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Electroweak symmetry breaking through bosonic seesaw mechanism in a classically conformal extension of the Standard Model

We suggest the so-called bosonic seesaw mechanism in the context of a classically conformal $U(1)_{B-L}$ extension of the Standard Model with two Higgs doublet fields. The $U(1)_{B-L}$ symmetry is radiatively broken via the Coleman-Weinberg mechanism, which also generates the mass terms for the two Higgs doublets through quartic Higgs couplings. Their masses are all positive but, nevertheless, the electroweak symmetry breaking is realized by the bosonic seesaw mechanism. We analyze the renormalization group evolutions for all model couplings, and find that a large hierarchy among the quartic Higgs couplings, which is crucial for the bosonic seesaw mechanism to work, is dramatically reduced toward high energies. Therefore, the bosonic seesaw is naturally realized with only a mild hierarchy, if some fundamental theory, which provides the origin of the classically conformal invariance, completes our model at some high energy, for example, the Planck scale. The requirements for the perturbativity of the running couplings and the electroweak vacuum stability in the renormalization group analysis as well as for the naturalness of the electroweak scale, we have identified the regions of model parameters. For example, the scale of the $U(1)_{B-L}$ gauge symmetry breaking is constrained to be $\lesssim 100$ TeV, which corresponds to the extra heavy Higgs boson masses to be $\lesssim 2$ TeV. Such heavy Higgs bosons can be tested at the Large Hadron Collider in the near future.