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We discuss at the first time ultraviolet behaviors of effective couplingconstants using renormalization group equations with mass dependent betafunctions in extended Higgs models, in which the scalar coupling constants mayblow up at a scale much below the Planck scale. It is important to preciselyevaluate such a scale as the upper bound where the models can hold. We findthat by the natural treatment of the threshold in our method the upper bound inextended Higgs models can be much higher than that obtained by using the massindependent beta function, especially when such an upper bound is relativelylow.

New application of the mass-dependent analysis for renormalization group equation to extended Higgs models