ELKO, Flagpole and Flag-Dipole Spinor Fields, and the Instanton Hopf Fibration

Abstract

In a previous paper we explicitly constructed a mapping that leads Dirac spinor fields to the dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields). ELKO spinor fields are prime candidates for describing dark matter, and belong to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class-(5), according to Lounesto spinor field classification, based on the relations and values taken by their associated bilinear covariants. Such a mapping between Dirac and ELKO spinor fields was obtained in an attempt to extend the Standard Model in order to encompass dark matter. Now we prove that such a mapping, analogous to the instanton Hopf fibration map $S^3... S^7\to S^4$, prevents ELKO to describe the instanton, giving a suitable physical interpretation to ELKO. We review ELKO spinor fields as type-(5) spinor fields under the Lounesto spinor field classification, explicitly computing the associated bilinear covariants. This paper is also devoted to investigate some formal aspects of the flag-dipole spinor fields, which correspond to the class-(4) under the Lounesto spinor field classification. In addition, we prove that type-(4) spinor fields (corresponding to flag-dipoles) and ELKO spinor fields (corresponding to flagpoles) can also be entirely described in terms of the Majorana and Weyl spinor fields. After all, by choosing a projection endomorphism of the spacetime algebra Cl(1,3) it is shown how to obtain ELKO, flagpole, Majorana and Weyl spinor fields, respectively corresponding to type-(5) and -(6) spinor fields, uniquely from limiting cases of a type-(4) (flag-dipole) spinor field, in a similar result obtained by Lounesto.