JCAP07(2005)012 ysics h $30.00 72 P 2 ed theoretical discovery stroparticle stroparticle cuss its relevance to the cosmological A [email protected] ced fermionic field is endowed with mass and D Grumiller and ime, and energy–momentum space. 1 .Webrieflydis erences with the WIMP and mirror matter proposals I he first details on the unexpect − = th/0412080 2 ) of matter is via Higgs, and with gravity. This aspect leads p- relic abundance to constrain the relevant cross-section. The he CPT on usual properties with respect to charge conjugation and parity, it non-standard Wigner class. Consequently, the theory exhibits non- We provide t dark matter, quantum field theory on curved space acks.iop.org/JCAP/2005/i=07/a=012 osmology and and osmology ur Theoretische Physik, University of Leipzig, Augustusplatz 10-11, atter are enumerated. In a critique of the theory we reveal a hint on on st ti tter cloud to set limits on the mass of the new particle and present [email protected] C la Ahluwalia-Khalilova cu lished 19 July 2005 celebrate the birthday of my wife Dr I S Ahluwalia-Khalilova and in memory al line at i:10.1088/1475-7516/2005/07/012 pb. Similarities and diff ASGBG/CIU, Department of Mathematics, Apartado Postal C-600, Institut f¨ eceived 8 December 2004 ub of a spin-one-halfcomplete matter set field ofDue with dual-helicity to mass its eigenspinors dimensionbelongs un to of a one. the chargelocality It with conjugation is ( operator. based upon a Keywords: ArXiv ePrint: Abstract. 1 2 do DV University of Zacatecas (UAZ), Zacatecas, Zac 98060, Mexico R Accepted 14 June 2005 P D-04109 Leipzig, Germany E-mail: dimension one, it canwith carry known a forms quarticus self-interaction. to Its contemplate the dominantthis interaction new suggestion fermion seriously asdark we a study ma prime a dark supernova-likeac explosion matter of candidate. a Taking galactic-mass ‘horizon problem’. Because the introdu On To of my father Shri B S Ahluwalia (1933–1977)—dva-k. analysis favours light mass2 (roughly 20 MeV) and relevant cross-section of about for dark m non-commutative aspects of spacet rnal of rnal

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An IOP and SISSA journal 2005 IOP Publishing Ltd 1475-7516/05/070012+ c matter one: theory, phenomenology, and dark Spin-half fermions with mass dimension J JCAP07(2005)012 2 3 8 39 21 43 27 33 35 18 15 11 ) ... 5 ..... 37 ...... 35 ...... 15 ...... 15 ...... 9 ...... 12 ...... 18 Spin-half fermions with mass dimension one ...... 36 ...... 18 stacks.iop.org/JCAP/2005/i=07/a=012 I physical amplitudes and cross-sections − rator: a brief review ...... 7 = parity asymmetry 2 ) ...... 39 ...... 6 07 (2005) 012 ( ...... 24 ...... 11 ,and CPT . . P for Elko ecting out antiparticles I ...... 27 ...... 8 ...... 13 ...... 14 oj and − ion between the spin sums and wave operators: C = ...... 25 quantum field 2 arge conjugation ope ...... 22 ) 1 2 O ...... 30 CPT ( ent with Wigner: ( for Dirac spinors for Elko O O ng lishi 8.1.2. Field–field, and momentum–momentum, anticommutators. 8.1.1. Field–momentum anticommutator. direction introducing dungskonjugationsoperators (Elko) 8.2. Massless limit8.3. and non-locality Signatures of Elko non-locality in the 6.1. The Elko6.2. propagator Mass dimension one: the Elko propagator in the absence of a preferred 8.1. Fundamental anticommutators for the Elko quantum field Spacetime evolution 5.1. Massive Elko5.2. do not satisfy When the wave5.3. operators Dirac and equation spin Non-trivial sums do connect not coincide: a pivotal observation 5.4. The 5.5. The 1.1. Genesis: from1.2. the Majorana The field new to spin- a call for a new dark matter field 1.3. On the presentation of the paper 2.1. The Dirac2.2. construct Dirac’s insight: not pr La 4.2. Agreem 4.1. Commutativity of 3.1. Formal structure3.2. of Elko Distinction between3.3. Elko and Majorana Explicit form spinors 3.4. of Elko A new3.5. dual for Orthonormality Elko and completeness relations for Elko 9. Identification of Elko with dark matter 6. Particle interpretation and mass dimensionality 8. Locality structure 7. Energy of vacuum and establishing the fermionic statistics 5. Journal of Cosmology and Astroparticle Physics 1. Introduction Contents 2. Emergence of the ch 3. Dual-helicity eigenspinors of charge conjugation operator, or Eigenspinoren des 4. Estab JCAP07(2005)012 3 67 68 62 55 57 62 44 63 68 ) his situation ...... 