Avoidance of a Landau pole by flat contributions in QED

•We present an approach to approximate both the ββ-function and the photon self-energy.•We find a sufficient criterion for the self-energy to entail the existence of a Landau pole.•We study non-perturbative ‘flat’ contributions that emerge within the context of our approach.•We discuss a toy model and how it is affected by flat contributions.

We consider massless Quantum Electrodynamics in the momentum scheme and carry forward an approach based on Dyson–Schwinger equations to approximate both the ββ-function and the renormalized photon self-energy (Yeats, 2011). Starting from the Callan–Symanzik equation, we derive a renormalization group (RG) recursion identity which implies a non-linear ODE for the anomalous dimension and extract a sufficient but not necessary criterion for the existence of a Landau pole. This criterion implies a necessary condition for QED to have no such pole. Solving the differential equation exactly for a toy model case, we integrate the corresponding RG equation for the running coupling and find that even though the ββ-function entails a Landau pole it exhibits a flat contribution capable of decreasing its growth, in other cases possibly to the extent that such a pole is avoided altogether. Finally, by applying the recursion identity, we compute the photon propagator and investigate the effect of flat contributions on both spacelike and timelike photons.