Elko Spinor Fields Research Articles

5D Elko spinor field non-minimally coupled to nonmetricity in f(Q) gravity

This paper aims to investigate the localization of the five-dimensional spinor field known as Elko (dual-helicity eigenspinors of the charge conjugation operator) by employing a Yukawa-like geometrical coupling in which the Elko field is non-minimally coupled to nonmetricity scalar Q. We adopt the braneworld scenarios in which the first-order formalism with sine-Gordon and linear superpotentials is employed to obtain the warp factors. A linear function supports the zero-mode trapping within the geometric coupling, leading to the same effective potential as the scalar field. Moreover, an exotic term must be added to obtain real-valued massive modes. Such modes are investigated through the Schrödinger-like approach.

Introduction to localization of Elko spinor fields on branes

In 2004, Dharam Vir Ahluwalia and Daniel Grumiller discovered a dark matter candidate called Elko: spin one-half fermions with mass dimension one. These fermions are described by a spinorial Klein-Gordon formalism and very different from electrons or neutrinos. In this review, we give a brief introduction to the localization of Elko spinor fields on branes. First, we briefly review some brane-world theories and the localization of bulk matter fields on branes. Then, after a brief introduction to Elko spinor fields, we focus on the localization of Elko fields on branes in three cases: the free Elko fields, the Elko fields with Yukawa-like coupling, and finally, the Elko fields with non-minimal coupling.

Localization of Elko spinor fields in tachyonic de Sitter braneworld models

In this paper, we investigate the localization of a five-dimensional (5D) Elko spinor field in de Sitter tachyonic braneworld models. These branes are generated by gravity coupled to a tachyonic bulk scalar field and include a de Sitter cosmological background induced on the 3D-brane. Using Yukawa-type coupling mechanism we show that a free massless Elko spinor field cannot be localized on tachyonic de Sitter branes while massive Elko fields are localized if their bulk mass obeys an upper bound. Also by introducing a tachyonic function F(T) as Yukawa interaction term between the Elko spinor and the background tachyonic scalar field we find that localized massless zero modes are given in terms of general Heun functions. Furthermore, it is shown that employing a new derivative coupling term in the action of Elko spinor fields leads to the Elko field localization on tachyonic de Sitter branes.

The massive Elko spinor field in the de Sitter braneworld model

The Elko spinor field is a spin 1/2 fermionic quantum field with a mass dimension introduced as a candidate of dark matter. In this work, we study the localization of Elko fields on a de Sitter thick brane constructed by a canonical or phantom scalar field. By presenting the mass-independent potentials of Kaluza-Klein (KK) modes in the corresponding Schrödinger equations, it is shown that the Elko spinor field with a five- dimensional mass term can be localized on de Sitter branes. Also, by using a Yukawa type coupling term, we find that the Elko spinor field can be localized on this brane with a particular coupling constant. Furthermore, it is shown that the zero mode of the Elko field is localized on the de Sitter brane through a Yukawa geometrical coupling term.

Localization of massless Elko spinor fields on de Sitter thick branes

In this work, we investigate the localization of a five-dimensional (5D) free massless Elko spinor field on de Sitter thick branes that are generated by a canonical or phantom scalar field. It is shown that the zero mode of Elko field cannot be localized on the de Sitter brane. In order to circumvent this problem, we employ a nonminimal coupling term in the 5D Elko spinor field action and find that the Elko field can be localized on the de Sitter brane with this mechanism.

The Clifford Algebra of Physical Space and Elko Spinors

Elko spinors are eigenspinors of the charge conjugation operator. In this work we use the Clifford algebra of the physical space in order to formulate the theory of Elko spinors and use a procedure analog to Ryder’s derivation of Dirac equation to come up with an equation for Elko spinor fields. Unlike other works in the literature where an equation for Elko spinor fields has been studied, in this work we obtain a first order differential equation for Elko spinor fields, which resembles but is different from the Dirac equation.

Exotic Elko on string-like defects in six dimensions

We analyse the trapping of eigenspinors of the charge conjugation operator with dual helicity (Elko), in thin and thick string-like models with codimension-2. Elko spinor fields describe mass-dimension-one fermions in four dimensions (and, correspondingly, mass-dimension-two fermions in six dimensions), that represent natural dark-matter prime candidates. This dark spinor has many applications, from particle physics to cosmology. On the other hand, six-dimensional brane-world models have, among other prominent features, the spontaneous confinement of free spin-1 fields and a mechanism that explains the mass hierarchy of fundamental fermions. In this paper, we use scalar couplings in order to confine the zero mode of Elko in six dimensions. Moreover, we use the Elko dark-spinor features to propose an exotic coupling in order to remove the complex-valued terms in the massive Kaluza-Klein modes. Hence, we show that six-dimensional models can resolve the main issues of Elko fields confinement presented in five dimensions.

