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In the book quantum field theory and standard model by Schwartz, there is a problem 9.4 that says by considering lorentz invariance of Compton scattering, you can prove that for spin 1 massless field there must be an $AA\phi\phi$ interaction, and for gravitons there must be a $hh\phi\phi$ interaction, although I have struggled with it for some time I could not proof this, any Ideas?

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I am a bit confused by this problem too.

My approach is the following. Let $m$ be the mass of the scalar particle, $p^\mu$ is the momentum of the incoming scalar particle and $q_1^\mu$ and $q_2^\mu$ the incoming and outgoing boson momenta, and $\epsilon_i^\mu$ the polarizations. I suppose the following 3-vertex $\Gamma^\mu (p, q) = p^\mu F(\frac{p^\mu q_\mu}{m^2})$, based on Lorentz invariance.

Now, taking just that vertex, Compton scattering should be given by two diagrams, the s-channel and the t-channel.

$iM_s = p_\mu F(\frac{p^\mu q_{1\mu}}{m^2})\frac{i}{(p+q_1)^2 - m^2 +i\epsilon}(p_\nu+q_{1\nu})p^\mu F(\frac{(p^\mu+ q_1^\mu)q_{1\mu}}{m^2})\epsilon_1^\mu\epsilon_2^{*\nu}$ $iM_t = p_\mu F(\frac{p^\mu q_{2\mu}}{m^2})\frac{i}{(p-q_2)^2 - m^2 +i\epsilon}(p_\nu-q_{2\nu})p^\mu F(\frac{(p^\mu+ q_2^\mu)q_{2\mu}}{m^2})\epsilon_2^\mu\epsilon_1^{*\nu}$

Making the substitution $\epsilon_1^\mu\rightarrow q_1^\mu$ to check Ward’s identity taking the soft limit, summing up the two diagrams gives me already 0. This is coherent with my understanding, since here we are working with an abelian theory. But then, I don't understand the need for the $\phi\phi A A$ interaction in the soft limit, which should be necessary only if we had more spin-1 particles and a non-abelian theory.

What am I missing?

Thank you

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  • $\begingroup$ I have no understanding of the quantum theory of a graviton but wouldn't a theory of gravitons be non-abelian given that diff invariance is the gauge invariance of such a theory? $\endgroup$ – Feynmans Out for Grumpy Cat 2 days ago
  • $\begingroup$ My point was one step before, just at the spin-1 massless particle. Take scalar qed, for example. You actually need the $\phi\phi A A$ interaction to ensure Lorentz invariance, if you work with a generic $q^\mu$ photon momentum. But you can't see it (at least in my understanding), just taking the soft limit, because in that limit the two diagrams cancel. For gravity, I am not sure that a spin-2 theory would need to be non-abelian. $\endgroup$ – Slz2718 yesterday

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