Research ArticleDark Spinors Hawking Radiation in String Theory Black Holes

R T Cavalcanti12 and Roldatildeo da Rocha34

1CCNH Universidade Federal do ABC 09210-580 Santo Andre SP Brazil2Dipartimento di Fisica e Astronomia Universita di Bologna Via Irnerio 46 40126 Bologna Italy3CMCC Universidade Federal do ABC 09210-580 Santo Andre SP Brazil4International School for Advanced Studies (SISSA) Via Bonomea 265 34136 Trieste Italy

Correspondence should be addressed to Roldao da Rocha roldaorochaufabcedubr

Received 26 August 2015 Accepted 24 November 2015

Academic Editor Enrico Lunghi

Copyright copy 2016 R T Cavalcanti and R da Rocha This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited The publication of this article was funded by SCOAP3

The Hawking radiation spectrum of Kerr-Sen axion-dilaton black holes is derived in the context of dark spinors tunnelling acrossthe horizon Since a black hole has a well defined temperature it should radiate in principle all the standardmodel particles similarto a black body at that temperature We investigate the tunnelling of mass dimension one spin-12 dark fermions which are beyondthe standardmodel and are prime candidates to the darkmatterTheir interactions with the standardmodel matter and gauge fieldsare suppressed by at least one power of unification scale being restricted just to the Higgs field and to the graviton likewise Thetunnelling method for the emission and absorption of mass dimension one particles across the event horizon of Kerr-Sen axion-dilaton black holes is shown here to provide further evidence for the universality of black hole radiation further encompassingparticles beyond the standard model

1 Introduction

Black hole tunnelling procedures have been placed as promi-nent methods to calculate the temperature of black holes [1ndash9] Tunnelling methods provide models for describing theblack hole radiation Various types of black holes have beeninvestigated in the context of tunnelling of fermions andbosons as well [1 2 9 10] Complementarily to the first resultsthat studied the tunnelling of particles across black holes[1 2] the Hamilton-Jacobi method was employed [3] andfurther generalized by applying the WKB approximation tothe Dirac equation [10] Tunnelling procedures are quite usedto investigate black holes radiation by taking into accountclassically forbidden paths that particles go through fromthe inside to the outside of black holes Moreover quantumWKB approaches were employed to calculate corrections tothe Bekenstein-Hawking entropy for the Schwarzschild blackhole [11]

The tunnelling method was employed to provide Hawk-ing radiation due to dark spinors for black strings [12]Moreover this method was also employed to model the

emission of spin-12 fermions and the Hawking radiationwas deeply analyzed as the tunnelling of Dirac particlesthroughout an event horizon where quantum correctionsin the single particle action are proportional to the usualsemiclassical contribution In addition the modifications tothe Hawking temperature and Bekenstein-Hawking entropywere derived for the Schwarzschild black holeWhen spin-12fermions are taken into account the effect of the spin of eachtype of fermion cancels out due to particles with the spin inany direction Hence the lowest WKB order implies that theblack hole intrinsic angular momentum remains constant intunnelling processes Hawking radiation emulates semiclas-sical quantum tunnelling methods wherein the Hamilton-Jacobi method is comprehensively used [3] Mass dimension32 fermions tunnelling has been studied in the chargeddilatonic black hole the rotating Einstein-Maxwell-Dilaton-Axion black hole and the rotating Kaluza-Klein black holelikewise [13] For a review see [9]

Our approach here is to employ an exact classical solutionin the low energy effective field theory describing heteroticstring theory the Kerr-Sen axion-dilaton black holes [14]

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 4681902 7 pageshttpdxdoiorg10115520164681902

2 Advances in High Energy Physics

They present charge magnetic dipole moment and angularmomentum involving the antisymmetric tensor field coupledto the Chern-Simons 3-form A myriad of black holes hasbeen considered for tunnelling methods of fermions andbosons as rotating and accelerating black holes and topo-logical BTZ Reissner-Nordstrom Kerr-Newman and Taub-NUT-AdS black holes including the tunnelling of higher spinfermions as well [15]

We shall study similar methods for spin-12 fermionsof mass dimension one different from the procedure forstandard mass dimension 32 fermions Elko dark spinorsnamely the dual-helicity eigenspinors of the charge conjuga-tion operator [16 17] are spin-12 fermions of mass dimen-sion one with novel features thatmake them capable of incor-porating both the Very Special Relativity (VSR) paradigmand the dark matter description as well [16 18] Such spinorsseem to be indeed a tip of the iceberg for a comprehensiveclass of nonstandard (singular) spinors [19 20] Moreovera mass generation mechanism has been introduced in [21]for such dark particles by a natural coupling to the kinksolution in field theory It provides exotic couplings amongscalar field topological solutions and Elko dark spinors [2122] Due to its very small coupling with the standard modelfields except the Higgs field dark spinors supply naturalself-interacting dark matter prime candidates Except forscalar fields and gravity Elko dark spinors interactions withthe standard model matter and gauge fields are suppressedby at least one power of unificationPlanck scale [16] Infact the Lagrangian of such a field contains a quartic self-interaction term and the interaction term of the new fieldwith spin-zero bosonic fields Moreover Elko framework isshown to be invariant under the action of the HOM(2) VSRgroup and covariant under SIM(2) VSR group [23] Elko darkspinor field is a representative of mass dimension one spin-12 fermions in the type-5 spinor field class in Lounestorsquosspinors classification however it is not the most generalsince Majorana spinors are also encompassed by such class[22] Some attempts to detect Elko at the LHC have beenproposed and important applications to cosmology havebeen widely investigated as well [12 16 18 24ndash26]

This paper is presented as follows the Kerr-Sen axion-dilaton black hole is briefly revisited in the next sectiontogether with the dark spinors framework We thus shallcalculate in Section 3 the probabilities of emission andabsorption of Elko dark particles across these black holesTherefore the WKB approximation is used to computethe tunnelling rate and thus the resulting tunnelling prob-ability Finally the associated Hawking temperature shallbe obtained corroborating the universal character of theHawking effect and further extending it to particles beyondthe standard model

2 Kerr-Sen Axion-Dilaton Black Holes andDark Spinors

String theory has solutions describing extra-dimensionalextended objects surrounded by event horizons presenting acausal structure associated with singularities in string theory

The low energy effective action of the heterotic string theoryis ruled by an action that up to higher derivative terms andother fields which are set to zero for the particular class ofbackgrounds considered [14] is given by

119878 = minusint1198894119909radicminus det119866119890

minusΦ

sdot (minus119877 +1

12119867120583]120588119867120583]120588

minus 119866120583]120597120583Φ120597]Φ +

1

8119865120583]119865120583])

(1)

where 119866120583] is the metric regarding a 120590-model [14] related to

the Einstein metric by 119890minusΦ

119866120583] Φ denotes the dilaton field 119877

stands for the scalar curvature 119865120583] = 120597

[120583119860]] is the Maxwell

field strength and119867120583]120588 = 120597

120583119861]120588+120597120588119861120583]+120597]119861120588120583minusΩ120583]120588 for the

Chern-Simons 3-formΩ120583]120588 minus (14)119860

(120583119865]120588) The above action

can be led to the one in [27] up to the 119867120583]120588119867120583]120588 term after

field redefinitionTheKerr-Sen dilaton-axion black holemetric is a solution

of the field equations derived from (1) In Boyer-Lindquistcoordinates it reads

1198891199042= minus

Δ minus 1198862sin2120579Σ

1198891199052+

Σ

Δ1198891199032+ Σ119889120579

2

+sin2120579Σ

[(1199032minus 2120573119903 minus 119886

2)2

minus Δ1198862sin2120579] 1198891205932

minus2119886 sin2120579

Σ[(1199032minus 2120573119903 minus 119886

2)2

minus Δ] 119889119905 119889120593

(2)

where

Σ = 1199032minus 2120573119903 + 119886

2cos2120579

Δ = 1199032minus 2120578119903 + 119886

2= (119903 minus 119903

+) (119903 minus 119903

minus)

120573 = 120578 sinh2 1205722

(3)

and 119903+[119903minus] are the coordinate outer [inner] singularities

Metric (2) describes a black hole solution with charge 119876mass119872 magnetic dipolemoment120583 and angularmomentum119869 given by

119876 =120578

radic2sinh120572

119872 =120578

2(1 + cosh120572)

120583 =1

radic2120578119886 sinh120572

119869 =120578119886

2(1 + cosh120572)

(4)

Advances in High Energy Physics 3

The associated 119892-factor can be expressed as 119892 = 2120583119872119876119869 = 2

[27] The parameters can be expressed in terms of genuinelyphysical quantities as

120578 = 119872 minus1198762

2119872

120572 = arcsinh(2radic2119876119872

21198722 minus 1198762)

119886 =119869

119872

(5)

The coordinate singularities thus read 119903plusmn

= 119872 minus 11987622119872 plusmn

radicminus11986921198722 + (119872 minus 11987622119872)2 which vanishes unless |119869| lt

1198722minus 11987622 The area of the outer event horizon is given by

119860 = 8120587119872(119872minus1198762

2119872+ radicminus

1198692

1198722+ (119872 minus

1198762

2119872)

2

) (6)

Thus in the extremal limit since |119869| rarr 119872minus11987622119872 it reads

119860 rarr 8120587|119869| In this limit the horizon is hence finite and thesurface gravity 120581 or equivalently the Hawking temperature119879119867

= 1205812120587 is provided by [14]

120581 =

radicminus41198692 + (21198722 minus 1198762)2

2119872(21198722 minus 1198762 + radicminus41198692 + (21198722 minus 1198762)2

)

(7)

Thus in the extremal limit we have the limit 120581 rarr 0 if 119869 = 0On the other hand if 119869 = 0 then 120581 = 14119872 in agreement withthe results of [27 28] For 119869 = 0 this black hole solution hasaspects analogous to the extremal rotating black hole ratherthan extremal charged black holes [27]

By performing the transformation 120601 = 120593minusΩ119905 whereΩ =

119886(1198862minus 2120573119903 + 119886

2minusΔ)((119903

2minus 2120573119903 + 119886

2)2minusΔ1198862sin2120579) the metric

(2) takes the form

1198891199042= minus

ΔΣ

(1199032 minus 2120573119903 minus 1198862)2

minus Δ1198862sin21205791198891199052+

Σ

Δ1198891199032

+ Σ1198891205792

+sin2120579Σ

[(1199032minus 2120573119903 minus 119886

2)2

minus Δ1198862sin2120579] 1198891206012

(8)

To study the Hawking radiation at the event horizon themetric is regarded near the horizon

1198891199042= minus119865 (119903

+) 1198891199052+

1

119866 (119903+)1198891199032+ Σ (119903

+) 1198891205792

+119867 (119903+)

Σ (119903+)1198891206012

(9)

where

119867(119903+) = sin2120579 (1199032

+minus 2120573119903

++ 1198862)2

119865 (119903+) =

2 (119903+minus 120578) (119903 minus 119903

+) Σ (119903+)

(1199032+minus 2120573119903

++ 1198862)2

119866 (119903+) =

2 (119903+minus 120578) (119903 minus 119903

+)

Σ (119903+)

(10)

In order to analyze the tunnelling of Elko dark particles acrossthe Kerr-Sen black hole event horizon we will study the rolethat Elko dark particles play in this backgroundThe essentialprominent Elko particles features are in short revisited [16]Elko dark spinors 120582(119901

120583) are eigenspinors of the charge

conjugation operator 119862 namely 119862120582(119901120583) = plusmn120582(119901120583) The

plus [minus] sign regards self-conjugate [anti-self-conjugate]spinors denoted by 120582119878(119901120583) [120582

119860(119901120583)] For spinors at rest 120582(119896120583)

the boosted spinors read 120582(119901120583) = 119890

119894120581sdot120593120582(119896120583) where 119896

120583=

(119898 lim119901rarr0

p|p|) where 119890119894120581sdot120593 denotes the boost operator

120601(119896120583) are defined to be eigenspinors of the helicity operator

as 120590 sdot p120601plusmn(119896120583) = plusmn120601plusmn(119896120583) where [16]

120601+(119896120583) = radic119898(

cos(120579

2) 119890minus1198941205932

sin(120579

2) 119890+1198941205932

) equiv (120572

120573)

120601minus(119896120583) = radic119898(

minus sin(120579

2) 119890minus1198941205932

cos(120579

2) 119890+1198941205932

) = (minus120573lowast

120572lowast)

(11)

Elko dark spinors 120582(119896120583) are constructed as

120582119878

plusmn(119896120583) = (

1205902(120601plusmn(119896120583))lowast

120601plusmn(119896120583)

)

120582119860

plusmn(119896120583) = plusmn(

minus1205902(120601∓(119896120583))lowast

120601∓(119896120583)

)

(12)

and have dual helicity as minus1198941205902(120601plusmn)lowast has helicity dual to that

of 120601plusmn The boosted terms

120582119860

plusmn(119901120583) = radic

119864 + 119898

2119898(1 plusmn

119901120583

119864 + 119898)120582119860

plusmn

120582119878

plusmn(119901120583) = radic

119864 + 119898

2119898(1 ∓

119901120583

119864 + 119898)120582119878

plusmn

(13)

are the expansion coefficients of a mass dimension onequantum field The Dirac operator does not annihilate the120582(119901120583) but instead the equations of motion read [16 17]

120574120583nabla120583120582119878

plusmn= plusmn119894

119898

ℎ120582119878

∓ (14)

120574120583nabla120583120582119860

∓= plusmn119894

119898

ℎ120582119860

plusmn (15)

Dark spinors nevertheless satisfy the Klein-Gordon equation

4 Advances in High Energy Physics

A mass dimension one quantum field can be thus con-structed as [17]

f (119909) = int1198893119901

(2120587)3

1

radic2119898119864 (p)

sdot sum

120588

[119887dagger

120588(p) 120582119860 (p) 119890119894119901120583119909

120583

+ 119886120588(p) 120582119878 (p) 119890minus119894119901120583119909

120583

]

(16)

The creation and annihilation operators 119886120588(p) 119886dagger120588(p) satisfy

the Fermi statistics [17] with similar anticommutators for119887120588(p) and 119887

dagger

120588(p) The mass dimensionality of f(119909) can be

realized from the adjoint

not

f (119909) = int1198893p

(2120587)3

1

radic2119898119864 (p)

sdot sum

120588

[119887120588(p)not

120582

119860

(p) 119890minus119894119901120583119909120583

+ 119886dagger

120588(p)not

120582

119878

(p) 119890119894119901120583119909120583

]

