Neutral vector boson from renormalizing IVB (intermediate vector boson) theory?

Question

Consider the intermediate vector boson (IVB) theory, with the interaction Lagrangian density $$\mathcal L_I = \sum_l g_W (\overline{\psi}_{\nu_l} \gamma^\alpha(1- \gamma_5)\psi_l ) W_\alpha^\dagger + \text{h.c.} \tag{1}$$ where $W_\alpha$ is a complex massive vector field, the $\psi_l$ are leptons and $\psi_{\nu_l}$ corresponding neutrinos. There are two "problems" with this theory. First, the lowest order diagrams contributing to $\nu_\mu e^-$ scattering are so 4:th order in the coupling constant $g_W$. (This figure and the one below reproduced from [M&S].) However, the observed cross sections are comparable to $\nu_e e^-$-scattering cross sections, for which the leading contributions are clearly second order in $g_W$. Thus one would be lead to an interaction with a neutral vector boson giving second order diagrams like the one below.

Second, the diagrams in Figure 16.11 are divergent; and since $W_\alpha$ is massive, the interaction (1) is not renormalizable.

Now, I think these are related. If we take the Wilsonian renormalization group view, then under RG flow all couplings are renormalized. If one is initially zero in the Lagrangian, RG flow will give it a non-zero value; hence the maxim that all interactions consistent with symmetry principles should be included in the Lagrangian.

One such interaction for the field content in IVB is $$\mathcal L_Z = g_Z \sum_l \overline{\psi}_{l}\gamma^\alpha(1-\gamma_5)\psi_l) \partial_\alpha (W_\beta^\dagger W^\beta) \; + \; (l \leftrightarrow \nu_l).$$ If we now put $Z^0 = \partial_\alpha (W_\beta W^\beta)$, this interaction gives the second diagram. This would be analogous to the pion-mediated scattering in effective theories from QCD, the pion operators being $\sim \overline{q}q$. However, I am not sure I have internalized the RG flow view completely or correctly, so my question is if this line of thinking is sound and one could produce an effective field theory from (1), including diagrams like that of the second figure. Perhaps such a calculation has already been done?

Reference

• [M&S] Franz Mandl and Graham Shaw. *Quantum Field Theory*. Second edition. John Wiley and Sons, 2010.