Mass renormalization in nonrelativistic QED

In nonrelativistic QED the charge of an electron equals its bare value, whereas the self-energy and the mass have to be renormalized. In our contribution we study perturbative mass renormalization, including second order in the fine structure constant $\alpha$, in the case of a single, spinless electron. As well known, if $m$ denotes the bare mass and $\mass$ the mass computed from the theory, to order $\alpha$ one has $$\frac{\mass}{m} =1+\frac{8\alpha}{3\pi} \log(1+\half (\Lambda/m))+O(\alpha^2)$$ which suggests that $\mass/m=(\Lambda/m)^{8\alpha/3\pi}$ for small $\alpha$. If correct, in order $\alpha^2$ the leading term should be $\displaystyle \half ((8\alpha/3\pi)\log(\Lambda/m))^2$. To check this point we expand $\mass/m$ to order $\alpha^2$. The result is $\sqrt{\Lambda/m}$ as leading term, suggesting a more complicated dependence of $m_{\mathrm{eff}}$ on $m$.