Spinors with torsion and matter$-$antimatter asymmetry

Abstract

The conservation law for the orbital plus spin angular momentum of a free Dirac particle in curved spacetime requires that the affine connection has the antisymmetric part: the torsion tensor, which extends general relativity to the Einstein$-$Cartan theory of gravity. In the presence of torsion, the Dirac equation becomes a nonlinear, cubic equation in the spinor wave function. We show that the energy eigenvalues of the corresponding Hamiltonian as functions of the momentum are different for the fermion and antifermion components of the spinor, violating charge conjugation symmetry, and also depend on the helicity. Consequently, particles of matter and antimatter have different dispersion relations and therefore different masses. This mass difference increases with density and becomes significant near the Cartan density, which existed in the early Universe. Because antimatter particles were more massive than matter particles, they were also slower during pair production in the early Universe and therefore had higher cross sections for gravitational capture by primordial black holes. This difference might have led to the matter$-$antimatter imbalance in the observable Universe.