54 ...... 58 ...... 57 tter and why is it invisible? 70%. That is, roughly 95% of ]hasdescribedt 4 ...... 63 ∼ ...... 59 ...... 60 Spin-half fermions with mass dimension one nors stacks.iop.org/JCAP/2005/i=07/a=012 )spi p ...... 63 ...... 64 ( hat is dark ma λ he relevant cross-section, mass, and a ∗ ...... 66 )) 0 he deficit is accounted for by non-baryonic dark ( } ...... 60 for Elko and non-standard dispersion relations ± L 07 (2005) 012 ( ...... 49 φ O η, η ...... 45 { . ]thatt 5 , ]. This inventory of cosmic baryons accounts for no more ...... 64 4 1 nors ...... 62 ...... 50 ...... 65 ) knows [ dependence of dark matter central cores for galaxies, and cosmic gamma-ray 0 )spi iliary details ncluding remarks ( φ p ...... 61 ± L ( ux φ ρ M 6 ...... 49 ison with WIMP .A galactic space [ ning the Elko mass and the relevant cross-section xA ]. One now 3 25%, and some form of all pervading dark energy, , ndi A.1. The A.2. Helicity properties of Θ( bursts 10.2.1. A brief run-through. 10.2.2. Details. compar B.1. Bi-orthonormality relations for B.2. The mass, 10 B.4. Spin sums B.5. Distributional part of B.3. Elko in the Majorana realization B.6. On the B.7. Some other anticommutators in the context of non-locality discussion 2 ∼ pe Appendix B. Elkology details 10.3. Similarities and differences from the mirror matter proposal 12.1. Elko as a generalization12.2. On of Lee–Wick Wigner–Weinberg non-locality, classes and12.3. A Snyder–Yang–Mendes algebra hint for non-commutative12.4. Generalization momentum space to higher12.5. spins A reference guide12.6. to Summary some of the key equations Acknowledgments Ap 10.1. Relic density of Elko: constraining t 10.2. Collapse of a primordial Elko cloud: independent constraint on the Elko References than 5% of theZwicky universe. [ The problem was first brought to attention as early as 1933 by Journal of Cosmology and Astroparticle Physics Stars, their remnants,cosmic and matter–energy content. gas Severalpervading in inter per cent galaxies, more contribute is accounted no for by more diffuse than material 1% of the total 1. Introduction 10. Constrai 11. Elko particles in a Thirring–Lense12. gravitational A field critique and co matter–energy content of the universe isprinciple invisible theoretical and has framework no known, for widely its accepted, first- description. Rees [ as ‘embarrassing’. The question we ask is: w matter, JCAP07(2005)012 ] ] e 4 ], ge 25 18 19 , , 6 24 char ) 1 2 rticle–antiparticle ]. Such a suggestion would ]. tors commute for bosons, and 27 , 21 0, its propagator is not that for 26 Spin-half fermions with mass dimension one ]). Scientific caution suggests [ ,opera ]= comes into existence. A condensed 17 P ]—with certain questions about Higgs stacks.iop.org/JCAP/2005/i=07/a=012 l-helicity eigenspinors of spin-one-half 8 ]–[ ssociated with the pa atter need not be confined to spin-one- icists’ focus to only those Wigner classes ality. This is done by constructing and C, P understand the particle content as implied ]–[ 12 ]—the dark matter may belong to the yet 6 o es on non-commutative spaces (for reviews, lement of non-locality which may be realized 10 spaper , 9 07 (2005) 012 ( look at the Majorana field. ](seealso[ find that the quantum field so constructed is rich in nfirms the LSND result. This combined circumstance ,andtheparity, 11 C ]co but it lacks in specific constructs. Yet, a focus on specifics ] 28 8 ales [ ab initio ]–[ 6 the property called for by the dark matter. In other words, we .Weshall ]). Furthermore, attempts to reconcile LSND excess events [ C of usual Wigner classes [ 23 ysical sc , ories. Our first step in that realm constitutes preserving the Poincar´ the physics community seems to be convinced of the