Kramers-Kr?nig approach to the electric permittivity of the vacuum in a strong constant electric field

We revisit the change of the electric permittivity of the QED vacuum by a strong constant electric field, motivated by the analogy between the dynamically assisted Schwinger effect in strong-field QED and the Franz-Keldysh effect in semiconductor physics. We develop a linear-response theory based on the nonequilibrium in-in formalism and the Furry-picture perturbation theory. Applying the developed formalism and also utilizing the Kramers-Kronig relation, we calculate the electric permittivity for wide values of the field strength and the probe frequency, including the supercritical field and/or high probe frequency that the previous research has not fully covered. We discover that the electric permittivity exhibits a characteristic oscillating feature in the high probe-frequency regime, which directly reflects the change of the QED-vacuum structure by the strong field. We also establish a quantitative correspondence between the electric permittivity and the number of electron-positron pairs produced by the dynamically assisted Schwinger effect.