ; 3) finally a negative minimum V_{min}<0 leads an eventually collapsing universe, even though we have a flat geometry.Therefore, a possible interpretation as Dark Matter or Dark Energy is allowed among the behaviors implicated in the model.