existence of dark 22 cetime symmetries. The literature on the subject has, so far, provided ormal theoretical point of view, building on the classic works of Wigner [ taining some of the key results is available as [ espa ample [ do envisage the possibility that dark m le general insights [ From a f Another reason for which we venture to make our research notes public is the following. We Initially, we did not set out to construct a field with the properties outlined above, ge conjugation operator, i.e., the operator a ]. It is in this latter spirit also that thi 20 anticommute for fermions. Yetgravitational attempts realms to immediately ask merge for the an quantum, e the relativistic,see and for the ex indirectly suggest abandoning the locality requirement [ for example in the framework of field theori gain strength if MiniBooNEshould [ encourage us to take aquantum cautious walk field outside the the boundariessymmetries set by local but relativistic– abandoning the demand of loc studying a quantum field based on the above-indicated eigenspinors of the spin- by Poincar´ valuab that existing data be viewedas with complementary dark contributors matter to and the modifications same of data. gravity at large scales for which the charge conjugation, The assumption of locality has confined the phys Journal of Cosmology and Astroparticle Physics matter, it isentirety, the important dark to matterconstant remain problem at open may astroph be to a the reflection possibility of that the growth in of part, the if Newtonian not in its particles being deferred to another place [ half alone, evenvast though majority the present of paper focuses on this spin. Furthermore, while a unexplored unusual Wigner classes. char symmetry, has precisely suggest that whateverin dark the matter low-energy is,Since limit one the it thing known that behavesrepresentation particles seems as spaces are reasonably one described assured of by is the quantum that representations fields of involving the finite-dimensional Lorentz group. Here we show that a quantum field based on dua can bring about[ important and unexpected insights which otherwise can be overlooked this paper provides an account of our attempt t version con structure: it belongs to the Wigner class with [ or a field whichstructure would when be we a took candidate an for dark matter. Instead, we were exposed to this conjugation operator the Dirac field, and its mass dimension is one. JCAP07(2005)012 ] 5 al 31 from (1.2) (1.1) ]–[ sign 29 ) σ DAMA- σ ], [ , 8 3 . µ x µ p +i )e rries mass dimension p ( h v ]. The unexpected results )ca ) x p 41 ( , † h b 40 physically indistinguishable + Spin-half fermions with mass dimension one inal Majorana idea—did not carry µ x µ ), is p i ollows. The Majorana field is obtained x stacks.iop.org/JCAP/2005/i=07/a=012 ups—that there exists a 6 − ( ]. c )e 32 )’, in the standard Dirac field [ p ]. The field Ψ( ( atisfy an appropriate completeness relation p iolated—there is no first-principle quantum h ( nfirm experimentally the realization of this 37 u † h noftheorig ) a ]–[ p × ( —for a Majorana particle. The initial 3- h 34 07 (2005) 012 ( a by other gro − , ), denoted by Ψ constitutes phenomenological realization of a quantum be asked to s x =+ he new field was based upon what are known as Majorana h ors has on the physical content of the resulting field, and bservation has inspired a whole generation of physicists , 0 ) dds to the excitement. This also asks for a quantum field m p x um field(s) which may attend to observations and experiments 3 ‘phase factor p ]a significance [ 3 π logical conclusion. It turned out, as the reader will read below, Ψ( d 2 β 43 σ i [ 2) representation space. In the context of generalization to higher / 1 , )=e )witha )= hawaythatt ted otherwise, the notation of Ryder [ x p (0 x ( ( ]totheir c fnew)quant † h Majorana particle ]. After decades of pioneering work, the Heidelberg–Moscow (HM) ⊕ b .Thissingleo Ψ Ψ( 42 R 33 0) : mass dimension one field came about as f ]insuc , 3 dark matter 2 ∈ 1 2 39 / fying , the