Is there literature on a continuous mass spectrum for the Higgs field? Various masses for the Higgs field are compatible with experiment, but is it possible that the Higgs field is not observable because it has a continuous mass spectrum?
Work in the 60s and 70s on free fields that have a continuous mass spectrum, often called generalized free fields, established that there is no simple particle interpretation if the mass spectrum is not delta function valued, which I take it can be described as due to destructive interference between the different mass components. [I'm taking that largely from R. F. Streater, Rep. Prog. Phys., 38, 771-846 (1975).]
Is the possibility that the Higgs has a continuous mass spectrum to be found in the literature, or is there some sufficiently obvious anomaly or other reason why it is not considered in any detail?
 A: The continuous mass spectrum is a possibility, but it is essentially equivalent to a large-extra-dimension model with an unconfined Higgs field in the extra dimensions.
If you have a scalar particle in 1 extra dimension, and it interacts with particles on a world which is 3+1 dimensional, it's propagator between interaction vertices is
$$\int {dk_5\over 2\pi} {i\over {k_5^2 + k^2 + m^2 + i\epsilon}} = {-1\over 2\sqrt{k^2 + m^2 +i\epsilon}}$$
The result is an unparticle spectrum with a continuous density of mass past m^2, the continuous density is exactly the possible $k_5^2$ values. These types of propagation functions give unitary quantum field theories when they are the dimensional reductions of higher dimensional theories with appropriate external potentials in the extra dimensions.
These types of theories are not renormalizable with scalars, because scalar interactions are already marginal in 4 dimensions. They are phenomenologically difficult to imagine because they are experimentally as easy to exclude as large extra dimensions.
These are theoretically interesting however as counterexamples to overgeneral theorems about possible types of quantum fields.
