  We examine a large class of scalar quantum field theories where vertices are
able to cancel adjacent propagators. These theories are obtained as
diffeomorphisms of the field variable of a free field. Their connected
correlations functions can be computed efficiently with an algebraic procedure
dubbed connected perspective in earlier work. Specifically,
  (1) We compare the Feynman rules in momentum-space to their position-space
counterparts, they agree. This is a a-posteriori justification for the Feynman
rules of the connected perspective.
  (2) We extend the connected perspective to also incorporate counterterm
vertices.
  (3) We examine a specific choice of diffeomorphism that assumes the vertices
at different valence proportional to each other. We find that these relations
lead to various simplifications.
  (4) We perturbatively compute the connected 2-point function to all orders
for an arbitrary diffeomorphism of a massless field. We thereby give a
systematic perturbative derivation of the exponential superpropagator known
from the literature.
  (5) We compute a number of one- and two-loop counterterms. We find that for
the specific diffeomorphism, the one-loop counterterms arise from the bare
vertices by a non-linear redefinition of the momentum.
  (6) We show that every diffeomorphism fulfils a set of infinitely many
Slavnov-Taylor-like identities which express diffeomorphism-invariance of the
S-matrix at loop-level.
  Finally, we comment on surprising similarities between the structure of
propagator-cancelling scalar theories and gauge theories .
