Hitoshi Murayama writes in his 221A Lecture Notes on Spin
How do we choose spin when you introduce a field, then? A consistent ( i.e. , renormalizable) quantum field theory can include only spin 0, 1/2, and 1. Renormalizable interactions are only those interactions that can appear without extra suppressions of $G_N E^2 / ̄\hbar c^5 5 = ( E/ 10^{19} \mathrm{\ GeV^2}) \ll 1$. Therefore, there is a reason why we see only particles of spin 1/2 and 1. Well, what about spin 0? We have not seen any fundamental particle of spin 0 yet. We are looking for one: the Higgs boson, which is expected to permeate our entire Universe, dragging the foot of all quarks, leptons, W and Z bosons, making it impossible for them to reach the speed of light. It is expected to be found this decade thanks to higher-energy accelerators being built. Once it is found, it exhausts all theoretical possibilities of spin 0, 1/2, and 1.
Is this really true? And if yes, why?
