a lagrangian for mass dimension one fermionic dark matter

Dodelson Keywords: Elko Fermionic dark matter Mass dimension one fermions ThemassdimensiononefermionicfieldassociatedwithElkosatisfiestheKlein–GordonbutnottheDirac equation.. Grumiller, Da

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www.elsevier.com/locate/physletb

Cheng-Yang Lee

Institute of Mathematics, Statistics and Scientific Computation, Unicamp, 13083-859 Campinas, São Paulo, Brazil

Article history:

Received 16 January 2016

Received in revised form 25 June 2016

Accepted 26 June 2016

Available online 30 June 2016

Editor: S Dodelson

Keywords:

Elko

Fermionic dark matter

Mass dimension one fermions

ThemassdimensiononefermionicfieldassociatedwithElkosatisfiestheKlein–GordonbutnottheDirac equation However,itspropagator isnotaGreen’sfunctionofthe Klein–Gordonoperator Wepropose

aninfinitesimaldeformationtothepropagatorsuchthatitadmitsanoperatorinwhichthe deformed propagatorisaGreen’sfunction.Thefieldisstillofmassdimensionone,buttheresultingLagrangianis modifiedinaccordancewiththeoperator

©2016TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense

(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3

1 Introduction

The theoretical discovery of Elko andthe associated mass

di-mension one fermions by [2,3] is a radical departure from the

Standard Model (SM) These fermions have renormalizable

self-interactionsandonlyinteractwiththeSMparticlesthrough

grav-ityandtheHiggsboson.Thesepropertiesmakethemnaturaldark

mattercandidates

Since their conceptions, Elko and its fermionic fields have

beenstudied inmanydisciplines The graviational interactions of

Elkohavereceivedmuchattention [8,9,11–18,20,23,29,36,46,47,49]

while its mathematical properties have been investigated by da

Rochaandcollaborators[10,22,24–28,38].Theseworksestablished

Elkoasaninflatoncandidateandthatitisaflagpolespinorofthe

Lounestoclassification [42] thus makingthem fundamentally

dif-ferentfromtheDiracspinor.Inparticlephysics,the signaturesof

thesemassdimension one fermionsatthe LargeHardonCollider

have been studied [7,30] In quantum field theory, much of the

attentionis focused on the foundations of the construction [5,6,

19,32–34,41,43–45].Theirsupersymmetricandhigher-spin

exten-sionshavealsobeencarriedoutby[40,50].Animportantresultis

that the fermionicfield andits higher-spin generalization violate

Lorentz symmetry due to the existence of a preferred direction

ThisledAhluwaliaandHorvathtosuggestthatthefermionicfield

satisfiesthesymmetryofveryspecialrelativity[4,21]

One question remains unanswered in the literature What is

thecorrectLagrangianofthemassdimension onefermion?Since

thefieldisconstructedusingElkoasexpansioncoefficientswhich

E-mail address:cylee@ime.unicamp.br

satisfy the Klein–Gordon equation, the naive answer would be the Klein–Gordon Lagrangian Butthis hastwo unsatisfactory as-pects Firstly, the resulting field-momentum anti-commutator is not givenby theDirac-deltafunction Secondly,thepropagatoris notaGreen’sfunctionoftheKlein–Gordonoperator

We propose an infinitesimally deformed propagatorsuch that

it isan Green’s function toan operator The resultingLagrangian determinedfromtheoperatordoesnothavetheabovementioned problemsandisstillofmassdimensionone

2 The Elko construct

We briefly review the construction of Elko and its fermionic field Formoredetails, pleaserefer tothereview article [1].Elko

isaGermanacronymfor Eigenspinoren des Ladungskonjugations-operators.Theyare acomplete setofeigenspinorsofthe charge-conjugation operator of the (12,0) ⊕ (0,12) representation of the Lorentzgroup.Thecharge-conjugationoperatorisdefinedas

