Pauli-type coupling of spinors and curved spacetime

Abstract

Abstract In this study we prove that the Pauli interaction---which is associated with a length parameter---emerges when the minimal coupling recipe is applied to the non-degenerate Dirac Lagrangian.
The conventional Dirac Lagrangian is rendered non-degenerate if supplemented by a particular term quadratic in the derivatives of the spinors.
For dimensional reasons, this non-degenerate Dirac Lagrangian is associated with a length parameter $\ell$.
It yields---just as the conventional degenerate Dirac Lagrangian---the standard free Dirac equation in Minkowski space as the $\ell$ terms cancel when setting up the Euler-Lagrange equations.
However, if the Dirac spinor is minimally coupled to gauge fields, then the length parameter~$\ell$ becomes a physical coupling constant yielding novel interactions.
For the U($1$) symmetry the Pauli coupling of fermions to electromagnetic fields arises, modifying the fermion's magnetic moment.
We discuss the impact of these findings on electrodynamics, and estimate the upper bound of the length parameter~$\ell$ from the yet existing discrepancy between the (SM) theory and measurement of the anomalous magnetic moment of light leptons.
This, and also recent studies of the renormalization theory, suggest that the Pauli coupling of leptons to the electromagnetic field is a necessary ingredient in QED, supporting the notion of a fundamental nature of the non-degenerate Dirac Lagrangian.

In a second step we then investigate how analogous ``Pauli-type'' couplings of gravity and matter arise if fermions are embedded in curved spacetime.
The diffeomorphism and Lorentz invariance of that system leads to an anomalous spin-torsion interaction and a curvature-dependent mass correction.
We calculate the mass correction in the De~Sitter geometry of vacuum with the cosmological constant $\Lambda$.
Possible implications for the existence of effective non-zero rest masses of neutrinos are addressed.
Finally, an outlook on the impact of mass correction on the physics of ``Big Bang'' cosmology, black holes, and of neutron stars is provided.