Will the $PT$ Symmetric and Non-Hermitian $\phi^4$ Theory Solve the Hierarchy and Triviality Problems in the Standard Model?

Abstract

Very recently, the Large Hadron Collider was turned on. There, the experiments are aiming to test different scenarios for elementary particles interactions from SUSY, Extra dimensions to others. In fact, SUSY was invented to kill the conceptual problems existing in the conventional Standard model \textit{i.e.} the Hierarchy problem. However, in recent years, certain theories which was rejected in the past like the wrong sign $\phi^{4}$ theory is now well known to be a truly physically acceptable theory. Here, we analyze the renormalization group flow of the different parameters in the theory. We find that, rather than the conventional $\phi^{4}$ theory and because the theory is asymptotically free, it does not sufferer from the catastrophic blow up of the mass parameter for UV scales. This feature greatly recommend that this theory is a plausible candidate to play the role of the $SU(2)\times U(1)$ symmetry breaking in the standard model. If this picture will agree with the experimental results from LHC, then the Higgs particle was massless in the far past. Moreover, the cosmological constant problem as a Hierarchy problem may be solved too. However, when trying to calculate the metric operator, we expected that the equivalent Hermitian theory may attain non-remormalizabe terms and thus the theory is incalculable. In fact, we show that there exists a Hermitian and renormalizable equivalent theory though of indefinite metric. We reformulated the Klein-Gordon equation in such a way that the wrong sign in the propagator attains a new physical meaning that the Higgs particle is made from exotic matter.