  We investigate the Landau-level structures encoded in the famous
Heisenberg-Euler (HE) effective action in constant electromagnetic fields. We
first discuss the HE effective actions for scalar and spinor QED, and then
extend it to the QCD analogue in the covariantly constant
chromo-electromagnetic fields. We identify all the Landau levels and the Zeeman
energies starting out from the proper-time representations at the one-loop
order, and derive the vacuum persistence probability for the Schwinger
mechanism in the summation form over independent contributions of the all-order
Landau levels. We find an enhancement of the Schwinger mechanism catalyzed by a
magnetic field for spinor QED and, in contrast, a stronger exponential
suppression for scalar QED due to the "zero-point energy" of the Landau
quantization. For QCD, we identify the discretized energy levels of the
transverse and longitudinal gluon modes on the basis of their distinct Zeeman
energies, and explicitly confirm the cancellation between the
longitudinal-gluon and ghost contributions in the Schwinger mechanism. We also
discuss the unstable ground state of the perturbative gluon excitations known
as the Nielsen-Olesen instability.