Λ(t) cosmology induced by a slowly varying Elko field

In this work the exact Friedmann-Robertson-Walker equations for an Elko spinor field coupled to gravity in an Einstein-Cartan framework are presented. The torsion functions coupling the Elko field spin-connection to gravity can be exactly solved and the FRW equations for the system assume a relatively simple form. In the limit of a slowly varying Elko spinor field there is a relevant contribution to the field equations acting exactly as a time varying cosmological model Λ(t)=Λ*+3β H2, where Λ* and β are constants. Observational data using distance luminosity from magnitudes of supernovae constraint the parameters Ωm and β, which leads to a lower limit to the Elko mass. Such model mimics, then, the effects of a dark energy fluid, here sourced by the Elko spinor field. The density perturbations in the linear regime were also studied in the pseudo-Newtonian formalism.

Could Elko Spinor Fields Induce VSR Symmetry in the DKP (Meson) Algebra?

It is shown that the meson algebra can be faced as the tensor product of Clifford algebras and, then, by constructing the DKP field by means of Elko spinors, we demonstrate that the symmetries of the so called very special relativity are inherited by the DKP field.

ELKO applications in cosmology

A brief review is presented about the ELKO spinor field applied to cosmology, with the main results and limitations of the theory. As a simple application, we have analyzed a model involving the interaction of the ELKO spinor field with dark matter in the universe from a dynamical system approach. When the system is rewritten in terms of the deceleration parameter q and under the assumption that such parameter is nearly constant for some different stages of the evolution, stability points were found for different types of interaction between dark matter and ELKO spinors. Within this new analysis several interesting scenarios are possible, depending on the interaction term. For example, if it is assumed that the equation of the state parameter of the ELKO field is of the phantom type, ωϕ < -1, the current acceleration of the universe can be driven by the decay of dark matter particles into ELKO field. Furthermore, even for ωϕ > 0, the inflationary period can be driven by the decay of the inflaton field (described by the ELKO spinor) into dark matter particles.

Classification of singular spinor fields and other mass dimension one fermions

In this paper, we investigate the constraint equations of the Lounesto spinor fields classification and show that it can be used to completely characterize all the singular classes, which can potentially accommodate further mass dimension one fermions, beyond the well known Elko spinor fields. This result can be useful for two purposes: Besides a great abridgement in the classification of a given spinor field, we provide a general form of each class of spinor fields, which can be used furthermore to search for a general classification of spinors dynamics.

VSR symmetries in the DKP algebra: The interplay between Dirac and Elko spinor fields

VSR symmetries are here naturally incorporated in the DKP algebra on the spin-0 and the spin-1 DKP sectors. We show that the Elko (dark) spinor fields structure plays an essential role on accomplishing this aim, unravelling hidden symmetries on the bosonic DKP fields under the action of discrete symmetries.

Some remarks on the attractor behaviour in ELKO cosmology

Recent results on the dynamical stability of a system involving the interaction of the ELKO spinor field with standard matter in the universe have been reanalysed, and the conclusion is that such system does not exhibit isolated stable points that could alleviate the cosmic coincidence problem. When a constant parameter δ related to the potential of the ELKO field is introduced in the system however, stable fixed points are found for some specific types of interaction between the ELKO field and matter. Although the parameter δ is related to an unknown potential, in order to satisfy the stability conditions and also that the fixed points are real, the range of the constant parameter δ can be constrained for the present time and the coincidence problem can be alleviated for some specific interactions. Such restriction on the ELKO potential opens possibility to apply the ELKO field as a candidate to dark energy in the universe, and so explain the present phase of acceleration of the universe through the decay of the ELKO field into matter.