(17)

where denoting hereupon by 120590120583 the Pauli matrices the con-jugate spinors are constructed by

not

120582120588(119901120583) = [Ξ120582

120588(119901120583)]dagger1205901otimes

I2 Here the operator Ξ = (12119898)(120582

119878

minus(119901120583)120582119878

minus(119901120583) minus

120582119860

minus(119901120583)120582119860

minus(119901120583) + 120582119878

+(119901120583)120582119878

+(119901120583) minus 120582119860

+(119901120583)120582119860

+(119901120583)) is an invo-

lution [17] where the standard Dirac conjugate 120582(119901120583) =

120582(119901120583)dagger1205740 is adopted The mass dimension of the new field is

determined by the SIM(2) covariant propagator [17]

119878 (119909 minus 1199091015840) = 119894⟨

10038161003816100381610038161003816100381610038161003816

T(f (119909)not

119891 (1199091015840))

10038161003816100381610038161003816100381610038161003816

⟩

= minus lim120598rarr0+

int1198894119901

(2120587)4119890minus119894119901120583(1199091015840

120583minus119909120583) (

I +G (120593)

119901120583119901120583minus 1198982 + 119894120598

)

(18)

where T is the canonical time-ordering operator and

G (120593) = (0 minus119894119890

minus119894120593

119894119890119894120593

0

) otimes 1205901

(19)

that respects symmetries of the theory of VSR [17 23]

3 Hawking Radiation from TunnellingDark Spinors

Hawking radiation from general black holes encompassesdistinguished charged and uncharged particles Tunnellingmethods can be employed for Elko dark particles across thehorizon of Kerr-Sen black holes From (8) the associated

tetrad can be chosen so that the following generators can beachieved

120574119905=

1

radic119865 (119903+)

(0 I2

I2

0)

120574120579=

1

radicΣ (119903+)

(0 1205902

minus1205902

0

)

120574119903= radic119866 (119903

+)(

0 1205901

minus1205901

0

)

120574119911= radic

Σ (119903+)

119867 (119903+)(

0 1205903

minus1205903

0

)

(20)

Elko dark spinors can be written as

120582119878

+= (

minus119894120573lowast

119894120572lowast

120572

120573

) exp (119894

ℏ)

120582119878

minus= (

minus119894120572

minus119894120573

minus120573lowast

120572lowast

) exp(119894

ℏ)

120582119860

+= (

119894120572

119894120573

minus120573lowast

120572lowast

) exp(119894

ℏ)

120582119860

minus= (

minus119894120573lowast

119894120572lowast

minus120572

minus120573

) exp(119894

ℏ)

(21)

where = (119905 119903 120579 119911) represents the classical action We usethe above forms for dark particles in each of (14) and (15) andthen solve this coupled system of equations By denoting

nabla120583= 120597120583+

1

8119894Γ120572120573

120583[120574120572 120574120573] (22)

where 120574120590 are the usual Clifford bundle generators for the

Minkowski spacetime By identifying 120582 [∘

120582] to the Elko spinoron the left [right] hand side of (14) and (15) then (14) reads120574120583(nabla120583+ 119890119860120583)120582 = 119894(119898ℎ)

∘

120582 Using the WKB approximationwhere = 119868 + O(ℎ) it yields

(119868120583+ 119890119860120583) 120574120583120582 = 119894119898

∘

120582 + O (ℎ) (23)

Advances in High Energy Physics 5

where 119868120583

equiv 120597119868120597119909120583 Taking merely the leading order terms

in the above equation from a general form 120582 = (119886 119887 119888 119889)⊤

∘

120582 = (∘

119886∘

119887∘

119888∘

119889)⊤ we have general Elko dynamic equations

governed by (23) The ansatz 119868(119905 119903 120579 120593) = minus(120596 minus 119895Ω)119905 + 119895120593 +

119882(119903) + Θ(120579) can be used where 120596 and 119895 denote the energyand magnetic quantum number of the particles respectivelyMoreover the parameters 119886 119887 119888 119889 are not independentIn fact (21) assert that for the self-conjugate spinors 120582

119878 wehave 119886 = minus119894119889

lowast and 119887 = 119894119888lowast whereas for the anti-self-

conjugate spinors 120582119860 it reads 119886 = 119894119889lowast and 119887 = minus119894119888

lowast Thus bycorresponding the 120582119878 [120582119860] spinors to the upper [lower] signbelow after awkward computation (23) yields

plusmnradic119866 (119903+)1198821015840119889lowast∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

119888lowast=∘

119889119898 (24)

plusmnradic119866 (119903+)1198821015840119888lowast∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

119889lowast= minus∘

119888119898 (25)

The angular function 119895120593 + Θ(120579) must be a complex functionand the same solution for it is achieved for both incomingand outgoing cases as well It implies that the contribution ofsuch function vanishes after dividing the outgoing probabilityby the incoming one Hence the angular function can beneglected hereupon

In the above system the equations for 120582119878+[120582119878

minus] are shown

to be equivalent to the ones for 120582119860

minus[120582119860

+] Thus we have to

deal solelywith the self-conjugate120582119878plusmnspinorsMoreover there

are more underlying equivalences In fact (24) for 120582119878

plusmnis

equivalent to (25) for 120582119878∓ Consequently there is just a couple

of equations for 120582119878plusmngiven by

radic119866 (119903+)1198821015840120572lowastminus

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120573lowast= 120573lowast119898

radic119866 (119903+)1198821015840120573 +

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120572 = 120572119898

(26)

radic119866 (119903+)1198821015840120572 ∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120573 = ∓120573119898

radic119866 (119903+)1198821015840120573lowast∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120572lowast= plusmn120572lowast119898

(27)

Combining either (26) or (27) implies equations for either120582119878

+or 120582119878minus respectively Hence for each 120582

119878 there is a systemof coupled equations for the dark spinor components 120572 and120573 and also another coupled system for 120572

lowast and 120573lowast which

are going to be solved separately We denote now the firstequation of each one of the systems below to be the equations

related to (120572 120573) whereas the second ones regard (120572lowast 120573lowast) We

can determine the above functions as

120582119878

+

1198821(119903) = plusmnint

radic1198982119865 minus (120596 minus 119895Ω

119867+ 119890119860+)2

119865119866119889119903

1198822(119903) = plusmnint

radic(119898radic119865 + 120596 minus 119895Ω

119867+ 119890119860+)2

119865119866119889119903

120582119878

minus

1198823(119903) = plusmnint

radic(119898radic119865 minus 120596 + 119895Ω

119867+ 119890119860+)2

119865119866119889119903

1198824(119903) = 119894119882

2(119903)

(28)

For massless particles the solutions for 120582119878

plusmnare equivalent

being given by

1198822(119903) = plusmnint

radic(120596 minus 119895Ω

119867+ 119890119860+)2

119865119866119889119903 = minus119894119882

1(119903)

(29)

The above expression for1198822(119903) is similar to the results in [29]

when119898 = 0 The solution for1198822(119903) is provided by

1198822(119903) = plusmn119894120587

1199032

+minus 2120573119903

++ 1198862

2 (119903++ 119898)

(120596 minus 119869Ω119867+ 119890119860+) (30)

Since 119865 rarr 0 near the black hole horizon the results holdboth for massive and their massless limit particles as well Infact for such limit (28) and (29) are equivalent It is worthemphasizing that (28) implies a real solution for 119882

1(119903) and

thus a null contribution for the tunnelling effectIn compliance with the WKB approximation the tun-

nelling rate reads Γ prop exp(minus2 Im 119868) where 119868 denotes theclassical action for the path Hence the imaginary part of theaction becomes a prominent goal for the tunnelling processThe imaginary part of the action yields

Im 119868plusmn= plusmn119894120587

1199032

+minus 2120573119903

++ 1198862

2 (119903++ 119898)

(120596 minus 119869Ω119867+ 119890119860+) (31)

Thus the resulting tunnelling probability is given by

Γ =119875(emission)

119875(absorption)

=119890minus2 Im 119868+

119890minus2 Im 119868minus

= minus21205871199032

+minus 2120573119903

++ 1198862

(119903++ 119898)

(120596 minus 119869Ω119867+ 119890119860+)

(32)

Finally the Hawking temperature of the Kerr-Sen dilaton-axion black hole is acquired

119879119867

=1

2120587

(119903++ 119898)

1199032+minus 2120573119903

++ 1198862

(33)

6 Advances in High Energy Physics

which is a universal formula also obtained by other methodsfor the Hawking temperature from fermions tunnelling [29]Obviously when the parameter 119886 tends to zero (33) providesthe well-known Hawking radiation associated with the staticblack hole

4 Concluding Remarks

We derived the Hawking radiation for spin-12 fermions ofmass dimension one represented by dark spinors tunnellingacross a Kerr-Sen dilaton-axion black hole horizonThe tem-perature of these solutions was computed and demonstratedto confirm the universal character of theHawking effect evenfor mass dimension one fermions of spin-12 that are beyondthe standard model The mass dimension one feature of suchspinor fields sharply suppresses the couplings to other fieldsof the standardmodel Indeed by power counting argumentsElko spinor fields can self-interact and further interact witha scalar (Higgs) field providing a renormalizable frameworkThis type of interaction means an unsuppressed quartic self-interaction The quartic self-interaction is essential to darkmatter observations [30 31] Therefore Elko spinor fieldsperform an adequate fermionic dark matter candidate Itcan represent a real model for dark matter tunnelling acrossblack holes Corrections of higher order in ℏ to the Hawkingtemperature (33) of type 119868 = 119868

0+ sum119899ge1

ℏ119899119868119899[11] can be still

implemented in the context of mass dimension one spin-12 fermions Moreover other mass dimension one fermions[20] and higher spin mass dimension one fermions can bestudied in the context of black hole tunnelling methodshowever these issues are beyond the scope of this paperwhich comprised dark particles tunnelling across Kerr-Sendilaton-axion black holes

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

R T Cavalcanti thanks CAPES and UFABC Roldao daRocha is grateful to CNPq Grants no 3030272012-6 no4516822015-7 and no 4733262013-2 for partial financialsupport and to FAPESP Grant no 201510270-0

References

[1] P Kraus and F Wilczek ldquoSelf-interaction correction to blackhole radiancerdquo Nuclear Physics B vol 433 no 2 pp 403ndash4201995

[2] M K Parikh and F Wilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000

[3] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 1999

[4] M Agheben M Nadalini L Vanzo and S Zerbini ldquoHawkingradiation as tunneling for extremal and rotating black holesrdquoJournal of High Energy Physics vol 2005 article 014 2005

[5] M Arzano A J Medved and E C Vagenas ldquoHawkingradiation as tunneling through the quantum horizonrdquo Journalof High Energy Physics vol 2005 no 9 p 37 2005

[6] Q-Q Jiang S-Q Wu and X Cai ldquoHawking radiation as tun-neling from the Kerr and Kerr-Newman black holesrdquo PhysicalReview D vol 73 no 6 Article ID 064003 2006

[7] J Zhang and Z Zhao ldquoCharged particlesrsquo tunnelling from theKerr-Newman black holerdquo Physics Letters B vol 638 no 2-3pp 110ndash113 2006

[8] R Li J-R Ren and S-W Wei ldquoHawking radiation of Diracparticles via tunneling from the Kerr black holerdquo Classical andQuantum Gravity vol 25 no 12 Article ID 125016 2008

[9] L Vanzo G Acquaviva and R Di Criscienzo ldquoTunnellingmethods and Hawkingrsquos radiation achievements andprospectsrdquo Classical and Quantum Gravity vol 28 no 18Article ID 183001 2011

[10] R Kerner and R B Mann ldquoFermions tunnelling from blackholesrdquo Classical and Quantum Gravity vol 25 no 9 Article ID095014 2008

[11] R Banerjee and B R Majhi ldquoHawking black body spectrumfrom tunneling mechanismrdquo Physics Letters B vol 675 no 2pp 243ndash245 2009

[12] R da Rocha and J M Hoff da Silva ldquoHawking radiationfrom Elko particles tunnelling across black-strings horizonrdquoEurophysics Letters vol 107 no 5 Article ID 50001 2014

[13] D Y Chen Q Q Jiang and X T Zu ldquoFermions tunnellingfrom the charged dilatonic black holesrdquo Classical and QuantumGravity vol 25 Article ID 205022 2008

[14] A Sen ldquoRotating charged black hole solution in heterotic stringtheoryrdquo Physical Review Letters vol 69 no 7 pp 1006ndash10091992

[15] A Yale andR BMann ldquoGravitinos tunneling fromblack holesrdquoPhysics Letters B vol 673 no 2 pp 168ndash172 2009

[16] D V Ahluwalia-Khalilova and D Grumiller ldquoSpin-halffermions with mass dimension one theory phenomenologyand dark matterrdquo Journal of Cosmology and AstroparticlePhysics vol 2005 no 02 12 pages 2005

[17] D V Ahluwalia ldquoOn a local mass dimension one Fermi fieldof spin one-half and the theoretical crevice that allows itrdquohttparxivorgabs13057509

[18] D V Ahluwalia C Y Lee D Schritt and T F Watson ldquoElkoas self-interacting fermionic dark matter with axis of localityrdquoPhysics Letters B vol 687 no 2-3 pp 248ndash252 2010

[19] R da Rocha L Fabbri J M H da Silva R T Cavalcanti andJ A Silva-Neto ldquoFlag-dipole spinor fields in ESK gravitiesrdquoJournal of Mathematical Physics vol 54 no 10 Article ID102505 2013

[20] R T Cavalcanti ldquoClassification of singular spinor fields andother mass dimension one fermionsrdquo International Journal ofModern Physics D vol 23 no 14 Article ID 1444002 2014

[21] A E Bernardini and R da Rocha ldquoDynamical dispersionrelation for ELKO dark spinor fieldsrdquo Physics Letters B vol 717no 1ndash3 pp 238ndash241 2012

[22] J M Hoff da Silva and R da Rocha ldquoUnfolding physics fromthe algebraic classification of spinor fieldsrdquoPhysics Letters B vol718 no 4-5 pp 1519ndash1523 2013

[23] A G Cohen and S L Glashow ldquoVery special relativityrdquo PhysicalReview Letters vol 97 no 2 Article ID 021601 3 pages 2006

[24] C G Bohmer J Burnett D F Mota and D J Shaw ldquoDarkspinor models in gravitation and cosmologyrdquo Journal of HighEnergy Physics vol 2010 no 7 article 053 2010

Advances in High Energy Physics 7

[25] R da Rocha A E Bernardini and J M da Silva ldquoExotic darkspinor fieldsrdquo Journal of High Energy Physics vol 2011 no 4article 110 2011

[26] A P dos Santos Souza S H Pereira and J F Jesus ldquoA newapproach on the stability analysis in ELKO cosmologyrdquo TheEuropean Physical Journal C vol 75 no 1 article 36 2015

[27] J H Horne and G T Horowitz ldquoRotating dilaton black holesrdquoPhysical Review D vol 46 no 4 pp 1340ndash1346 1992

