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

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 colour 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 present many attractive points such as the unified picture of the gauge and Higgs field as the generalized connection on the discrete spaceM4×ZN. This approach leads us to unified picture of gauge and Higgs fields as the generalized connection. The standard model needsN=2 discrete space for reconstruction in this formalism. LRSM is still alive as a model with the intermediate symmetry of the spontaneously brokenSO(10) grand unified theory (GUT).N=3 discrete space is needed for the reconstruction of LRSM to include two Higgs bosonsφ andξ which are as usual transformed as (2, 2*, 0) and (1, 3, −2) underSU(2)L×SU(2)R×U(1)Y, respectively.ξ is responsible to makeνR Majorana fermion and so well explains the seesaw mechanism. Up and down quarks have different masses through the vacuum expectation value ofφ.