  We employ resurgent transseries as algebraic tools to investigate two
self-consistent Dyson-Schwinger equations, one in Yukawa theory and one in
quantum electrodynamics. After a brief but pedagogical review, we derive fixed
point equations for the associated anomalous dimensions and insert a moderately
generic log-free transseries ansatz to study the possible strictures imposed.
While proceeding in various stages, we develop an algebraic method to keep
track of the transseries' coefficients. We explore what conditions must be
violated in order to stay clear of fixed point theorems to eschew a unique
solution, if so desired, as we explain. An interesting find is that the flow of
data between the different sectors of the transseries shows a pattern typical
of resurgence, ie the phenomenon that the perturbative sector of the
transseries talks to the nonperturbative ones in a one-way fashion. However,
our ansatz is not exotic enough as it leads to trivial solutions with vanishing
nonperturbative sectors, even when logarithmic monomials are included. We see
our result as a harbinger of what future work might reveal about the
transseries representations of observables in fully renormalised
four-dimensional quantum field theories and adduce a tentative yet to our mind
weighty argument as to why one should not expect otherwise.
  This paper is considerably self-contained. Readers with little prior
knowledge are let in on the basic reasons why perturbative series in quantum
field theory eventually require an upgrade to transseries. Furthermore, in
order to acquaint the reader with the language utilised extensively in this
work, we also provide a concise mathematical introduction to grid-based
transeries.
