# Questing mass dimension 1 spinor fields

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## Abstract

This work deals with new classes of spinors of mass dimension 1 in Minkowski spacetime. In order to accomplish it, Lounesto’s classification scheme and the inversion theorem are going to be used. The algebraic framework shall be revisited by explicating the central point performed by the Fierz aggregate. Then the spinor classification is generalized in order to encompass the new mass dimension 1 spinors. The spinor operator is shown to play a prominent role to engender the new mass dimension 1 spinors, accordingly.

## Keywords

Spinor Operator Minkowski Spacetime Mass Dimension Spinor Field Dirac Particle## 1 Introduction

There is a spinor classification due to Lounesto [1], which is particularly interesting for physicists due to its twofold ubiquitous aspect: on the one hand it is based upon bilinear covariants, and thus upon physical observables. On the other hand, by a peculiar multivector structure—the Fierz aggregate—that leads to the so-called boomerang [1], a quite elegant geometrical interpretation may be added to the classification. Moreover, with the aid of the boomerang it is possible likewise to prove that there are precisely six different classes of spinors in Lounesto’s classification [1]. The most general forms of the respective spinors in each class were introduced in [2]. Lounesto’s spinor classification was further employed to derive all the Lagrangians for gravity from the quadratic spinor Lagrangian [3]. Higher dimensional spaces have a similar spinor classification [4], however, the so-called geometric Fierz identities [5] obstruct the proliferation of new spinors classes in higher dimensions [4].

Within the Lounesto classification, a specific bilinear covariant plays a crucial role, since it cannot be zero. This bilinear represents the current density, at least for the case of a regular spinor describing the electron. Its components read \(\mathbf{J}=J_\mu e^\mu = \psi ^\dagger \gamma _0\gamma _\mu \psi e^\mu \), where \(\psi \) denotes a spinor and \(e^\mu \) is a dual basis in \(\mathcal{{C}}{\ell }_{1,3}\). Additionally, it is valuable to remark that \(\mathbf{J}=J_\mu e^\mu \) is essential for the definition of the boomerang structure. Regarding the electron theory, it is straightforward to realize the physical argument to explain why \(\mathbf{J}\) must not vanish. Indeed, \(\mathbf{J}\) is the conserved current in this case and therefore if \(\mathbf{J}=0\) there is no associated particle [6]. In particular the time component \(J_0 = \psi ^\dagger \psi \) provides the probability density of the electron, and when integrated over the spacetime it should obviously be non-null.

One of the main points that shall be pursued in this work is that \(\mathbf{J}\) can be understood as a conserved current solely when the considered spinor obeys the usual dynamics rules by the Dirac equation, namely, it is an eigenspinor of the Dirac operator or, equivalently, it is described by the Dirac Lagrangian. The canonical mass dimension in this case is the same mass dimension 3 / 2 associated to usual spin-1/2 fermions in the standard model. Since we are looking for possible manifestations of mass dimension 1 fermions in Minkowski spacetime, it is indeed possible to set \(\mathbf{J}=0\), accordingly. In fact, by accomplishing it, even the previously mentioned algebraic argument precluding new spinor classes may be circumvented. Nevertheless, in this novel context, we should emphasize that the underlying dynamics shall not be dictated by the well-known Dirac equation. As the construction is relativistic, the spinors arising from the analysis with \({\mathbf J}=0\) shall respect a priori merely the Klein–Gordon equation. Actually, in a very conventional scheme, they must do so. Hence, the epigraph is now explained: the resulting spinors must have mass dimension 1. Clearly by “mass dimension” we mean the canonical mass dimension of the associated quantum field, which inherits this property from the dynamics respected by its expansion coefficients.

Mass dimension 1 spinors have attracted attention mainly due to the fact that they can be coupled only to gravity, and to scalar fields as well, in a perturbatively renormalizable way. It thus makes it suitable for exploration under the ensign of dark matter. Mass dimension 1 spinors in Minkowski spacetime known in the literature are the so-called Elko spinors, which have been studied in a comprehensive context. They comprise prominent applications in 4D gravity and cosmology [3, 7, 8, 9, 10, 11], and in brane-world models as well [12, 13], besides their exotic counterparts [14, 15]. Moreover, in spite of the robust and rich framework already developed [16, 17, 18, 19, 20], Elko has been predicted to be measured in Higgs processes at LHC [21, 22] and explored in tunneling methods concerning black holes [23]. Massive spin-1/2 fields of mass dimension were obtained by constructing quantum fields from higher-spin Elkos, however, these fields are still linked to the Elko construct. We stress, however, that the spinors to be found here are intrinsically different from the Elkos by the simple fact that \(\mathbf{J}\ne 0\) in the Elko case.

