Chiral Anomaly, Schwinger Effect, Euler-Heisenberg Lagrangian, and
  application to axion inflation
  Particle production in strong electromagnetic fields is a recurring theme in
solid state physics, heavy ion collisions, early universe cosmology and formal
quantum field theory. In this paper we discuss the Dirac equation in a
background of parallel electric and magnetic fields. We review the Schwinger
particle production rate, clarify the emergence of the chiral anomaly equation
and compute the induced current of charged fermions. We distinguish the
contributions from non-perturbative particle production, from the running of
the gauge coupling constant and from non-linearities in the effective QED
Lagrangian, and clarify how these contributions arise within a single
framework. We apply these results to axion inflation. A Chern-Simons coupling
between the pseudoscalar particle driving cosmic inflaton and an abelian gauge
group induces a dual production of gauge fields and charged fermions. We show
that the resulting scalar and gravitational wave power spectra strongly depend
on the fermion mass.
