Quark anomalous magnetic moment in an extreme magnetic background from perturbative QCD

We compute the one-loop QCD correction to the photon-quark-antiquark vertex in an extremely strong magnetic background, i.e., one in which $\sqrt{eB}$ is much larger than all other mass scales. We resort to the lowest-Landau level approximation, and consider on shell fermions. We find that the total magnetic moment is such that the anomalous magnetic moment contributes to the electric part $\sim iE_3\sigma_3$, the other contributions to $\boldsymbol{B}\cdot\boldsymbol{\sigma}$ and $i\boldsymbol{E}\cdot\boldsymbol{\sigma}_\perp$ being completely suppressed. We show results for the anomalous magnetic moment of quarks up and down, scaled by their values in the vacuum, as a function of the external magnetic field, considering dressed gluons and a renormalization scale $\mu_\mathrm{\tiny \overline{ MS}}=\sqrt{|eB|}$.