Beyond unitarity: new on-shell representations for loop amplitudes

It has been very fruitful to learn to rephrase loop computations, in quantum field theory, in terms of tree-level building blocks. In this talk I present a novel representation, which combines the best features from the Feynman tree theorem and modern unitarity-based approaches.
It exists to all loop orders in any theory, is fully constructive, involves only on-shell trees, and is explicitly ghost-free. This comes at the cost of slightly unfamiliar linear propagators, but which can nonetheless be term-wise integrated using standard techniques. Based on 1509.02169