and (\[oep2\]) will have a small effect on the physics. The philosophical limitation is that the vertex functions are supposed to only contain effect of a range shorter than $\Lambda_s^{-1}$. Longer range effects are to be included by explicit dynamics of the lighter degrees of freedom in the problem.

Thus, the issue of whether the two models are equivalent comes down to whether both $g_{\rm v,t}(q^2)$ and $\hat{g}_{\rm v,t}(q^2)$ can be analytic for $q^2<\Lambda_s^2$ while satisfying eq. (\[condition\]). In effect, the question is whether $$f(q^2) =  \left (\frac{q^2 - m_\omega^2}{q^2 - m_\omega^2 +
\pi_{\omega}(q^2)} \right )^{1/2}$$ is analytic for $q^2 < \Lambda_s^2$. Non-analyticity can occur when either the numerator or denominator vanishes or when $\pi_{\omega}$ is non-analytic. In fact, we should relax this restriction slightly—the non-analyticity associated with the $\omega$ coupling to three low energy pions which slightly broadens the pole and gives a small non-resonant contribution is, as discussed above, innocuous. In any event, this issue does not arise in the context of the models in refs. [@GHT92]-[@MTRC94].

Clearly, the analytic structure of $f(q^2)$ depends in detail on the choice of model. The simplest way to make the physics explicit is to make a spectral representation[@spectralref] for the propagator: 