  Usual treatment of the Nambu--Jona-Lasinio (NJL) model using loop momentum
cutoff suffers from ambiguities in choosing the loop momenta to be cut off and
violation of (external) gauge invariance. We define the NJL model from the
starting Lagrangian level by using a higher derivative fermion kinetic term
with a cutoff parameter $\Lambda$. This definition is free from such
ambiguities and manifestly keeps the chiral symmetry as well as the gauge
invariance. Quantization of this higher derivative system, current operators
and calculational method are discussed in some detail. Calculating the pion
decay constant and $\pi^0\rightarrow 2\gamma$ decay amplitude, we explicitly
demonstrate that the low energy theorem holds. It is observed that the NJL mass
relation $m_\sigma = 2 m_0$ between the $\sigma$ meson and fermion masses no
longer holds in this model. We also present a simplified calculational method
which is valid when the cutoff parameter $\Lambda $ is much larger than the
energy scale of physics.
