A Lagrangian For Mass Dimension One Fermionic Dark Matter

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Physics

Letters

B

www.elsevier.com/locate/physletb

A

Lagrangian

for

mass

dimension

one

fermionic

dark

matter

Cheng-Yang Lee

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

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Article history:

Received16January2016

Receivedinrevisedform25June2016 Accepted26June2016

Availableonline30June2016 Editor: S.Dodelson Keywords: Elko

Fermionicdarkmatter Massdimensiononefermions

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:[email protected].

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. TheElkoconstruct

We briefly review the construction of Elko and its fermionic field. Formoredetails, pleaserefer tothereview article [1].Elko isaGermanacronymfor Eigenspinorendes Ladungskonjugations-operators.Theyare acomplete setofeigenspinorsofthe charge-conjugation operator of the

(

1₂

,

0

)

⊕ (

0

,

1₂

)

representation of the Lorentzgroup.Thecharge-conjugationoperatorisdefinedas

C

=



O

−

i



−1

−

i



O



K (1)

where K complex conjugates anything to its right and



is the spin-halfWignertime-reversalmatrix



=



0

−

1 1 0



.

(2)

ItsactiononthePaulimatrices

σ

= (

σ

1

,

σ

2

,

σ

3

)

is



σ



−1

= −

σ

∗

.

(3)

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

0370-2693/©2016TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

ThecompletesetofElkoisconstructedfromafour-component 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|→0p and

ˆ

1

2

σ

· ˆ

p

φ (



,

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-conjugationoperator

C

withthefollowingchoiceofphases

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

C

ξ(

p

,

α

)

= ξ(

p

,

α

),

(7a)

C

ζ (

p

,

α

)

= −ζ(

p

,

α

).

(7b)

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]

¬

ξ (

p

,

α

)

= [(

p

)ξ(

p

,

α

)

]

†

,

(8a)

¬

ζ (

p

,

α

)

= [(

p

)ζ (

p

,

α

)

]

†



(8b)

where † represents Hermitian conjugation and



is a block-off-diagonalmatrixcomprisedof2

×

2 identitymatrix



=



O I I O



.

(9)

Thematrix

(p)

isdefinedas

(

p

)

=

1 2m



α



ξ(

p

,

α

) ¯ξ(

p

,

α

)

− ζ(

p

,

α

) ¯ζ(

p

,

α

)



.

(10)

ThebaroverthespinorsdenotestheDiracdual.Thedualensures thattheElkonormsareorthonormal

¬

ξ (

p

,

α

)ξ(

p

,

α



)

= −

¬

ζ (

p

,

α

)ζ (

p

,

α



)

=

2m

δ

αα (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)

where

G(φ)

isanoff-diagonalmatrix

G

(φ)

=

i

⎛

⎜

⎜

⎜

⎜

⎝

0 0 0

−

e−iφ 0 0 eiφ ₀ 0

−

e−iφ _{0 0} eiφ _{0 0 0}

⎞

⎟

⎟

⎟

⎟

⎠

.

(13)

The angle

φ

is defined via the following parametrization of the momentum

p

= |

p

|(

sin

θ

cos

φ,

sin

θ

sin

φ,

cos

θ )

(14)

where 0

≤ θ ≤

π

and0

≤ φ <

2

π

.Multiply eqs. (12a) and(12b)

with

ξ(p,

α



)

and

ζ (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-figurationspacefor

λ(

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

)

= (

2

π

)

−3/2

_d3_p



2mEp



α

[

e−ip·x

ξ(

p

,

α

)

a

(

p

,

α

)

+

eip·x

ζ (

p

,

α

)

b‡

(

p

,

α

)

],

(16a) ¬

(

x

)

= (

2

π

)

−3/2

d3p



2mEp



α

[

eip·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

)

}

= δ

αα

δ

3

(

p

−

p

).

(17)

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. TheLagrangian:definingtheproblem

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

i

δ

3

(x

−

y)I.Instead,itisgivenby1

{λ(

t

,

x

),

π

kg

(

t

,

y

)

} =

i

_d3_p

(

2

π

)

3e−

ip·(x−y)

_[

_I

₊

_G

_(φ)

_]

₍₁₈₎

1 _{Fortherestofthepaper,wewillbeworkingwith}

λ(x),buttheresultsholdfor

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

)

]|

=

i

_d4_p

(

2

π

)

4e− ip·(x−y)



I

+

G

(φ)

p

·

p

−

m2

₊

_i

_



.

