Renormalization group equation, the naturalness problem, and the understanding of the Higgs mass term

Abstract

The naturalness problem might be studied on the complex two dimensional plane with the technique of dimensional regularization(DREG). The Renormalization group equation(RGE) of the Higgs mass on the plane suggests the Higgs mass approaches zero at ultraviolate (UV) scale, the scale can be Planck scale when the top quark pole mass $M_{t}=168$ GeV. The real issue of the naturalness problem in the sense of Wilsonian renormalization group method is not about quadratic divergences but the rescaling effect. The Higgs mass can be considered to be one composed mass. All terms in the lagrangian in this scenario are marginal terms and no relevant terms are left, thus no rescaling effect to cause the naturalness problem anymore. RGE of the vacuum expectation value (VEV) in the Landau gauge up to two-loop order is studied. Scale-dependent behavior of the composed Higgs mass shows that we can have one tiny Higgs mass at high energy scale, even around the Planck scale, when $M_{t}\leq170.7$ GeV.