charge-conjugated Ψ( β llow, unless sta 38 nd the Standard Model. If the Majorana field was theoretically known since 1937, In the meantime, there has been progress on the experimental front. While the Now, whether one is considering the Majorana field or the Dirac field, both are )itself al for x yo We fo field which wasdiscovery—awaiting due confirmation never before known to have been used by Nature, the concurrent evidence for a the property required for the identification with neutrinos. sign be that a field based onoperator—constituting the a significant dual-helicity extensio eigenspinors of the spin-one-half charge conjugation on dark matter. based upon the Diracfield spinors. [ In 1957, there was an effort to reformulate the Majorana three-halves. spinors. It seemsstructure to of have the remained Majoranawhy unasked the spin as same to spinorsin should what not the effect (1 the choice for the helicity now has better than 4- that we present herecontained arose in [ when the present authors decided to take the research notes then, within the frameworkoperation of of known charge spacetime conjugation symmetries—with and parity, parity, and v combined field which fitsdecades the later, are 1933 the Zwicky experiments,anew(oraseto the call observations, and of the theory dark merging with matter a discovery. call for Only now, some seven Journal of Cosmology and Astroparticle Physics where suggestion [ collaboration has, in thesome last may few prefer years, to presented say, tantalizing the hints first experimental evidence—or, as to devote their entire academic lives to co spins, a preliminary exploration of these issues emerged in [ Ψ( 3 1.1. Genesis: from the MajoranaThe field spin- to a call for a new dark matter field by identi so that JCAP07(2005)012 6 1 2 and tors. (1.3) ile the ]. The ) doublets pera 2) charge 51 o L ,wh )identified / ]); the most 1 p and 1 spin- , ( (2) 46 † β 10.3 3 2 (0 c for the following , ]–[ SU ent from that of ⊕ µ utrino mass generation, 44 x be of importance. These 0) µ Elko )and p , onjugations date for identification p 2 k +i ( / † β )e )as d p p )isdiffer ( ( x A β A ( derations are confined to the physical / adungs λ η S si na field. However, an extended ) L ssible phase, it yields a neutral ood candi λ terms may also p on tical importance of the results ), with Spin-half fermions with mass dimension one ( particles of mass dimension three- x † β rc ( 1 2 d η Elko ou + stacks.iop.org/JCAP/2005/i=07/a=012 ll utrino oscillations and ne µ often take liberty of just saying ‘dimension’ rather x ,a µ p 11 i ningful and phonetically viable acronym, we ]). But in doing that, one goes beyond the )isnotag − eld, whether charged or neutral, carries mass )uptoapo x field ( 49 p )e ( η 1 2 p † β igenspinoren des ugation operators do not commute [ ( ]–[ c S β E 07 (2005) 012 ( on. These are postponed to section 47 λ : ) ark matter candidates (see, e.g., [ p ( es. As such, it cannot be part of the es not have a charge with respect to any of the Standard Model gauge ). We shall abbreviate β Elko 3 c )with he understanding of ne . p ecent review on dark matter. 5 icle. When coupling to the left-handed sector with Yukawa-like terms it is ( luded that rt † β unting suggests that quartic β ar d eting candidate for the Majora ) or ately, results in the mixing of mass dimension p the dual-helicity eigenspinors of the (1 ( ]f ,becauseitdo igenstates of the charge operator. This fact, combined with the 50 1 quantum field mE Elko 1 2 ssible phenomenological and theore 2 )are p .So,weconc ( )asacomp 4 2. Naive power co 3 A x / ) / p ( S 3 π η identification of λ d (2 ass dimension’. With a minor exception in section the electroweak neutrinos ough we will not consider this possibility in the current work it is to be noted that, in principle, a right- In the context of extended spacetime symmetries, supersymmetric partners of the Given the po As will be shown in detail below, the charged field )= th x In this paper, while referring to a quantum field, we shall Al ( reader is referred to [ experimentally