C =



O −i−1



where K complex conjugates anything to its right and  is the spin-halfWignertime-reversalmatrix

 =



0 −1

1 0



ItsactiononthePaulimatrices σ = ( σ1, σ2, σ3)is

http://dx.doi.org/10.1016/j.physletb.2016.06.064

0370-2693/©2016 The Author Published by Elsevier B.V This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ) Funded by 3

spinoroftheform

χ (p, α ) =



ϑφ∗(p, σ )

φ (p, σ )



(4)

where φ (  , σ ) is a left-handed Weyl spinor in the helicity basis

with  =lim|p|→0 p andˆ

1

2σ · ˆpφ (  , σ ) = σ φ (  , σ ) (5)

so that σ = ±1

2 denotes the helicity Here α = ∓ σ denotes the

dual-helicitynature ofthe spinor withthetop andbottom signs

denoting the helicity of the right- and left-handed Weyl spinors

respectively The spinor χ (p, α ) becomes the eigenspinor of the

charge-conjugationoperatorCwiththefollowingchoiceofphases

C χ (p, α ) |ϑ=±i= ± χ (p, α ) |ϑ=±i (6)

thusgiving usfourElkos Spinors withthe positive andnegative

eigenvalues are called the self-conjugate and anti-self-conjugate

spinors.Theyaredenotedas

Therearesubtletiesinvolvedinchoosingthelabellingsandphases

fortheself-conjugateandanti-self-conjugate spinors.The details,

includingthesolutionsofthespinorscanbefoundin[6,sec II.A]

The Elkodual which yields the invariant inner-product is

de-finedas[1,48]

¬

¬

where † represents Hermitian conjugation and  is a

block-off-diagonalmatrixcomprisedof2×2 identitymatrix

 =



O I

I O



Thematrix(p)isdefinedas

(p) = 1

2m



α



ξ(p, α ) ¯ξ(p, α ) − ζ(p, α ) ¯ζ(p, α ) 

ThebaroverthespinorsdenotestheDiracdual.Thedualensures

thattheElkonormsareorthonormal

¬

ξ (p, α )ξ(p, α) = −¬

ζ (p, α )ζ (p, α) =2mδαα (11)

andtheirspin-sumsread



α

ξ(p, α ) ξ (¬ p, α ) =m[ G (φ) +I], (12a)



α

ζ (p, α ) ζ (¬ p, α ) =m[ G (φ) −I] (12b)

whereG(φ)isanoff-diagonalmatrix

G (φ) =i

⎛

⎜

⎜

⎝

0 0 0 −e−i φ

0 0 e i φ 0

0 −e−i φ 0 0

e i φ 0 0 0

⎞

⎟

⎟

The angle φ is defined via the following parametrization of the

momentum

p= |p|(sinθcosφ,sinθsinφ,cosθ ) (14)

where 0≤ θ ≤ π and0≤ φ <2π.Multiply eqs (12a) and(12b)

withξ(p, α)andζ (p, α)fromtherightandapplythe orthonor-malrelations,weobtain

[G (φ) −I]ξ(p, α ) =0, (15a) [G (φ) +I]ζ (p, α ) =0. (15b)

Sincetheseidentitieshavenoexplicitenergydependence,the cor-responding equation in the configurationspace has no dynamics andthereforecannotbethefieldequationforthemassdimension onefermions.Nevertheless,writingtheaboveidentitiesinthe con-figurationspaceforλ(x)isnon-trivialandisa taskthat mustbe accomplishedinordertoderivetheHamiltonian Thisissueis ad-dressedinthenextsection

Identifying the self-conjugate and anti-self-conjugate spinors withtheexpansioncoefficientsforparticlesandanti-particles,the twomass dimensiononefermionicfieldsandtheir adjoints,with theappropriatenormalizationare

(x) = (2π )−3/2 d3p



2mEp



α

[e−ip·xξ(p, α )a(p, α )

