Reconstruction of the standard model with classical conformal invariance in noncommutative geometry

Abstract

In this paper, we derive the standard model with classical conformal invariance from the Yang--Mills--Higgs model in noncommutative geometry (NCG). In the ordinary context of the NCG, the {\it distance matrix} $M_{nm}$ which corresponds to the vacuum expectation value of Higgs fields is taken to be finite. However, since $M_{nm}$ is arbitrary in this formulation, we can take all $M_{nm}$ to be zero. In the original composite scheme, the Higgs field itself vanishes with the condition $M_{nm} = 0$. Then, we adopt the elemental scheme, in which the gauge and the Higgs bosons are regarded as elemental fields. By these assumptions, all scalars do not have vevs at tree level. The symmetry breaking mechanism will be implemented by the Coleman--Weinberg mechanism. As a result, we show a possibility to solve the hierarchy problem in the context of NCG. Unfortunately, the Coleman--Weinberg mechanism does not work in the SM Higgs sector, because the Coleman--Weinberg effective potential becomes unbounded from below for $m_{t} > m_{Z}$. However, viable models can be possible by proper extensions.