# Type-4 spinors: transmuting from Elko to single-helicity spinors

## Abstract

In this communication we briefly report an unexpected theoretical discovery which emerge from the mapping of Elko mass-dimension-one spinors into single helicity spinors. Such procedure unveils a class of spinor which is classified as type-4 spinor field within Lounesto classification. In this paper we explore the underlying physical and mathematical contents of the type-4 spinor.

## 1 Introduction

According to the Lounesto classification there are six disjoint classes of spinors [1]. The first three classes stand for regular spinors fields for spin 1 / 2 fermions and the remaining three classes stand for singular spinors like type-4, Majorana and Weyl spinors fields. Until the present moment, all the above mentioned spinors (and the associated quantum fields) are well-established except for the type-4 spinor. Since it was proposed by Lounesto [1], no physical or experimental evidence was found, therefore, we do not know what kind of particle is described by the mentioned class of spinors. The classical path that several physicists and mathematicians normally use to construct such spinors is based on the so-called *inversion theorem* [2, 3], however, the last mentioned protocol do not explicitly provide the spinor’s components. In other words, it only shows how components might be connected, therefore, we do not have enough information about how it emerges from the space-time symmetries. It does not bring information about the related dynamics and the behaviour under certain discrete symmetries like parity (*P*), charge conjugation (*C*) and time-reversal (*T*). Consequently, all the above mentioned features remain open for investigation. The focus of the present communication is to find explicitly a spinor to fill this gap in the Lounesto classification and shed some light on several related areas.

Although there is no quantum field operator constructed based on type-4 spinors, it is expected that it does not respect the full Lorentz symmetries. This spinors are potential candidates to describe dark matter, dark energy and to construct mass-dimension-one fermions [4]. While there is compelling evidence in astrophysics and cosmology that most of the mass of the Universe is composed of a new form of non baryonic dark matter, there is a lack of evidence of the existence of new physics at LHC (Large Hadron Collider) and other particle physics experiments. On the theory side, many specific models with new particles and interactions beyond the standard model have been proposed to account for dark matter [5]. The type-4 spinors fields are wealthy regarding their mathematical structure [1, 6] and from the Physics point of view have been found to be the particle corresponding to the solution of the Dirac equation in *f*(*R*) gravity with torsion [7].

In this paper we report an unexpected theoretical discovery which comes from mapping of Mass-Dimension-One (MDO) spinors into single helicity spinors. MDO spinors were firstly introduced in the literature around the 90’s. Elko spinor fields compose a complete set of dual helicity spinor that are neutral under charge conjugation operator. Because of this, Elko fields have suppressed interactions with the Standard Model particles. In such a way, these fields are dark with respect to the matter and gauge fields of the Standard Model, interacting only with gravity and the Higgs boson [8]. The net result of such mapping protocol surprisingly enough give rise to the type-4 spinors and also to regular spinors (it can be accomplished by fixing an arbitrary phase parameter). This given protocol must be understood as an attempt to incorporate dark matter to the Standard Model. The present paper give allowance to transmute from the MDO field to another kind of MDO field and also from MDO field to Dirac field. Evidently given task was obscure until the present days because type-4 spinors are “born” from MDO spinors. Therefore, only Elko spinors were considered to belong to a class of mass-dimension-one fields, however, type-4 spinors carry the same mass-dimensionality as Elko’s do, in the meantime, they are single helicity spinors. Interestingly enough, it is the first time that we observe a single helicity spinor endowed with mass-dimension-one feature. These new objects endowed with mass-dimension-one are strong candidates to describe dark matter and perhaps dark energy. The difference among type-4 spinors and Elko spinors lies on the fact that the first type have no suppressed interaction with the Standard Model of particle physics. So, the last statement translates into the fact that it does not exhibit neutral character under charge conjugation operator.

In the present work we explore the underlying details related to the construction of the type-4 spinors and their physical information. Here we report what we have deciphered about the associated dynamics, spin sums calculation and the encoded physical information, looking towards a possible quantization. The paper is organized as follows: Sect. 2 is an in-depth overview about MDO and single helicity spinors, highlighting the main aspects of both spinors. In the Sect. 3 we define and establish the mapping protocol employed throughout this work. In the Sect. 4 we explore the physical information, i.e., we evaluate the bilinear covariants. We reserved Sect. 5 to study the dynamics associated with type-4 spinors and compute the spin sums, based on the new observed aspects inherited from Elko spinors. Finally, Sect. 6 we conclude.

