  We compute non-perturbative contributions to the Adler function, the
derivative of the vacuum polarization function in gauge theory, using
resurgence methods and Borel-summed gauge field propagators. At 2-loop, to
order $1/N_f$, we construct the full 2-parameter transseries and perform the
sum over the non-perturbative sectors. We then introduce a convolution-based
method to derive the transseries structure of product series, which can also be
used to study higher orders in the expansion in $1/N_f$. We compute 3-loop
planar diagrams, at order $1/N_f^2$, and for each diagram study the asymptotic
behavior and resulting non-perturbative information in the transseries. A
structure emerges that, from a resurgence point of view, is quite different
from toy models hitherto studied. We study in particular the first and second
non-perturbative sectors, their relation to UV and IR renormalons, and how
their presence influences the perturbative expansions in neighbouring sectors.
Finally, finding that many non-perturbative sectors have asymptotic series, we
derive relations among all of them, thus providing an interesting new
perspective on the alien lattice for the Adler function.
