Mass renormalization in nonrelativistic quantum electrodynamics

In nonrelativistic quantum electrodynamics the charge of an electron equals its bare value, whereas the self-energy and the mass must be renormalized. In our contribution we study perturbative mass renormalization, including second order in the fine structure constant $α$⁠, in the case of a single, spinless electron. As is well known, if $m$ denotes the bare mass and $meff$ the mass computed from the theory, to order $α$ one has $meff∕m=1+(8α∕3π)log(1+12(Λ∕m))+O(α2)$ which suggests that $meff∕m=(Λ∕m)8α∕3π$ for small $α$⁠. If correct, in order $α2$ the leading term should be $12((8α∕3π)log(Λ∕m))2$⁠. To check this point we expand $meff∕m$ to order $α2$⁠. The result is $Λ∕m$ as leading term, suggesting a more complicated dependence of $meff$ on $m$⁠.