  We discuss two problems in complexified auxiliary fields in fermionic
effective models, the auxiliary sign problem associated with the repulsive
vector-field and the choice of the cut for the scalar field appearing from the
logarithmic function. In the fermionic effective models with attractive scalar
and repulsive vector-type interaction, the auxiliary scalar and vector fields
appear in the path integral after the bosonization of fermion ilinears. When we
make the path integral well-defined by the Wick rotation of the vector field,
the oscillating Boltzmann weight appears in the partition function. This
"auxiliary" sign problem can be solved by using the Lefschetz-thimble
path-integral method, where the integration path is constructed in the complex
plane. Another serious obstacle in the numerical construction of Lefschetz
thimbles is caused by singular points and cuts induced by multivalued functions
of the complexified scalar field in the momentum integration. We propose a new
prescription which fixes gradient flow trajectories on the same Riemann sheet
in the flow evolution by performing the momentum integration in the complex
domain.
