S -duality and helicity amplitudes

We examine interacting abelian theories at low energies and show that holomorphically normalized photon helicity amplitudes transform into dual amplitudes under sl(2, $$ mathrm{mathbb{z}} $$ ) as modular forms with weights that depend on the number of positive and negative helicity photons and on the number of internal photon lines. moreover, canonically normalized helicity amplitudes transform by a phase, so that even though the amplitudes are not duality invariant, their squares are duality invariant. we explicitly verify the duality transformation at one loop by comparing the amplitudes in the case of an electron and the dyon that is its sl(2, $$ mathrm{mathbb{z}} $$ ) image, and extend the invariance of squared amplitudes order by order in perturbation theory. we demonstrate that s-duality is a property of all low-energy effective abelian theories with electric and/or magnetic charges and see how the duality generically breaks down at high energies.