+e ip·xζ (p, α )b‡(p, α ) ], (16a)

¬

(x) = (2π )−3/2 d3p



2mEp



α

[e ip·x¬

ξ (p, α )a‡(p, α )

+e−ip·x¬

ζ (p, α )b(p, α ) ], (16b)

λ(x) = (x) |b‡=a‡, (16c)

¬

λ(x) =¬

(x) |b‡=a‡. (16d)

Here a(p, α ) and b‡(p, α ) are the annihilation and creation op-erators for particles and anti-particles Theysatisfy the standard anti-commutationrelations

{a(p, α),a‡(p, α ) } = {b(p, α),b‡(p, α ) }

Notethatforthecreationoperators,wehaveintroducedanew op-erator‡ inplaceoftheusualHermitianconjugation†.Thisfollows fromtheobservationthatsincetheDiracandElkodualare differ-ent,itsuggeststhat thecorresponding adjointsforthe respective particle states may be different Assuming they are different, it maythenbecome necessarytodevelop anewformalismfor par-ticlesstates withthe new‡ adjoint inparallelto [31] Thisisan importantissuethat deserves furtherstudybutsince itdoesnot affectourobjectiveofderivingtheLagrangian,weshallleaveitfor futureinvestigation

3 The Lagrangian: defining the problem

There are two reasons why the Klein–Gordon Lagrangian are unsatisfactory for the mass dimension one fermions Firstly, the field does not satisfy the canonical anti-commutation relations (CARs)sincethefield-momentumanti-commutatorisnotequalto

iδ3(x −y)I.Instead,itisgivenby1

{λ(t,x), πkg(t,y) } =i d

3p (2π )3e−ip ·(x−y)[I+ G (φ) ] (18)

1 For the rest of the paper, we will be working withλ( x ), but the results hold for

x also.

where πkg = ∂¬

λ/∂t is the conjugate momentum of the Klein–

Gordon Lagrangian Secondly, the propagator obtained from the

fermionictime-orderproductis

S(x,y) =  |T[λ(x) λ(¬ y) ]|

4p

(2π )4e−ip ·( x−y )



I+ G (φ)

p·p−m2+i



ThisisnotaGreen’sfunctionoftheKlein–Gordonoperator

(∂μ∂μ+m2)S(x,y) = −i d

4p (2π )4e−ip ·( x−y )[I+ G (φ) ]. (20)

In eqs (18)and (20)the problemresides inthe three and

four-dimensionalFouriertransformofG(φ)whicharenon-vanishing

3.1 The inverse of I+ G(φ)

Inaconsistent fieldtheory,we expectthe fieldstosatisfythe

canonicalanti-commutationrelationsandthattheLagrangianand

propagatorto havethe samesymmetry In thisrespect,it is

evi-dentthataKlein–GordonLagrangianisinadequatetodescribethe

massdimensiononefermions.HerewederivetheLagrangianthat

providesa completedescription ofλ(x) Westart bydetermining

theoperator O(x)in whichthepropagatorgivenby eq.(19)isa

Green’sfunction

O (x)S(x,y) = −iδ4(x−y). (21)

According to eq (20), this can be achieved by determining the

inverse of I+ G(φ) and the corresponding operator in the

con-figurationspace.Howeverthismatrixisnon-invertiblesince

det[I+ G (φ) ] =0. (22)

Thisproblemcanbebypassedbyconsideringamoregeneral

ma-trixI+ τ G(φ)where τ isarealconstant.Itsinverseis

[I+ τ G (φ) ]−1=I− τ G (φ)

Thesingularitiescanbeavoidedbytakingthelimit τ → ±1.This

is possible since a simple calculation shows that the inverse is

well-definedforallvaluesof τ

[I+ τ G (φ) ]−1[I+ τ G (φ) ] =



1− τ2

1− τ2



wherewe have used theidentity G2(φ) =I.Therefore, toobtain

theinverseofI+ G(φ),wemustfirstperforma τ-deformation

I+ G (φ) →I+ τ G (φ). (25)

