  To reveal the Schwinger effect for quarks, i.e., pair creation process of
quarks and antiquarks, we derive the vacuum decay rate at strong coupling using
AdS/CFT correspondence. Magnetic fields, in addition to the electric field
responsible for the pair creation, causes prominent effects on the rate, and is
important also in experiments such as RHIC/LHC heavy ion collisions. In this
paper, through the gravity dual we obtain the full Euler-Heisenberg Lagrangian
of N=2 supersymmetric QCD and study the Schwinger mechanism with not only a
constant electric field but also a constant magnetic field as external fields.
We determine the quark mass and temperature dependence of the Lagrangian. In
sharp contrast with the zero magnetic field case, we find that the imaginary
part, and thus the vacuum decay rate, diverges in the massless zero-temperature
limit. This may be related to a strong instability of the QCD vacuum in strong
magnetic fields. The real part of the Lagrangian serves as a generating
function for non-linear electro-magnetic responses, and is found such that the
Cotton-Mouton effect vanishes. Interestingly, our results of the Schwinger /
Cotton-Mouton effects coincide precisely with those of N=2 supersymmetric QED.