observed spacetime symmetries.in a If few supersymmetrydark years is matter. then discovered our atthe Obviously, proposal LHC it construct will presented is compete herematter also for may require conceivable a a be that more ‘natural the detailed both source status’ discussi supersymmetric of as dark a partners matter. candidate for The remarks on mirror four-dimensional spacetime of special relativity. Standard Model fields also provide d discussed being neutralino (see, e.g., [ than ‘m 5 4 fermionic fields In general, the charge and charge conj 1.2. The new spin- Dirac particles arecircumstances e summarized above,classes. suggests The simplest studying of these in is detail the spin- the unexplored Wigner η halves. In other words, a description of neutrinos by handed neutrino may be reason. At the end of our path to obtain a mea eventually settled for the German Dirac. On field which is differentwas from to that offer ofand Majorana. detailed analysis revealed As thatdimension already the one, new noted, and fi our not initial three-halv motivation with each other appropri with where the considerations may be of relevance fo groups and thus isemphasized a that truly the neutral coupling part mass constant dimension in 1 such terms will cease to be dimensionless; rather, it will have positive of the Standard Model which necessarily include spin- obtained, a natural question may arise in the mind of our reader as to why such a Journal of Cosmology and Astroparticle Physics and deserve a separate study. conjugation operator (see section JCAP07(2005)012 . 7 ]. 7 58 Elko .An ]–[ .The wthat ) heoretic 55 3.5 Elko discussion next section es elements of ,wesho is devoted to a .The ile the associated 6 5 .The 2 fanyfield-t outlin Elko establishing that the ation, is recognized as ,wh 8.3 shows that while acting 3.4 viol is far from trivial, and we 4.2 ]. For instance, an argument ection 54 CPT , Elko Spin-half fermions with mass dimension one .S 53 8 , .Apartfrom is focused on constraining the 27 stacks.iop.org/JCAP/2005/i=07/a=012 , is devoted to a possible identification 4.1 ge conjugation operator, i.e., 10 ]. Yet, for the present authors, that has 26 9 and gravitational realms spacetime must 7 tbeanintegralparto ibit non-locality: typically, what localizes One reason may be that any student of quantum field is the subject of section rgence of non-locality it is not completely basis is not an identity operator; instead it Elko fthechar ons are the subject of a short section, rguments can be found, for example, in [ is presented in detail in section 07 (2005) 012 ( the identity operator. Section Elko ese remarks in the concluding section. Elko minus 2) parity operator on the at the representation-space level. The dimension one aspect / 1 , and briefly discusses the relevance of the obtained non-locality (0 Elko ⊕ Elko also serves another useful purpose. It sheds additional light on the ]). In the absence of such charges there is nothing that protects the rrently, it may be noted that today the conventional wisdom has 0) the identity operator. Similarly, section , 6 52 2 / ore concrete about th and framework to dark matter. Section minus 5 .Concu the parity operator on the 6 the square of the combined operation of charge conjugation, spatial parity, and Elko n-commutative, and that non-locality mus ledgeable physicists citing an important 1966 paper of Lee and Wick which essentially rising ge conjugation and parity operators commute while acting on ections Elko ow -matrix theory for However, it is emphasized that in the context of solitons ‘non-locality’ refers to a classical field configuration, structure. The simplest of these early a Journal of Cosmology and Astroparticle Physics while the non-locality encountered in the present work appears at the level of field (anti) commutators. 