Theinverseisthengivenbyeq.(23).Detailsonthemathematical

foundationoftheproposeddeformationandinversecanbefound

in[44]

To determine the form of [I+ τ G(φ)]−1 in the configuration

space,wefirstneedtodefineanoperatorG whichcorrespondsto

thematrixG(φ)intheconfigurationspace.Forthispurpose,

frac-tional derivatives must be introduced since the matrix elements

ofG(φ) are proportional to e±i φ andthat itcan be expressed in

termsofcomplexmomentap±= (p1±ip2)/ √

2 as

e±i φ= [p±(p∓)−1]1/2. (26)

Therearemanydefinitionsoffractionalderivatives.Thefractional

derivativethat isappropriateforourtaskistheFourierfractional

derivative [35, pg 562] [37] The general properties of fractional

derivatives,includingFourier’sdefinition,aregivenin Appendix A

TheoperatorG isdefinedas

G =i

⎛

⎜

⎜

⎝

+ ∂−−1/2

0 0 ∂−1/2

+ ∂−1/2 0

0 −∂1/2

∂−1/2

⎞

⎟

⎟

⎠

(27)

where∂±actonthecomplexcoordinatesx±= (x1±ix2)/ √

2

An interesting property of the Fourier fractional derivative, which turns out to be important for our construct is its non-uniqueness.Toseewhatthismeans,letusconsideri∂+12∂−−12f(x),

anelementofGf(x)where f(x)isatestfunctionwiththeFourier transform

f(x) = d3p

(2π )3e−ip·x F(p). (28)

Usingeq.(A.6),weobtain

∂−1/2

− f(x) = d3p

(2π )3e−ip·x e−i π /4e−in− p−1/2

where we haveused the fact that i−1 2=e−i π / e−in−π has two rootswithn−=0,1.Actingontheexpressionagainwithi∂+12 we obtain

i∂+1/2∂−−1/2f(x) = d3p

(2π )3(iω )e−ip·x e−i φ F(p) (30)

where ω =e i ( n+−n−) π with n±=0,1.Duetotheambiguityofthe phases, in order toobtain G(φ) in the momentum space, all the elementsofGf(x)musthavethesamephasessothat

Gf(x) = d3p

(2π )3ωe−ip·xG (φ)F(p). (31)

Forfunctionssuchas f(x)comprisedofasingle Fourier trans-form, ω isaglobalphase anditsvalue isunimportant.But since

λ(x) isa sum ofthe Fourier transformof the self-conjugateand anti-self-conjugatespinors,theactionofG onthefieldyields2

G λ(x) = (2π )−3 d3p



2mEp



α

[e−ip·xωξξ(p, α )a(p, α )

Sincethephasescantakethevaluesof±1,G λ(x)hasfour possi-blesolutions.Outofthefourpossibilities,aswe willshowinthe nextsection,theonlysolutionwhichyieldsapositive-definitefree Hamiltonianis

Thisthengivesustheequation

whichisthecounterpartofeqs.(15a)and (15b)inthe configura-tionspace

HavingdefinedG,theinverseofI+ τ G(φ)intheconfiguration spaceisgivenby

A = I− τ G

NowweapplyAtoeq.(20).The τ-deformedpropagatoris

2 In this paper, we assume that the fermionic fieldλ( x )and its dual¬λ( x ), both

of which are a sum of the Fourier transform of the self-conjugate and anti-self-conjugate spinors and its dual, are well-defined.

