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

I revisit the 1-loop self-energy of an electron of mass m in a strong, constant and uniform external magnetic field B. First, I show, after Tsai [1], how, for an electron in the lowest Landau level, Schwinger’s techniques [2] explained by Dittrich and Reuter [3] lead to the same integral deduced by Demeur [4] and used later by Jancovici [5]. Then, I calculate the Demeur-Jancovici integral in the range 75 ≤ L ≡ |e|B m2 ≤ 10 000, which yields δm ' αm 4π [( lnL− γE − 3 2 )2 − 94 + π β−1 + π 6 + π Γ[1−β] Lβ−1 + 1 L ( π 2−β − 5)+O( 1 L≥2 )] with β ' 1.175, very close numerically to Jancovici’s last estimate δm ' αm 4π [( ln 2L− γE − 32 )2 +A+ . . . ] with A ' 3.5 (undetermined in [5]). The (lnL), only contribution to be ever considered, gets largely damped, in particular by the large lnL which arises from the counterterm implementing suitable renormalization conditions. The former exceeds by 45% the true estimate at L = 100 and by more at lower L. The addition to existing literature is small but some consequences may be worth deeper investigations. PACS: 12.15.Lk 12.20.Ds 1Sorbonne Universités, UPMC Univ Paris 06, UMR 7589, LPTHE, F-75005, Paris, France 2CNRS, UMR 7589, LPTHE, F-75005, Paris, France. 3Postal address: LPTHE tour 13-14, 4ème étage, UPMC Univ Paris 06, BP 126, 4 place Jussieu, F-75252 Paris Cedex 05 (France) [email protected]