region. In principle, those models could be related to QCD by integrating out gluonic fields and reparameterizing fermionic degrees of freedom (the bosonization procedure). In effect, the physically observed particles should appear as fundamental fields of the model.

One of the first models of this type that has become very popular was the Skyrme model \[\]. It is an example of a fully bosonized effective model. During many years of intensive investigations the Skyrme model has proven to be theoretically quite interesting, though comparison with experiment has not always been good.

The simplest purely fermionic Lorentz invariant model with spontaneous chiral symmetry breaking is the Nambu–Jona-Lasinio (NJL) model \[\] that has followed the example of the BCS model of superconductivity. It contains chirally invariant local four-fermion interaction terms and it is non-renormalizable in 4-dimensional space-time. The NJL Lagrangian in its simplest $SU(2)$ form has the following structure:

$${\cal L}_{NJL} = \bar q \, (i \slash\partial - m_0)\, q +
{1\over 2} G \,
\,[\,(\bar q q )^2 + (\bar q i \vec\tau \gamma_{5} q )^2\,                ]
\ ,  \eqno(\NJLLAG)$$

where $G$ is the coupling constant, $m_0$ is the current quark mass and $\vec\tau$ are the Pauli matrices in the isospin 