  Motivated by the discovery of a new scalar field and amelioration of the
electroweak vacuum stability ascribed to a singlet scalar field embedded in the
standard model (SM), we examine the implication of the perturbative unitarity
in the SM with a singlet scalar field. Taking into account the full
contributions to the scattering amplitudes, we derive unitarity conditions on
the scattering matrix which can be translated into bounds on the masses of the
scalar fields. In the case that the singlet scalar field develops vacuum
expectation value (VEV), we get the upper bound on the singlet scalar mass
varying with the mixing between the singlet and Higgs scalars. On the other
hand, the mass of the Higgs scalar can be constrained by the unitarity
condition in the case that the VEV of the singlet scalar is not generated.
Applying the upper bound on the Higgs mass to the scenario of the unitarized
Higgs inflation, we discuss how the unitarity condition can constrain the Higgs
inflation. The singlet scalar mass is not constrained by the unitarity itself
when we impose Z_2 in the model because of no mixing with the Higgs scalar.
But, regarding the singlet scalar field as a cold dark matter candidate, we
derive upper bound on the singlet scalar mass by combining the observed relic
abundance with the unitarity condition.
