Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification

TL;DR

This paper explores a noncommutative geometric framework that naturally leads to a Pati-Salam unification model, extending the Standard Model by unifying leptons and quarks and predicting new Higgs fields and symmetry breaking mechanisms.

Contribution

It classifies product noncommutative spaces without the first order condition, resulting in a Pati-Salam model that unifies leptons and quarks within a geometric framework.

Findings

Derives a Pati-Salam type model from noncommutative geometry.

Predicts additional Higgs fields responsible for symmetry breaking.

Unifies leptons and quarks in a four-color scheme.

Abstract

The assumption that space-time is a noncommutative space formed as a product of a continuous four dimensional manifold times a finite space predicts, almost uniquely, the Standard Model with all its fermions, gauge fields, Higgs field and their representations. A strong restriction on the noncommutative space results from the first order condition which came from the requirement that the Dirac operator is a differential operator of order one. Without this restriction, invariance under inner automorphisms requires the inner fluctuations of the Dirac operator to contain a quadratic piece expressed in terms of the linear part. We apply the classification of product noncommutative spaces without the first order condition and show that this leads immediately to a Pati-Salam SU(2)_{R}x SU(2)_{L}x SU(4) type model which unifies leptons and quarks in four colors. Besides the gauge fields, there are 16 fermions in the (2,1,4)+(1,2,4) representation, fundamental Higgs fields in the (2,2,1), (2,1,4) and (1,1,1+15) representations. Depending on the precise form of the order one condition or not there are additional Higgs fields which are either composite depending on the fundamental Higgs fields listed above, or are fundamental themselves. These additional Higgs fields break spontaneously the Pati-Salam symmetries at high energies to those of the Standard Model.