Left-right symmetric gauge model in a noncommutative geometry on $M_4\times Z_4$

TL;DR

This paper reconstructs a left-right symmetric gauge model within a novel noncommutative geometry framework on a discrete space, representing all particles and fields in matrix form, and explains symmetry breaking and mass generation mechanisms.

Contribution

It introduces a new scheme of noncommutative differential geometry that allows a unified matrix representation of gauge, Higgs fields, and fermions in a left-right symmetric model.

Findings

Successful representation of all fields in 24x24 matrices.

Derivation of Higgs potential and interaction terms within the new NCG scheme.

Explanation of symmetry breaking and mass generation consistent with the standard model.

Abstract

The left-right symmetric gauge model with the symmetry of $SU(3)_c\times SU(2)_L\times SU(2)_R\times U(1)$ is reconstructed in a new scheme of the noncommutative differential geometry (NCG) on the discrete space $M_4\times Z_4$ which is a product space of Minkowski space and four points space. The characteristic point of this new scheme is to take the fermion field to be a vector in a 24-dimensional space which contains all leptons and quarks. Corresponding to this specification, all gauge and Higgs boson fields are represented in $24\times 24$ matrix forms. We incorporate two Higgs doublet bosons $h$ and $SU(2)_R$ adjoint Higgs $\xi_R$ which are as usual transformed as $(2,2^\ast,0)$ and $(1,3,-2)$ under $SU(2)_L\times SU(2)_R\times U(1)$, respectively. Owing to the revise of the algebraic rules in a new NCG, we can obtain the necessary potential and interacting terms between these Higgs bosons which are responsible for giving masses to the particles included. Among the Higgs doublet bosons, one CP-even scalar boson survives in the weak energy scale and other four scalar bosons acquire the mass of the $SU(2)_R\times U(1)$ breaking scale, which is similar to the situation in the standard model. $\xi_R$ is responsible to spontaneously break $SU(2)\ma{R} \times U(1)$ down to $U(1)\ma{Y}$ and so well explains the seesaw mechanism. Up and down quarks have the different masses through the vacuum expectation value of $h$.