S ( τ )(x,y) =i d

4p (2π )4e−ip ·( x−y )



I+ τ G (φ)

p·p−m2+i



WhenAactsontheS ( τ )(x,y),wemusttake ω =1 sothatinthe

limit τ →1,weobtain

lim

τ→1A (∂μ∂μ+m2)S ( τ )(x,y) = −iδ4(x−y)I (37)

Theoperator O(x),inwhich thepropagatorisa Green’sfunction

ofistherefore

BasedontheformofO(x),theLagrangianforλ(x)is

L = Aab(∂μλ¬a∂μλb−m2λ¬aλb) (39)

wherewe sumover the repeatedindices The operator A is

di-mensionlesssoλ(x)remainsamassdimensiononefield.Thefield

equationis

TheoperatorAdoesnotaffectthesolutionsofthefieldequation

butaswewillshowinthesubsequentsection,itensuresthatthe

fieldsatisfiesthecanonicalanti-commutationrelations

3.2 Canonical anti-commutation relations

TheCARs of λ(x) are determined by {λ(t x),λ(t y)}, { π (t x),

π (t y)}and{λ(t x), π (t y)}.Thefirstanti-commutatoridentically

vanishes[6, sec III.B].When theLagrangian isKlein–Gordon, the

secondanti-commutatoridenticallyvanishesandthethirdisgiven

byeq (18).We show that thefield associated witheq (39)

sat-isfiesthe CARsby computingtherelevantanti-commutators The

conjugatemomentumis

πb(y) = Aab(y) ∂

¬

λa

which differs from πkg (y) by a factor of A(y) Since fractional

derivativescommute,theanti-commutatorbetweentheconjugate

momentumis

A direct evaluation of the τ-deformed field-momentum

anti-commutatornowyields

lim

τ→1{λ(t,x), π (t,y) }( τ )=iδ3(x−y)I (43)

We now show that the free Hamiltonian is positive-definite

Thisisachievedby showingthatthefree Hamiltonian,asa

func-tionofλ(x)and¬λ(x)isidenticaltotheonegivenin[40,eq (97)]

InordertoobtainthecorrectHamiltonian,wemusttakethe

con-jugatemomentumassociatedwithbothλ(x)andλ(¬ x)intoaccount

wherethelaterisdefinedas

¬

πa= − ∂ L

∂ λ¬a/∂t = − Aab

∂λb

TheHamiltonianisthengivenbytheLegendretransformation

H= d3x



∂ λ¬a

∂t



Aab

∂λb

∂t



−



Aab

∂λb

∂t



∂ λ¬a

∂t − L



. (45)

Inobtainingthefirstterm, wehaveusedthedefinitionofA and

theintegration by parts rulefor theFourier fractional derivative

The first two terms can be simplified further since in the limit

τ →1,wehaveasimpleidentity

lim

τ→1Aabλb(x) =1

2λa(x). (46)

Usingtheidentity

d3xAab(∂μλ¬a∂μλb−m2λ¬aλb)

2 d

3x(∂μλ¬a∂μλa−m2λ¬aλa), (47)

theHamiltoniansimplifiesto

H=1

2 d

3x



− ∂λa

∂t

∂ λ¬a

∂t − ∂i¬

λa∂iλa+m2λ¬aλa



Thisisidenticalto[40,eq (89)]andisthereforepositive-definite

4 Conclusions

The propagator and Lagrangian proposed in this paper ad-dressed the outstanding problems of the mass dimension one fermions.Theypreserve themassdimensionalityand renormaliz-ableself-interaction.ThefieldsatisfiestheCARsandthe propaga-torisaGreen’sfunctiontotheoperatorgivenineq.(38) Whilethemassdimensiononefermionicfield violatesLorentz symmetry,inlightofthenewLagrangian,itisneverthelessa well-definedquantum field inthe sense thatit hasa positive-definite free Hamiltonian, it satisfies the CARs and furnishes fermionic statistics These properties are highly non-trivial They require careful choices of expansion coefficients, adjoints These results strongly suggest that the mass dimension one fermions have a well-definedspace-timesymmetry