(19)

ThisisnotaGreen’sfunctionoftheKlein–Gordonoperator

(∂

μ

∂

μ

+

m2

)

S

(

x

,

y

)

= −

i

_d4_p

(

2

π

)

4e−

ip·(x−y)

_[

_I

₊

_G

_(φ)

_].

₍₂₀₎ In eqs. (18)and (20)the problemresides inthe three and four-dimensionalFouriertransformof

G(φ)

whicharenon-vanishing.

3.1. TheinverseofI

+

G(φ)

Inaconsistent fieldtheory,we expectthe fieldstosatisfythe canonicalanti-commutationrelationsandthattheLagrangianand propagatorto havethe samesymmetry. In thisrespect,it is evi-dentthataKlein–GordonLagrangianisinadequatetodescribethe massdimensiononefermions.HerewederivetheLagrangianthat providesa completedescription of

λ(

x

)

. Westart bydetermining theoperator

O(

x

)

in whichthepropagatorgivenby eq.(19)isa Green’sfunction

O

(

x

)

S

(

x

,

y

)

= −

i

δ

4

(

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)

1

−

τ

2

.

(23)

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

−

τ

2 1

−

τ

2



I

=

I (24)

wherewe have used theidentity

G

2

(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,wefirstneedtodefineanoperator

G

whichcorrespondsto thematrix

G(φ)

intheconfigurationspace.Forthispurpose, frac-tional derivatives must be introduced since the matrix elements of

G(φ)

are proportional to e±iφ _{andthat itcan be expressed in} termsofcomplexmomentap±

= (

p1

±

ip2

)/

√

2 as

e±iφ

= [

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

. Theoperator

G

isdefinedas

G

=

i

⎛

⎜

⎜

⎜

⎜

⎝

0 0 0

−∂

₊1/2

∂

₋−1/2 0 0

∂

₊−1/2

∂

₋1/2 0 0

−∂

₊1/2

∂

₋−1/2 0 0

∂

₊−1/2

∂

₋1/2 0 0 0

⎞

⎟

⎟

⎟

⎟

⎠

(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

∂

₊1/2

∂

₋−1/2f

(

x

)

, anelementof

G

f

(

x

)

where f

(

x

)

isatestfunctionwiththeFourier transform f

(

x

)

=

d3p

(

2

π

)

3e− ip·x_F

₍

_p

_).

₍₂₈₎

Usingeq.(A.6),weobtain

∂

₋−1/2f

(

x

)

=

_d3_p

(

2

π

)

3e−

ip·x_e−iπ/4_e−in−π_p−1/2

+ F

(

p

)

(29)

where we haveused the fact that i−1/2

₌

_e−iπ/4_e−in_{−π has two}

rootswithn₋

=

0

,

1.Actingontheexpressionagainwithi

∂

₊1/2 we obtain i

∂

₊1/2

∂

₋−1/2f

(

x

)

=

d3p

(

2

π

)

3

(

i

ω

)

e− ip·x_e−iφ_F

₍

_p

₎

₍₃₀₎ where

ω

=

ei(n₊−n₋)_{π withn}

±

=

0

,

1.Duetotheambiguityofthe phases, in order toobtain

G(φ)

in the momentum space, all the elementsof

G

f

(

x

)

musthavethesamephasessothat

G

f

(

x

)

=

_d3_p

(

2

π

)

3

ω

e

−ip·x

_G

_(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,theactionof

G

onthefieldyields2

G

λ(

x

)

= (

2

π

)

−3

d3p



2mEp



α

[

e−ip·x

ω

ξ

ξ(

p

,

α

)

a

(

p

,

α

)

−

eip·x

ω

ζ

ζ (

p

,

α

)

a‡

(

p

,

α

)

].

(32)

Sincethephasescantakethevaluesof

±

1,

G λ(

x

)

hasfour possi-blesolutions.Outofthefourpossibilities,aswe willshowinthe nextsection,theonlysolutionwhichyieldsapositive-definitefree Hamiltonianis

ω

ξ

= −

ω

ζ

=

1

.

(33)

Thisthengivesustheequation

G

λ(

x

)

= λ(

x

),

(34)

whichisthecounterpartofeqs.(15a)and

(15b)

inthe configura-tionspace.

