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High-energy theory

Turning off a graviton mass

About 10 min if you have seen this material before, 35 min or so if it is new to you.

Background
Assumed background: linearised gravity and the exchange amplitude between two conserved sources TμνT^{\mu\nu} and T′μνT'^{\mu\nu}. A The massless particle of helicity 2 whose exchange produces gravity. mass of 10−3310^{-33} eV is smaller than every other scale in any experiment, so the natural expectation is that predictions differ by something of order mm and are otherwise identical.
For a massless graviton, A∝κ2q2(TμνTμν′−12TT′),\mathcal{A} \propto \frac{\kappa^2}{q^2} \left(T^{\mu\nu}T'_{\mu\nu} - \tfrac{1}{2} T T'\right), and for a graviton with a Fierz-Pauli mass mm, A∝κ2q2+m2(TμνTμν′−13TT′).\mathcal{A} \propto \frac{\kappa^2}{q^2 + m^2} \left(T^{\mu\nu}T'_{\mu\nu} - \tfrac{1}{3} T T'\right). The 12\tfrac{1}{2} counts the two A massless particle's spin measured along its direction of motion, taking whole-number or half-integer values of either sign. a massless graviton has and the 13\tfrac{1}{3} counts the five a massive one has. For a static source Tμν=ρ δ0μδ0νT^{\mu\nu} = \rho\,\delta^\mu_0 \delta^\nu_0. For a photon, T=0T = 0.
Work at q2≫m2q^2 \gg m^2, so the propagator poles agree. Fix κ\kappa in the massive theory by requiring the Newtonian force between two static masses to match the massless theory's. With that normalisation, what is the ratio of the massive theory's prediction for the deflection of light to the massless theory's?

Past 1 hr on this? That usually means a missing prerequisite rather than a lack of effort — worth reading up before pushing further, starting with vDVZ discontinuity (Wikipedia).

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