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We compute non-perturbative contributions to the Adler function, thederivative of the vacuum polarization function in gauge theory, usingresurgence methods and Borel-summed gauge field propagators. At 2-loop, toorder $1/N_f$, we construct the full 2-parameter transseries and perform thesum over the non-perturbative sectors. We then introduce a convolution-basedmethod to derive the transseries structure of product series, which can also beused to study higher orders in the expansion in $1/N_f$. We compute 3-loopplanar diagrams, at order $1/N_f^2$, and for each diagram study the asymptoticbehavior and resulting non-perturbative information in the transseries. Astructure emerges that, from a resurgence point of view, is quite differentfrom toy models hitherto studied. We study in particular the first and secondnon-perturbative sectors, their relation to UV and IR renormalons, and howtheir presence influences the perturbative expansions in neighbouring sectors.Finally, finding that many non-perturbative sectors have asymptotic series, wederive relations among all of them, thus providing an interesting newperspective on the alien lattice for the Adler function.

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