Extensions of the noncommutative Standard Model and the weak order one condition

TL;DR

This paper explores a weaker form of the first-order condition in noncommutative geometry extensions of the Standard Model, reducing fine-tuning and impacting neutrino mass terms in specific models.

Contribution

It introduces a weaker first-order condition applicable to extended models, addressing fine-tuning issues and the presence of Majorana mass terms.

Findings

Reduces fine-tuning in Yukawa matrices.

Eliminates Majorana mass terms in the Pati-Salam model.

Supports the consistency of noncommutative gauge theories.

Abstract

In the derivation of the Standard Model from the axioms of Noncommutative Geometry, the scalar sector is given by a finite Dirac operator which has to satisfy the so-called \emph{first-order condition}. However, the general solution to this constraint still has unphysical terms which must be fine-tuned to zero. Moreover, the first-order condition generally does not survive in extensions to models with gauge groups larger that $U(1)\times SU(2)\times SU(3)$. In this paper we show that in the $U(1)_{\rm B-L}$-extension one can implement a weaker form of the first-order condition which we argue is necessary in order for Noncommutative Gauge Theory to make sense at all, and that this condition reduce the amount of fine-tuning to the off-diagonal terms in the Yukawa mass matrices for the leptons and quarks. We also show that this condition eliminates the Majorana mass terms for right-handed neutrinos when it is applied to the Pati-Salam model.