  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.
