Higher order effective interactions and effective bosonized model for 2-N particle states

In this work, an effective fermion model with particular higher order interactions given by: $I_{II}={\sum }_n^N{g}_{2^n}({\bar{\psi }}_a{\psi }_a{)}^{2^n}$, for finite $N$, is investigated by means of the auxiliary field method by taking into account 2$n$-particle states as proposed in Ref. [1]. The bosonized model was found to exhibit an approximate symmetry when expanded for weak field fluctuations around the mean field solutions. In the present work, the role of the mean field solutions for the corresponding auxiliary fields is investigated. With the integration of fermions, the resulting determinant is expanded in a polynomial boson model in the weak field approximation and the approximate symmetry found for this series does not appear if the boson mean fields are set to zero.