Self-duality, Helicity and Higher-loop Euler-Heisenberg Effective Actions∗

The Euler-Heisenberg effective action in a self-dual background is remarkably simple at two-loop. This simplicity is due to the inter-relationship between self-duality, helicity and supersymmetry. Applications include two-loop helicity amplitudes, beta-functions and nonperturbative effects. The two-loop EulerHeisenberg effective Lagrangian for QED in a self-dual background field is naturally expressed in terms of one-loop quantities. This mirrors similar behavior recently found in two-loop amplitudes in N=4 SUSY Yang-Mills theory. 1 One-Loop Euler-Heisenberg Effective Action In computing effective actions in gauge theories and in gravity, one can use the external field as a probe of the vacuum structure of the quantum theory[1]. A great deal has been learned from the seminal work of Heisenberg and Euler[2], who already in 1936 produced the paradigm for this entire approach by computing the nonperturbative, renormalized, one-loop effective action for QED in a background of constant field strength Fμν . This special soluble case of a constant field strength leads to several important insights and applications: • Light-Light scattering. The Euler-Heisenberg effective action is nonlinear in the electromagnetic fields, and the quartic and higher terms represent new nonlinear interactions, the first of which is light-light scattering, which does not occur in the tree level Maxwell action. Expanding the Euler-Heisenberg answer to quartic order we find