Unveiling a spinor field classification with non-Abelian gauge symmetries

A spinor fields classification with non-Abelian gauge symmetries is introduced, generalizing the the U(1) gauge symmetries-based Lounesto's classification. Here, a more general classification, contrary to the Lounesto's one, encompasses spinor multiplets, corresponding to non-Abelian gauge fields. The particular case of SU(2) gauge symmetry, encompassing electroweak and electromagnetic conserved charges, is then implemented by a non-Abelian spinor classification, now involving 14 mixed classes of spinor doublets. A richer flagpole, dipole, and flag-dipole structure naturally descends from this general classification. The Lounesto's classification of spinors is shown to arise as a Pauli's singlet, into this more general classification.


I. INTRODUCTION
The Lounesto's spinor fields classification [1] represents an assortment of all spinor field in Minkowski spacetime that has been shown to be complementary to the Cartan's and the Wigner's classifications of spinors. In the Lounesto's spinor fields classification the standard Majorana, Weyl, and Dirac spinor fields are representatives of very particular subsets in different classes of spinors, classified according to their bilinear covariants. Several non-standard spinors, charged and neutral as well, have been studied. Refs. [2,3] encode an up-to-date on the Lounesto's classification. Besides, concrete examples of non-standard spinor fields were provided in, e. g., [4][5][6]. This classification has been extended, in order to further encompass new classes of spinors on higher dimensional spacetimes. For example, new spinors were constructed on 7d manifolds, that in particular arise as new solutions of the Euler-Lagrange equations in the AdS 4 ×S 7 compactifications in string theory [7], as well as for AdS 5 × S 5 compactifications [8].
Both from the formal and the pragmatic points of view, the Lounesto's classification of spinors is well established and successful, for its huge variety of applications and exploratory features on the search of new fermions fields [2]. However, it is remarkably limited in the context of gauge symmetries, just holding for the case of U(1) (abelian) gauge symmetries. In fact, spinors in the Lounesto's classification can be split into classes of charged and neutral spinors, under the conserved electric charge that is evinced from the Noether's theorem, due to the U(1) gauge symmetry underlying the equations for motion ruling all the spinor fields. Therefore, the Lounesto's classification is not able to encompass spinor multiplets, corresponding to non-Abelian gauge fields.
In particular, it does not encode electroweak and strong conserved charges. In fact, the Standard Model (SM) of elementary particles is described by a gauge theory, effectively governed by the gauge group SU(3) × SU(2) × U(1), describing strong, weak and electromagnetic interactions. Those interactions are implemented when the corresponding bosonic gauge fields, that include 8 massless gluons, 3 massive bosons, W ± and Z, and 1 photon, are exchanged, to respectively describe strong and electroweak. The fermionic sectors of the theory describe matter and it is encoded into 3-fold families of quarks and leptons, together with their antiparticles. The Lounesto's classification can solely encompass Abelian gauge symmetries, with conserved electric charge. Exploring algebraic solutions for the U(1) electromagnetic potential appearing in the Dirac equation, Refs. [16,17] showed the prototypical inversion theorem for the real vector potential. The SU(2) case was scrutinized in Ref. [18].
