Derivative corrections to the Heisenberg-Euler effective action

Felix Karbstein

doi:10.1007/jhep09(2021)070

Abstract

We show that the leading derivative corrections to the Heisenberg-Euler effective action can be determined efficiently from the vacuum polarization tensor evaluated in a homogeneous constant background field. After deriving the explicit parameter-integral representation for the leading derivative corrections in generic electromagnetic fields at one loop, we specialize to the cases of magnetic- and electric-like field configurations characterized by the vanishing of one of the secular invariants of the electromagnetic field. In these cases, closed-form results and the associated all-orders weak- and strong-field expansions can be worked out. One immediate application is the leading derivative correction to the renowned Schwinger-formula describing the decay of the quantum vacuum via electron-positron pair production in slowly-varying electric fields.

Highlights

We show that the leading derivative corrections to the Heisenberg-Euler effective action can be determined efficiently from the vacuum polarization tensor evaluated in a homogeneous constant background field
After deriving the explicit parameter-integral representation for the leading derivative corrections in generic electromagnetic fields at one loop, we specialize to the cases of magnetic- and electric-like field configurations characterized by the vanishing of one of the secular invariants of the electromagnetic field
For the special cases of magnetic- and electric-like field configurations characterized by the vanishing of one of the secular invariants of the electromagnetic field, we obtain closed-form expressions and work out all-orders weak- and strong-field expansions

Summary

Our approach

We demonstrate that the leading derivative correction to the Heisenberg-Euler effective action can efficiently be determined from the vacuum polarization tensor evaluated in a generic constant and homogeneous background field F. In position space, this correction contains exactly two derivatives but arbitrary powers of the electromagnetic field F. The contributions to Πμν(k, k |F) beyond quartic order, which translate into higherorder n derivative terms are not helpful for the purpose of a systematic derivation of higher-order derivative corrections to ΓHE This is a direct consequence of the fact that there is no unambiguous way in assigning the additional derivatives to any of the two inhomogeneous fields F (x) before invoking the substitution F → F. As the determination of the n-derivative contribution only requires knowledge of the term scaling as k2n ∼ kσ1 . . . kσ2n of the n-rank polarization tensor, aiming at the evaluation of the respective contribution in cases where the required polarization tensor has not yet been determined, for this endeavor it suffices to determine this tensor only at an accuracy of order k2n

Explicit calculation

Magnetic- and electric-like field configurations

Conclusions and outlook

A Identities