Perturbation theory of transformed quantum fields

We consider a scalar quantum field $\phi$ with arbitrary polynomial self-interaction in perturbation theory. If the field variable $\phi$ is repaced by a local diffeomorphism $\phi(x) = \rho(x) + a_1 \rho^2(x) +\ldots$, this field $\rho$ obtains infinitely many additional interaction vertices. We show that the $S$-matrix of $\rho$ coincides with the one of $\phi$ without using path-integral arguments. This result holds even if the underlying field has a propagator of higher than quadratic order in the momentum. If tadpole diagrams vanish, the diffeomorphism can be tuned to cancel all contributions of an underlying $\phi^s$-type self interaction at one fixed external offshell momentum, rendering $\rho$ a free theory at this momentum. Finally, we propose one way to extend the diffeomorphism to a non-local transformation involving derivatives without spoiling the combinatoric structure of the local diffeomorphism.