  Very recently, the Lee-Wick standard model has been introduced as a non-SUSY
extension of the Standard model which solves the Hierarchy problem. In this
model, each field kinetic term attains a higher derivative term. Like any
Lee-Wick theory, this model suffers from existence of Ghost states. In this
work, we consider a prototype scalar field theory with its kinetic term has a
higher derivative term, which mimics the scalar sector in the Lee-Wick Standard
model. We introduced an imaginary auxiliary field to have an equivalent
non-Hermitian two-field scalar field theory. We were able to calculate the
positive definite metric operator $\eta$ in quantum mechanical and quantum
field versions of the theory in a closed form. While the Hamiltonian is
non-Hermitian in a Hilbert space with the Dirac sense inner product, it is
Hermitian in a Hilbert space endowed by the inner product $<n|\eta|m>$ as well
as having a correct-sign propagator (no Lee-Wick fields). Besides, the obtained
metric operator also diagonalizes the Hamiltonian in the two fields (no
mixing). Moreover, the Hermiticity of $\eta$ constrained the two Higgs masses
to be related as $M>2m$, which has been obtained in another work using a very
different regime and thus supports our calculation. Also, an equivalent
Hermitian (in the Dirac sense) Hamiltonian is obtained which has no Ghost
states at all, which is a forward step to make the Lee-Wick theories more
popular among the Physicists.
