Non-commutative geometry and the standard model vacuum

TL;DR

This paper explores the space of Dirac operators in non-commutative geometry for the standard model, considering extended neutrino states and general fluctuations, revealing restrictive conditions on vacuum solutions.

Contribution

It extends the spectral action framework by including right-handed neutrinos and general fluctuations, analyzing their impact on the standard model vacuum structure.

Findings

Non-trivial vacua with Majorana-like lepton masses are possible.

Equations of motion are too restrictive for standard model mass matrices.

General fluctuations lead to new insights into the vacuum structure.

Abstract

The space of Dirac operators for the Connes-Chamseddine spectral action for the standard model of particle physics coupled to gravity is studied. The model is extended by including right-handed neutrino states, and the S0-reality axiom is not assumed. The possibility of allowing more general fluctuations than the inner fluctuations of the vacuum is proposed. The maximal case of all possible fluctuations is studied by considering the equations of motion for the vacuum. Whilst there are interesting non-trivial vacua with Majorana-like mass terms for the leptons, the conclusion is that the equations are too restrictive to allow solutions with the standard model mass matrix.