Havingdefined

G

,theinverseofI

+

τ

G(φ)

intheconfiguration spaceisgivenby

A

=

I

−

τ

G

1

−

τ

2 (35)

Nowweapply

A

toeq.(20).The

τ

-deformedpropagatoris

2 _{Inthispaper,weassumethatthefermionicfieldλ(x)anditsdual}¬_λ(x),both ofwhichareasumoftheFouriertransformofthe self-conjugateand anti-self-conjugatespinorsanditsdual,arewell-defined.

S(τ)

(

x

,

y

)

=

i

_d4_p

(

2

π

)

4e −ip·(x−y)



I

+

τ

G

(φ)

p

·

p

−

m2

₊

_i

_



.

(36)

When

A

actsontheS(τ)

(

x

,

y

)

,wemusttake

ω

=

1 sothatinthe limit

τ

→

1,weobtain

lim

τ→1

A

(∂

μ

_∂

μ

+

m2

)

S(τ)

(

x

,

y

)

= −

i

δ

4

(

x

−

y

)

I

.

(37)

Theoperator

O(

x

)

,inwhich thepropagatorisa Green’sfunction ofistherefore

O

(

x

)

=

A

(∂

μ

∂

μ

+

m2

).

(38)

Basedontheformof

O(

x

)

,theLagrangianfor

λ(

x

)

is

L

=

A

ab

(∂

μ

¬

λ

a

∂

μ

λ

b

−

m2

¬

λ

a

λ

b

)

(39)

wherewe sumover the repeatedindices. The operator

A

is di-mensionlessso

λ(

x

)

remainsamassdimensiononefield.Thefield equationis

A

ab

(∂

μ

∂

μ

+

m2

)λ

b

=

0

.

(40)

Theoperator

A

doesnotaffectthesolutionsofthefieldequation butaswewillshowinthesubsequentsection,itensuresthatthe fieldsatisfiesthecanonicalanti-commutationrelations.

3.2.Canonicalanti-commutationrelations

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

π

b

(

y

)

=

A

ab

(

y

)

∂

λ

¬a

∂

t

(

y

)

(41)

which differs from

πkg

(

y

)

by a factor of

A(

y

)

. Since fractional derivativescommute,theanti-commutatorbetweentheconjugate momentumis

{

π

(

t

,

x

),

π

(

t

,

y

)

} =

O

.

(42)

A direct evaluation of the

τ

-deformed field-momentum anti-commutatornowyields

lim

τ→1

{λ(

t

,

x

),

π

(

t

,

y

)

}

(τ)

₌

_i

_δ

3

₍

_x

₋

_y

₎

_I

_.

₍₄₃₎

We now show that the free Hamiltonian is positive-definite. Thisisachievedby showingthatthefree Hamiltonian,asa func-tionof

λ(

x

)

and¬

λ(

x

)

isidenticaltotheonegivenin[40,eq. (97)]. InordertoobtainthecorrectHamiltonian,wemusttakethe con-jugatemomentumassociatedwithboth

λ(

x

)

and

λ(

¬ x

)

intoaccount wherethelaterisdefinedas

¬

π

a

= −

∂

L

∂

λ

¬a

/∂

t

= −

A

ab

∂λ

b

∂

t

.

(44)

TheHamiltonianisthengivenbytheLegendretransformation

H

=

d3x



∂

λ

¬a

∂

t



A

ab

∂λ

b

∂

t



−



A

ab

∂λ

b

∂

t



∂

λ

¬a

∂

t

−

L



.

(45)

Inobtainingthefirstterm, wehaveusedthedefinitionof

A

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→1

A

ab

λ

b

(

x

)

=

1

2

λ

a

(

x

).

(46)

Usingtheidentity

d3x

A

ab

(∂

μ ¬

λ

a

∂

μ

λ

b

−

m2 ¬

λ

a

λ

b

)

=

1 2

d3x

(∂

μ

λ

¬a

∂

μ

λ

a

−

m2

λ

¬a

λ

a

),

(47)

theHamiltoniansimplifiesto

H

=

1 2

d3x



−

∂λ

a

∂

t

∂

λ

¬a

∂

t

− ∂i

¬

λ

a

∂

i

λ

a

+

m2 ¬

λ

a

λ

a



.

(48)

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. Fractionalderivatives

Allfractionalderivativessatisfy thefollowingproperties.Let

α

beanarbitraryrealnumber,inthelimit

α

→

n wheren isa posi-tiveinteger, lim α→n dα dxα f

(

x

)

=

dn dxnf

(

x

).

