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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?

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  • $\begingroup$ 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?). $\endgroup$ – Cosmas Zachos Feb 7 at 1:46
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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

https://arxiv.org/abs/1209.6213

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.

As long as you do your calculations consistently, you are assured that the amplitudes for physical processes will always be gauge independent, so you can choose the gauge you prefer.

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