Defining the Nambu--Jona-Lasinio Model by Higher Derivative Kinetic Term

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.