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 $α$, 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 $α$ one has $$\frac{\mass}{m} =1+\frac{8α}{3π} \log(1+\half (Λ/m))+O(α^2)$$ which suggests that $\mass/m=(Λ/m)^{8α/3π}$ for small $α$. If correct, in order $α^2$ the leading term should be $\displaystyle \half ((8α/3π)\log(Λ/m))^2$. To check this point we expand $\mass/m$ to order $α^2$. The result is $\sqrt{Λ/m}$ as leading term, suggesting a more complicated dependence of $m_{\mathrm{eff}}$ on $m$.

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