Fields with continuously distributed mass

We describe local field theories with continuously distributed mass. Such models can be realized as models in d > 4 space-time with Poincare invariance only in four-dimensional space-time. We also discuss some possible phenomenological consequences. Namely, we show that the Higgs boson phenomenology in the SM extension with continuously distributed Higgs boson mass can differ in a drastic way from the standard Higgs boson phenomenology.