Can Electroweak Theory Without A Higgs Particle Be Renormalizable?

Whether there exists a massive electroweak (EW) theory, without a Higgs spontaneous symmetry breaking mechanism, that is gauge invariant and renormalizable is investigated. A Stueckelberg formalism for massive $W$ and $Z$ bosons is used to derive a gauge invariant EW theory. Negative energy scalar fields that emerge from the gauge invariance of the Lagrangian are removed by invoking an indefinite metric in Hilbert space. A unitary S-matrix and a positive energy spectrum can be obtained by using the PT symmetric formulation of the pseudo-Hermitian Hamiltonian. The theory predicts that if for a system of particles the scalar boson energy $E_s < μ=λ^{1/2}M_W$, where $λ$ is a gauge parameter and $M_W$ is the $W$ boson mass, then as $λ\rightarrow\infty$ the scalar boson mass $μ=λ^{1/2}M_W$ tends to infinity. The theory is perturbatively renormalizable and does not violate longitudinally polarized $W_L W_L\rightarrow W_L W_L$ scattering in the energy range $E < μ=λ^{1/2}M_W$ for which the scalar bosons have decoupled and they have an undetected mass. This means that with this scenario the EW theory can only be treated as an effective renomalizable theory and not as a UV complete theory.

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