## 2 Elementary overview on mass-dimension-one and single-helicity spinors

Since they were proposed, the Dirac spinors are well known to describe a specific particle: the electron. Most text-books spend dozen of pages defining the Dirac spinor main features, e.g., the related dynamics, the quantum field operator, bilinear structures, interactions and couplings, etc. Dirac spinors emerge naturally from the symmetries of the full Poincaré group,^{1} and the particle interpretation depends on some given properties under transformations by certain group transformations [9].

*S*and

*A*stand for self-conjugated and anti-self-conjugated under charge conjugation operation (\(\mathcal {C}\lambda ^{S/A}_{h}=\pm \lambda ^{S/A}_{h}\)) and the lower index

*h*stands for the helicity of each component [14].

## 3 Mapping Elko spinors into single helicity spinors: the rise of a hidden class

^{2}

^{3}

^{4}However, it is possible to keep the right-hand helicity and then change the helicity of the left-hand component, performing the following transformation

## 4 Classifying the \(\psi \) spinors

Consider the Minkowski spacetime \((M,\eta _{\mu \nu })\) and its tangent bundle *TM*. Denoting sections of the exterior bundle by \(\sec \Lambda (TM)\), and given a *k*-vector \(a \in \sec \Lambda ^k(TM)\), the reversion is defined by \(\tilde{a}=(-1)^{|k/2|}a\), and the grade involution by \(\hat{a}=(-1)^{k}a\), where |*k*| stands for the integral part of *k*. By extending the Minkowski metric from \(\sec \Lambda ^1(TM)=\sec T^*M\) to \(\sec \Lambda (TM)\), and considering \(a_1,a_2 \in \sec \Lambda (V)\), the left contraction is given by \({g}(a \lrcorner a_1,a_2)={g}(a_1 ,\tilde{a}\wedge a_2 ). \) The well-known Clifford product between the dual of a vector field \( v \in \sec \Lambda ^1(TM)\) and a multivector is given by \( v a = v \wedge a+ v a \), defining thus the spacetime Clifford algebra \(C\ell _{1,3}\). The set \(\{{e}_{\mu }\}\) represents sections of the frame bundle \(\mathbf {P}_{{\mathrm {SO}}_{1,3}^{e}}(M)\) and \(\{\gamma ^{\mu }\}\) can be further thought as being the dual basis \(\{{e}_{\mu }\}\), namely, \(\gamma ^{\mu }({e}_{\mu })=\delta ^\mu _{\;\nu }\). Classical spinors are objects of the space that carries the usual \(\tau =(1/2,0)\oplus (0,1/2)\) representation of the Lorentz group, that can be thought as being sections of the vector bundle \(\mathbf {P}_{{\mathrm {Spin}}_{1,3}^{e}}(M)\times _{\tau }\mathbb {C}^{4}\).

*Z*given by

- 1.
\(\sigma \ne 0,\;\;\;\;\;\omega \ne 0.\)

- 2.
\(\sigma \ne 0, \;\;\;\;\;\omega =0.\)

- 3.
\(\sigma =0,\;\;\;\;\;\omega \ne 0.\)

- 4.
\(\sigma =0=\omega ,\;\;\;\;\;K_{\mu }\ne 0,\;\;\;\;\;S_{\mu \nu }\ne 0.\)

- 5.
\(\sigma =0=\omega ,\;\;\;\;\;K_{\mu }=0,\;\;\;\;\;S_{\mu \nu }\ne 0.\)

- 6.
\(\sigma =0=\omega ,\;\;\;\;\;K_{\mu }\ne 0,\;\;\;\;\;S_{\mu \nu }=0.\)

As showed in Ref. [15], Elko mass-dimension-one spinors do not belong to the Lounesto classification due to the fact that they are endowed with different dual structure and dual-helicity features, a specific classification for these spinors remains open. In this communication we constructed the first mass-dimension-one fermion which carry single-helicity feature. We have developed a connection between the Lounesto classification and a special spinor type within the mass-dimension-one classification, it is the only known fermion that compose this new classification. If more mass-dimension-one fermions (like Elko) exist, another mapping treatment can be developed and perhaps a general, robust and complete connection betwen Lounesto and MDO classification could be established.

