  We revisit the problem of the helicity-0 ghost mode of massive graviton in
the de Sitter background. In general, the presence of a ghost particle, which
has negative energy, drives the vacuum to be unstable through pair production
of ghost particles and ordinary particles. In the case that the vacuum state
preserves the de Sitter invariance, the number density created by the pair
production inevitably diverges due to unsuppressed ultra-violet(UV)
contributions. In such cases one can immediately conclude that the model is not
viable. However, in the massive gravity theory we cannot construct a vacuum
state which respects the de Sitter invariance. Therefore the presence of a
ghost does not immediately mean the breakdown of the model. Explicitly
estimating the number density and the energy density of particles created by
the pair production of two conformal scalar particles and one helicity-0 ghost
graviton, we find that these densities both diverge. However, since models with
helicity-0 ghost graviton have no de Sitter invariant vacuum state, it is
rather natural to consider a UV cutoff scale in the three-dimensional momentum
space. Then, even if we take the cutoff scale as large as the Planck scale, the
created number density and energy density are well suppressed. In many models
the cutoff scale is smaller than the Planck scale. In such models the created
number density and the energy density are negligiblly small as long as only the
physics below the cutoff scale is concerned.
