  In this paper we calculate the non-perturbative Euler-Heisenberg Lagrangian
for massless QED in a strong magnetic field $H$, where the breaking of the
chiral symmetry is dynamically catalyzed by the external magnetic field via the
formation of an electro-positron condensate. This chiral condensate leads to
the generation of dynamical parameters that have to be found as solutions of
non-perturbative Schwinger-Dyson equations. Since the electron-positron pairing
mechanism leading to the breaking of the chiral symmetry is mainly dominated by
the contributions from the infrared region of momenta much smaller than
$\sqrt{eH}$, the magnetic field introduces a dynamical ultraviolet cutoff in
the theory that also enters in the non-perturbative Euler-Heisenberg action.
Using this action, we show that the system exhibits a significant
paraelectricity in the direction parallel to the magnetic field. The
nonperturbative nature of this effect is reflected in the non-analytic
dependence of the obtained electric susceptibility on the fine-structure
constant. The strong paraelectricity in the field direction is linked to the
orientation of the electric dipole moments of the pairs that form the chiral
condensate. The large electric susceptibility can be used to detect the
realization of the magnetic catalysis of chiral symmetry breaking in physical
systems.
