# When can you simplify the $W$ boson propagator?

I have seen in several sources that the propagator of the $$W$$ boson is: $$\frac{- i \left( g^{\mu\nu} - \frac{P^\mu P^\nu}{m_W^2} \right)}{p^2 - m_W^2} .$$ But then in some calculations (usually approximations) I have seen that only $$\frac{-i}{p^2 - m_W^2} .$$ is used as the propagator. So my question is why is this allowed and when is it allowed to do this?

• Acting on a 4-vector orthogonal to p, it reproduces it times the scalar you write. Acting on a vector parallel to p it reproduces it times something else (what?). – Cosmas Zachos Feb 7 at 1:46

You have to be careful, there are many subtleties on the quantization of a spontaneously broken gauge theory like the Standard Model (SM) electroweak sector. I recommend you to learn about the quantization of a gauge theory via path integral methods. To quantize a gauge theory, you need to introduce a gauge fixing term (and the associated ghost Lagrangian). For the SM, people usually consider a generic $$R_\xi$$ guage fixing term. You can find Feynman rules for the SM in a generic $$R_\xi$$ gauge fixing in this article
In particular look into Eq. 52 for the W-boson propagator. $$\xi_W$$ in this Eq. is a free real parameter that comes from the gauge fixing. When you do calculations, you are free to specify the gauge fixing terms, for instance $$\xi_W=1$$ corresponds to Feynman Gauge while $$\xi_W=0$$ to the Landau gauge. Note that in the Feynman gauge you get rid of the $$k_\mu k_\nu$$ part.