\,
+  \, g_\omega^{ \rm t}(q^2)   q^\alpha \sigma^{(1)}_{\alpha \mu}
 )
(g^{\mu \nu} - q^{\mu}q^{\nu}/m_{\omega}^2) (g_{\omega}^{\rm
v}(q^2)
\gamma^{(2)}_{\nu} + g_{\omega}^{\rm t}(q^2) \sigma_{\nu \beta}
q^\beta
)  }{q^2 - m_\omega^2}
 \label{oep}$$ where $g_{\omega}^{\rm v} (q^2)$ and $g_{\omega}^{\rm t}(q^2)$ are the vector and tensor couplings of the omega to the nucleons. The superscripts 1 and 2 label the nucleon. These couplings are analytic functions of $q^2$ for $q^2 < \Lambda_s^2$; the propagator is clearly analytic for $q^2 > \Lambda_s^2$.

In principle, we could consider a more sophisticated model consistent with the philosophy outlined above. For example, one could explicitly include the exchange of three low-energy pions (with the quantum numbers of the rho) along with an omega self-energy due to its coupling with the three pion channel and a longer-range part of the $\omega$-N vertex due to three pion exchange. In practice, one expects such effects to be small: in part they serve to simply widen the omega pole by an amount of no practical significance to the potential; other effects of coupling to the three pion channel are small because they are weakly coupled. In any event, we will stick to the conventional assumptions underlying meson exchange models and neglect such effects. In 