The τ-deformed Lagrangian differs from the original Klein– GordonLagrangianproposedbyAhluwaliaandGrumiller[2,3].On the one hand, it resolves the problem of the CARs But on the other hand,the new Lagrangian suggests that the associated in-teractionsforthe fermions mustnow befunctionsof A(x) Con-structingwell-definedinteractionsmaybedifficultastheoperator hasapoleat τ =1.Ifitisnotremoved orcancelled,itcanmake thescatteringamplitudesdivergentandthusnon-physical.Despite the difficulties,theproposed τ-deformation maystill be ofvalue

to the theory Recently, it was suggested that by constructing a

τ-deformed field adjoint,it is possible to obtain a fully Lorentz-covarianttheory[45]

Ifthemassdimensionone fermionicfieldsareinvariant under very specialrelativityasproposed byAhluwalia andHorvath, the effects of Lorentz violationwould be minimal since very special relativity is compatible with the null results of the Michelson– Morley experiments and other well-known relativistic effects [4, 39].Wewouldexpectdiscretesymmetryviolationsandscattering cross-sectionsinvolvingmassdimensiononefermionstohave de-pendenceonapreferreddirection

Onabroaderpicture,thistheoreticalconstructpresentsan in-terestingnewparadigm.Space-timesymmetrymaybeamere re-flection ofthesymmetryoftherodsandclockscomprisedofthe

SMparticles.Space-time,accordingtotherodsandclocksmadeof darkmatter,maypaintacompletelydifferentpicture

Acknowledgements

IwouldliketothankN FaustinoandG.S deSouzafor discus-sionsattheearlystage ofthiswork.IamgratefultoR da Rocha forsuggestionsandreadingtheinitialmanuscript Duringthe re-visionofthe manuscript,I havebenefited greatlyfromnumerous discussionswithD.V Ahluwalia.Thisresearchissupportedbythe CNPqgrant313285/2013-6

Appendix A Fractional derivatives

Allfractionalderivativessatisfy thefollowingproperties.Let α

beanarbitraryrealnumber,inthelimit α →n where n isa

posi-tiveinteger,

lim

α→n

d α

dx α f(x) = d n

dx n f(x). (A.1)

Theoperationsarelinear

d α

dx α[c f(x) ] =c d

α

dx α f(x), (A.2a)

d α

dx α[f(x) +g(x) ] = d α

dx α f(x) + d α

dx α g(x). (A.2b)

TheLeibnizruleis

d α

dx α(f g) =

∞



j=0



α

j

 

d α−j

dx α f

 

d j

dx j g



(A.3)

wherethebinomialcoefficientisgeneralizedtoarbitraryreal

num-bersbythe( α )function



α

j



(j+1)( α −j+1) . (A.4)

TheFourierfractionalderivative isdefinedasfollows.Let f(x)

and F(k) be two single-variable functions relatedby the Fourier

transform

f(x) = 1

2π

∞

−∞

dk e−ikx F(k), (A.5a)

F(k) = 1

2π

∞

−∞

dx e ikx f(x). (A.5b)

Thedefinitionofthefractionalderivative on f(x)isa

straightfor-wardgeneralisationoftheusualderivative

d α

dx α f(x) = 1

2π

∞

−∞

dk( −ik)α e−ikx F(k). (A.6)

Thisformulaisvalidforallvaluesof α.Theonlyconditionsneeded

are the existence of the Fouriertransform for f(x) andits

frac-tionalderivative.Usingeq.(A.6),weobtaintheintegrationbyparts

rule

dx



f(x)d

α

dx α g(x)



= (−1)α dx



g(x) d α

dx α f(x)



. (A.7)

References

[1] D.V Ahluwalia, On a local mass dimension one Fermi field of spin one-half and

the theoretical crevice that allows it, 2013.

[2] D.V Ahluwalia, D Grumiller, Dark matter: a spin one half fermion field with

mass dimension one?, Phys Rev D 72 (2005) 067701.