Elko Spinor Fields and Massive Magnetic Like Monopoles

In this paper we recall that by construction Elko spinor fields of λ and ρ types satisfy a coupled system of first order partial differential equations (csfopde) that once interacted leads to Klein-Gordon equations for the λ and ρ type fields. Since the csfopde is the basic one and since the Klein-Gordon equations for λ and ρ possess solutions that are not solutions of the csfopde for λ and ρ we infer that it is legitimate to attribute to those fields mass dimension 3/2 (as is the case of Dirac spinor fields) and not mass dimension 1 as previously suggested in recent literature (see list of references). A proof of this fact is offered by deriving the csfopde for the λ and ρ from a Lagrangian where these fields have indeed mass dimension 3/2. Taking seriously the view that Elko spinor fields due to its special properties given by their bilinear invariants may be the description of some kind of particles in the real world a question then arises: what is the physical meaning of these fields? Here we proposed that the fields λ and ρ serve the purpose of building the fields $\mathcal {K}\in \mathcal {C}\ell ^{0} (M,\eta )\otimes {\mathbb {R}}_{1,3}^{0}$ and $\mathcal {M}\in \sec \mathcal {C}\ell ^{0}(M,\eta ) \otimes {\mathbb {R}}_{1,3}^{0}$ (see (37)). These fields are electrically neutral but carry magnetic like charges which permit them to couple to a $su(2)\simeq {\textit {spin}}_{3,0}\subset {\mathbb {R}}_{3,0}^{0}$ valued potential $\mathcal {A}\in \sec {\textstyle \bigwedge \nolimits ^{1}} T^{\ast }M\otimes {\mathbb {R}}_{3,0}^{0}$ . If the field $\mathcal {A}$ is of short range the particles described by the $\mathcal {K}$ and $\mathcal {M}$ fields may be interacting and forming condensates of zero spin particles analogous to dark matter, in the sense that they do not couple with the electromagnetic field (generated by charged particles) and are thus invisible. Also, since according to our view the Elko spinor fields as well as the $\mathcal {K}$ and $\mathcal {M}$ fields are of mass dimension 3/2 we show how to calculate the correct propagators for the $\mathcal {K}$ and $\mathcal {M}$ fields. We discuss also the main difference between Elko and Majorana spinor fields, which are kindred since both belong to class five in Lounesto classification of spinor fields. Most of our presentation uses the representation of spinor fields in the Clifford bundle formalism, which makes very clear the meaning of all calculations.

Comment on “Self-interacting Elko dark matter with an axis of locality”

In this comment we show that the statement in Ahluwalia et al. [Phys. Rev. D 83, 065017 (2011)] that the existence of Elko spinor fields imply in an axis of locality is equivocated. The anticommutator ${\ensuremath{\Lambda}(\mathbf{x},t),\ensuremath{\Pi}(\mathbf{x},t)}$ is strictly local.

Spinor dark energy and cosmological coincidence problem

Recently, the so-called Elko spinor field has been proposed to be a candidate of dark energy. It is a non-standard spinor and has unusual properties. When the Elko spinor field is used in cosmology, its unusual properties could bring some interesting consequences. In the present work, we discuss the cosmological coincidence problem in the spinor dark energy models by using the dynamical system method. Our results show that the cosmological coincidence problem should be taken to heart in the investigations of spinor dark energy models.

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

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.

FROM DIRAC ACTION TO ELKO ACTION

A fundamental action, representing a mass dimension-transmuting operator between Dirac and ELKO spinor fields, is performed on the Dirac Lagrangian, in order to lead it into the ELKO Lagrangian. Such a dynamical transformation can be seen as a natural extension of the Standard Model that incorporates dark matter fields. The action of the mass dimension-transmuting operator on a Dirac spinor field, that defines and introduces such a mapping, is shown to be a composition of the Dirac operator and the nonunitary transformation that maps Dirac spinor fields into ELKO spinor fields, defined in J. Math. Phys.48, 123517 (2007). This paper gives allowance for ELKO, as a candidate to describe dark matter, to be incorporated in the Standard Model. It is intended to present for the first time, up to our knowledge, the dynamical character of a mapping between Dirac and ELKO spinor fields, transmuting the mass dimension of spin one-half fermionic fields from 3/2 to 1 and from 1 to 3/2.

ELKO SPINOR FIELDS: LAGRANGIANS FOR GRAVITY DERIVED FROM SUPERGRAVITY

Dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields) belong — together 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 Einstein–Palatini, 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.

WHERE ARE ELKO SPINOR FIELDS IN LOUNESTO SPINOR FIELD CLASSIFICATION?

This paper proves that from the algebraic point of view ELKO spinor fields belong together with Majorana spinor fields to a wider class, the so-called flagpole spinor fields, corresponding to the class 5, according to Lounesto spinor field classification. We show moreover that algebraic constraints imply that any class 5 spinor field is such that the 2-component spinor fields entering its structure have opposite helicities. The proof of our statement is based on Lounesto general classification of all spinor fields, according to the relations and values taken by their associated bilinear covariants, and can eventually shed some new light on the algebraic investigations concerning dark matter.