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The question is in the title really. I know gravity + QFT is on shaky ground but I'm not looking for Feynman rules, just a schematic, if that is possible.

Edit 1: added possible diagram for clarification.

Possible diagram

Edit 2: Actually, isn't this diagram the most important? enter image description here G represents a graviton and m represents some matter, e.g. proton, or quark.

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Your diagram looks OK - encapsulates the fact that gravitons can couple to themselves and to matter. – twistor59 Jan 27 at 19:43
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You would probably be interested in Feynman's lectures on gravitation (not to be confused with his popular level lectures "On the Character of Physical Law" where he discusses gravity at length). He derives GR from the quantum field theory of spin 2 particles. – Michael Brown Mar 29 at 13:53
@MichaelBrown Thanks for the suggestion! Do you know if there are gluon-matter scattering diagrams in that book? – user12345 Mar 30 at 20:15
@user16307 It's been a while since I've read it but I'm certain there are. I need to check it out again. :) – Michael Brown Apr 1 at 0:23
@MichaelBrown Fancy scanning the image, added with some words to make an answer? :) – user12345 Apr 1 at 11:27
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First of all, let me comment on the "gravity + QFT" statement. For sufficiently small curvatures, where we can neglect the effects of quantum gravity, we can treat excitations of gravitational field as normal spin-2 particles. Exactly in this spirit the field of QFT in curved space was created. This theory describes well the interactions of ordinary particles with gravitons. It is an effective field theory that breaks down at some very high energy, when higher order terms come into play and cannot be neglected. The problem only occurs when we try to interpret the theory as a fundamental and take this cut-off energy to infinity, as this results in divergences at high energies.

Returning to your question, in the approximation of small curvatures i.e. in QFT in curved background, the interactions of gravitons are just like interactions of any other gauge field e.g. gluon, except that if couples to everything that has mass.

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To be absolutely certain, could I ask you to inspect the image I have drawn and uploaded. The black line represents, say, a proton, and the red line is the graviton. – user12345 Jan 27 at 16:54
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Yes, this diagram will contribute to physical processes. Just a comment: please remember that feynman diagrams are just perturbative calculation tools, this particular one will depend on the arbitrary gauge choise, so it self does not represent the full physical process. – Jakub Jan 27 at 17:01

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