At tree level there is no momenum integral because there are no propagators. Heuristically, we have a particle at rest (say, a Higgs) with a given 4 momentum, that decays into a particle and antiparticle. By symmetry the decay products have equal masses/energies and opposite 3momenta, hence all momenta in the system are constrained. the amplitude is just proportional to a dimensionless coupling times the external fermion factors. When this is summed for spins, the fermion factors become a trace of something with dimensions of mass squared.
