Dimensional Regularization is Generic

Abstract

The absence of the quadratic divergence in the Higgs sector of the Standard Model in the dimensional regularization is usually regarded to be an exceptional property of a specific regularization. To understand what is going on in the dimensional regularization, we illustrate how to reproduce the results of the dimensional regularization for the λφ 4 theory in the more conventional regularization such as the higher derivative regularization; the basic postulate involved is that the quadratically divergent induced mass, which is independent of the scale change of the physical mass, is kinematical and unphysical. This is consistent with the derivation of the Callan-Symanzik equation, which is a comparison of two theories with slightly different masses, for the λφ 4 theory without encountering the quadratic divergence. We thus suggest that the dimensional regularization is generic in a bottom-up approach starting with a successful low-energy theory. We also define a modified version of the mass independent renormalization for a scalar field which leads to the homogeneous renormalization group equation. Implications of the present analysis on the Standard Model at high energies and the presence or absence of SUSY at LHC energies are briefly discussed.

Naturalness in see-saw mechanism and Bogoliubov transformation

“…[13] are working. As a resolution of the issue of quadraric divergences in such a scheme, one may argue for the generic nature of the dimensional regularization (i.e., a suitably formulated higher derivative regularization reproduces the main results of the dimensional regularization) [14], which is also related to the derivation of the Callan-Symanzik equation [15,16] without encountering quadratic divergences [15]. The fact that the quadratic divergence is not important in a properly formulated Wilsonian renormalization group flow has also been shown in Ref.…”

“…In the picture of Bogoliubov quasi-fermions, the starting theory has the very large mass scale M as in (14) and the neutrino mass is given by the enormous cancellation of large masses. This suggests that the fine tuning is a relevant issue.…”

“…where v is the vacuum value of H. This neutrino mass term may be compared with Weinberg's dimension five operator [21] if one chooses M ∆ around the GUT mass scale such as ∼ 10 15 GeV. The fine tuning in this model appears if one considers a oneloop tadpole correction (with heavy ∆-field drawing a loop) to the quartic coupling [14]. To maintain the Standard Model Higgs mechanism with the ordinary vacuum value v so that the neutrino mass m L is kept small, one needs to satisfy |λ 3 M 2 ∆ | < v 2 , namely, |λ 3 | < v 2 /M 2 ∆ = 10 −26 , which is also disposed of by a fine tuning of the induced term minus a finite local counter term in the Higgs mass term m 2 H H 2 , expressed by the symbolic notation M 2 ∆ − M 2 ∆ = v 2 , without making |λ 3 | very small.…”

“…Instead of making |λ 4 | very small, this condition is replaced by a tuning of the tree-level Higgs mass term m 2 LR Trϕ † LR ϕ LR to absorb D 2 in a symbolic notation D 2 − D 2 = v 2 . At the one-loop level, the tadpole with heavy ϕ R drawing a loop induces [14]. Thus the fine tuning in the symbolic notation D 2 − D 2 = v 2 in the Higgs mass term is required at the one-loop level also, by noting M ϕ R ∼ D in the model [2].…”

Naturalness made easy: two-loop naturalness bounds on minimal SM extensions

“…Refs. [24][25][26][27][28][29][30][31][32][33]. In particular, the large cancellation between the unmeasurable bare mass and the cutoff regulator contribution is considered an unphysical artifact of the regularisation and renormalisation procedure for the EFT.…”