[28] G W Gibbons and K-I Maeda ldquoBlack holes and membranesin higher-dimensional theories with dilaton fieldsrdquo NuclearPhysics B vol 298 no 4 pp 741ndash775 1988

[29] D-Y Chen and X-T Zu ldquoHawking radiation of fermions forthe Kerr-Sen dilaton-axion black holerdquo Modern Physics LettersA vol 24 no 14 pp 1159ndash1165 2009

[30] P K Samal R Saha P Jain and J P Ralston ldquoSignals ofstatistical anisotropy in WMAP foreground-cleaned mapsrdquoMonthly Notices of the Royal Astronomical Society vol 396 no1 pp 511ndash522 2009

[31] M Frommert andTA Ensslin ldquoThe axis of evilmdasha polarizationperspectiverdquoMonthly Notices of the Royal Astronomical Societyvol 403 no 4 pp 1739ndash1748 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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FluidsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in Condensed Matter Physics

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AstronomyAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

2 Advances in High Energy Physics

They present charge magnetic dipole moment and angularmomentum involving the antisymmetric tensor field coupledto the Chern-Simons 3-form A myriad of black holes hasbeen considered for tunnelling methods of fermions andbosons as rotating and accelerating black holes and topo-logical BTZ Reissner-Nordstrom Kerr-Newman and Taub-NUT-AdS black holes including the tunnelling of higher spinfermions as well [15]

We shall study similar methods for spin-12 fermionsof mass dimension one different from the procedure forstandard mass dimension 32 fermions Elko dark spinorsnamely the dual-helicity eigenspinors of the charge conjuga-tion operator [16 17] are spin-12 fermions of mass dimen-sion one with novel features thatmake them capable of incor-porating both the Very Special Relativity (VSR) paradigmand the dark matter description as well [16 18] Such spinorsseem to be indeed a tip of the iceberg for a comprehensiveclass of nonstandard (singular) spinors [19 20] Moreovera mass generation mechanism has been introduced in [21]for such dark particles by a natural coupling to the kinksolution in field theory It provides exotic couplings amongscalar field topological solutions and Elko dark spinors [2122] Due to its very small coupling with the standard modelfields except the Higgs field dark spinors supply naturalself-interacting dark matter prime candidates Except forscalar fields and gravity Elko dark spinors interactions withthe standard model matter and gauge fields are suppressedby at least one power of unificationPlanck scale [16] Infact the Lagrangian of such a field contains a quartic self-interaction term and the interaction term of the new fieldwith spin-zero bosonic fields Moreover Elko framework isshown to be invariant under the action of the HOM(2) VSRgroup and covariant under SIM(2) VSR group [23] Elko darkspinor field is a representative of mass dimension one spin-12 fermions in the type-5 spinor field class in Lounestorsquosspinors classification however it is not the most generalsince Majorana spinors are also encompassed by such class[22] Some attempts to detect Elko at the LHC have beenproposed and important applications to cosmology havebeen widely investigated as well [12 16 18 24ndash26]

This paper is presented as follows the Kerr-Sen axion-dilaton black hole is briefly revisited in the next sectiontogether with the dark spinors framework We thus shallcalculate in Section 3 the probabilities of emission andabsorption of Elko dark particles across these black holesTherefore the WKB approximation is used to computethe tunnelling rate and thus the resulting tunnelling prob-ability Finally the associated Hawking temperature shallbe obtained corroborating the universal character of theHawking effect and further extending it to particles beyondthe standard model

2 Kerr-Sen Axion-Dilaton Black Holes andDark Spinors

String theory has solutions describing extra-dimensionalextended objects surrounded by event horizons presenting acausal structure associated with singularities in string theory

The low energy effective action of the heterotic string theoryis ruled by an action that up to higher derivative terms andother fields which are set to zero for the particular class ofbackgrounds considered [14] is given by

119878 = minusint1198894119909radicminus det119866119890

minusΦ

sdot (minus119877 +1

12119867120583]120588119867120583]120588

minus 119866120583]120597120583Φ120597]Φ +

1

8119865120583]119865120583])

(1)

where 119866120583] is the metric regarding a 120590-model [14] related to

the Einstein metric by 119890minusΦ

119866120583] Φ denotes the dilaton field 119877

stands for the scalar curvature 119865120583] = 120597

[120583119860]] is the Maxwell

field strength and119867120583]120588 = 120597

120583119861]120588+120597120588119861120583]+120597]119861120588120583minusΩ120583]120588 for the

Chern-Simons 3-formΩ120583]120588 minus (14)119860

(120583119865]120588) The above action

can be led to the one in [27] up to the 119867120583]120588119867120583]120588 term after

field redefinitionTheKerr-Sen dilaton-axion black holemetric is a solution

of the field equations derived from (1) In Boyer-Lindquistcoordinates it reads

1198891199042= minus

Δ minus 1198862sin2120579Σ

1198891199052+

Σ

Δ1198891199032+ Σ119889120579

2

+sin2120579Σ

[(1199032minus 2120573119903 minus 119886

2)2

minus Δ1198862sin2120579] 1198891205932

minus2119886 sin2120579

Σ[(1199032minus 2120573119903 minus 119886

2)2

minus Δ] 119889119905 119889120593

(2)

where

Σ = 1199032minus 2120573119903 + 119886

2cos2120579

Δ = 1199032minus 2120578119903 + 119886

2= (119903 minus 119903

+) (119903 minus 119903

minus)

120573 = 120578 sinh2 1205722

(3)

and 119903+[119903minus] are the coordinate outer [inner] singularities

Metric (2) describes a black hole solution with charge 119876mass119872 magnetic dipolemoment120583 and angularmomentum119869 given by

119876 =120578

radic2sinh120572

119872 =120578

2(1 + cosh120572)

120583 =1

radic2120578119886 sinh120572

119869 =120578119886

2(1 + cosh120572)

(4)

Advances in High Energy Physics 3

The associated 119892-factor can be expressed as 119892 = 2120583119872119876119869 = 2

[27] The parameters can be expressed in terms of genuinelyphysical quantities as

120578 = 119872 minus1198762

2119872

120572 = arcsinh(2radic2119876119872

21198722 minus 1198762)

119886 =119869

119872

(5)

The coordinate singularities thus read 119903plusmn

= 119872 minus 11987622119872 plusmn

radicminus11986921198722 + (119872 minus 11987622119872)2 which vanishes unless |119869| lt

1198722minus 11987622 The area of the outer event horizon is given by

119860 = 8120587119872(119872minus1198762

2119872+ radicminus

1198692

1198722+ (119872 minus

1198762

2119872)

2

) (6)

Thus in the extremal limit since |119869| rarr 119872minus11987622119872 it reads

119860 rarr 8120587|119869| In this limit the horizon is hence finite and thesurface gravity 120581 or equivalently the Hawking temperature119879119867

= 1205812120587 is provided by [14]

120581 =

radicminus41198692 + (21198722 minus 1198762)2

2119872(21198722 minus 1198762 + radicminus41198692 + (21198722 minus 1198762)2

)

(7)

Thus in the extremal limit we have the limit 120581 rarr 0 if 119869 = 0On the other hand if 119869 = 0 then 120581 = 14119872 in agreement withthe results of [27 28] For 119869 = 0 this black hole solution hasaspects analogous to the extremal rotating black hole ratherthan extremal charged black holes [27]

By performing the transformation 120601 = 120593minusΩ119905 whereΩ =

119886(1198862minus 2120573119903 + 119886

2minusΔ)((119903

2minus 2120573119903 + 119886

2)2minusΔ1198862sin2120579) the metric

(2) takes the form

1198891199042= minus

ΔΣ

(1199032 minus 2120573119903 minus 1198862)2

minus Δ1198862sin21205791198891199052+

Σ

Δ1198891199032

+ Σ1198891205792

+sin2120579Σ

[(1199032minus 2120573119903 minus 119886

2)2

minus Δ1198862sin2120579] 1198891206012

(8)

To study the Hawking radiation at the event horizon themetric is regarded near the horizon

1198891199042= minus119865 (119903

+) 1198891199052+

1

119866 (119903+)1198891199032+ Σ (119903

+) 1198891205792

+119867 (119903+)

Σ (119903+)1198891206012

(9)

where

119867(119903+) = sin2120579 (1199032

+minus 2120573119903

++ 1198862)2

119865 (119903+) =

2 (119903+minus 120578) (119903 minus 119903

+) Σ (119903+)

(1199032+minus 2120573119903

++ 1198862)2

119866 (119903+) =

2 (119903+minus 120578) (119903 minus 119903

+)

Σ (119903+)

(10)

In order to analyze the tunnelling of Elko dark particles acrossthe Kerr-Sen black hole event horizon we will study the rolethat Elko dark particles play in this backgroundThe essentialprominent Elko particles features are in short revisited [16]Elko dark spinors 120582(119901

120583) are eigenspinors of the charge

conjugation operator 119862 namely 119862120582(119901120583) = plusmn120582(119901120583) The

plus [minus] sign regards self-conjugate [anti-self-conjugate]spinors denoted by 120582119878(119901120583) [120582

119860(119901120583)] For spinors at rest 120582(119896120583)

the boosted spinors read 120582(119901120583) = 119890

119894120581sdot120593120582(119896120583) where 119896

120583=

(119898 lim119901rarr0

p|p|) where 119890119894120581sdot120593 denotes the boost operator

120601(119896120583) are defined to be eigenspinors of the helicity operator

as 120590 sdot p120601plusmn(119896120583) = plusmn120601plusmn(119896120583) where [16]

120601+(119896120583) = radic119898(

cos(120579

2) 119890minus1198941205932

sin(120579

2) 119890+1198941205932

) equiv (120572

120573)

120601minus(119896120583) = radic119898(

minus sin(120579

2) 119890minus1198941205932

cos(120579

2) 119890+1198941205932

) = (minus120573lowast

120572lowast)

(11)

Elko dark spinors 120582(119896120583) are constructed as

120582119878

plusmn(119896120583) = (

1205902(120601plusmn(119896120583))lowast

120601plusmn(119896120583)

)

120582119860

plusmn(119896120583) = plusmn(

minus1205902(120601∓(119896120583))lowast

120601∓(119896120583)

)

(12)

and have dual helicity as minus1198941205902(120601plusmn)lowast has helicity dual to that

of 120601plusmn The boosted terms

120582119860

plusmn(119901120583) = radic

119864 + 119898

2119898(1 plusmn

119901120583

119864 + 119898)120582119860

plusmn

120582119878

plusmn(119901120583) = radic

119864 + 119898

2119898(1 ∓

119901120583

119864 + 119898)120582119878

plusmn

(13)

are the expansion coefficients of a mass dimension onequantum field The Dirac operator does not annihilate the120582(119901120583) but instead the equations of motion read [16 17]

120574120583nabla120583120582119878

plusmn= plusmn119894

119898

ℎ120582119878

∓ (14)

120574120583nabla120583120582119860

∓= plusmn119894

119898

ℎ120582119860

plusmn (15)

Dark spinors nevertheless satisfy the Klein-Gordon equation

4 Advances in High Energy Physics

A mass dimension one quantum field can be thus con-structed as [17]

f (119909) = int1198893119901

(2120587)3

1

radic2119898119864 (p)

sdot sum

120588

[119887dagger

120588(p) 120582119860 (p) 119890119894119901120583119909

120583

+ 119886120588(p) 120582119878 (p) 119890minus119894119901120583119909

120583

]

(16)

The creation and annihilation operators 119886120588(p) 119886dagger120588(p) satisfy

the Fermi statistics [17] with similar anticommutators for119887120588(p) and 119887

dagger

120588(p) The mass dimensionality of f(119909) can be

realized from the adjoint

not

f (119909) = int1198893p

(2120587)3

1

radic2119898119864 (p)

sdot sum

120588

[119887120588(p)not

120582

119860

(p) 119890minus119894119901120583119909120583

+ 119886dagger

120588(p)not

120582

119878

(p) 119890119894119901120583119909120583

]

(17)

where denoting hereupon by 120590120583 the Pauli matrices the con-jugate spinors are constructed by

not

120582120588(119901120583) = [Ξ120582

120588(119901120583)]dagger1205901otimes

I2 Here the operator Ξ = (12119898)(120582

119878

minus(119901120583)120582119878

minus(119901120583) minus

120582119860

minus(119901120583)120582119860

minus(119901120583) + 120582119878

+(119901120583)120582119878

+(119901120583) minus 120582119860

+(119901120583)120582119860

+(119901120583)) is an invo-

lution [17] where the standard Dirac conjugate 120582(119901120583) =

120582(119901120583)dagger1205740 is adopted The mass dimension of the new field is

determined by the SIM(2) covariant propagator [17]

119878 (119909 minus 1199091015840) = 119894⟨

10038161003816100381610038161003816100381610038161003816

T(f (119909)not

119891 (1199091015840))

10038161003816100381610038161003816100381610038161003816

⟩

= minus lim120598rarr0+

int1198894119901

(2120587)4119890minus119894119901120583(1199091015840

120583minus119909120583) (

I +G (120593)

119901120583119901120583minus 1198982 + 119894120598

)

(18)

where T is the canonical time-ordering operator and

G (120593) = (0 minus119894119890

minus119894120593

119894119890119894120593

0

) otimes 1205901

(19)

that respects symmetries of the theory of VSR [17 23]

3 Hawking Radiation from TunnellingDark Spinors

Hawking radiation from general black holes encompassesdistinguished charged and uncharged particles Tunnellingmethods can be employed for Elko dark particles across thehorizon of Kerr-Sen black holes From (8) the associated

tetrad can be chosen so that the following generators can beachieved

120574119905=

1

radic119865 (119903+)

(0 I2

I2

0)

120574120579=

1

radicΣ (119903+)

(0 1205902

minus1205902

0

)

120574119903= radic119866 (119903

+)(

0 1205901

minus1205901

0

)

120574119911= radic

Σ (119903+)

119867 (119903+)(

0 1205903

minus1205903

0

)

(20)

Elko dark spinors can be written as

120582119878

+= (

minus119894120573lowast

119894120572lowast

120572

120573

) exp (119894

ℏ)

120582119878

minus= (

minus119894120572

minus119894120573

minus120573lowast

120572lowast

) exp(119894

ℏ)

120582119860

+= (

119894120572

119894120573

minus120573lowast

120572lowast

) exp(119894

ℏ)

120582119860

minus= (

minus119894120573lowast

119894120572lowast

minus120572

minus120573

) exp(119894

ℏ)

