Rewording Theoretical Predictions at Colliders with Vacuum Amplitudes

We propose multiloop vacuum amplitudes as the optimal building blocks for efficiently assembling theoretical predictions at high-energy colliders. This hypothesis is strongly supported by the manifestly causal properties of the loop-tree duality (LTD) representation of a vacuum amplitude. The vacuum amplitude, acting as a kernel, encodes all the final states contributing to a given scattering or decay process through residues in the on-shell energies of the internal propagators. It also naturally implements gauge invariance and the wave function renormalisation of the external legs. This methodological approach, dubbed LTD causal unitary, leads to a novel representation of differential cross sections and decay rates that is locally free of ultraviolet and infrared singularities at all orders in perturbation theory. Threshold singularities also match between different phase-space residues. Most notably, it allows us to conjecture for the first time the local functional form of initial-state collinear singularities. The fulfillment of all these properties provides a theoretical description of differential observables at colliders that is well defined in the four physical dimensions of the space-time.