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On wikipedia one can read the following statement:

The photon and gluon do not get a mass through renormalization because gauge symmetry protects them from getting a mass. This is a consequence of the Ward identity.

Can someone briefly outline why the forbidden mass is a consequence from the Ward identity?

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  • $\begingroup$ This quote (I believe from the Wikipedia article "Self-energy") is a bit misleading anyway, because there are no massless gluons in nature. Also there are Ward identities for the W and Z bosons, and this does not prevent them from acquiring a mass through the Higgs mechanism. $\endgroup$ – M.Jo Sep 23 '19 at 13:01
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Without gauge invariance, the masses of vector bosons would be affected by contributions from higher order Feynman diagrams. As a result, even if the bosons have zero mass in the fundamental theory, their masses can become nonzero due to these contributions.

In the presence of gauge invariance, those diagrams that would have contributed to these changes in the mass cannot contribute thanks to the Ward-Takahashi identities. They cause those diagrams to become zero.

For more detail, one would need to look at the self-energy diagrams that would produce contributions to the boson mass and then see what the effect of the Ward-Takahashi identities are on those diagrams.

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