Rewording Theoretical Predictions at Colliders with Vacuum Amplitudes

We propose multiloop vacuum amplitudes in the loop-tree duality (LTD) 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 LTD representation of a vacuum amplitude. The vacuum amplitude in LTD, 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. Gauge invariance and the wave function renormalization of the external legs are naturally incorporated. This methodological approach, dubbed LTD causal unitary, leads to a novel differential representation of 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 splitting functions. 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.