Resurgence analysis of the Adler function at order $1/N_f^2$

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.