(21)

where = (119905 119903 120579 119911) represents the classical action We usethe above forms for dark particles in each of (14) and (15) andthen solve this coupled system of equations By denoting

nabla120583= 120597120583+

1

8119894Γ120572120573

120583[120574120572 120574120573] (22)

where 120574120590 are the usual Clifford bundle generators for the

Minkowski spacetime By identifying 120582 [∘

120582] to the Elko spinoron the left [right] hand side of (14) and (15) then (14) reads120574120583(nabla120583+ 119890119860120583)120582 = 119894(119898ℎ)

∘

120582 Using the WKB approximationwhere = 119868 + O(ℎ) it yields

(119868120583+ 119890119860120583) 120574120583120582 = 119894119898

∘

120582 + O (ℎ) (23)

Advances in High Energy Physics 5

where 119868120583

equiv 120597119868120597119909120583 Taking merely the leading order terms

in the above equation from a general form 120582 = (119886 119887 119888 119889)⊤

∘

120582 = (∘

119886∘

119887∘

119888∘

119889)⊤ we have general Elko dynamic equations

governed by (23) The ansatz 119868(119905 119903 120579 120593) = minus(120596 minus 119895Ω)119905 + 119895120593 +

119882(119903) + Θ(120579) can be used where 120596 and 119895 denote the energyand magnetic quantum number of the particles respectivelyMoreover the parameters 119886 119887 119888 119889 are not independentIn fact (21) assert that for the self-conjugate spinors 120582

119878 wehave 119886 = minus119894119889

lowast and 119887 = 119894119888lowast whereas for the anti-self-

conjugate spinors 120582119860 it reads 119886 = 119894119889lowast and 119887 = minus119894119888

lowast Thus bycorresponding the 120582119878 [120582119860] spinors to the upper [lower] signbelow after awkward computation (23) yields

plusmnradic119866 (119903+)1198821015840119889lowast∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

119888lowast=∘

119889119898 (24)

plusmnradic119866 (119903+)1198821015840119888lowast∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

119889lowast= minus∘

119888119898 (25)

The angular function 119895120593 + Θ(120579) must be a complex functionand the same solution for it is achieved for both incomingand outgoing cases as well It implies that the contribution ofsuch function vanishes after dividing the outgoing probabilityby the incoming one Hence the angular function can beneglected hereupon

In the above system the equations for 120582119878+[120582119878

minus] are shown

to be equivalent to the ones for 120582119860

minus[120582119860

+] Thus we have to

deal solelywith the self-conjugate120582119878plusmnspinorsMoreover there

are more underlying equivalences In fact (24) for 120582119878

plusmnis

equivalent to (25) for 120582119878∓ Consequently there is just a couple

of equations for 120582119878plusmngiven by

radic119866 (119903+)1198821015840120572lowastminus

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120573lowast= 120573lowast119898

radic119866 (119903+)1198821015840120573 +

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120572 = 120572119898

(26)

radic119866 (119903+)1198821015840120572 ∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120573 = ∓120573119898

radic119866 (119903+)1198821015840120573lowast∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120572lowast= plusmn120572lowast119898

(27)

Combining either (26) or (27) implies equations for either120582119878

+or 120582119878minus respectively Hence for each 120582

119878 there is a systemof coupled equations for the dark spinor components 120572 and120573 and also another coupled system for 120572

lowast and 120573lowast which

are going to be solved separately We denote now the firstequation of each one of the systems below to be the equations

related to (120572 120573) whereas the second ones regard (120572lowast 120573lowast) We

can determine the above functions as

120582119878

+

1198821(119903) = plusmnint

radic1198982119865 minus (120596 minus 119895Ω

119867+ 119890119860+)2

119865119866119889119903

1198822(119903) = plusmnint

radic(119898radic119865 + 120596 minus 119895Ω

119867+ 119890119860+)2

119865119866119889119903

120582119878

minus

1198823(119903) = plusmnint

radic(119898radic119865 minus 120596 + 119895Ω

119867+ 119890119860+)2

119865119866119889119903

1198824(119903) = 119894119882

2(119903)

(28)

For massless particles the solutions for 120582119878

plusmnare equivalent

being given by

1198822(119903) = plusmnint

radic(120596 minus 119895Ω

119867+ 119890119860+)2

119865119866119889119903 = minus119894119882

1(119903)

(29)

The above expression for1198822(119903) is similar to the results in [29]

when119898 = 0 The solution for1198822(119903) is provided by

1198822(119903) = plusmn119894120587

1199032

+minus 2120573119903

++ 1198862

2 (119903++ 119898)

(120596 minus 119869Ω119867+ 119890119860+) (30)

Since 119865 rarr 0 near the black hole horizon the results holdboth for massive and their massless limit particles as well Infact for such limit (28) and (29) are equivalent It is worthemphasizing that (28) implies a real solution for 119882

1(119903) and

thus a null contribution for the tunnelling effectIn compliance with the WKB approximation the tun-

nelling rate reads Γ prop exp(minus2 Im 119868) where 119868 denotes theclassical action for the path Hence the imaginary part of theaction becomes a prominent goal for the tunnelling processThe imaginary part of the action yields

Im 119868plusmn= plusmn119894120587

1199032

+minus 2120573119903

++ 1198862

2 (119903++ 119898)

(120596 minus 119869Ω119867+ 119890119860+) (31)

Thus the resulting tunnelling probability is given by

Γ =119875(emission)

119875(absorption)

=119890minus2 Im 119868+

119890minus2 Im 119868minus

= minus21205871199032

+minus 2120573119903

++ 1198862

(119903++ 119898)

(120596 minus 119869Ω119867+ 119890119860+)

(32)

Finally the Hawking temperature of the Kerr-Sen dilaton-axion black hole is acquired

119879119867

=1

2120587

(119903++ 119898)

1199032+minus 2120573119903

++ 1198862

(33)

6 Advances in High Energy Physics

which is a universal formula also obtained by other methodsfor the Hawking temperature from fermions tunnelling [29]Obviously when the parameter 119886 tends to zero (33) providesthe well-known Hawking radiation associated with the staticblack hole

4 Concluding Remarks

We derived the Hawking radiation for spin-12 fermions ofmass dimension one represented by dark spinors tunnellingacross a Kerr-Sen dilaton-axion black hole horizonThe tem-perature of these solutions was computed and demonstratedto confirm the universal character of theHawking effect evenfor mass dimension one fermions of spin-12 that are beyondthe standard model The mass dimension one feature of suchspinor fields sharply suppresses the couplings to other fieldsof the standardmodel Indeed by power counting argumentsElko spinor fields can self-interact and further interact witha scalar (Higgs) field providing a renormalizable frameworkThis type of interaction means an unsuppressed quartic self-interaction The quartic self-interaction is essential to darkmatter observations [30 31] Therefore Elko spinor fieldsperform an adequate fermionic dark matter candidate Itcan represent a real model for dark matter tunnelling acrossblack holes Corrections of higher order in ℏ to the Hawkingtemperature (33) of type 119868 = 119868

0+ sum119899ge1

ℏ119899119868119899[11] can be still

implemented in the context of mass dimension one spin-12 fermions Moreover other mass dimension one fermions[20] and higher spin mass dimension one fermions can bestudied in the context of black hole tunnelling methodshowever these issues are beyond the scope of this paperwhich comprised dark particles tunnelling across Kerr-Sendilaton-axion black holes

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

R T Cavalcanti thanks CAPES and UFABC Roldao daRocha is grateful to CNPq Grants no 3030272012-6 no4516822015-7 and no 4733262013-2 for partial financialsupport and to FAPESP Grant no 201510270-0

References

[1] P Kraus and F Wilczek ldquoSelf-interaction correction to blackhole radiancerdquo Nuclear Physics B vol 433 no 2 pp 403ndash4201995

[2] M K Parikh and F Wilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000

[3] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 1999

[4] M Agheben M Nadalini L Vanzo and S Zerbini ldquoHawkingradiation as tunneling for extremal and rotating black holesrdquoJournal of High Energy Physics vol 2005 article 014 2005

[5] M Arzano A J Medved and E C Vagenas ldquoHawkingradiation as tunneling through the quantum horizonrdquo Journalof High Energy Physics vol 2005 no 9 p 37 2005

[6] Q-Q Jiang S-Q Wu and X Cai ldquoHawking radiation as tun-neling from the Kerr and Kerr-Newman black holesrdquo PhysicalReview D vol 73 no 6 Article ID 064003 2006

[7] J Zhang and Z Zhao ldquoCharged particlesrsquo tunnelling from theKerr-Newman black holerdquo Physics Letters B vol 638 no 2-3pp 110ndash113 2006

[8] R Li J-R Ren and S-W Wei ldquoHawking radiation of Diracparticles via tunneling from the Kerr black holerdquo Classical andQuantum Gravity vol 25 no 12 Article ID 125016 2008

[9] L Vanzo G Acquaviva and R Di Criscienzo ldquoTunnellingmethods and Hawkingrsquos radiation achievements andprospectsrdquo Classical and Quantum Gravity vol 28 no 18Article ID 183001 2011

[10] R Kerner and R B Mann ldquoFermions tunnelling from blackholesrdquo Classical and Quantum Gravity vol 25 no 9 Article ID095014 2008

[11] R Banerjee and B R Majhi ldquoHawking black body spectrumfrom tunneling mechanismrdquo Physics Letters B vol 675 no 2pp 243ndash245 2009

[12] R da Rocha and J M Hoff da Silva ldquoHawking radiationfrom Elko particles tunnelling across black-strings horizonrdquoEurophysics Letters vol 107 no 5 Article ID 50001 2014

[13] D Y Chen Q Q Jiang and X T Zu ldquoFermions tunnellingfrom the charged dilatonic black holesrdquo Classical and QuantumGravity vol 25 Article ID 205022 2008

[14] A Sen ldquoRotating charged black hole solution in heterotic stringtheoryrdquo Physical Review Letters vol 69 no 7 pp 1006ndash10091992

[15] A Yale andR BMann ldquoGravitinos tunneling fromblack holesrdquoPhysics Letters B vol 673 no 2 pp 168ndash172 2009

[16] D V Ahluwalia-Khalilova and D Grumiller ldquoSpin-halffermions with mass dimension one theory phenomenologyand dark matterrdquo Journal of Cosmology and AstroparticlePhysics vol 2005 no 02 12 pages 2005

[17] D V Ahluwalia ldquoOn a local mass dimension one Fermi fieldof spin one-half and the theoretical crevice that allows itrdquohttparxivorgabs13057509

[18] D V Ahluwalia C Y Lee D Schritt and T F Watson ldquoElkoas self-interacting fermionic dark matter with axis of localityrdquoPhysics Letters B vol 687 no 2-3 pp 248ndash252 2010

[19] R da Rocha L Fabbri J M H da Silva R T Cavalcanti andJ A Silva-Neto ldquoFlag-dipole spinor fields in ESK gravitiesrdquoJournal of Mathematical Physics vol 54 no 10 Article ID102505 2013

[20] R T Cavalcanti ldquoClassification of singular spinor fields andother mass dimension one fermionsrdquo International Journal ofModern Physics D vol 23 no 14 Article ID 1444002 2014

[21] A E Bernardini and R da Rocha ldquoDynamical dispersionrelation for ELKO dark spinor fieldsrdquo Physics Letters B vol 717no 1ndash3 pp 238ndash241 2012

[22] J M Hoff da Silva and R da Rocha ldquoUnfolding physics fromthe algebraic classification of spinor fieldsrdquoPhysics Letters B vol718 no 4-5 pp 1519ndash1523 2013

[23] A G Cohen and S L Glashow ldquoVery special relativityrdquo PhysicalReview Letters vol 97 no 2 Article ID 021601 3 pages 2006

[24] C G Bohmer J Burnett D F Mota and D J Shaw ldquoDarkspinor models in gravitation and cosmologyrdquo Journal of HighEnergy Physics vol 2010 no 7 article 053 2010

Advances in High Energy Physics 7

[25] R da Rocha A E Bernardini and J M da Silva ldquoExotic darkspinor fieldsrdquo Journal of High Energy Physics vol 2011 no 4article 110 2011

[26] A P dos Santos Souza S H Pereira and J F Jesus ldquoA newapproach on the stability analysis in ELKO cosmologyrdquo TheEuropean Physical Journal C vol 75 no 1 article 36 2015

[27] J H Horne and G T Horowitz ldquoRotating dilaton black holesrdquoPhysical Review D vol 46 no 4 pp 1340ndash1346 1992

[28] G W Gibbons and K-I Maeda ldquoBlack holes and membranesin higher-dimensional theories with dilaton fieldsrdquo NuclearPhysics B vol 298 no 4 pp 741ndash775 1988

[29] D-Y Chen and X-T Zu ldquoHawking radiation of fermions forthe Kerr-Sen dilaton-axion black holerdquo Modern Physics LettersA vol 24 no 14 pp 1159ndash1165 2009

[30] P K Samal R Saha P Jain and J P Ralston ldquoSignals ofstatistical anisotropy in WMAP foreground-cleaned mapsrdquoMonthly Notices of the Royal Astronomical Society vol 396 no1 pp 511ndash522 2009

[31] M Frommert andTA Ensslin ldquoThe axis of evilmdasha polarizationperspectiverdquoMonthly Notices of the Royal Astronomical Societyvol 403 no 4 pp 1739ndash1748 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 3

The associated 119892-factor can be expressed as 119892 = 2120583119872119876119869 = 2

[27] The parameters can be expressed in terms of genuinelyphysical quantities as

120578 = 119872 minus1198762

2119872

120572 = arcsinh(2radic2119876119872

21198722 minus 1198762)

119886 =119869

119872

(5)

The coordinate singularities thus read 119903plusmn

= 119872 minus 11987622119872 plusmn

radicminus11986921198722 + (119872 minus 11987622119872)2 which vanishes unless |119869| lt

1198722minus 11987622 The area of the outer event horizon is given by

119860 = 8120587119872(119872minus1198762

2119872+ radicminus

1198692

1198722+ (119872 minus

1198762

2119872)

2

) (6)

Thus in the extremal limit since |119869| rarr 119872minus11987622119872 it reads

119860 rarr 8120587|119869| In this limit the horizon is hence finite and thesurface gravity 120581 or equivalently the Hawking temperature119879119867

= 1205812120587 is provided by [14]

120581 =

radicminus41198692 + (21198722 minus 1198762)2

2119872(21198722 minus 1198762 + radicminus41198692 + (21198722 minus 1198762)2

)

(7)

Thus in the extremal limit we have the limit 120581 rarr 0 if 119869 = 0On the other hand if 119869 = 0 then 120581 = 14119872 in agreement withthe results of [27 28] For 119869 = 0 this black hole solution hasaspects analogous to the extremal rotating black hole ratherthan extremal charged black holes [27]

