Vector Boson Propagator and gauge

[ChrisVer]

Well I'm trying to understand the difference between these propagators:

$\frac{g_{\mu \nu}}{k^{2} - m^2 + i \epsilon}$

and

$\frac{g_{\mu \nu}+ \frac{ k_{\mu} k_{\nu}}{m^{2}}}{k^{2} - m^2 + i \epsilon}$

My professor told me that they are different gauges, and the from the second you rule out the goldstone modes. Can someone explain it to me a little better or refer me to some textbook? I don't also get the meaning of goldstone modes- if you have a massive vector boson then you don't have goldstone modes [they become the longitudial dofs]

Also I am having one more question. If the first doesn't kill out the goldstone bosons, then can someone -after arriving at the end result- kill them? Maybe by a gauge transformation? and what gauge transformation?

 

[nrqed]

Well I'm trying to understand the difference between these propagators:

$\frac{g_{\mu \nu}}{k^{2} - m^2 + i \epsilon}$

and

$\frac{g_{\mu \nu}+ \frac{ k_{\mu} k_{\nu}}{m^{2}}}{k^{2} - m^2 + i \epsilon}$

My professor told me that they are different gauges, and the from the second you rule out the goldstone modes. Can someone explain it to me a little better or refer me to some textbook? I don't also get the meaning of goldstone modes- if you have a massive vector boson then you don't have goldstone modes [they become the longitudial dofs]

Also I am having one more question. If the first doesn't kill out the goldstone bosons, then can someone -after arriving at the end result- kill them? Maybe by a gauge transformation? and what gauge transformation?

I am not sure what your prof meant by "killing off the Goldstone modes". The two expressions correspond to different gauges. If you put a parameter $\xi$ in front of the $k^\mu k^\nu$ term, upon summing up any gauge invariant combination of Feynman diagrams (in particular the sum of the diagrams of a given loop order), the $\xi$ dependent terms will cancel out, showing that adding that term does not change anything to the final result. I guess it is possible to think of these extra terms as being associate to the Goldstone modes (like some pieces of the $g^{\mu \nu}$ piece in QED correspond to the longitudinal modes) .

 

[Avodyne]

You need to look at a QFT text for details; try Srednicki.