## 5 Type-4 spinors underlying features: dynamic and spin sums

As mentioned before, type-4 spinors do not satisfy the Dirac equation as previosly demonstrated by Lounesto [1]. Thus, we concluded that they are not eigenspinors of the parity operator (\(P=m^{-1}\gamma _{\mu }p^{\mu }\)), they also do not belong to a set of eigenspinors of the charge conjugation operator due to the fact that their representation spaces are not connected by the Wigner time-reversal operator (\(\Theta \)). Moreover, as a characteristic inherited from Elko spinors, type-4 spinors only satisfy the Klein–Gordon wave equation.

^{5}

*S*(

*p*) matrix is defined as

*S*(

*p*) is not Lorentz invariant. The

*S*(

*p*) operator present the following properties \(S^{2}(p)=\mathbb {1}\) and \(S^{-1}(p)\) exist. An interesting fact that must be stressed is that

Note that it is also possible to redefine the dual structure of the above spinors, ensuring a Lorentz invariant spin sum and providing new physical information. Such redefinition give allowance for a Lorentz invariant theory but the new bilinear structure do not respect FPK identities so we have chosen to abandon the dual redefinition.

## 6 Final remarks

The present paper reports the discovery of type-4 spinors. We constructed the first case of mass-dimension-one fermions endowed with single helicity. Our theoretical results create a new class of single-helicity spinors, that are not eigenspinor of charge conjugation, parity or time-reversal symmetry. In additon, we have shown that the type-4 spinors only carry relevant physical information and satisfy the FPK identities if we chooses to set \(\kappa =1\), otherwise, we can not guarantee it. We also showed that type-4 spinors do not fulfil the Dirac dynamic and found a non-conventional dynamics, which is given on the right hand side of the spin sums presented in (26) and (28).

It is also possible to redefine the dual structure which lead us to an invariant and non-vanishing norm under the orthonormal relation. This reveals new dynamics and new physical observables codified in this structure. Nevertheless, the price to be paid is that the related bilinear structures do not respect the FPK identities and should be deformed as was shown in [15]. In this scenario, these spinors would not belong to the Lounesto classification anymore.

From the physical point of view we highlight that this mapping procedure shows that once we break the chirality symmetry of the Elko spinor we are breaking the link between the representation spaces of the resulting spinors. Therefore, these new spinors only inherit some characteristics of the originating spinor, and, in addition, such symmetry break brings a breach between the representation spaces, making the resulting spinors even more exotic.

## Footnotes

- 1.
By full Poincaré group we mean boosts, rotations, space-time translations, parity and time-reversal symmetries.

- 2.
A very same mathematical treatment is also valid for the \(\lambda ^{A}\) spinors

*mutatis mutandis*. - 3.
To reduce the notation we have defined the boost parameters as \(\mathcal {B}^{\pm }(p)\equiv \sqrt{\frac{E+m}{2m}}\big (1\pm \frac{p}{E+m}\big )\).

- 4.
The symbols \(\uparrow \) and \(\downarrow \) stand for the positive and negative helicity of the components.

- 5.
Please note that here we have fixed \(\kappa _1=\kappa _2= \kappa _3=\kappa _4=1\), in agreement with the discussions made in Sect. 4.

## Notes

### Acknowledgements

The authors express their gratitude to Prof. Julio Marny Hoff da Silva and Prof. José Abdalla Helayël-Neto for careful reading the entire first draft of the manuscript, providing many insightful suggestions and questions. We also thanks to Dr. Oswaldo Miranda for fruitful discussions in the final stage of the work. RJBR thanks CAPES and CNPq (Grant Number 155675/2018-4) for the financial support and CHCV thanks CNPq (PCI Grant Number 300381/2018-2) for the financial support.

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