New Incorporation of the Strong Interaction in NCG and Standard Model

TL;DR

This paper reconstructs the Standard Model within a non-commutative geometric framework, incorporating strong interactions and generation mixing, resulting in GUT-like predictions without an actual GUT.

Contribution

It introduces a novel method to embed strong interactions and fermion mixing into non-commutative geometry, reproducing GUT-scale relations without a grand unified theory.

Findings

Reproduces GUT-scale Weinberg angle $ heta_W$

Predicts Higgs mass relation $m_H = rac{4}{\sqrt{3}} m_W \\sin\theta_W$

Achieves GUT-like coupling unification without GUT

Abstract

The standard model is reconstructed by new method to incorporate strong interaction into our previous scheme based on the non-commutative geometry. The generation mixing is also taken into account. Our characteristic point is to take the fermion field so as to contain quarks and leptons all together which is almost equal to that of SO(10) grand unified theory(GUT). The space-time $M_4\times Z_2$; Minkowski space multiplied by two point discrete space is prepared to express the left-handed and right-handed fermion fields. The generalized gauge field $A(x,y)$ written in one-differential form extended on $M_4\times Z_2$ is well built to give the correct Dirac Lagrangian for fermion sector. The fermion field is a vector in 24-dimensional space and gauge and Higgs fields are written in $24\times24$ matrices. At the energy of the equal coupling constants for both sheets $y=\pm$ expected to be amount to the energy of GUT scale, we can get $\sin^2\theta_{_{W}}=3/8$ and $m_{_{H}}=\sqrt{2}m_{_{W}}$. In general, the equation $m\ma{H}=(4/\sqrt {3})m\ma{W}\sin\theta\ma{W}$ is followed. Then, it should be noticed that the same result as that of the grand unified theory such as SU(5) or SO(10) GUT is obtained without GUT but with the approach based on the non-commutative geometry and in addition the Higgs mass is related to other physical quantities as stated above.