[3] D.V Ahluwalia, D Grumiller, Spin half fermions with mass dimension one:

the-ory, phenomenology, and dark matter, J Cosmol Astropart Phys 0507 (2005)

012.

[4] D.V Ahluwalia, S.P Horvath, Very special relativity as relativity of dark matter:

the Elko connection, J High Energy Phys 1011 (2010) 078.

[5] D.V Ahluwalia, C.-Y Lee, D Schritt, Elko as self-interacting fermionic dark

mat-ter with axis of locality, Phys Lett B 687 (2010) 248–252.

[6] D.V Ahluwalia, C.-Y Lee, D Schritt, Self-interacting Elko dark matter with an

axis of locality, Phys Rev D 83 (2011) 065017.

[7] A Alves, F de Campos, M Dias, J.M Hoff da Silva, Searching for Elko dark matter spinors at the CERN LHC, Int J Mod Phys A 30, no 01 (2015) 1550006 [8] A Basak, J.R Bhatt, S Shankaranarayanan, K Prasantha Varma, Attractor be-haviour in ELKO cosmology, J Cosmol Astropart Phys 1304 (2013) 025 [9] A Basak, S Shankaranarayanan, Super-inflation and generation of first order vector perturbations in ELKO, J Cosmol Astropart Phys 1505 (05) (2014) 034 [10] A Bernardini, R da Rocha, Dynamical dispersion relation for ELKO dark spinor fields, Phys Lett B 717 (2012) 238–241.

[11] C.G Boehmer, The Einstein–Cartan–Elko system, Ann Phys 16 (2007) 38–44 [12] C.G Boehmer, The Einstein–Elko system: can dark matter drive inflation?, Ann Phys 16 (2007) 325–341.

[13] C.G Boehmer, Dark spinor inflation: theory primer and dynamics, Phys Rev D

77 (2008) 123535.

[14] C.G Boehmer, J Burnett, Dark spinors with torsion in cosmology, Phys Rev D

78 (2008) 104001.

[15] C.G Boehmer, J Burnett, Dark energy with dark spinors, Mod Phys Lett A 25 (2010) 101–110.

[16] C.G Boehmer, J Burnett, Dark spinors, arXiv:1001.1141 [gr-qc], 2010 [17] C.G Boehmer, J Burnett, D.F Mota, D.J Shaw, Dark spinor models in gravitation and cosmology, J High Energy Phys 1007 (2010) 053.

[18] C.G Boehmer, D.F Mota, CMB anisotropies and inflation from non-standard spinors, Phys Lett B 663 (2008) 168–171.

[19] R Cavalcanti, J.M.H da Silva, R da Rocha, VSR symmetries in the DKP algebra: the interplay between Dirac and Elko spinor fields, Eur Phys J Plus 129 (11) (2014) 246.

[20] G Chee, Stability of de Sitter solutions sourced by dark spinors, arXiv:1007.

0554 [gr-qc], 2010.

[21] A.G Cohen, S.L Glashow, Very special relativity, Phys Rev Lett 97 (2006) 021601.

[22] R da Rocha, A.E Bernardini, J Hoff da Silva, Exotic dark spinor fields, J High Energy Phys 1104 (2011) 110.

[23] R da Rocha, L Fabbri, J Hoff da Silva, R Cavalcanti, J Silva-Neto, Flag-dipole spinor fields in ESK gravities, J Math Phys 54 (2013) 102505.

[24] R da Rocha, J Hoff da Silva, ELKO spinor fields: Lagrangians for gravity derived from supergravity, Int J Geom Methods Mod Phys 6 (2009) 461–477 [25] R da Rocha, J Hoff da Silva, A.E Bernardini, Elko spinor fields as a tool for probing exotic topological spacetime features, Int J Mod Phys Conf Ser 3 (2011) 133–142.

[26] R da Rocha, J.M Hoff da Silva, From Dirac spinor fields to ELKO, J Math Phys.

