at $q^2=0$. One determines the nucleon-nucleon potential generated by the propagator of Eq.(\[prop\]) by taking $q^2$ to be space-like $q^2=-Q^2<0$; the potential is proportional to the integral $$\int  {\rm d}Q Q{\sin (Qr)\over r}
\frac{1}{-Q^2 - m_\omega^2 -(Q^2+m_\omega^2)^2 BQ^2 }.$$ One does the contour integration by identifying the poles. There is always a pole at $Q^2=-m_\omega^2$ which is the standard term expected from the exchange of an $\omega^2$-meson. There are other poles at positions determined by the value of B. One finds that if $ m_\omega^4>  m_\omega^4+4/B\ge0$ there will be poles with $Q^2$ real and negative. At least one of the poles must be at $|Q^2|<m_\omega^2$, which is unrealistic in our philosophy. If $m_\omega^4+4/B >  m_\omega^4 $, the poles occur for $Q^2>0$ which are physically un-allowable tachyonic excitations. Similarly, if $m_\omega^4+4/B<0$, there are poles off the real axis, which violates the spectral representation and also renders the model for $ \pi_{\omega}
(q^2)$ as useless. This analysis demonstrates that a spectral function of the form in Eq.(\[exam\]) is not viable.

Let us now summarize the effects of the momentum dependence of the $\omega$ self energy on the meson exchange potential. In any realistic model, ([*i.e.*]{} any model without unphysical low $q^2$ spectral 