On the low-energy limit of the QED N-photon amplitudes: part 2

In recent work, Gies and Karbstein have discovered that the two-loop Euler-Heisenberg Lagrangians for scalar and spinor QED have non-vanishing reducible contributions in addition to the well-studied irreducible ones. This invalidates previous applications of those Lagrangians to the computation of the two-loop $N$-photon amplitudes in the low energy limit. Here we compute the corrections to those amplitudes due to the reducible contributions.