48 (2007) 123517.

[27] R da Rocha, J.M Hoff da Silva, ELKO, flagpole and flag-dipole spinor fields, and the instanton Hopf fibration, Adv Appl Clifford Algebras 20 (2010) 847–870 [28] R da Rocha, W.A Rodrigues Jr., Where are ELKO spinor fields in Lounesto spinor field classification?, Mod Phys Lett A 21 (2006) 65–74.

[29] J.M.H da Silva, S Pereira, Exact solutions to Elko spinors in spatially flat Friedmann–Robertson–Walker spacetimes, J Cosmol Astropart Phys 1403 (2014) 009.

[30] M Dias, F de Campos, J Hoff da Silva, Exploring Elko typical signature, Phys Lett B 706 (2012) 352–359.

[31] P.A.M Dirac, The Principles of Quntum Mechanics, 4th edition, Univ Pr., Oxford,

UK, 1958.

[32] L Fabbri, Causal propagation for ELKO fields, Mod Phys Lett A 25 (2010) 151–157.

[33] L Fabbri, Causality for ELKOs, Mod Phys Lett A 25 (2010) 2483–2488 [34] L Fabbri, Zero energy of plane-waves for ELKOs, Gen Relativ Gravit 43 (2011) 1607–1613.

[35] J.B.J Fourier, Théorie analytique de la chaleur, Univ Pr., Cambridge, UK, 1822,

p 643.

[36] D Gredat, S Shankaranarayanan, Modified scalar and tensor spectra in spinor driven inflation, J Cosmol Astropart Phys 1001 (2010) 008.

[37] R Herrmann, Fractional Calculus: An Introduction for Physicists, World Scien-tific, 2011, 261 pp.

[38] J.M Hoff da Silva, R da Rocha, From Dirac action to ELKO action, Int J Mod Phys A 24 (2009) 3227–3242.

[39] S.P Horvath, On the relativity of elko dark matter, Master’s thesis, University

of Canterbury, 2011, 106 pp.

[40] C.-Y Lee, Self-interacting mass-dimension one fields for any spin, Int J Mod Phys A 30 (2015) 1550048.

[41] C.-Y Lee, Symmetries of Elko and massive vector fields, Doctoral thesis, Univer-sity of Canterbury, 2012, 198 pp.

[42] P Lounesto, Clifford algebras and spinors, Lond Math Soc Lect Note Ser 286 (2001) 1–338.

[43] A Nikitin, Non-standard Dirac equations for non-standard spinors, Int J Mod Phys D 23 (14) (2014) 1444007.

[44] R.J.B Rogério, J.M.H da Silva, The local vicinity of spins sum for certain mass dimension one spinors, arXiv:1602.05871 [hep-th].

[45] D.V Ahluwalia, A story of phases, duals, and adjoints for a local Lorentz co-variant theory of mass dimension one fermion, arXiv:1601.03188 [hep-th] [46] S Shankaranarayanan, What-if inflaton is a spinor condensate?, Int J Mod Phys D 18 (2009) 2173–2179.

[47] S Shankaranarayanan, Dark spinor driven inflation, arXiv:1002.1128

[astro-ph.CO], 2010.

[48] L Sperança, An identification of the Dirac operator with the parity operator,

Int J Mod Phys D 23 (14) (2014) 1444003.

[49] H Wei, Spinor dark energy and cosmological coincidence problem, Phys Lett.

B 695 (2011) 307–311.

[50] K.E Wunderle, R Dick, A supersymmetric Lagrangian for fermionic fields with mass dimension one, Can J Phys 90 (2012) 1185.

[45] D.V Ahluwalia, A story of phases, duals, and adjoints for a local Lorentz co-variant theory of mass. .. class="text_page_counter">Trang 6

[47] S Shankaranarayanan, Dark spinor driven inflation, arXiv:1002.1128

[astro-ph.CO],