  A quadratic action in even Grassmann variables with the quantum numbers of
the pions has been studied. It includes the $\sigma$-field in order to be
invariant under SU(2)_L times SU(2)_R transformations over the quarks. This
action exhibits the Goldstone phenomenon reducing its symmetry to the O(3)
isospin invariance. The model has been investigated in the
Stratonovitch-Hubbard representation, in which form it is reminiscent of the
Gell-Mann-Levy model. By the saddle point method a renormalizable expansion in
inverse powers of the index of nilpotency of the mesonic fields (which is 24),
is generated. The way it might be used in a new perturbative approach to QCD is
outlined.
