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Classically conformal U(1)$^\prime$ extended Standard Model and Higgs vacuum stability

We consider the minimal U(1)$^\prime$ extension of the Standard Model (SM) with conformal invariance at the classical level, where in addition to the SM particle contents, three generations of right-handed neutrinos and a U(1)$^\prime$ Higgs field are introduced. In the presence of the three right-handed neutrinos, which are responsible for the seesaw mechanism, this model is free from all the gauge and gravitational anomalies. The U(1)$^\prime$ gauge symmetry is radiatively broken via the Coleman-Weinberg mechanism, by which the U(1)$^\prime$ gauge boson ($Z^\prime$ boson) mass as well as the Majorana mass for the right-handed neutrinos are generated. The radiative U(1)$^\prime$ symmetry breaking also induces a negative mass squared for the SM Higgs doublet to trigger the electroweak symmetry breaking. In this context, we investigate a possibility to solve the SM Higgs vacuum instability problem. The model includes only three free parameters (U(1)$^\prime$ charge of the SM Higgs doublet, U(1)$^\prime$ gauge coupling and $Z^\prime$ boson mass), for which we perform parameter scan, and identify a parameter region resolving the SM Higgs vacuum instability. We also examine naturalness of the model. The heavy states associated with the U(1)$^\prime$ symmetry breaking contribute to the SM Higgs self-energy. We find an upper bound on $Z^\prime$ boson mass, $m_{Z^\prime} \lesssim 6$ TeV, in order to avoid a fine-tuning severer than 10 % level. The $Z^\prime$ boson in this mass range can be discovered at the LHC Run-2 in the near future.