The classification of mass dimension 1 spinors is performed by a possible and consistent modification in the Lounesto classification. However, in order to have an explicit form for them it is necessary the use of the so-called inversion theorem [24, 25].

This paper is organized as follows: in the next section the main steps of the framework which supports our analysis shall be revisited, namely the standard Lounesto classification and the inversion theorem. In Sect. 3 we show the existence of three new classes of mass dimension 1 spinors, obtaining the algebraic form in each case accordingly. In the last section we make our concluding remarks and present a brief outlook.

## 2 The framework

In order to properly address the problem to be approached and solved, it is pivotal to review some key aspects of the standard formalism, highlighting the structures to be studied and generalized. To start, Lounesto’s spinor classification shall be revisited, and subsequently the inversion theorem algorithm shall be thereafter employed, accordingly.

### 2.1 The Lounesto’s spinors classification and generalizations

Consider the Minkowski spacetime \((M,\eta _{\mu \nu })\) and its tangent bundle *TM*. Denoting sections of the exterior bundle by \(\sec \varLambda (TM)\), given a *k*-vector \(a \in \sec \varLambda ^k(TM)\), the reversion is defined by \(\tilde{a}=(-1)^{|k/2|}a\), while the grade involution reads \(\hat{a}=(-1)^{k}a\), where |*k*| stands for the integral part of *k*. By extending the Minkowski metric from \(\sec \varLambda ^1(TM)=\sec T^*M\) to \(\sec \varLambda (TM)\), and considering \(a_1,a_2 \in \sec \varLambda (V)\), 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 for (the dual of) a vector field \( v \in \sec \varLambda ^1(TM)\) and a multivector is prescribed by \( v a = v \wedge a+ v \lrcorner a \), defining thus the spacetime Clifford algebra \(\mathcal{{C}} \ell _{1,3}\). 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 of 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, which can be thought of as being sections of the vector bundle \(\mathbf {P}_{\mathrm {Spin}_{1,3}^{e}}(M)\times _{\tau }\mathbb {C}^{4}\).

The above bilinear covariants in the Dirac theory are interpreted, respectively, as the mass of the particle (\(\sigma \)), the pseudo-scalar (\(\omega \)) relevant for parity-coupling, the current of probability (\({\mathbf {J}}\)), the direction of the electron spin (\({\mathbf {K}}\)), and the probability density of the intrinsic electromagnetic moment (\({\mathbf {S}}\)) associated to the electron. The most important bilinear covariant for our goal here is \({\mathbf {J}}\), although with a different meaning. In fact, in the next section we shall set \({\mathbf {J}}=0\), enabling the extension of the standard Lounesto classification to this case.

*h*is a real number. The multivector as expressed in Eq. (7) is a boomerang [19]. By inspecting the condition (6) we see that for singular spinors \({\mathbf {Z}}^2=0\). However, in order for the FPK identities to hold it is also necessary that both conditions

^{1}\({\mathbf {J}}^2=0\) and \((\mathbf {s}+h\gamma _{0123})^2=-1\) must be satisfied. These considerations are important in order to constrain the possible spinor classes.

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

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

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

- 4.
\(\sigma =0=\omega ,\) \(\mathbf K \ne 0,\) \(\quad \mathbf S \ne 0\);

- 5.
\(\sigma =0=\omega ,\) \(\mathbf K =0,\) \(\quad \mathbf S \ne 0\);

- 6.
\(\sigma =0=\omega ,\) \(\mathbf K \ne 0,\) \(\quad \mathbf S =0\).

As is clear from the above reasoning, \({\mathbf {J}}\ne 0\) is much more a matter of taste. There is instead algebraic necessity of demonstrating the existence of six different classes. In fact, however, a non-vanishing \({\mathbf {J}}\) is indispensable only for the regular spinor case. As mentioned, the above classification makes use of this constraint in all the cases, since the very idea of the classification was to categorize spinors which could be related to Dirac particles in some respect. As far as we leave this (physical) concept, more spinors can be found.

