Fumio Hiroshima and Herbert Spohn
Mass renormalization in nonrelativistic QED
(65K, latex)
ABSTRACT. 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 $m$.