Strong coupling and the infrared structure of QED

The construction of a strong-coupling expansion in continuum QED by means of the extraction of infrared structure associated with all the virtual-photon exchanges on a single fermion loop is shown to lead to a $\beta$ function of the unrenormalized theory $\beta \mathrm{}\left(e\right)\mathrm{}\sim -\left(\frac{45}{4}\right)\times e{\left(\ln e\right)}^{-1\mathrm{}}$, $e\gg 1$, in contrast to the conventional perturbation form $\beta \mathrm{}\left(e\right)\mathrm{}\sim b_1{e}^3$, $b_1>0\mathrm{}$, for $e\ll 1$. Various approximations are introduced for computational simplicity, such as the neglect of fermion spin dependence, but gauge invariance is strictly enforced at every stage of the calculation. This result suggests that scalar QED has a strong-coupling phase similar to the confining phase of QCD.