Causal perturbation theory and scattering amplitudes

Motivated by the limited interaction between the mathematical physics community and theoretical physicists - particularly in high-energy theory - we present a computation that is typically the first example in QFT courses but, to our knowledge, does not appear in the literature: the scattering amplitudes of the $λϕ^4$ model in four-dimensional space-time, derived within the framework of causal perturbation theory. Our aim is to introduce this mathematically rigorous formalism in the most accessible way possible. To that end, we emphasize general aspects of the theory while deliberately avoiding overly sophisticated mathematics. Finally, we briefly discuss how the divergent integrals encountered in quantum field theory can be reinterpreted as issues concerning the domain of distributions.