two equivalent ways of representing the same physics. One way for this to occur is if the vertex functions in the two models are related by $$g_\omega^{\rm v,t} \, = \, \hat{g}_\omega^{\rm v,t} \, \left
(\frac{q^2
- m_\omega^2}{q^2 - m_\omega^2 + \pi_{\omega}(q^2)} \right
)^{1/2}.
\label{condition}$$ Note that the square root factor is unity for $q^2=m_\rho^2$ due to the renormalization of $ \pi_{\omega}(q^2)$. The result (\[condition\]) is not surprising in light of the formal analysis of Sect.II. Neither the propagator off-shell nor the vertex function are separately meaningful.

Given that vertex functions are fit to some set of data, the only reason the condition in eq. (\[condition\]) would not be satisfied would be due to practical and philosophical limitations in the forms used in the fitting of the vertex functions. The practical limitation is that one must take some limited trial form for the phenomenological coupling. To the extent that meson exchange models make sense in the regime where they are used, the trial forms must be rich enough to describe the data with reasonable precision. Thus, apart from the philosophical concerns discussed below, eq. (\[condition\]) can be satisfied well enough so that any difference between the potentials of eqs. (\[oep\]) 