By performing the transformation 120601 = 120593minusΩ119905 whereΩ =

119886(1198862minus 2120573119903 + 119886

2minusΔ)((119903

2minus 2120573119903 + 119886

2)2minusΔ1198862sin2120579) the metric

(2) takes the form

1198891199042= minus

ΔΣ

(1199032 minus 2120573119903 minus 1198862)2

minus Δ1198862sin21205791198891199052+

Σ

Δ1198891199032

+ Σ1198891205792

+sin2120579Σ

[(1199032minus 2120573119903 minus 119886

2)2

minus Δ1198862sin2120579] 1198891206012

(8)

To study the Hawking radiation at the event horizon themetric is regarded near the horizon

1198891199042= minus119865 (119903

+) 1198891199052+

1

119866 (119903+)1198891199032+ Σ (119903

+) 1198891205792

+119867 (119903+)

Σ (119903+)1198891206012

(9)

where

119867(119903+) = sin2120579 (1199032

+minus 2120573119903

++ 1198862)2

119865 (119903+) =

2 (119903+minus 120578) (119903 minus 119903

+) Σ (119903+)

(1199032+minus 2120573119903

++ 1198862)2

119866 (119903+) =

2 (119903+minus 120578) (119903 minus 119903

+)

Σ (119903+)

(10)

In order to analyze the tunnelling of Elko dark particles acrossthe Kerr-Sen black hole event horizon we will study the rolethat Elko dark particles play in this backgroundThe essentialprominent Elko particles features are in short revisited [16]Elko dark spinors 120582(119901

120583) are eigenspinors of the charge

conjugation operator 119862 namely 119862120582(119901120583) = plusmn120582(119901120583) The

plus [minus] sign regards self-conjugate [anti-self-conjugate]spinors denoted by 120582119878(119901120583) [120582

119860(119901120583)] For spinors at rest 120582(119896120583)

the boosted spinors read 120582(119901120583) = 119890

119894120581sdot120593120582(119896120583) where 119896

120583=

(119898 lim119901rarr0

p|p|) where 119890119894120581sdot120593 denotes the boost operator

120601(119896120583) are defined to be eigenspinors of the helicity operator

as 120590 sdot p120601plusmn(119896120583) = plusmn120601plusmn(119896120583) where [16]

120601+(119896120583) = radic119898(

cos(120579

2) 119890minus1198941205932

sin(120579

2) 119890+1198941205932

) equiv (120572

120573)

120601minus(119896120583) = radic119898(

minus sin(120579

2) 119890minus1198941205932

cos(120579

2) 119890+1198941205932

) = (minus120573lowast

120572lowast)

(11)

Elko dark spinors 120582(119896120583) are constructed as

120582119878

plusmn(119896120583) = (

1205902(120601plusmn(119896120583))lowast

120601plusmn(119896120583)

)

120582119860

plusmn(119896120583) = plusmn(

minus1205902(120601∓(119896120583))lowast

120601∓(119896120583)

)

(12)

and have dual helicity as minus1198941205902(120601plusmn)lowast has helicity dual to that

of 120601plusmn The boosted terms

120582119860

plusmn(119901120583) = radic

119864 + 119898

2119898(1 plusmn

119901120583

119864 + 119898)120582119860

plusmn

120582119878

plusmn(119901120583) = radic

119864 + 119898

2119898(1 ∓

119901120583

119864 + 119898)120582119878

plusmn

(13)

are the expansion coefficients of a mass dimension onequantum field The Dirac operator does not annihilate the120582(119901120583) but instead the equations of motion read [16 17]

120574120583nabla120583120582119878

plusmn= plusmn119894

119898

ℎ120582119878

∓ (14)

120574120583nabla120583120582119860

∓= plusmn119894

119898

ℎ120582119860

plusmn (15)

Dark spinors nevertheless satisfy the Klein-Gordon equation

4 Advances in High Energy Physics

A mass dimension one quantum field can be thus con-structed as [17]

f (119909) = int1198893119901

(2120587)3

1

radic2119898119864 (p)

sdot sum

120588

[119887dagger

120588(p) 120582119860 (p) 119890119894119901120583119909

120583

+ 119886120588(p) 120582119878 (p) 119890minus119894119901120583119909

120583

]

(16)

The creation and annihilation operators 119886120588(p) 119886dagger120588(p) satisfy

the Fermi statistics [17] with similar anticommutators for119887120588(p) and 119887

dagger

120588(p) The mass dimensionality of f(119909) can be

realized from the adjoint

not

f (119909) = int1198893p

(2120587)3

1

radic2119898119864 (p)

sdot sum

120588

[119887120588(p)not

120582

119860

(p) 119890minus119894119901120583119909120583

+ 119886dagger

120588(p)not

120582

119878

(p) 119890119894119901120583119909120583

]

(17)

where denoting hereupon by 120590120583 the Pauli matrices the con-jugate spinors are constructed by

not

120582120588(119901120583) = [Ξ120582

120588(119901120583)]dagger1205901otimes

I2 Here the operator Ξ = (12119898)(120582

119878

minus(119901120583)120582119878

minus(119901120583) minus

120582119860

minus(119901120583)120582119860

minus(119901120583) + 120582119878

+(119901120583)120582119878

+(119901120583) minus 120582119860

+(119901120583)120582119860

+(119901120583)) is an invo-

lution [17] where the standard Dirac conjugate 120582(119901120583) =

120582(119901120583)dagger1205740 is adopted The mass dimension of the new field is

determined by the SIM(2) covariant propagator [17]

119878 (119909 minus 1199091015840) = 119894⟨

10038161003816100381610038161003816100381610038161003816

T(f (119909)not

119891 (1199091015840))

10038161003816100381610038161003816100381610038161003816

⟩

= minus lim120598rarr0+

int1198894119901

(2120587)4119890minus119894119901120583(1199091015840

120583minus119909120583) (

I +G (120593)

119901120583119901120583minus 1198982 + 119894120598

)

(18)

where T is the canonical time-ordering operator and

G (120593) = (0 minus119894119890

minus119894120593

119894119890119894120593

0

) otimes 1205901

(19)

that respects symmetries of the theory of VSR [17 23]

3 Hawking Radiation from TunnellingDark Spinors

Hawking radiation from general black holes encompassesdistinguished charged and uncharged particles Tunnellingmethods can be employed for Elko dark particles across thehorizon of Kerr-Sen black holes From (8) the associated

tetrad can be chosen so that the following generators can beachieved

120574119905=

1

radic119865 (119903+)

(0 I2

I2

0)

120574120579=

1

radicΣ (119903+)

(0 1205902

minus1205902

0

)

120574119903= radic119866 (119903

+)(

0 1205901

minus1205901

0

)

120574119911= radic

Σ (119903+)

119867 (119903+)(

0 1205903

minus1205903

0

)

(20)

Elko dark spinors can be written as

120582119878

+= (

minus119894120573lowast

119894120572lowast

120572

120573

) exp (119894

ℏ)

120582119878

minus= (

minus119894120572

minus119894120573

minus120573lowast

120572lowast

) exp(119894

ℏ)

120582119860

+= (

119894120572

119894120573

minus120573lowast

120572lowast

) exp(119894

ℏ)

120582119860

minus= (

minus119894120573lowast

119894120572lowast

minus120572

minus120573

) exp(119894

ℏ)

(21)

where = (119905 119903 120579 119911) represents the classical action We usethe above forms for dark particles in each of (14) and (15) andthen solve this coupled system of equations By denoting

nabla120583= 120597120583+

1

8119894Γ120572120573

120583[120574120572 120574120573] (22)

where 120574120590 are the usual Clifford bundle generators for the

Minkowski spacetime By identifying 120582 [∘

120582] to the Elko spinoron the left [right] hand side of (14) and (15) then (14) reads120574120583(nabla120583+ 119890119860120583)120582 = 119894(119898ℎ)

∘

120582 Using the WKB approximationwhere = 119868 + O(ℎ) it yields

(119868120583+ 119890119860120583) 120574120583120582 = 119894119898

∘

120582 + O (ℎ) (23)

Advances in High Energy Physics 5

where 119868120583

equiv 120597119868120597119909120583 Taking merely the leading order terms

in the above equation from a general form 120582 = (119886 119887 119888 119889)⊤

∘

120582 = (∘

119886∘

119887∘

119888∘

119889)⊤ we have general Elko dynamic equations

governed by (23) The ansatz 119868(119905 119903 120579 120593) = minus(120596 minus 119895Ω)119905 + 119895120593 +

119882(119903) + Θ(120579) can be used where 120596 and 119895 denote the energyand magnetic quantum number of the particles respectivelyMoreover the parameters 119886 119887 119888 119889 are not independentIn fact (21) assert that for the self-conjugate spinors 120582

119878 wehave 119886 = minus119894119889

lowast and 119887 = 119894119888lowast whereas for the anti-self-

conjugate spinors 120582119860 it reads 119886 = 119894119889lowast and 119887 = minus119894119888

lowast Thus bycorresponding the 120582119878 [120582119860] spinors to the upper [lower] signbelow after awkward computation (23) yields

plusmnradic119866 (119903+)1198821015840119889lowast∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

119888lowast=∘

119889119898 (24)

plusmnradic119866 (119903+)1198821015840119888lowast∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

119889lowast= minus∘

119888119898 (25)

The angular function 119895120593 + Θ(120579) must be a complex functionand the same solution for it is achieved for both incomingand outgoing cases as well It implies that the contribution ofsuch function vanishes after dividing the outgoing probabilityby the incoming one Hence the angular function can beneglected hereupon

In the above system the equations for 120582119878+[120582119878

minus] are shown

to be equivalent to the ones for 120582119860

minus[120582119860

+] Thus we have to

deal solelywith the self-conjugate120582119878plusmnspinorsMoreover there

are more underlying equivalences In fact (24) for 120582119878

plusmnis

equivalent to (25) for 120582119878∓ Consequently there is just a couple

of equations for 120582119878plusmngiven by

radic119866 (119903+)1198821015840120572lowastminus

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120573lowast= 120573lowast119898

radic119866 (119903+)1198821015840120573 +

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120572 = 120572119898

(26)

radic119866 (119903+)1198821015840120572 ∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120573 = ∓120573119898

radic119866 (119903+)1198821015840120573lowast∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120572lowast= plusmn120572lowast119898

(27)

Combining either (26) or (27) implies equations for either120582119878

+or 120582119878minus respectively Hence for each 120582

119878 there is a systemof coupled equations for the dark spinor components 120572 and120573 and also another coupled system for 120572

lowast and 120573lowast which

are going to be solved separately We denote now the firstequation of each one of the systems below to be the equations

related to (120572 120573) whereas the second ones regard (120572lowast 120573lowast) We

can determine the above functions as

120582119878

+

1198821(119903) = plusmnint

radic1198982119865 minus (120596 minus 119895Ω

119867+ 119890119860+)2

119865119866119889119903

1198822(119903) = plusmnint

radic(119898radic119865 + 120596 minus 119895Ω

119867+ 119890119860+)2

119865119866119889119903

120582119878

minus

1198823(119903) = plusmnint

radic(119898radic119865 minus 120596 + 119895Ω

119867+ 119890119860+)2

119865119866119889119903

1198824(119903) = 119894119882

2(119903)

(28)

For massless particles the solutions for 120582119878

plusmnare equivalent

being given by

1198822(119903) = plusmnint

radic(120596 minus 119895Ω

119867+ 119890119860+)2

119865119866119889119903 = minus119894119882

1(119903)

(29)

The above expression for1198822(119903) is similar to the results in [29]

when119898 = 0 The solution for1198822(119903) is provided by

1198822(119903) = plusmn119894120587

1199032

+minus 2120573119903

++ 1198862

2 (119903++ 119898)

(120596 minus 119869Ω119867+ 119890119860+) (30)

Since 119865 rarr 0 near the black hole horizon the results holdboth for massive and their massless limit particles as well Infact for such limit (28) and (29) are equivalent It is worthemphasizing that (28) implies a real solution for 119882

1(119903) and

thus a null contribution for the tunnelling effectIn compliance with the WKB approximation the tun-

nelling rate reads Γ prop exp(minus2 Im 119868) where 119868 denotes theclassical action for the path Hence the imaginary part of theaction becomes a prominent goal for the tunnelling processThe imaginary part of the action yields

Im 119868plusmn= plusmn119894120587

1199032

+minus 2120573119903

++ 1198862

2 (119903++ 119898)

(120596 minus 119869Ω119867+ 119890119860+) (31)

Thus the resulting tunnelling probability is given by

Γ =119875(emission)

119875(absorption)

=119890minus2 Im 119868+

119890minus2 Im 119868minus

= minus21205871199032

+minus 2120573119903

++ 1198862

(119903++ 119898)

(120596 minus 119869Ω119867+ 119890119860+)

(32)

Finally the Hawking temperature of the Kerr-Sen dilaton-axion black hole is acquired

119879119867

=1

2120587

(119903++ 119898)

1199032+minus 2120573119903

++ 1198862

(33)

6 Advances in High Energy Physics

which is a universal formula also obtained by other methodsfor the Hawking temperature from fermions tunnelling [29]Obviously when the parameter 119886 tends to zero (33) providesthe well-known Hawking radiation associated with the staticblack hole

4 Concluding Remarks

We derived the Hawking radiation for spin-12 fermions ofmass dimension one represented by dark spinors tunnellingacross a Kerr-Sen dilaton-axion black hole horizonThe tem-perature of these solutions was computed and demonstratedto confirm the universal character of theHawking effect evenfor mass dimension one fermions of spin-12 that are beyondthe standard model The mass dimension one feature of suchspinor fields sharply suppresses the couplings to other fieldsof the standardmodel Indeed by power counting argumentsElko spinor fields can self-interact and further interact witha scalar (Higgs) field providing a renormalizable frameworkThis type of interaction means an unsuppressed quartic self-interaction The quartic self-interaction is essential to darkmatter observations [30 31] Therefore Elko spinor fieldsperform an adequate fermionic dark matter candidate Itcan represent a real model for dark matter tunnelling acrossblack holes Corrections of higher order in ℏ to the Hawkingtemperature (33) of type 119868 = 119868

0+ sum119899ge1

ℏ119899119868119899[11] can be still

implemented in the context of mass dimension one spin-12 fermions Moreover other mass dimension one fermions[20] and higher spin mass dimension one fermions can bestudied in the context of black hole tunnelling methodshowever these issues are beyond the scope of this paperwhich comprised dark particles tunnelling across Kerr-Sendilaton-axion black holes

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

R T Cavalcanti thanks CAPES and UFABC Roldao daRocha is grateful to CNPq Grants no 3030272012-6 no4516822015-7 and no 4733262013-2 for partial financialsupport and to FAPESP Grant no 201510270-0

References

[1] P Kraus and F Wilczek ldquoSelf-interaction correction to blackhole radiancerdquo Nuclear Physics B vol 433 no 2 pp 403ndash4201995

