The 1-loop self-energy of an electron in a strong external magnetic field revisited

Bruno Machet (1)

1 LPTHE - Laboratoire de Physique Théorique et Hautes Energies

I revisit the 1-loop self-energy of an electron in a strong, constant and uniform external magnetic field B. First, I show, after Tsai (1974), how, for an electron in the lowest Landau level, Schwinger's techniques, as explained by Dittrich and Reuter (1985) lead to the same integral deduced by Demeur (1953) and used later by Jancovici (1969). Then, I calculate the Demeur-Jancovici integral in the range $75 \leq L\equiv\frac{|e|B}{m^2} \leq 10\,000$, which yields $\delta m \simeq \frac{\alpha m}{4\pi} \left[\left(\ln L-\gamma_E-\frac32\right)^2 -\frac94 +\frac{\pi}{\beta-1} +\frac{\pi^2}{6} +\frac{\pi\;\Gamma[1-\beta]}{L^{\beta-1}} +\frac{1}{L}\left(\frac{\pi}{2-\beta}-5\right) +{\cal O}(\frac{1}{L^{\geq 2}})\right],\ \beta \simeq 1.175$, close to Jancovici's last estimate $\delta m \simeq \frac{\alpha m}{4\pi}\left[\left(\ln 2L-\gamma_E-\frac32\right)^2 +A+\ldots\right]$ with $A\simeq 3.5$ (previously undetermined). The term proportional to $(\ln\frac{|e|B}{m^2})^2$ can never be considered to be leading and gets largely damped, in particular by the large $\ln \frac{|e|B}{m^2}$ which arises from the counterterm implementing suitable renormalization conditions. The addition with respect to existing litterature is small but some consequences may be worth deeper investigations.

Mots clés: Quantum Electrodynamics; external magnetic field; electron self-energy

Domaines: Physique des Hautes Energies - Phénoménologie [hep-ph]

Bruno Machet. The 1-loop self-energy of an electron in a strong external magnetic field revisited. 2015. ⟨hal-01215168v1⟩