By taking \({\mathbf {J}}=0\), we cannot describe Dirac particles anymore. Therefore, the spinors arising from this consideration must be merely ruled by the Klein–Gordon dynamics and, therefore, they must have mass dimension 1. We finalize by stressing that the resulting spinors (see Sect. 3) have to be singular, as in the contrary case they would violate the FPK identities and, besides, the geometrical aspects underlying the algebraic structure need to be reconsidered.

### 2.2 The inversion theorem

It is important to stress that the alluded inversion is not unique, since we can choose an arbitrary phase \(\varphi \), and the constant spinor \(\xi \). Thus, concerning the inversion program, it is fairly important to bear in mind that it is useful within the formal algebraic context. In the next section, we shall apply the inversion theorem in order to recover mass dimension 1 spinors coming from a suitable modification of Lounesto’s scheme.

## 3 Algebraic construction of new spinors

*f*and \({e}_{0123}\). This yields the equality \( q_1f + q_2{e}_{0123}f = fq_1 + {e}_{0123}fq_2, \) evincing that the left ideal \(\mathcal{{C}} \ell _{1,3}f\) is in fact a right module over \(\mathbb {K}\) with a basis \(\{f,{e}_{0123}f\}\). Moreover, the orthonormal basis \(\{e_{\mu }\}\) has an immediate standard representation,

*f*and the multivector \({e}_{0123}f\):

^{2}Taking into account Eq. (15), it is usual, in order to reduce the degrees of freedom of \(\varPsi \), to define the following relation:

As remarked in Sect. 2, the Lounesto classification is based upon the FPK identities. As far as these relations are satisfied, novel possibilities involving spinors can be considered. We propose a classification of new spinors, arising from considering that the bilinear covariant \({\mathbf {J}}\) is always null and the aggregate associated (\({\mathbf {Z}}\)) is no longer a boomerang as well. On the other hand, the bilinear covariants still satisfy the identities (2). As emphasized by the previous analysis, this last requirement is important, since we shall express the new algebraic spinors functional form.

The consideration that the bilinear covariants must satisfy the FPK identities with \({\mathbf {J}}=0\) reveals the existence of three new spinors. We shall finalize this section by evincing their bilinears and their algebraic structure.

*Case 1:*\(\sigma =0=\omega \), \({\mathbf {J}}=0\), \({\mathbf {K}}\ne 0\) and \({\mathbf {S}}\ne 0\). It can be verified that all the FPK identities (2) are satisfied. Moreover, the aggregate (not a boomerang) associated with this spinor reads

*Case 2:*\(\sigma =0=\omega \), \({\mathbf {J}}=0\), \({\mathbf {K}}=0\) and \({\mathbf {S}}\ne 0\). Here, the FPK identities are also satisfied and the aggregate associated is simply given by

*Case 3:*\(\sigma =0=\omega \), \({\mathbf {J}}=0\), \({\mathbf {K}}\ne 0\), and \({\mathbf {S}}=0\), again the FPK identities hold, and the associated spinor operator has the following form:

## 4 Concluding remarks and outlook

We have shown the existence of three new spinors of mass dimension 1, via the inversion theorem and a consistent modification of the Lounesto spinor field classification. This has been achieved considering the specific bilinear covariant \(\mathbf{J}\) to be equal to zero. Physically, it means that the new spinors cannot respect the Dirac dynamics, only the Klein–Gordon one, enabling thus the canonical mass dimension to be equal to 1.

A word of caution may be added to these final remarks. As remarked along in the text, the adopted procedure is consistent; and bearing in mind the precedent opened by previous mass dimension 1 spinors (the Elkos), the spinors found may have several physically relevant aspects to be explored [21]. This is, in fact, our belief concerning the generalization presented here. However, one must take into account that the classification and the algebraic functional form do not say much about the emergence of these spinors in nature. As it is, the quantities described in the cases 1, 2, and 3 of the previous section are mathematically well-defined structures whose associated physical field would have interesting properties. The possibility of a physical manifestation of such spinors is currently under investigation.

## Footnotes

## Notes

### Acknowledgments

CHCV thanks to CAPES (PEC-PG) for the financial support, JMHS thanks to CNPq (308623/2012-6; 445385/2014-6) for partial support. RdR is grateful to CNPq Grants 473326/2013-2 and 303027/2012-6. RdR is grateful to Prof. Loriano Bonora, for the fruitful discussions.

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