Real Z2-bigradings, Majorana modules, and the standard model action

Abstract

The action functional of the standard model of particle physics is intimately related to a specific class of first order differential operators called Dirac operators of Pauli type (“Pauli–Dirac operators”). The aim of this article is to carefully analyze the geometrical structure of this class of Dirac operators on the basis of real Dirac operators of simple type. On the basis of simple type Dirac operators, it is shown how the standard model action (STM action) may be viewed as generalizing the Einstein–Hilbert action in a similar way that the Einstein–Hilbert action is generalized by a cosmological constant. Furthermore, we demonstrate how the geometrical scheme presented allows to naturally incorporate also Majorana mass terms within the standard model. For reasons of consistency, these Majorana mass terms are shown to dynamically contribute to the Einstein–Hilbert action by a “true” cosmological constant. Due to its specific form, this cosmological constant can be very small. Nonetheless, this cosmological constant may provide a significant contribution to dark matter/energy. In the geometrical description presented, this possibility arises from a subtle interplay between Dirac and Majorana masses.