Two-Loop Self-Dual QED

We present explicit closed-form expressions for the two-loop Euler-Heisenberg Lagrangians in a constant self-dual field, for both spinor and scalar QED. The simplicity of these representations allows us to examine in detail the asymptotic properties of these Lagrangians, and to construct their imaginary part using Borel dispersion relations. In particular, for this self-dual case we obtain the explicit form of the Lebedev-Ritus functions appearing in the Schwinger representation of the imaginary part at two loops. Using the connection between self-duality and helicity, we also obtain explicit formulas for the low energy limits of the `all + helicity' N photon amplitudes, in scalar and spinor QED at one and two loops.