[2] M K Parikh and F Wilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000

[3] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 1999

[4] M Agheben M Nadalini L Vanzo and S Zerbini ldquoHawkingradiation as tunneling for extremal and rotating black holesrdquoJournal of High Energy Physics vol 2005 article 014 2005

[5] M Arzano A J Medved and E C Vagenas ldquoHawkingradiation as tunneling through the quantum horizonrdquo Journalof High Energy Physics vol 2005 no 9 p 37 2005

[6] Q-Q Jiang S-Q Wu and X Cai ldquoHawking radiation as tun-neling from the Kerr and Kerr-Newman black holesrdquo PhysicalReview D vol 73 no 6 Article ID 064003 2006

[7] J Zhang and Z Zhao ldquoCharged particlesrsquo tunnelling from theKerr-Newman black holerdquo Physics Letters B vol 638 no 2-3pp 110ndash113 2006

[8] R Li J-R Ren and S-W Wei ldquoHawking radiation of Diracparticles via tunneling from the Kerr black holerdquo Classical andQuantum Gravity vol 25 no 12 Article ID 125016 2008

[9] L Vanzo G Acquaviva and R Di Criscienzo ldquoTunnellingmethods and Hawkingrsquos radiation achievements andprospectsrdquo Classical and Quantum Gravity vol 28 no 18Article ID 183001 2011

[10] R Kerner and R B Mann ldquoFermions tunnelling from blackholesrdquo Classical and Quantum Gravity vol 25 no 9 Article ID095014 2008

[11] R Banerjee and B R Majhi ldquoHawking black body spectrumfrom tunneling mechanismrdquo Physics Letters B vol 675 no 2pp 243ndash245 2009

[12] R da Rocha and J M Hoff da Silva ldquoHawking radiationfrom Elko particles tunnelling across black-strings horizonrdquoEurophysics Letters vol 107 no 5 Article ID 50001 2014

[13] D Y Chen Q Q Jiang and X T Zu ldquoFermions tunnellingfrom the charged dilatonic black holesrdquo Classical and QuantumGravity vol 25 Article ID 205022 2008

[14] A Sen ldquoRotating charged black hole solution in heterotic stringtheoryrdquo Physical Review Letters vol 69 no 7 pp 1006ndash10091992

[15] A Yale andR BMann ldquoGravitinos tunneling fromblack holesrdquoPhysics Letters B vol 673 no 2 pp 168ndash172 2009

[16] D V Ahluwalia-Khalilova and D Grumiller ldquoSpin-halffermions with mass dimension one theory phenomenologyand dark matterrdquo Journal of Cosmology and AstroparticlePhysics vol 2005 no 02 12 pages 2005

[17] D V Ahluwalia ldquoOn a local mass dimension one Fermi fieldof spin one-half and the theoretical crevice that allows itrdquohttparxivorgabs13057509

[18] D V Ahluwalia C Y Lee D Schritt and T F Watson ldquoElkoas self-interacting fermionic dark matter with axis of localityrdquoPhysics Letters B vol 687 no 2-3 pp 248ndash252 2010

[19] R da Rocha L Fabbri J M H da Silva R T Cavalcanti andJ A Silva-Neto ldquoFlag-dipole spinor fields in ESK gravitiesrdquoJournal of Mathematical Physics vol 54 no 10 Article ID102505 2013

[20] R T Cavalcanti ldquoClassification of singular spinor fields andother mass dimension one fermionsrdquo International Journal ofModern Physics D vol 23 no 14 Article ID 1444002 2014

[21] A E Bernardini and R da Rocha ldquoDynamical dispersionrelation for ELKO dark spinor fieldsrdquo Physics Letters B vol 717no 1ndash3 pp 238ndash241 2012

[22] J M Hoff da Silva and R da Rocha ldquoUnfolding physics fromthe algebraic classification of spinor fieldsrdquoPhysics Letters B vol718 no 4-5 pp 1519ndash1523 2013

[23] A G Cohen and S L Glashow ldquoVery special relativityrdquo PhysicalReview Letters vol 97 no 2 Article ID 021601 3 pages 2006

[24] C G Bohmer J Burnett D F Mota and D J Shaw ldquoDarkspinor models in gravitation and cosmologyrdquo Journal of HighEnergy Physics vol 2010 no 7 article 053 2010

Advances in High Energy Physics 7

[25] R da Rocha A E Bernardini and J M da Silva ldquoExotic darkspinor fieldsrdquo Journal of High Energy Physics vol 2011 no 4article 110 2011

[26] A P dos Santos Souza S H Pereira and J F Jesus ldquoA newapproach on the stability analysis in ELKO cosmologyrdquo TheEuropean Physical Journal C vol 75 no 1 article 36 2015

[27] J H Horne and G T Horowitz ldquoRotating dilaton black holesrdquoPhysical Review D vol 46 no 4 pp 1340ndash1346 1992

[28] G W Gibbons and K-I Maeda ldquoBlack holes and membranesin higher-dimensional theories with dilaton fieldsrdquo NuclearPhysics B vol 298 no 4 pp 741ndash775 1988

[29] D-Y Chen and X-T Zu ldquoHawking radiation of fermions forthe Kerr-Sen dilaton-axion black holerdquo Modern Physics LettersA vol 24 no 14 pp 1159ndash1165 2009

[30] P K Samal R Saha P Jain and J P Ralston ldquoSignals ofstatistical anisotropy in WMAP foreground-cleaned mapsrdquoMonthly Notices of the Royal Astronomical Society vol 396 no1 pp 511ndash522 2009

[31] M Frommert andTA Ensslin ldquoThe axis of evilmdasha polarizationperspectiverdquoMonthly Notices of the Royal Astronomical Societyvol 403 no 4 pp 1739ndash1748 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Advances in High Energy Physics

A mass dimension one quantum field can be thus con-structed as [17]

f (119909) = int1198893119901

(2120587)3

1

radic2119898119864 (p)

sdot sum

120588

[119887dagger

120588(p) 120582119860 (p) 119890119894119901120583119909

120583

+ 119886120588(p) 120582119878 (p) 119890minus119894119901120583119909

120583

]

(16)

The creation and annihilation operators 119886120588(p) 119886dagger120588(p) satisfy

the Fermi statistics [17] with similar anticommutators for119887120588(p) and 119887

dagger

120588(p) The mass dimensionality of f(119909) can be

realized from the adjoint

not

f (119909) = int1198893p

(2120587)3

1

radic2119898119864 (p)

sdot sum

120588

[119887120588(p)not

120582

119860

(p) 119890minus119894119901120583119909120583

+ 119886dagger

120588(p)not

120582

119878

(p) 119890119894119901120583119909120583

]

(17)

where denoting hereupon by 120590120583 the Pauli matrices the con-jugate spinors are constructed by

not

120582120588(119901120583) = [Ξ120582

120588(119901120583)]dagger1205901otimes

I2 Here the operator Ξ = (12119898)(120582

119878

minus(119901120583)120582119878

minus(119901120583) minus

120582119860

minus(119901120583)120582119860

minus(119901120583) + 120582119878

+(119901120583)120582119878

+(119901120583) minus 120582119860

+(119901120583)120582119860

+(119901120583)) is an invo-

lution [17] where the standard Dirac conjugate 120582(119901120583) =

120582(119901120583)dagger1205740 is adopted The mass dimension of the new field is

determined by the SIM(2) covariant propagator [17]

119878 (119909 minus 1199091015840) = 119894⟨

10038161003816100381610038161003816100381610038161003816

T(f (119909)not

119891 (1199091015840))

10038161003816100381610038161003816100381610038161003816

⟩

= minus lim120598rarr0+

int1198894119901

(2120587)4119890minus119894119901120583(1199091015840

120583minus119909120583) (

I +G (120593)

119901120583119901120583minus 1198982 + 119894120598

)

(18)

where T is the canonical time-ordering operator and

G (120593) = (0 minus119894119890

minus119894120593

119894119890119894120593

0

) otimes 1205901

(19)

that respects symmetries of the theory of VSR [17 23]

3 Hawking Radiation from TunnellingDark Spinors

Hawking radiation from general black holes encompassesdistinguished charged and uncharged particles Tunnellingmethods can be employed for Elko dark particles across thehorizon of Kerr-Sen black holes From (8) the associated

tetrad can be chosen so that the following generators can beachieved

120574119905=

1

radic119865 (119903+)

(0 I2

I2

0)

120574120579=

1

radicΣ (119903+)

(0 1205902

minus1205902

0

)

120574119903= radic119866 (119903

+)(

0 1205901

minus1205901

0

)

120574119911= radic

Σ (119903+)

119867 (119903+)(

0 1205903

minus1205903

0

)

(20)

Elko dark spinors can be written as

120582119878

+= (

minus119894120573lowast

119894120572lowast

120572

120573

) exp (119894

ℏ)

120582119878

minus= (

minus119894120572

minus119894120573

minus120573lowast

120572lowast

) exp(119894

ℏ)

120582119860

+= (

119894120572

119894120573

minus120573lowast

120572lowast

) exp(119894

ℏ)

120582119860

minus= (

minus119894120573lowast

119894120572lowast

minus120572

minus120573

) exp(119894

ℏ)

(21)

where = (119905 119903 120579 119911) represents the classical action We usethe above forms for dark particles in each of (14) and (15) andthen solve this coupled system of equations By denoting

nabla120583= 120597120583+

1

8119894Γ120572120573

120583[120574120572 120574120573] (22)

where 120574120590 are the usual Clifford bundle generators for the

Minkowski spacetime By identifying 120582 [∘

120582] to the Elko spinoron the left [right] hand side of (14) and (15) then (14) reads120574120583(nabla120583+ 119890119860120583)120582 = 119894(119898ℎ)

∘

120582 Using the WKB approximationwhere = 119868 + O(ℎ) it yields

(119868120583+ 119890119860120583) 120574120583120582 = 119894119898

∘

120582 + O (ℎ) (23)

Advances in High Energy Physics 5

where 119868120583

equiv 120597119868120597119909120583 Taking merely the leading order terms

in the above equation from a general form 120582 = (119886 119887 119888 119889)⊤

∘

120582 = (∘

119886∘

119887∘

119888∘

119889)⊤ we have general Elko dynamic equations

governed by (23) The ansatz 119868(119905 119903 120579 120593) = minus(120596 minus 119895Ω)119905 + 119895120593 +

119882(119903) + Θ(120579) can be used where 120596 and 119895 denote the energyand magnetic quantum number of the particles respectivelyMoreover the parameters 119886 119887 119888 119889 are not independentIn fact (21) assert that for the self-conjugate spinors 120582

119878 wehave 119886 = minus119894119889

lowast and 119887 = 119894119888lowast whereas for the anti-self-

conjugate spinors 120582119860 it reads 119886 = 119894119889lowast and 119887 = minus119894119888

lowast Thus bycorresponding the 120582119878 [120582119860] spinors to the upper [lower] signbelow after awkward computation (23) yields

plusmnradic119866 (119903+)1198821015840119889lowast∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

119888lowast=∘

119889119898 (24)

plusmnradic119866 (119903+)1198821015840119888lowast∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

119889lowast= minus∘

119888119898 (25)

The angular function 119895120593 + Θ(120579) must be a complex functionand the same solution for it is achieved for both incomingand outgoing cases as well It implies that the contribution ofsuch function vanishes after dividing the outgoing probabilityby the incoming one Hence the angular function can beneglected hereupon

In the above system the equations for 120582119878+[120582119878

minus] are shown

to be equivalent to the ones for 120582119860

minus[120582119860

+] Thus we have to

deal solelywith the self-conjugate120582119878plusmnspinorsMoreover there

are more underlying equivalences In fact (24) for 120582119878

plusmnis

equivalent to (25) for 120582119878∓ Consequently there is just a couple

of equations for 120582119878plusmngiven by

radic119866 (119903+)1198821015840120572lowastminus

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120573lowast= 120573lowast119898

radic119866 (119903+)1198821015840120573 +

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120572 = 120572119898

(26)

radic119866 (119903+)1198821015840120572 ∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120573 = ∓120573119898

radic119866 (119903+)1198821015840120573lowast∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120572lowast= plusmn120572lowast119898

(27)

Combining either (26) or (27) implies equations for either120582119878

+or 120582119878minus respectively Hence for each 120582

119878 there is a systemof coupled equations for the dark spinor components 120572 and120573 and also another coupled system for 120572

lowast and 120573lowast which

are going to be solved separately We denote now the firstequation of each one of the systems below to be the equations

related to (120572 120573) whereas the second ones regard (120572lowast 120573lowast) We

can determine the above functions as

120582119878

+

1198821(119903) = plusmnint

radic1198982119865 minus (120596 minus 119895Ω

119867+ 119890119860+)2

119865119866119889119903

1198822(119903) = plusmnint

radic(119898radic119865 + 120596 minus 119895Ω

119867+ 119890119860+)2

119865119866119889119903

120582119878

minus

1198823(119903) = plusmnint

radic(119898radic119865 minus 120596 + 119895Ω

119867+ 119890119860+)2

119865119866119889119903

1198824(119903) = 119894119882

2(119903)

(28)

For massless particles the solutions for 120582119878

plusmnare equivalent

being given by

1198822(119903) = plusmnint

radic(120596 minus 119895Ω

119867+ 119890119860+)2

119865119866119889119903 = minus119894119882

1(119903)

(29)

The above expression for1198822(119903) is similar to the results in [29]

when119898 = 0 The solution for1198822(119903) is provided by

1198822(119903) = plusmn119894120587

1199032

+minus 2120573119903

++ 1198862

2 (119903++ 119898)

(120596 minus 119869Ω119867+ 119890119860+) (30)

Since 119865 rarr 0 near the black hole horizon the results holdboth for massive and their massless limit particles as well Infact for such limit (28) and (29) are equivalent It is worthemphasizing that (28) implies a real solution for 119882

1(119903) and

thus a null contribution for the tunnelling effectIn compliance with the WKB approximation the tun-

nelling rate reads Γ prop exp(minus2 Im 119868) where 119868 denotes theclassical action for the path Hence the imaginary part of theaction becomes a prominent goal for the tunnelling processThe imaginary part of the action yields

Im 119868plusmn= plusmn119894120587

1199032

+minus 2120573119903

++ 1198862

2 (119903++ 119898)

(120596 minus 119869Ω119867+ 119890119860+) (31)

Thus the resulting tunnelling probability is given by

Γ =119875(emission)

119875(absorption)

=119890minus2 Im 119868+

119890minus2 Im 119868minus

= minus21205871199032

+minus 2120573119903

++ 1198862

(119903++ 119898)

(120596 minus 119869Ω119867+ 119890119860+)

