Euler-Heisenberg Lagrangian under an axial gauge field

Augmentations to the Euler-Heisenberg Lagrangian (QED one-loop effective action in homogeneous electromagnetic fields) under a constant background axial gauge are examined. Two special configurations admit an exact eigendecomposition, and hence effective action as a spectral sum, of the augmented Dirac operator: one with a magnetic field with chiral chemical potential, and the other with an electric field with spatial axial gauge, which resembles an emergent vorticity. An enhancement to Schwinger pair production is found for the latter, which is more fully analyzed using the worldline instanton formalism. There it is found the overall enhancement is due to the spatial axial gauge serving as a negative mass shift. Finally, we remark on the exactly solvable massless case for arbitrary electromagnetic and axial gauges.

Figure 1

Imaginary part of the effective action, $\mathrm{Im}\mathrm{\Gamma }(E,{\omega }_5)$, to $\mathcal{O}({\omega }_5^4)$ as a function of normalized mass, $\overline{m}$, and spatial axial gauge, ${\overline{\omega }}_5$, as given in Eq. (54). Notice for constant $\overline{m}$, there is an enhancement of pair production for increasing ${\overline{\omega }}_5$. Also lines of approximately constant values of the imaginary part are seen for linear relationships between $\overline{m}$ and ${\overline{\omega }}_5$.