  We study the renormalization flow of axion electrodynamics, concentrating on
the non-perturbative running of the axion-photon coupling and the mass of the
axion (like) particle. Due to a non-renormalization property of the
axion-photon vertex, the renormalization flow is controlled by photon and axion
anomalous dimensions. As a consequence, momentum-independent axion
self-interactions are not induced by photon fluctuations. The non-perturbative
flow towards the ultraviolet exhibits a Landau-pole-type behavior, implying
that the system has a scale of maximum UV extension and that the renormalized
axion-photon coupling in the deep infrared is bounded from above. Even though
gauge invariance guarantees that photon fluctuations do not decouple in the
infrared, the renormalized couplings remain finite even in the deep infrared
and even for massless axions. Within our truncation, we also observe the
existence of an exceptional RG trajectory, which is extendable to arbitrarily
high scales, without being governed by a UV fixed point.