(A.1)

Theoperationsarelinear

dα dxα

[

c f

(

x

)

] =

c dα dxαf

(

x

),

(A.2a) dα dxα

[

f

(

x

)

+

g

(

x

)

] =

dα dxαf

(

x

)

+

dα dxαg

(

x

).

(A.2b)

TheLeibnizruleis

dα dxα

(

f g

)

=

∞



j=0



α

j

 

dα−j dxα f

 

dj dxjg



(A.3)

wherethebinomialcoefficientisgeneralizedtoarbitraryreal num-bersbythe

(

α

)

function



α

j



=

(

α

+

1

)

(

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 2

π

∞

−∞ dk e−ikxF

(

k

),

(A.5a) F

(

k

)

=

1 2

π

∞

−∞ dx eikxf

(

x

).

(A.5b)

Thedefinitionofthefractionalderivative on f

(

x

)

isa straightfor-wardgeneralisationoftheusualderivative

dα dxαf

(

x

)

=

1 2

π

∞

−∞ dk

(

−

ik

)

αe−ikxF

(

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 dxαg

(

x

)



= (−

1

)

α

dx



g

(

x

)

d α dxαf

(

x

)



.

(A.7) References

[1] D.V.Ahluwalia,OnalocalmassdimensiononeFermifieldofspinone-halfand thetheoreticalcrevicethatallowsit,2013.

[2]D.V.Ahluwalia,D.Grumiller,Darkmatter:a spinonehalffermionfieldwith massdimensionone?,Phys.Rev.D72(2005)067701.

[3]D.V.Ahluwalia,D.Grumiller,Spinhalffermionswithmassdimensionone: the-ory,phenomenology,anddarkmatter,J.Cosmol.Astropart.Phys.0507(2005) 012.

[4]D.V.Ahluwalia,S.P.Horvath,Veryspecialrelativityasrelativityofdarkmatter: theElkoconnection,J.HighEnergyPhys.1011(2010)078.

[5]D.V.Ahluwalia,C.-Y.Lee,D.Schritt,Elkoasself-interactingfermionicdark mat-terwithaxisoflocality,Phys.Lett.B687(2010)248–252.

[6]D.V.Ahluwalia,C.-Y.Lee,D.Schritt,Self-interactingElkodarkmatterwithan axisoflocality,Phys.Rev.D83(2011)065017.

[7]A.Alves,F.deCampos,M.Dias,J.M.Hoff daSilva,SearchingforElkodark matterspinorsattheCERNLHC,Int.J.Mod.Phys.A30,no.01(2015)1550006. [8]A.Basak,J.R.Bhatt,S.Shankaranarayanan,K.Prasantha Varma,Attractor

be-haviourinELKOcosmology,J.Cosmol.Astropart.Phys.1304(2013)025. [9]A.Basak,S.Shankaranarayanan,Super-inflationand generationoffirstorder

vectorperturbationsinELKO,J.Cosmol.Astropart.Phys.1505 (05)(2014)034. [10]A.Bernardini,R.da Rocha,DynamicaldispersionrelationforELKOdarkspinor

fields,Phys.Lett.B717(2012)238–241.

[11]C.G.Boehmer,TheEinstein–Cartan–Elkosystem,Ann.Phys.16(2007)38–44. [12]C.G.Boehmer,TheEinstein–Elkosystem:candarkmatterdriveinflation?,Ann.

Phys.16(2007)325–341.

[13]C.G.Boehmer,Darkspinorinflation:theoryprimeranddynamics,Phys.Rev.D 77(2008)123535.

[14]C.G.Boehmer,J.Burnett,Darkspinorswithtorsionincosmology,Phys.Rev.D 78(2008)104001.

[15]C.G.Boehmer,J.Burnett,Darkenergywithdarkspinors,Mod.Phys.Lett.A25 (2010)101–110.

[16]C.G.Boehmer,J.Burnett,Darkspinors,arXiv:1001.1141[gr-qc],2010. [17]C.G.Boehmer,J.Burnett,D.F.Mota,D.J.Shaw,Darkspinormodelsingravitation

andcosmology,J.HighEnergyPhys.1007(2010)053.

[18]C.G.Boehmer, D.F.Mota, CMBanisotropiesand inflationfrom non-standard spinors,Phys.Lett.B663(2008)168–171.

[19]R.Cavalcanti,J.M.H.daSilva,R.daRocha,VSRsymmetriesintheDKPalgebra: theinterplaybetweenDiracandElkospinorfields,Eur.Phys.J.Plus129 (11) (2014)246.