(32)

Finally the Hawking temperature of the Kerr-Sen dilaton-axion black hole is acquired

119879119867

=1

2120587

(119903++ 119898)

1199032+minus 2120573119903

++ 1198862

(33)

6 Advances in High Energy Physics

which is a universal formula also obtained by other methodsfor the Hawking temperature from fermions tunnelling [29]Obviously when the parameter 119886 tends to zero (33) providesthe well-known Hawking radiation associated with the staticblack hole

4 Concluding Remarks

We derived the Hawking radiation for spin-12 fermions ofmass dimension one represented by dark spinors tunnellingacross a Kerr-Sen dilaton-axion black hole horizonThe tem-perature of these solutions was computed and demonstratedto confirm the universal character of theHawking effect evenfor mass dimension one fermions of spin-12 that are beyondthe standard model The mass dimension one feature of suchspinor fields sharply suppresses the couplings to other fieldsof the standardmodel Indeed by power counting argumentsElko spinor fields can self-interact and further interact witha scalar (Higgs) field providing a renormalizable frameworkThis type of interaction means an unsuppressed quartic self-interaction The quartic self-interaction is essential to darkmatter observations [30 31] Therefore Elko spinor fieldsperform an adequate fermionic dark matter candidate Itcan represent a real model for dark matter tunnelling acrossblack holes Corrections of higher order in ℏ to the Hawkingtemperature (33) of type 119868 = 119868

0+ sum119899ge1

ℏ119899119868119899[11] can be still

implemented in the context of mass dimension one spin-12 fermions Moreover other mass dimension one fermions[20] and higher spin mass dimension one fermions can bestudied in the context of black hole tunnelling methodshowever these issues are beyond the scope of this paperwhich comprised dark particles tunnelling across Kerr-Sendilaton-axion black holes

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

R T Cavalcanti thanks CAPES and UFABC Roldao daRocha is grateful to CNPq Grants no 3030272012-6 no4516822015-7 and no 4733262013-2 for partial financialsupport and to FAPESP Grant no 201510270-0

References

[1] P Kraus and F Wilczek ldquoSelf-interaction correction to blackhole radiancerdquo Nuclear Physics B vol 433 no 2 pp 403ndash4201995

[2] M K Parikh and F Wilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000

[3] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 1999

[4] M Agheben M Nadalini L Vanzo and S Zerbini ldquoHawkingradiation as tunneling for extremal and rotating black holesrdquoJournal of High Energy Physics vol 2005 article 014 2005

[5] M Arzano A J Medved and E C Vagenas ldquoHawkingradiation as tunneling through the quantum horizonrdquo Journalof High Energy Physics vol 2005 no 9 p 37 2005

[6] Q-Q Jiang S-Q Wu and X Cai ldquoHawking radiation as tun-neling from the Kerr and Kerr-Newman black holesrdquo PhysicalReview D vol 73 no 6 Article ID 064003 2006

[7] J Zhang and Z Zhao ldquoCharged particlesrsquo tunnelling from theKerr-Newman black holerdquo Physics Letters B vol 638 no 2-3pp 110ndash113 2006

[8] R Li J-R Ren and S-W Wei ldquoHawking radiation of Diracparticles via tunneling from the Kerr black holerdquo Classical andQuantum Gravity vol 25 no 12 Article ID 125016 2008

[9] L Vanzo G Acquaviva and R Di Criscienzo ldquoTunnellingmethods and Hawkingrsquos radiation achievements andprospectsrdquo Classical and Quantum Gravity vol 28 no 18Article ID 183001 2011

[10] R Kerner and R B Mann ldquoFermions tunnelling from blackholesrdquo Classical and Quantum Gravity vol 25 no 9 Article ID095014 2008

[11] R Banerjee and B R Majhi ldquoHawking black body spectrumfrom tunneling mechanismrdquo Physics Letters B vol 675 no 2pp 243ndash245 2009

[12] R da Rocha and J M Hoff da Silva ldquoHawking radiationfrom Elko particles tunnelling across black-strings horizonrdquoEurophysics Letters vol 107 no 5 Article ID 50001 2014

[13] D Y Chen Q Q Jiang and X T Zu ldquoFermions tunnellingfrom the charged dilatonic black holesrdquo Classical and QuantumGravity vol 25 Article ID 205022 2008

[14] A Sen ldquoRotating charged black hole solution in heterotic stringtheoryrdquo Physical Review Letters vol 69 no 7 pp 1006ndash10091992

[15] A Yale andR BMann ldquoGravitinos tunneling fromblack holesrdquoPhysics Letters B vol 673 no 2 pp 168ndash172 2009

[16] D V Ahluwalia-Khalilova and D Grumiller ldquoSpin-halffermions with mass dimension one theory phenomenologyand dark matterrdquo Journal of Cosmology and AstroparticlePhysics vol 2005 no 02 12 pages 2005

[17] D V Ahluwalia ldquoOn a local mass dimension one Fermi fieldof spin one-half and the theoretical crevice that allows itrdquohttparxivorgabs13057509

[18] D V Ahluwalia C Y Lee D Schritt and T F Watson ldquoElkoas self-interacting fermionic dark matter with axis of localityrdquoPhysics Letters B vol 687 no 2-3 pp 248ndash252 2010

[19] R da Rocha L Fabbri J M H da Silva R T Cavalcanti andJ A Silva-Neto ldquoFlag-dipole spinor fields in ESK gravitiesrdquoJournal of Mathematical Physics vol 54 no 10 Article ID102505 2013

[20] R T Cavalcanti ldquoClassification of singular spinor fields andother mass dimension one fermionsrdquo International Journal ofModern Physics D vol 23 no 14 Article ID 1444002 2014

[21] A E Bernardini and R da Rocha ldquoDynamical dispersionrelation for ELKO dark spinor fieldsrdquo Physics Letters B vol 717no 1ndash3 pp 238ndash241 2012

[22] J M Hoff da Silva and R da Rocha ldquoUnfolding physics fromthe algebraic classification of spinor fieldsrdquoPhysics Letters B vol718 no 4-5 pp 1519ndash1523 2013

[23] A G Cohen and S L Glashow ldquoVery special relativityrdquo PhysicalReview Letters vol 97 no 2 Article ID 021601 3 pages 2006

[24] C G Bohmer J Burnett D F Mota and D J Shaw ldquoDarkspinor models in gravitation and cosmologyrdquo Journal of HighEnergy Physics vol 2010 no 7 article 053 2010

Advances in High Energy Physics 7

[25] R da Rocha A E Bernardini and J M da Silva ldquoExotic darkspinor fieldsrdquo Journal of High Energy Physics vol 2011 no 4article 110 2011

[26] A P dos Santos Souza S H Pereira and J F Jesus ldquoA newapproach on the stability analysis in ELKO cosmologyrdquo TheEuropean Physical Journal C vol 75 no 1 article 36 2015

[27] J H Horne and G T Horowitz ldquoRotating dilaton black holesrdquoPhysical Review D vol 46 no 4 pp 1340ndash1346 1992

[28] G W Gibbons and K-I Maeda ldquoBlack holes and membranesin higher-dimensional theories with dilaton fieldsrdquo NuclearPhysics B vol 298 no 4 pp 741ndash775 1988

[29] D-Y Chen and X-T Zu ldquoHawking radiation of fermions forthe Kerr-Sen dilaton-axion black holerdquo Modern Physics LettersA vol 24 no 14 pp 1159ndash1165 2009

[30] P K Samal R Saha P Jain and J P Ralston ldquoSignals ofstatistical anisotropy in WMAP foreground-cleaned mapsrdquoMonthly Notices of the Royal Astronomical Society vol 396 no1 pp 511ndash522 2009

[31] M Frommert andTA Ensslin ldquoThe axis of evilmdasha polarizationperspectiverdquoMonthly Notices of the Royal Astronomical Societyvol 403 no 4 pp 1739ndash1748 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 5

where 119868120583

equiv 120597119868120597119909120583 Taking merely the leading order terms

in the above equation from a general form 120582 = (119886 119887 119888 119889)⊤

∘

120582 = (∘

119886∘

119887∘

119888∘

119889)⊤ we have general Elko dynamic equations

governed by (23) The ansatz 119868(119905 119903 120579 120593) = minus(120596 minus 119895Ω)119905 + 119895120593 +

119882(119903) + Θ(120579) can be used where 120596 and 119895 denote the energyand magnetic quantum number of the particles respectivelyMoreover the parameters 119886 119887 119888 119889 are not independentIn fact (21) assert that for the self-conjugate spinors 120582

119878 wehave 119886 = minus119894119889

lowast and 119887 = 119894119888lowast whereas for the anti-self-

conjugate spinors 120582119860 it reads 119886 = 119894119889lowast and 119887 = minus119894119888

lowast Thus bycorresponding the 120582119878 [120582119860] spinors to the upper [lower] signbelow after awkward computation (23) yields

plusmnradic119866 (119903+)1198821015840119889lowast∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

119888lowast=∘

119889119898 (24)

plusmnradic119866 (119903+)1198821015840119888lowast∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

119889lowast= minus∘

119888119898 (25)

The angular function 119895120593 + Θ(120579) must be a complex functionand the same solution for it is achieved for both incomingand outgoing cases as well It implies that the contribution ofsuch function vanishes after dividing the outgoing probabilityby the incoming one Hence the angular function can beneglected hereupon

In the above system the equations for 120582119878+[120582119878

minus] are shown

to be equivalent to the ones for 120582119860

minus[120582119860

+] Thus we have to

deal solelywith the self-conjugate120582119878plusmnspinorsMoreover there

are more underlying equivalences In fact (24) for 120582119878

plusmnis

equivalent to (25) for 120582119878∓ Consequently there is just a couple

of equations for 120582119878plusmngiven by

radic119866 (119903+)1198821015840120572lowastminus

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120573lowast= 120573lowast119898

radic119866 (119903+)1198821015840120573 +

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120572 = 120572119898

(26)

radic119866 (119903+)1198821015840120572 ∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120573 = ∓120573119898

radic119866 (119903+)1198821015840120573lowast∓

(120596 minus 119895Ω119867+ 119890119860+)

radic119865 (119903+)

120572lowast= plusmn120572lowast119898

(27)

Combining either (26) or (27) implies equations for either120582119878

+or 120582119878minus respectively Hence for each 120582

119878 there is a systemof coupled equations for the dark spinor components 120572 and120573 and also another coupled system for 120572

lowast and 120573lowast which

are going to be solved separately We denote now the firstequation of each one of the systems below to be the equations

related to (120572 120573) whereas the second ones regard (120572lowast 120573lowast) We

can determine the above functions as

120582119878

+

1198821(119903) = plusmnint

radic1198982119865 minus (120596 minus 119895Ω

119867+ 119890119860+)2

119865119866119889119903

1198822(119903) = plusmnint

radic(119898radic119865 + 120596 minus 119895Ω

119867+ 119890119860+)2

119865119866119889119903

120582119878

minus

1198823(119903) = plusmnint

radic(119898radic119865 minus 120596 + 119895Ω

119867+ 119890119860+)2

119865119866119889119903

1198824(119903) = 119894119882

2(119903)

(28)

For massless particles the solutions for 120582119878

plusmnare equivalent

being given by

1198822(119903) = plusmnint

radic(120596 minus 119895Ω

119867+ 119890119860+)2

119865119866119889119903 = minus119894119882

1(119903)

(29)

The above expression for1198822(119903) is similar to the results in [29]

when119898 = 0 The solution for1198822(119903) is provided by

1198822(119903) = plusmn119894120587

1199032

+minus 2120573119903

++ 1198862

2 (119903++ 119898)

(120596 minus 119869Ω119867+ 119890119860+) (30)

Since 119865 rarr 0 near the black hole horizon the results holdboth for massive and their massless limit particles as well Infact for such limit (28) and (29) are equivalent It is worthemphasizing that (28) implies a real solution for 119882

1(119903) and

thus a null contribution for the tunnelling effectIn compliance with the WKB approximation the tun-

nelling rate reads Γ prop exp(minus2 Im 119868) where 119868 denotes theclassical action for the path Hence the imaginary part of theaction becomes a prominent goal for the tunnelling processThe imaginary part of the action yields

Im 119868plusmn= plusmn119894120587

1199032

+minus 2120573119903

++ 1198862

2 (119903++ 119898)

(120596 minus 119869Ω119867+ 119890119860+) (31)

Thus the resulting tunnelling probability is given by

Γ =119875(emission)

119875(absorption)

=119890minus2 Im 119868+

119890minus2 Im 119868minus

= minus21205871199032

+minus 2120573119903

++ 1198862

(119903++ 119898)

(120596 minus 119869Ω119867+ 119890119860+)

(32)

Finally the Hawking temperature of the Kerr-Sen dilaton-axion black hole is acquired

119879119867

=1

2120587

(119903++ 119898)

1199032+minus 2120573119903

++ 1198862

(33)

6 Advances in High Energy Physics

which is a universal formula also obtained by other methodsfor the Hawking temperature from fermions tunnelling [29]Obviously when the parameter 119886 tends to zero (33) providesthe well-known Hawking radiation associated with the staticblack hole

4 Concluding Remarks

We derived the Hawking radiation for spin-12 fermions ofmass dimension one represented by dark spinors tunnellingacross a Kerr-Sen dilaton-axion black hole horizonThe tem-perature of these solutions was computed and demonstratedto confirm the universal character of theHawking effect evenfor mass dimension one fermions of spin-12 that are beyondthe standard model The mass dimension one feature of suchspinor fields sharply suppresses the couplings to other fieldsof the standardmodel Indeed by power counting argumentsElko spinor fields can self-interact and further interact witha scalar (Higgs) field providing a renormalizable frameworkThis type of interaction means an unsuppressed quartic self-interaction The quartic self-interaction is essential to darkmatter observations [30 31] Therefore Elko spinor fieldsperform an adequate fermionic dark matter candidate Itcan represent a real model for dark matter tunnelling acrossblack holes Corrections of higher order in ℏ to the Hawkingtemperature (33) of type 119868 = 119868

0+ sum119899ge1

ℏ119899119868119899[11] can be still

implemented in the context of mass dimension one spin-12 fermions Moreover other mass dimension one fermions[20] and higher spin mass dimension one fermions can bestudied in the context of black hole tunnelling methodshowever these issues are beyond the scope of this paperwhich comprised dark particles tunnelling across Kerr-Sendilaton-axion black holes

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

R T Cavalcanti thanks CAPES and UFABC Roldao daRocha is grateful to CNPq Grants no 3030272012-6 no4516822015-7 and no 4733262013-2 for partial financialsupport and to FAPESP Grant no 201510270-0

References

[1] P Kraus and F Wilczek ldquoSelf-interaction correction to blackhole radiancerdquo Nuclear Physics B vol 433 no 2 pp 403ndash4201995

