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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 SMparticle contents, three generations of right-handed neutrinos and aU(1)$^\prime$ Higgs field are introduced. In the presence of the threeright-handed neutrinos, which are responsible for the seesaw mechanism, thismodel is free from all the gauge and gravitational anomalies. The U(1)$^\prime$gauge symmetry is radiatively broken via the Coleman-Weinberg mechanism, bywhich the U(1)$^\prime$ gauge boson ($Z^\prime$ boson) mass as well as theMajorana mass for the right-handed neutrinos are generated. The radiativeU(1)$^\prime$ symmetry breaking also induces a negative mass squared for the SMHiggs doublet to trigger the electroweak symmetry breaking. In this context, weinvestigate a possibility to solve the SM Higgs vacuum instability problem. Themodel includes only three free parameters (U(1)$^\prime$ charge of the SM Higgsdoublet, U(1)$^\prime$ gauge coupling and $Z^\prime$ boson mass), for which weperform parameter scan, and identify a parameter region resolving the SM Higgsvacuum instability. We also examine naturalness of the model. The heavy statesassociated with the U(1)$^\prime$ symmetry breaking contribute to the SM Higgsself-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 thenear future.

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