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I understand that the EM mediator is the photon and is absorbed and emitted by electrons. I understand that the strong force mediator is the gluon and is absorbed and emitted by quarks. Both electrons and quarks are pointlike elementary particles with rest mass.

If there are gravitons and they are the mediators for Gravity, which pointlike elementary particle with rest mass would absorb and emit them?

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All particles that are able to interact electromagnetically (not just electrons)can emit or absorb photons. So by definition if there was a graviton all particles that interact gravitationally could emit and absorb them. Your question gives the constraint of which particles with rest mass would emit/absorb gravitons. Mass is a measure of the particles involvement in gravitational interaction (like charge is a measure of involvement in the electromagnetic interaction). So all particles with rest mass would emit and absorb gravitons.

The question asks only about particles with rest mass, but in fact all particles would emit and absorb gravitons, not just those with rest mass. The source of gravitational fields is the stress-energy tensor, not rest mass. Roughly speaking, mass and energy and equivalent ($E=mc^2$), so energy participates in gravitational forces as well.

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    $\begingroup$ Anything with electric charge can emit photons. That includes quarks, Z bosons, any chemical ion, and so on. Similarly anything with "gravitational charge" can emit gravitons but that really means anything with energy. $\endgroup$ – knzhou Apr 7 '18 at 12:08
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    $\begingroup$ @knzhou Z are neutral, but in higher orders they can when interacting emit a photon. $\endgroup$ – anna v Apr 7 '18 at 12:35
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    $\begingroup$ So all particles with rest mass would emit and absorb gravitons. This is wrong. All particles would emit and absorb gravitons, not just those with rest mass. The source of gravitational fields is the stress-energy tensor, not rest mass. $\endgroup$ – Ben Crowell Apr 7 '18 at 14:22
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    $\begingroup$ @BenCrowell The question is about particles with rest mass, so I have constrained myself to them. I didn't claim particles with no rest mass will not emit/absorb gravitons. $\endgroup$ – K. Kirilov Apr 7 '18 at 14:26
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    $\begingroup$ @K.Kirilov: In that case, how about editing the answer? It's misleading in its present form. $\endgroup$ – Ben Crowell Apr 7 '18 at 16:14

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