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Let's look at the analog of the massive photon. The propagator for this is $$\frac{ g^{\mu\nu} - \frac{p^\mu p^\nu}{m^2} }{ p^2 + m^2 }$$ At large energies, this scales like $\frac{p^\mu p^\nu }{m^2 p^2} \sim \frac{1}{m^2}$. In the same way, the massive graviton propagator takes the general form  \frac{g^{\mu\alpha} g^{\nu\beta} + g^{\mu\beta} ...
A mass term in the photon self-energy tensor would look like $Ag^{\mu\nu}$, where $A$ approaches a constant as $q^{2}\rightarrow0$. What is important is not the degree of divergence of $A$, but the Lorentz structure of the term. In particular, a term of the form $Bq^{2}g^{\mu\nu}$ is not a mass term; when contracted with the external fields $A^{\mu}$ and ...