  In de Sitter space, the current induced by an electric field in vacuum is
known to feature certain peculiarities, such as infrared hyperconductivity for
light bosons in weak electric fields. Moreover, negative conductivity has been
claimed to occur for light bosons in moderate electric fields, and for fermions
of any mass in electric fields below a certain threshold. Furthemore, in the
limit of large mass and weak electric field, the current contains terms which
are not exponentially suppressed, contrary to the semiclassical intuition. Here
we explain these behaviors, showing that most of the reported negative
conductivity is spurious. First, we show that the terms which are not
exponentially suppressed follow precisely from the local Euler-Heisenberg
Lagrangian (suitably generalized to curved space). Thus, such terms are
unrelated to pair creation or to the transport of electric charge. Rather, they
correspond to non-linearities of the electric field (responsible in particular
for vacuum birefringence). The remaining contributions are exponentially
suppressed and correspond to the creation of Schwinger pairs. Second, we argue
that for light carriers the negative term in the regularized current does not
correspond to a negative conductivity, but to the logarithmic running of the
electric coupling constant, up to the high energy Hubble scale. We conclude
that none of the above mentioned negative contributions can cause an
instability such as the spontaneous growth of an electric field in de Sitter,
at least within the weak coupling regime. Third, we provide a heuristic
derivation of infrared hyperconductivity, which clarifies its possible role in
magnetogenesis scenarios.
