$$\frac{1}{q^2 - m_\omega^2 + \pi_{\omega} (q^2)} = \int \, {\rm
d}s \,
\frac{\rho(s) }{q^2 - s + i \epsilon} + {\rm subtraction \quad
terms}.
 \label{prop}$$ Different models will give rise to different spectral functions. However, if the model is realistic, the only substantial spectral strength for $q^2< \Lambda_s^2$ occurs at or near the omega pole. Accordingly any model which gives significant amounts of spectral strength below $\Lambda_s^2$ (apart from the $\omega$ pole), can be considered as unrealistic in our philosophy. If, however, all of the spectral strength is either at the $\omega$ pole or above $\Lambda_s^2$, then $f(q^2)$ is analytic for $q^2$ below $\Lambda_s^2$. The apparent non-analyticity due to the denominator vanishing at $q^2=
m_\omega^2$ is precisely canceled by a vanishing numerator. (Recall, all renormalization effects are included in $\pi_\omega$ so that the position of the $\omega$ pole does not shift.) In this case, one may re-define the vertex functions according to eq. (\[condition\]).

To see how the spectral representation constrains the allowable forms of the self energies consider consider the following simple example in which the self energy has the form

$$\pi_{\omega} (q^2)=(q^2-m_\omega^2)^2 Bq^2. \label{exam}$$

This form is motivated by the renormalization requirements, that $\pi_\omega(q^2)$ and its derivative vanish 