[2] M K Parikh and F Wilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000

[3] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 1999

[4] M Agheben M Nadalini L Vanzo and S Zerbini ldquoHawkingradiation as tunneling for extremal and rotating black holesrdquoJournal of High Energy Physics vol 2005 article 014 2005

[5] M Arzano A J Medved and E C Vagenas ldquoHawkingradiation as tunneling through the quantum horizonrdquo Journalof High Energy Physics vol 2005 no 9 p 37 2005

[6] Q-Q Jiang S-Q Wu and X Cai ldquoHawking radiation as tun-neling from the Kerr and Kerr-Newman black holesrdquo PhysicalReview D vol 73 no 6 Article ID 064003 2006

[7] J Zhang and Z Zhao ldquoCharged particlesrsquo tunnelling from theKerr-Newman black holerdquo Physics Letters B vol 638 no 2-3pp 110ndash113 2006

[8] R Li J-R Ren and S-W Wei ldquoHawking radiation of Diracparticles via tunneling from the Kerr black holerdquo Classical andQuantum Gravity vol 25 no 12 Article ID 125016 2008

[9] L Vanzo G Acquaviva and R Di Criscienzo ldquoTunnellingmethods and Hawkingrsquos radiation achievements andprospectsrdquo Classical and Quantum Gravity vol 28 no 18Article ID 183001 2011

[10] R Kerner and R B Mann ldquoFermions tunnelling from blackholesrdquo Classical and Quantum Gravity vol 25 no 9 Article ID095014 2008

[11] R Banerjee and B R Majhi ldquoHawking black body spectrumfrom tunneling mechanismrdquo Physics Letters B vol 675 no 2pp 243ndash245 2009

[12] R da Rocha and J M Hoff da Silva ldquoHawking radiationfrom Elko particles tunnelling across black-strings horizonrdquoEurophysics Letters vol 107 no 5 Article ID 50001 2014

[13] D Y Chen Q Q Jiang and X T Zu ldquoFermions tunnellingfrom the charged dilatonic black holesrdquo Classical and QuantumGravity vol 25 Article ID 205022 2008

[14] A Sen ldquoRotating charged black hole solution in heterotic stringtheoryrdquo Physical Review Letters vol 69 no 7 pp 1006ndash10091992

[15] A Yale andR BMann ldquoGravitinos tunneling fromblack holesrdquoPhysics Letters B vol 673 no 2 pp 168ndash172 2009

[16] D V Ahluwalia-Khalilova and D Grumiller ldquoSpin-halffermions with mass dimension one theory phenomenologyand dark matterrdquo Journal of Cosmology and AstroparticlePhysics vol 2005 no 02 12 pages 2005

[17] D V Ahluwalia ldquoOn a local mass dimension one Fermi fieldof spin one-half and the theoretical crevice that allows itrdquohttparxivorgabs13057509

[18] D V Ahluwalia C Y Lee D Schritt and T F Watson ldquoElkoas self-interacting fermionic dark matter with axis of localityrdquoPhysics Letters B vol 687 no 2-3 pp 248ndash252 2010

[19] R da Rocha L Fabbri J M H da Silva R T Cavalcanti andJ A Silva-Neto ldquoFlag-dipole spinor fields in ESK gravitiesrdquoJournal of Mathematical Physics vol 54 no 10 Article ID102505 2013

[20] R T Cavalcanti ldquoClassification of singular spinor fields andother mass dimension one fermionsrdquo International Journal ofModern Physics D vol 23 no 14 Article ID 1444002 2014

[21] A E Bernardini and R da Rocha ldquoDynamical dispersionrelation for ELKO dark spinor fieldsrdquo Physics Letters B vol 717no 1ndash3 pp 238ndash241 2012

[22] J M Hoff da Silva and R da Rocha ldquoUnfolding physics fromthe algebraic classification of spinor fieldsrdquoPhysics Letters B vol718 no 4-5 pp 1519ndash1523 2013

[23] A G Cohen and S L Glashow ldquoVery special relativityrdquo PhysicalReview Letters vol 97 no 2 Article ID 021601 3 pages 2006

[24] C G Bohmer J Burnett D F Mota and D J Shaw ldquoDarkspinor models in gravitation and cosmologyrdquo Journal of HighEnergy Physics vol 2010 no 7 article 053 2010

Advances in High Energy Physics 7

[25] R da Rocha A E Bernardini and J M da Silva ldquoExotic darkspinor fieldsrdquo Journal of High Energy Physics vol 2011 no 4article 110 2011

[26] A P dos Santos Souza S H Pereira and J F Jesus ldquoA newapproach on the stability analysis in ELKO cosmologyrdquo TheEuropean Physical Journal C vol 75 no 1 article 36 2015

[27] J H Horne and G T Horowitz ldquoRotating dilaton black holesrdquoPhysical Review D vol 46 no 4 pp 1340ndash1346 1992

[28] G W Gibbons and K-I Maeda ldquoBlack holes and membranesin higher-dimensional theories with dilaton fieldsrdquo NuclearPhysics B vol 298 no 4 pp 741ndash775 1988

[29] D-Y Chen and X-T Zu ldquoHawking radiation of fermions forthe Kerr-Sen dilaton-axion black holerdquo Modern Physics LettersA vol 24 no 14 pp 1159ndash1165 2009

[30] P K Samal R Saha P Jain and J P Ralston ldquoSignals ofstatistical anisotropy in WMAP foreground-cleaned mapsrdquoMonthly Notices of the Royal Astronomical Society vol 396 no1 pp 511ndash522 2009

[31] M Frommert andTA Ensslin ldquoThe axis of evilmdasha polarizationperspectiverdquoMonthly Notices of the Royal Astronomical Societyvol 403 no 4 pp 1739ndash1748 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Advances in High Energy Physics

which is a universal formula also obtained by other methodsfor the Hawking temperature from fermions tunnelling [29]Obviously when the parameter 119886 tends to zero (33) providesthe well-known Hawking radiation associated with the staticblack hole

4 Concluding Remarks

We derived the Hawking radiation for spin-12 fermions ofmass dimension one represented by dark spinors tunnellingacross a Kerr-Sen dilaton-axion black hole horizonThe tem-perature of these solutions was computed and demonstratedto confirm the universal character of theHawking effect evenfor mass dimension one fermions of spin-12 that are beyondthe standard model The mass dimension one feature of suchspinor fields sharply suppresses the couplings to other fieldsof the standardmodel Indeed by power counting argumentsElko spinor fields can self-interact and further interact witha scalar (Higgs) field providing a renormalizable frameworkThis type of interaction means an unsuppressed quartic self-interaction The quartic self-interaction is essential to darkmatter observations [30 31] Therefore Elko spinor fieldsperform an adequate fermionic dark matter candidate Itcan represent a real model for dark matter tunnelling acrossblack holes Corrections of higher order in ℏ to the Hawkingtemperature (33) of type 119868 = 119868

0+ sum119899ge1

ℏ119899119868119899[11] can be still

implemented in the context of mass dimension one spin-12 fermions Moreover other mass dimension one fermions[20] and higher spin mass dimension one fermions can bestudied in the context of black hole tunnelling methodshowever these issues are beyond the scope of this paperwhich comprised dark particles tunnelling across Kerr-Sendilaton-axion black holes

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

R T Cavalcanti thanks CAPES and UFABC Roldao daRocha is grateful to CNPq Grants no 3030272012-6 no4516822015-7 and no 4733262013-2 for partial financialsupport and to FAPESP Grant no 201510270-0

References

[1] P Kraus and F Wilczek ldquoSelf-interaction correction to blackhole radiancerdquo Nuclear Physics B vol 433 no 2 pp 403ndash4201995

[2] M K Parikh and F Wilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000

[3] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 1999

[4] M Agheben M Nadalini L Vanzo and S Zerbini ldquoHawkingradiation as tunneling for extremal and rotating black holesrdquoJournal of High Energy Physics vol 2005 article 014 2005

[5] M Arzano A J Medved and E C Vagenas ldquoHawkingradiation as tunneling through the quantum horizonrdquo Journalof High Energy Physics vol 2005 no 9 p 37 2005

[6] Q-Q Jiang S-Q Wu and X Cai ldquoHawking radiation as tun-neling from the Kerr and Kerr-Newman black holesrdquo PhysicalReview D vol 73 no 6 Article ID 064003 2006

[7] J Zhang and Z Zhao ldquoCharged particlesrsquo tunnelling from theKerr-Newman black holerdquo Physics Letters B vol 638 no 2-3pp 110ndash113 2006

[8] R Li J-R Ren and S-W Wei ldquoHawking radiation of Diracparticles via tunneling from the Kerr black holerdquo Classical andQuantum Gravity vol 25 no 12 Article ID 125016 2008

[9] L Vanzo G Acquaviva and R Di Criscienzo ldquoTunnellingmethods and Hawkingrsquos radiation achievements andprospectsrdquo Classical and Quantum Gravity vol 28 no 18Article ID 183001 2011

[10] R Kerner and R B Mann ldquoFermions tunnelling from blackholesrdquo Classical and Quantum Gravity vol 25 no 9 Article ID095014 2008

[11] R Banerjee and B R Majhi ldquoHawking black body spectrumfrom tunneling mechanismrdquo Physics Letters B vol 675 no 2pp 243ndash245 2009

[12] R da Rocha and J M Hoff da Silva ldquoHawking radiationfrom Elko particles tunnelling across black-strings horizonrdquoEurophysics Letters vol 107 no 5 Article ID 50001 2014

[13] D Y Chen Q Q Jiang and X T Zu ldquoFermions tunnellingfrom the charged dilatonic black holesrdquo Classical and QuantumGravity vol 25 Article ID 205022 2008

[14] A Sen ldquoRotating charged black hole solution in heterotic stringtheoryrdquo Physical Review Letters vol 69 no 7 pp 1006ndash10091992

[15] A Yale andR BMann ldquoGravitinos tunneling fromblack holesrdquoPhysics Letters B vol 673 no 2 pp 168ndash172 2009

[16] D V Ahluwalia-Khalilova and D Grumiller ldquoSpin-halffermions with mass dimension one theory phenomenologyand dark matterrdquo Journal of Cosmology and AstroparticlePhysics vol 2005 no 02 12 pages 2005

[17] D V Ahluwalia ldquoOn a local mass dimension one Fermi fieldof spin one-half and the theoretical crevice that allows itrdquohttparxivorgabs13057509

[18] D V Ahluwalia C Y Lee D Schritt and T F Watson ldquoElkoas self-interacting fermionic dark matter with axis of localityrdquoPhysics Letters B vol 687 no 2-3 pp 248ndash252 2010

[19] R da Rocha L Fabbri J M H da Silva R T Cavalcanti andJ A Silva-Neto ldquoFlag-dipole spinor fields in ESK gravitiesrdquoJournal of Mathematical Physics vol 54 no 10 Article ID102505 2013

[20] R T Cavalcanti ldquoClassification of singular spinor fields andother mass dimension one fermionsrdquo International Journal ofModern Physics D vol 23 no 14 Article ID 1444002 2014

[21] A E Bernardini and R da Rocha ldquoDynamical dispersionrelation for ELKO dark spinor fieldsrdquo Physics Letters B vol 717no 1ndash3 pp 238ndash241 2012

[22] J M Hoff da Silva and R da Rocha ldquoUnfolding physics fromthe algebraic classification of spinor fieldsrdquoPhysics Letters B vol718 no 4-5 pp 1519ndash1523 2013

[23] A G Cohen and S L Glashow ldquoVery special relativityrdquo PhysicalReview Letters vol 97 no 2 Article ID 021601 3 pages 2006

[24] C G Bohmer J Burnett D F Mota and D J Shaw ldquoDarkspinor models in gravitation and cosmologyrdquo Journal of HighEnergy Physics vol 2010 no 7 article 053 2010

Advances in High Energy Physics 7

[25] R da Rocha A E Bernardini and J M da Silva ldquoExotic darkspinor fieldsrdquo Journal of High Energy Physics vol 2011 no 4article 110 2011

[26] A P dos Santos Souza S H Pereira and J F Jesus ldquoA newapproach on the stability analysis in ELKO cosmologyrdquo TheEuropean Physical Journal C vol 75 no 1 article 36 2015

[27] J H Horne and G T Horowitz ldquoRotating dilaton black holesrdquoPhysical Review D vol 46 no 4 pp 1340ndash1346 1992

[28] G W Gibbons and K-I Maeda ldquoBlack holes and membranesin higher-dimensional theories with dilaton fieldsrdquo NuclearPhysics B vol 298 no 4 pp 741ndash775 1988

[29] D-Y Chen and X-T Zu ldquoHawking radiation of fermions forthe Kerr-Sen dilaton-axion black holerdquo Modern Physics LettersA vol 24 no 14 pp 1159ndash1165 2009

[30] P K Samal R Saha P Jain and J P Ralston ldquoSignals ofstatistical anisotropy in WMAP foreground-cleaned mapsrdquoMonthly Notices of the Royal Astronomical Society vol 396 no1 pp 511ndash522 2009

[31] M Frommert andTA Ensslin ldquoThe axis of evilmdasha polarizationperspectiverdquoMonthly Notices of the Royal Astronomical Societyvol 403 no 4 pp 1739ndash1748 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 7

[25] R da Rocha A E Bernardini and J M da Silva ldquoExotic darkspinor fieldsrdquo Journal of High Energy Physics vol 2011 no 4article 110 2011

[26] A P dos Santos Souza S H Pereira and J F Jesus ldquoA newapproach on the stability analysis in ELKO cosmologyrdquo TheEuropean Physical Journal C vol 75 no 1 article 36 2015

[27] J H Horne and G T Horowitz ldquoRotating dilaton black holesrdquoPhysical Review D vol 46 no 4 pp 1340ndash1346 1992

[28] G W Gibbons and K-I Maeda ldquoBlack holes and membranesin higher-dimensional theories with dilaton fieldsrdquo NuclearPhysics B vol 298 no 4 pp 741ndash775 1988

[29] D-Y Chen and X-T Zu ldquoHawking radiation of fermions forthe Kerr-Sen dilaton-axion black holerdquo Modern Physics LettersA vol 24 no 14 pp 1159ndash1165 2009

[30] P K Samal R Saha P Jain and J P Ralston ldquoSignals ofstatistical anisotropy in WMAP foreground-cleaned mapsrdquoMonthly Notices of the Royal Astronomical Society vol 396 no1 pp 511ndash522 2009

[31] M Frommert andTA Ensslin ldquoThe axis of evilmdasha polarizationperspectiverdquoMonthly Notices of the Royal Astronomical Societyvol 403 no 4 pp 1739ndash1748 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of