In Standard Model every fundamental interaction is described by means of exchanging gluons of particular kind. It is very natural as gluons has spin with values given by inteegers, and can share the same quantum state, which gives us interactions with reasonable energy transfers.
On the other side, I cannot remember anyone proposed even theoretical toy model where interactions are exchanged by means of fermion particles of some kind.
Q: Could you provide any useful reference for such toy model?
It is remarkable that such hypothetical interactions should be much weaker than any governed by gluons, as "exchanging fermions" cannot stay at the same quantum states, so number of possible exchanges ( various effective exchange rates per seconds or per area or volume) should be very small.
I even can imagine, that with certain coupling with topological space properties, requirement of particular energy transfer level gives as results curved space, just because particular number of "exchanging fermions" have to be exchanged and the only curved space would allow required "exchanging paths" big enough for "exchanging fermions" ( bigger area, volume etc with given borders).
So if we're able to discover fundamental force, much much weaker than the others, it may be an example of such interaction! Oh, wait....
