Constant electric fields in de Sitter require a tachyonic photon mass ~H, yielding finite positive Schwinger currents for massless fermions and scalars after on-shell renormalization.
Schwartz, Quantum Field Theory and the Standard Model , Cambridge University Press (3, 2014)
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Classical constant electric fields and the Schwinger effect in de Sitter
Constant electric fields in de Sitter require a tachyonic photon mass ~H, yielding finite positive Schwinger currents for massless fermions and scalars after on-shell renormalization.