**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 $\alpha^2$, enlightening 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 a joint work with Christian Hainzl.