Possible treatment of the ghost states in the Lee-Wick standard model

In this work, we employ the techniques used to cure the indefinite norm problem in pseudo-Hermitian Hamiltonians to show that the ghost states in a higher derivative scalar field theory are not real ghosts. For the model under investigation, an imaginary auxiliary field is introduced to have an equivalent non-Hermitian two-field scalar theory. We were able to calculate exactly the positive definite metric operator η for the quantum mechanical as well as the quantum field versions of the theory. While the equivalent Hamiltonian is non-Hermitian in a Hilbert space characterized by the Dirac sense inner product, it is, however, a Hermitian in a Hilbert space endowed with the inner product . The main feature of the latter Hilbert space is that the propagator has the correct sign (no Lee-Wick fields). Moreover, the calculated metric operator diagonalizes the Hamiltonian in the two fields (no mixing). We found that the Hermiticity of the calculated metric operator to lead to the constrain M>2m for the two Higgs masses, in agreement with other calculations in the literature. Besides, our mass formulas coincide with those obtained in other works (obtained by a very different regime but with the existence of ghost states), which means that our positive normed Hamiltonian form preserves the mass spectra.