A Complete Set of 4-Point Amplitudes in the Constructive Standard Model

Abstract

We present a complete set of 4-point amplitudes in the constructive Standard Model at tree level. Any 4-point amplitude can be obtained from the results presented here by a suitable choice of masses, a permutation of the particles (by crossing symmetry), and a reversal of the momenta of the outgoing particles. We have validated all of these amplitudes by comparing with Feynman diagrams for a variety of masses, scattering energy and angles, and helicities of the photons and gluons, when they are in the initial states. The standard constructive techniques work for these amplitudes without the need for any contact terms and indeed, contact terms are not allowed. Only three 4-point vertices are used (allowed), involving the Higgs boson and the W and Z bosons. When external photons or gluons are present, the amplitude simplifies to a single spinor-product structure, present in the numerator. In a few cases, however, the propagator structure is more complex, with different terms depending on the charge or color structure. In the case of internal photons or gluons, we find that the massless limit of a massive photon or gluon diagram gives the correct result in all cases. We have additionally found that using the $x$ factor directly gives the correct result in all cases and agrees with the massless limit calculation.