Reconstruction of the spontaneously broken gauge theory in non-commutative geometry

TL;DR

This paper extends non-commutative geometry methods to reconstruct the standard model and left-right symmetric models without extra discrete spaces, unifying gauge and Higgs fields and explaining neutrino masses.

Contribution

It introduces a modified scheme incorporating strong interactions via direct product internal symmetry, eliminating the need for extra discrete spaces for certain models.

Findings

Reconstruction of the standard model with N=2 discrete space.

Reconstruction of left-right symmetric model with N=3 discrete space.

Explanation of neutrino Majorana masses and seesaw mechanism.

Abstract

The scheme previously proposed by the present authors is modified to incorporate the strong interaction by affording the direct product internal symmetry. We do not need to prepare the extra discrete space for the color gauge group responsible for the strong interaction to reconstruct the standard model and the left-right symmetric gauge model(LRSM). The approach based on non-commutative geometry leads us to presents many attractive points such as the unified picture of the gauge and Higgs field as the generalized connection on the discrete space; Minkowski space multipied by N-points discrete space. This approach leads us to unified picture of gauge and Higgs fields as the generalized connection. The standard model needs N=2 discrete space for reconstruction in this formalism. \lr is still alive as a model with the intermediate symmetry of the spontaneously broken SO(10) grand unified theory(GUT). N=3 discrete space is needed for the reconstruction of LRSM to include two Higgs bosons $\phi$ and $\xi$ which are as usual transformed as (2,2*,0)$ and (1,3,-2) under left-handed SU(2)x right-handed SU(2)x U(1), respectively. xi is responsible to make the right handed-neutrino Majorana fermion and so well explains the seesaw mechanism. Up and down quarks have the different masses through the vacuum expectation value of phi.