The main aim here is to propose a spinor field classification that further encompasses non-this extended classification, to a Pauli's singlet, also encompassing electroweak and electromagnetic conserved charges corresponding to the SU(2) × U(1) symmetry. The extended, non-Abelian, Fierz identities are then here scrutinized. This paper is organized as follows: after reviewing the Lounesto's classification, the Fierz identities and the Fierz aggregate in Sect. II, Sect. III is devoted to analyze the inversion of the Dirac equation, showing that it holds for the case of type-1 regular spinors. The electromagnetic potential can be also expressed as spinor fields for the case of type-2 and type-3 regular spinors, and we show that the inversion can not be implemented for singular spinors. In Sect. IV a non-Abelian  i=0 Ω i (M ) denotes the exterior bundle, the bilinear covariants are sections of Ω(M ), whose splitting is then represented by [9] sec Ω 0 (M ) sec where J µ =ψγ µ ψ, S µν =ψ[γ µ , γ ν ]ψ, and K µ =ψγ 5 γ µ ψ are the respective coefficients of the Lorentz bilinear covariants, in the above table. Also, γ 5 = iγ 0 γ 1 γ 2 γ 3 is the chiral operator implemented by the volume element (for the Clifford product denoted by juxtaposition); the Diracconjugated spinor readsψ = ψ † γ 0 , and hereon γ µν := i 2 [γ µ , γ ν ]. Besides, γ µ γ ν + γ ν γ µ = 2η µν 1, where η µν denotes the Minkowski metric. The physical observables, exclusively for the Dirac's theory describing the electron, are realized by the bilinear covariants. In fact, the 1-form current density J, the 2-form spin density S, and the 1-form chiral current density K, satisfy, together to the scalar and pseudoscalar bilinears, the Fierz identities [1] −ωS µν + σǫ αβ The Lounesto's classification reads [1]: The condition J = 0 holds for all spinors into the above classes (2a -2f). Other classes corresponding to J = 0 have been derived in Ref. [11], whose representative spinors have been conjectured to be ghost spinors. The most general representative spinor fields of each Lounesto's spinor class, were listed in Ref. [13]. Moreover, a gauge spinor field classification have also been proposed in Ref. [15].
Singular spinors consist of flag-dipole, flagpole, and dipole spinors, respectively in the fourth, fifth, and sixth classes in the just mentioned six classes (2a -2f). The standard Dirac spinor is an element of the set of regular spinors in class 1. Moreover, Majorana spinors are neutral spinors that embrace particular realizations of flagpole type-5 spinors. The chiral Weyl spinors consist of a tiny subset of dipole spinors. In fact, in Ref. [13] one sees that chiral spinors are in the classes 6 that consists of dipole spinors, however only chiral spinors that satisfy the Weyl equation are Weyl spinors. Since type-5 spinors phenomenologically accommodate mass dimension one spinors [12,14], the class 6 might also accommodate mass dimension one spinors, whose dynamics, of course, is not ruled by the Weyl equation. Nevertheless, the classes (2a -2f) provide a comprehensive sort of new possibilities that have not been explored yet [5].
The Fierz identities (1a) do not hold for singular spinors. Based on a Fierz aggregate, the Fierz identities (1a) can be replaced by the most general equations The above equations are reduced to Eqs. (1a), in the case where both σ and ω are not equal zero, e. g., for type-1 spinor regular spinor fields in the (2a) Lounesto's class. When γ 0 Z † γ 0 = Z, then the Fierz aggregate is a self-conjugated structure called a boomerang [1].
The 1-form field J is interpreted as being a pole, and flagpoles are consequently elements of the class 5 in Lounesto classification. In fact, for this one has K = 0 and S = 0, being the flagpole hence characterized by the non-vanishing S and K. Besides, as type-4 spinors have the 2-form field S and the 1-form fields J and K non null, together they corresponding to a flag-dipole structure.
Hence, the Fierz identities (4a -4i) can be then generalized for the non-Abelian case, yielding, Emulating the Fierz identity S µν = 1 σ 2 −ω 2 [σǫ µν ρχ − i ωǫ αρχ ǫ αµν ]J ρ K χ , [9,10] for the i = 0 case that corresponds to a Pauli's singlet equivalent to the Lounesto's classification, one can further calculate other Fierz identities for the non-Abelian case, as Hence, adding the term ǫ µνρχ J 0ρ K 0χ to ǫ µναβ J aα K a β yields [18] following that Moreover a generalized Fierz identity holds for the non-Abelian density, written as a function of the non-Abelian chiral current and the non-Abelian spin density [18]: ρχ +iK a ǫ αρχ ǫ αµν ] J i ρ K iχ