Abstract: We investigate the self-energy of one electron coupled to a quantized radiation field for an arbitrary cut-off parameter $\Lambda$ and for small values of $\alpha$, where $\alpha$ is the coupling parameter representing the fine structure constant. We explain the strategy to develop the self-energy in powers of $\alpha$ up to the second term, in $\alpha^2$, enlighting the fact that the method would actually allow to expand the self-energy up to any power of $\alpha$. Enhanced binding for electrons then follows as an immediate consequence. This is joint work with Christian Hainzl.