[20]G.Chee,StabilityofdeSittersolutionssourcedbydarkspinors,arXiv:1007. 0554[gr-qc],2010.

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

[22]R.da Rocha,A.E.Bernardini,J.Hoff da Silva,Exoticdarkspinorfields,J.High EnergyPhys.1104(2011)110.

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

[24]R.daRocha,J.HoffdaSilva,ELKOspinorfields:Lagrangiansforgravityderived fromsupergravity,Int.J.Geom.MethodsMod.Phys.6(2009)461–477. [25]R.daRocha,J.HoffdaSilva,A.E.Bernardini,Elkospinorfieldsasatoolfor

probingexotictopologicalspacetimefeatures,Int.J.Mod.Phys.Conf. Ser.3 (2011)133–142.

[26]R.daRocha,J.M.HoffdaSilva,FromDiracspinorfieldstoELKO,J.Math.Phys. 48(2007)123517.

[27]R.daRocha,J.M.HoffdaSilva,ELKO,flagpoleandflag-dipolespinorfields,and theinstantonHopffibration,Adv.Appl.CliffordAlgebras20(2010)847–870. [28]R.da Rocha,W.A. RodriguesJr., WhereareELKOspinor fieldsinLounesto

spinorfieldclassification?,Mod.Phys.Lett.A21(2006)65–74.

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

[30]M.Dias,F.deCampos,J.HoffdaSilva,ExploringElkotypicalsignature,Phys. Lett.B706(2012)352–359.

[31]P.A.M.Dirac,ThePrinciplesofQuntumMechanics,4thedition,Univ.Pr.,Oxford, UK,1958.

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

[33]L.Fabbri,CausalityforELKOs,Mod.Phys.Lett.A25(2010)2483–2488. [34]L.Fabbri,Zeroenergyofplane-wavesforELKOs,Gen.Relativ.Gravit.43(2011)

1607–1613.

[35]J.B.J.Fourier,Théorieanalytiquedelachaleur,Univ.Pr.,Cambridge,UK,1822, p. 643.

[36]D.Gredat,S.Shankaranarayanan,Modifiedscalarandtensorspectrainspinor driveninflation,J.Cosmol.Astropart.Phys.1001(2010)008.

[37]R.Herrmann,FractionalCalculus:An IntroductionforPhysicists,World Scien-tific,2011,261 pp.

[38]J.M.HoffdaSilva,R.daRocha,FromDiracactiontoELKOaction,Int.J.Mod. Phys.A24(2009)3227–3242.

[39]S.P.Horvath,Ontherelativityofelkodarkmatter,Master’sthesis,University ofCanterbury,2011,106 pp.

[40]C.-Y.Lee,Self-interactingmass-dimensiononefieldsforanyspin,Int.J.Mod. Phys.A30(2015)1550048.

[41]C.-Y.Lee,SymmetriesofElkoandmassivevectorfields,Doctoralthesis, Univer-sityofCanterbury,2012,198 pp.

[42]P.Lounesto,Cliffordalgebrasandspinors,Lond.Math.Soc.Lect.NoteSer.286 (2001)1–338.

[43]A.Nikitin,Non-standardDiracequationsfornon-standardspinors,Int.J.Mod. Phys.D23 (14)(2014)1444007.

[44]R.J.B.Rogério,J.M.H.daSilva,Thelocalvicinityofspinssumforcertainmass dimensiononespinors,arXiv:1602.05871[hep-th].

[45]D.V.Ahluwalia,Astoryofphases,duals,andadjointsforalocalLorentz co-varianttheoryofmassdimensiononefermion,arXiv:1601.03188[hep-th]. [46]S.Shankaranarayanan, What-ifinflatonis aspinorcondensate?,Int.J.Mod.

[47]S. Shankaranarayanan, Darkspinor driven inflation, arXiv:1002.1128 [astro-ph.CO],2010.

[48]L.Sperança,AnidentificationoftheDiracoperatorwiththeparityoperator, Int.J.Mod.Phys.D23 (14)(2014)1444003.

[49]H.Wei,Spinordarkenergyandcosmologicalcoincidenceproblem,Phys.Lett. B695(2011)307–311.

[50]K.E.Wunderle,R.Dick,AsupersymmetricLagrangianforfermionicfieldswith massdimensionone,Can.J.Phys.90(2012)1185.