From Dipole spinors to a new class of mass dimension one fermions
aa r X i v : . [ phy s i c s . g e n - ph ] S e p From Dipole spinors to a new class of mass dimension one fermions
R. J. Bueno Rogerio ∗ Institute of Physics and Chemistry, Federal University of Itajub´a , Itajub´a, Minas Gerais, 37500-903, Brazil.
Abstract.
In this letter, we investigate a quite recent new class of spin one-half fermions, namely
Ahluwalia class-7 spinors , endowed with mass dimensionality 1 rather than 3 /
2, being candidatesto describe dark matter. Such spinors, under the Dirac adjoint structure, belongs to the Lounesto’sclass-6, namely dipole spinors. Up to our knowledge, dipole spinor fields have Weyl spinor fields astheir most known representative, nonetheless, here we explore the dark counterpart of the dipolespinors, which represents eigenspinors of the chirality operator.
PACS numbers: 11.10.-z, 03.65.Fd, 03.70.+kKeywords: Mass dimension one, dipole spinors, Lounesto classification
I. INTRODUCTION
The Dirac and Majorana fields stand for a small partof a wide realm of the spinors fields. So it is naturalto look for more possibilities to strengthen the pillars ofQuantum Field Theory, and a method to accomplish thatis extracting as much physical information relevant tothe particles, and also to seek new candidates to explainphysical phenomenon that are still open in physics — asone of the most important open problem: dark matter.Dark matter does not interact with the electromagneticforce, thus, it can not be directly experienced. Regard-less of its evident gravitational effects [1], up to now, noassociated dark particle has ever been successfully de-tected, however, we have some strong candidates whichmay shed some light towards dark matter interpretation[2–4].The work deals with a particular new class of spinorswhich describe eigenspinors of chirality operator (chi-ral spinors), namely dipole spinors, in the lights ofLounesto’s [5]. Weyl spinors, that are a particular class ofdipole spinors with U (1) gauge symmetry [6] were shownto belong to the Lounesto’s class 6, being, thus, dipolespinor fields governed by the Weyl equation. However,the class 6 of dipole spinor fields further allocates massdimension one spinors, whose dynamics, of course, is notruled by the Weyl equation [7].The unveiled Ahluwalia class-7 spinors stand for an en-tirely new class of spin one-half fermions endowed withmass dimensionality one, providing some new dark mat-ter candidates [3]. Such a recently discovered fermionsemerge from some (specific) set of arrangement of theClifford algebra basis. The algorithm used to define thesenew spinors lies in an investigation of a linearly indepen-dent sets of the square roots of the 4 × j [8]. In addiction, we also high-light the quite new unexpected mass dimension three-half bosons of spin one-half [9]. We emphasize that thespinors at hands stand for a complete set of eigenspinors ∗ Electronic address: [email protected] of Γ and since Γ commutes with the chirality operator,they are also eigenspinors of the chirality operator.According to Lounesto, dipole spinors are described by(massless) neutrinos [5]. Nonetheless, we show the possi-bility to enlarge the dipole spinors understanding, sinceAhluwalia class-7 spinors brings to light the possibilityof obtaining dark and massive dipole spinors. Generallyspeaking, mass dimension one fermions compose what isusually known as “ Beyond the Standard Model ” of par-ticle physics — such as Elko spinors [10] and flag-dipolespinors [4]. Interestingly enough, a possible path to deepunderstand and also look towards describing
Beyond theStandard Model physics is accomplished by exploring un-expected ideas, especially taking into account the actualstatus of theoretical physics.As it can be directly seen, the Ahluwalia class-7 spinorsopens up the possibility of providing a natural self-interacting dark matter candidate. The dark featureencoded on these new spin one-half matter fields arisesfrom the following observation: the mass dimension in-compatibility with the usual fermions of the StandardModel of particle physics, moreover, as highlighted in[3], Ahluwalia class-7 fermions cannot enter the stan-dard model doublets — standing for natural dark mat-ter candidates with unsuppressed quartic self-interaction,with mass dimensionality found to be 1 rather than 3 / II. OUTLINE AND DISCUSSIONS
The well known Lounesto’s spinor classification standfor a comprehensive spinor categorization based on thebilinear covariants, disclosing the possibility of a large va-riety of spinors. Comprising regular and singular spinorswhich include the cases of Dirac, flag-dipole, Majoranaand Weyl spinors [11].The aforementioned classification stands for a geomet-rical classification and usually classify spinors accordingto their physical information, the criterion lies on theso-called bi-spinorial densities [12, 13].The mass dimension one theory is still an open issue inQFT [4, 10]. Then, we believe that a wide and interestingcontent still “hidden” in the mass dimension one theory.We start the discussions bringing to scene the new classof mass dimension one fermions recently developed in [3].Such set of mass dimension one fermions arise from thedirect computation of the eigenspinors of a specific set ofmatrices, recalling the well known linearly independentsquare roots of identity [8], given by I (1) iγ iγ iγ γ (2) iγ γ iγ γ iγ γ γ γ γ γ γ γ (3) iγ γ γ iγ γ γ iγ γ γ γ γ γ (4) iγ γ γ γ (5)denoting the above set of matrices by Γ j , j = 1 , · · · , = I and Γ = iγ γ γ γ . The factor i ensurethe following: Γ j = + I , in addition, ensuring real eigen-values. For completeness, note that { Γ j , Γ k } = 2 δ jk I for j, k = 2 , · · · ,
