Consider the two Feynman diagrams below:
I know for the second that we should use the Breit-wigner curve, but what about the first? If we go about using the propagator: \[\frac{1}{\tilde q^2-m_z^2c^2}\text{, (+ - - -) signature}\] then we a singularity when $\tilde q^2=m_z^2c^2$ which is subverted when the Breit-Wigner curve is used. Thus my question is as follows: When is it appropriate to use the propagator as given above and when to use the Breit-Wigner curve.
Your definition of the propagator is only valid for stable particles. In the case of unstable particles, an extra term arises which contains the width of the unstable particle, e.g. for your Z-boson (in natural units):
$$ \frac{1}{\tilde{q}^{2}-m_{Z}^{2}} \rightarrow \frac{1}{\tilde{q}^{2}-m_{Z}^{2}+i m_{Z}\Gamma_{Z}} $$
That additional part in the propagator will lead to the typical Breit-Wigner shape in the invariant mass of the two final state particles in your case.
Details on the proper treatment of unstable particles (=resonances) in the propagators of quantum field theories can be found e.g. in this article.
