Elko Spinor Fields: Lagrangians for Gravity Derived From Supergravity

Abstract

Dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields) belongtogether with Majorana spinor fields -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. There exists only six such disjoint classes: the first three corresponding to Dirac spinor fields, and the other three respectively corresponding to flagpole, flag-dipole and Weyl spinor fields. Using the mapping from ELKO spinor fields to the three classes Dirac spinor fields, it is shown that the Einstein-Hilbert, the EinsteinPalatini, and the Holst actions can be derived from the Quadratic Spinor Lagrangian (QSL), as the prime Lagrangian for supergravity. The Holst action is related to the Ashtekar's quantum gravity formulation. To each one of these classes, there corresponds a unique kind of action for a covariant gravity theory. Furthermore we consider the necessary and sufficient conditions to map Dirac spinor fields (DSFs) to ELKO, in order to naturally extend the Standard Model to spinor fields possessing mass dimension one. As ELKO is a prime candidate to describe dark matter and can be obtained from the DSFs, via a mapping explicitly constructed that does not preserve spinor field classes, we prove that -in particular -the Einstein-Hilbert, Einstein-Palatini, and Holst actions can be derived from the QSL, as a fundamental Lagrangian for supergravity, via ELKO spinor fields. The geometric meaning of the mass dimension-transmuting operator -leading ELKO Lagrangian into the Dirac Lagrangian -is also pointed out, together with its relationship to the instanton Hopf fibration.

Spinor fields classification in arbitrary dimensions and new classes of spinor fields on 7-manifolds

“…A complete overview of this classification with further applications in field theory and gravitation can be found in [27], being also further explored in the context of black hole thermodynamics [28]. Indeed, black hole tunnelling methods were studied for Elko spinor fields as special type of flagpoles [28], which play an essential role in constructing various theories of gravity naturally arising from supergravity [29,30]. Flagpoles spinor fields and Lounesto spinor field classification are moreover discussed in the context of the instanton Hopf fibration [31], and experimental signatures of the type-5 spinors in such a classification are related to the Higgs boson at LHC [32].…”

An identification of the Dirac operator with the parity operator

“…Elko spinor fields have been investigated in the last decade [8][9][10][11][12][13][14][15][16][17][18] and has been further applied to gravity, cosmology, and field theory. [8][9][10][11][12][13][14][15][16][17][18] We recall that its starting point, in Ref.…”

“…[8][9][10][11][12][13][14][15][16][17][18] We recall that its starting point, in Ref. [2], is to take eigenspinors of the charge conjugation operator instead of the parity operator.…”

On the bilinear covariants associated to mass dimension one spinors

“…However, the spin sums are invariant under transformations of the very special relativity (VSR) [2][3][4]. There are several areas in which Elko spinors have been studied, from accelerator physics [5][6][7][8] to cosmology [9][10][11][12][13][14][15][16][17][18][19]. In para e-mail: hoff@feg.unesp.br b e-mail: ccoronado@feg.unesp.br c e-mail: rodolforogerio@feg.unesp.br d e-mail: e.scatena@ufsc.br ticular, the appreciation of new dual structures brings about interesting possibilities within the algebraic scope [20,21].…”