I’m afraid you have not made a case for a “No” answer. We know of well defined mathematical criteria for self-interacting fields which works both classically and quantum mechanically: a field theory is said to be self-interacting if, in the absence of sources, the fields satisfy non-linear (can be coupled) second order partial differential equations. And this applies to all self-interacting theories known to us:
(1) In the [itex]\Phi^{4}[/itex] theory, we have [itex]\partial_{\mu}(\partial^{\mu}\Phi ) \sim - \lambda \Phi (\Phi^{2})[/itex].
(2) For Yang-mills field, you have [itex]\partial_{\nu}F_{a}^{\nu}{}_{\mu} = - f_{abc}A^{\nu}_{b} (F_{c \nu \mu})[/itex].
And (3) in free space, the gravitational field satisfies [itex]\partial_{\nu}(\sqrt{-g}G^{\nu}{}_{\mu}) = - (1/2)\partial_{\mu}g^{\nu \rho} \ (\sqrt{-g} G_{\nu \rho})[/itex].
So, why should (1) and (2) but not (3) be self-interacting?
