$N$-Photon Amplitudes in EFT from Recursion Relations and Effective Vertices

We present a recursive framework for computing arbitrary tree-level helicity amplitudes in the general effective field theory (EFT) of electromagnetism. We show that photon amplitudes can be constructed using a CSW-like recursion, whose building blocks consist of purely local subamplitudes. For a fixed number of external legs, only a finite number of these contact amplitudes need to be computed, and these admit compact expressions in terms of Hafnians. We further show that the contact amplitudes are encoded in an effective contact Lagrangian, that can be computed systematically from the general EFT Lagrangian using a suitable generating functional. We provide this contact Lagrangian up to $N=14$ photons. We argue that the contact Lagrangian reveals the hidden simplicity of helicity-conserving amplitudes, providing a simple proof of the equivalence between off-shell electromagnetic duality and helicity conservation. We further show how to resum the contact Lagrangian of a helicity-conserving theory to all $N$, and illustrate the procedure for Born-Infeld (BI) electromagnetism. Building on these methods and results, we compute the complete 6-point and 8-point tree-level helicity amplitudes generated by generic EFT operators, and the tree amplitudes up to 10-point in BI electromagnetism.