Standard Model in Differential Geometry on Discrete Space M4*Z3

TL;DR

This paper reconstructs the Standard Model within a generalized differential geometric framework on a discrete space $M_4 imes Z_3$, integrating strong interaction and spontaneous symmetry breaking through non-commutative geometry.

Contribution

It introduces a novel approach to formulate the Standard Model using generalized differential calculus on a discrete space, unifying gauge and Higgs fields as connections in non-commutative geometry.

Findings

Unified gauge and Higgs fields as generalized connections.

Reproduction of Yang-Mills-Higgs and Dirac Lagrangians.

Two models with different Higgs particle assignments.

Abstract

Standard model is reconstructed using the generalized differential calculus extended on the discrete space $M_4\times Z_3$. $Z_3$ is necessary for the inclusion of strong interaction. Our starting point is the generalized gauge field expressed as $A(x,y)=\!\sum_{i}a^\dagger_{i}(x,y){\bf d}a_i(x,y), (y=0,\pm)$, where $a_i(x,y)$ is the square matrix valued function defined on $M_4\times Z_3$ and ${\bf d}=d+{\chi}$ is generalized exterior derivative. We can construct the consistent algebra of $d_{\chi}$ with the introduction of the symmetry breaking function $M(y)$ and the spontaneous breakdown of gauge symmetry is coded in ${d_{\chi}}$. The gauge field $A_\mu(x,y)$ and Higgs field $\Phi(x,y)$ are written in terms of $a_i(x,y)$ and $M(y)$, which might suggest $a_i(x,y)$ to be more fundamental object. The unified picture of the gauge field and Higgs field as the generalized connection in non-commutative geometry is realized. Not only Yang-Mills-Higgs lagrangian but also Dirac lagrangian, invariant against the gauge transformation, are reproduced through the inner product between the differential forms. Two model constructions are presented, which are distinguished in the particle assignment of Higgs field $\Phi(x,y)$.