CP violation, massive neutrinos, and its chiral condensate: new results from Snyder noncommutative geometry

TL;DR

This paper explores how Snyder noncommutative geometry influences CP violation and neutrino mass generation, proposing a novel mechanism that extends beyond the Standard Model by linking chiral condensates to fundamental spacetime structures.

Contribution

It demonstrates that a modified Dirac equation within Snyder geometry can generate neutrino masses via CP violation and describes a gauge theory for massive neutrino fields, introducing a new mass generation mechanism.

Findings

Neutrinos acquire mass through CP violation in Snyder geometry.

The model predicts a chiral condensate of massive neutrinos.

The approach extends the Standard Model framework.

Abstract

The Snyder model of a noncommutative geometry due to a minimal scale $\ell$, e.g. the Planck or the Compton scale, yields $\ell^2$-shift within the Einstein Hamiltonian constraint, and $\gamma^5$-term in the free Dirac equation violating CP symmetry manifestly. In this paper the Dirac equation is reconsidered. In fact, there is no any reasonable cause for modification of the Minkowski hyperbolic geometry of a momentum space. It is the consistency -- in physics phase space, spacetime (coordinates), and momentum space (dynamics) are independent mathematical structures. It is shown that the modified Dirac equation yields the kinetic mass generation mechanism for the left- and right-handed Weyl chiral fields, and realizes the idea of neutrinos receiving mass due to CP violation. It is shown that the model is equivalent to the gauge field theory of composed two 2-flavor massive fields. The global chiral symmetry spontaneously broken into the isospin group leads to the chiral condensate of massive neutrinos. This result is beyond the Standard Model, but in general can be included into the theory of elementary particles and fundamental interactions.