16. This representation is irreducible, anyother representation can be expressed in terms of the el-ements above [8]. Such a set of matrices may provide arange of possibilities to define new mass dimension onefermions and also mass dimension three-halves bosons[3, 9].The method developed in [3] lies on the complete setof eigenspinors provided by Γ = iγ γ . Thus, its restspinors are λ ( ) = − i , λ ( ) = i , (6)and λ ( ) = − i , λ ( ) = i . (7)Notice that λ and λ correspond to eigenvalue +1and the remaining two spinors correspond to − Ahluwaliaspinors of class-7 . So far, we introduced the rest-framespinors. To define the spinors for an arbitrary momen-tum, λ j ( p ) = κλ j ( ), it suffices acting with the boostoperator κ = r E + m m I + ~σ · ~ p E + m I − ~σ · ~ p E + m ! , (8) in which σ stands for the Pauli matrices. Thus, the λ ’sspinors under the Dirac dual ¯ λ j = λ † j γ , furnish the fol-lowing: σ = ¯ λλ , ω = i ¯ λγ λ and S µν = i ¯ λγ µ γ ν λ iden-tically vanishing. And the set of non-vanishing bilinearquantities: J µ = ¯ λγ µ λ and K µ = − ¯ λγ γ µ λ , standing forthe conserved current and the the axial-vector current,respectively. Thus, the λ spinor yield J λ = − K λ = ( E + m ) − p y ( E + m ) + p m ( E + m ) ,J λ = − K λ = − p x ( E + m − p y ) m ( E + m ) ,J λ = − K λ = ( E + m ) − p y ( E + m ) − p x + p y − p z m ( E + m ) ,J λ = − K λ = − p z ( E + m − p y ) m ( E + m ) . (9)The λ spinors provide J λ = − K λ = ( E + m ) + 2 p y ( E + m ) + p m ( E + m ) ,J λ = − K λ = − p x ( E + m + p y ) m ( E + m ) ,J λ = − K λ = − ( E + m ) + 2 p y ( E + m ) − p x + p y − p z m ( E + m ) ,J λ = − K λ = − p z ( E + m + p y ) m ( E + m ) . (10)The λ furnish the following bilinear forms J λ = K λ = ( E + m ) + 2 p y ( E + m ) + p m ( E + m ) ,J λ = K λ = − p x ( E + m + p y ) m ( E + m ) ,J λ = K λ = − ( E + m ) + 2 p y ( E + m ) − p x + p y − p z m ( E + m ) ,J λ = K λ = − p z ( E + m + p y ) m ( E + m ) . (11)Finally, for λ spinors we have J λ = K λ = ( E + m ) − p y ( E + m ) + p m ( E + m ) ,J λ = K λ = − p x ( E + m − p y ) m ( E + m ) ,J λ = K λ = − ( E + m ) − p y ( E + m ) − p x + p y − p z m ( E + m ) ,J λ = K λ = − p z ( E + m − p y ) m ( E + m ) , (12)belonging, thus, to the Lounesto class-6, standing for adipole spinor. A straightforward examination of all bi-linear forms introduced above, shows that Fierz-Pauli-Kofink identities are automatically reached: J µ J µ = 0, K µ K µ = 0, both clearly being (null) invariants, more-over, we also have J µ K µ = 0 and J µ K ν − K µ J ν = 0. No-tice the following K µ = − J µ for λ ( p ) and λ ( p ) spinorsand K µ = J µ for λ ( p ) and λ ( p ) spinors, such resultsare consistent with expectations of Lounesto [5].Under the action of the charge-conjugation operator,the Ahluwalia class-7 spinors behave like C λ ( p ) = − λ ( p ) , C λ ( p ) = λ ( p ) , (13) C λ ( p ) = − λ ( p ) , C λ ( p ) = λ ( p ) , (14)in which C stands for the charge-conjugation opera-tor. The right observance of the last set of equationsevinces that the charge-conjugation operator plugs the λ ’s particle-antiparticle counterparts. Thus, Ahluwaliaclass-7 spinors — up to a constant multiplicative factorand an appropriate dual re-definition, it may lead to amore involved physical framework— being expansion co-efficient functions of a quantum field, bringing to the lighta new set of dark fields.Notice because γ commutes with Γ , the Ahluwaliaclass-7 spinors also becomes eigenspinors of the chiralityoperator, − iγ ≡ γ , furnishing the relations − iγ λ j = − λ j , j = 1 , − iγ λ j = + λ j , j = 3 , . (16) Interesting enough, accordingly to Ref [14] and laterverified in [15], driven by an exhaustive analysis of ir-reducible representations and the Clifford (sub)algebra,it is possible to write a Dirac spinor ( ψ D ) from dipolespinors ψ D = λ + + λ − , (17)namely Weyl condition . The sub indexes refers to thebehaviour of the λ spinors under action of the chiral op-erator in (15) and (16). A straightforward calculation,shows up two possibilities to define Dirac spinor ψ D ( p ) = λ ( p ) + λ ( p ) , (18) ψ D ( p ) = λ ( p ) + λ ( p ) , (19)the above set of Dirac spinors belong to class-2 withinLounesto’s classification. Moreover, from dipole spinorsit is also possible to define another interesting set, stand-ing for the followingΨ ( p ) = λ ( p ) + λ ( p ) , (20)Ψ ( p ) = λ ( p ) + λ ( p ) . (21)The above spinors holds the following properties C Ψ ( p ) = +Ψ ( p ) and C Ψ ( p ) = − Ψ ( p ). III. CONCLUDING REMARKS
We finalize by elucidating how useful tool shown to bethe roots of the identity matrix in the (1 / , ⊕ (0 , / Beyond the Standard Model ”.The focus of this work are the
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