6 presents the dual-helicity eigenspinors o The general plan ofmay the be paper in isof apparent order. the from particle–antiparticle the To symmetry tablereview establish with on of the our the contents. emergence spacetime notation, Yet of symmetries, a the and charge few we conjugation to specifics present operator a remind in brief section the reader of the relation 1.3. On the presentation of the paper construct has not been undertaken before. physics who wished tokn venture on suchassures a that journey any would suchnot be theory been will immediately be a discouragedconstruct non-local by discouragement. for [ reasons We weargue simply have naively already set why mentioned. out suchotherwise The to fields feature extended are look of field likely neutrality at(see configurations allows to for an one like exh example to alternative solitons [ to is the a Dirac conserved (topological) charge evolved to a position where non-locality, and at times even We shall be m can be made thatbe no at the interface of quantum orthonormality and completenessaction relati of the (1 appropriate new dual for these spinors is introduced in section surp ‘particle’ from spreading and thus the eme an expected part of a theory of quantum gravity [ take some time to present the details in section char square of is given by on time reversal operators yields of s Dirac construct. The statistics for the mass and theand relevant important cross-section. differences, The from presented the construct WIMP carries and some mirror similarities, matter proposals. This is the to the horizon problem of cosmology. Section detailed examination of of the quantum field based upon S of the Locality structure of the theory is obtained in section JCAP07(2005)012 8 4) )& 0 ends (2.2) (2.3) (2.5) (2.1) ( R φ ) 12 0) , 2 / (1 ll phrase our κ structed follows )= p ( R . , φ 2) representation-space 0) representation-space non-locality is discussed / , )thuscon 1 gives rise to an important 2 .Yet,wesha , / p 1 2 m m p ;(2. p ( | · Elko · ψ )and + + p p | 0 σ σ ( E E states Spin-half fermions with mass dimension one L = he two-fold generalization, i.e., to φ − + ϕ ergent I I ± codes in it the dispersion relation stacks.iop.org/JCAP/2005/i=07/a=012 Elko mension 1 neutral, rather than charged, m m )= auxiliary details of calculations and some , m m 0 rthatt | provides a reference guide to some of the + + ( transform as (0 2 2 =1en transform as (1 p R m | E E .Theem θ ) φ ) 2 p w. The unconventionally long section 11 12.5 ( p ( )= L om prematurely invoking any folklore. 07 (2005) 012 ( = = R φ ϕ sinh φ − dual-helicity of ϕ ϕ θ · · sinh( , 2 terplay of 2 2 σ σ ains a detailed critique, and discussion pointing towards fined as ) ) − + .The . p p , 2 ( ( L R m E m 10.3 ,isde φ φ – + ϕ properties. 2 =exp =exp )= llowing these equations a reader should be able to construct a rough ), where p ve Weyl spinors identity cosh 10.1 ϕ ]fromthein 2) 0) 0 , / )= ( 2 = 1 , / 60 L p yfo 2 , ( (0 (1 φ κ cosh( κ ψ E 2) 59 rameter, ,whichalsocont / , 1 he reader is requested to reserve judgment until having read the entire paper , 12 elkological ubject matter at hand requires a somewhat pedagogic approach to the (0 32 κ ommutative energy momentum space on the one hand and a non-commutative es )= In order not to allow the discussion to spread over too large a technical landscape we Th p mmetry. This is discussed in section ( L Journal of Cosmology and Astroparticle Physics Both the Dirac and Majoranarepresentation, fields is are built upon Dirac spinors. A Dirac spinor, in Weyl 2.1. The Dirac construct subject of sections and quick overview of the theoretical flo with a summary. A set of appendices provides additional have chosen to confine ourselves to the mass di field. For a similar reason we shall confine ourselves to spin- arguments and presentation in such a manne higher spins and to charged fields, will be rendered obvious. The boost pa in section anon-c spacetime on the other hand. Section key equations; b asy where the massive Weyl spinors and because of the φ presentation. Wepedantic. T follow this demand without apology, even at the cost of seeming and is, in particular, asked to refrain fr 2. Emergence of the charge conjugation operator: a brief review objects, and massi objects. The momentum–space wave equation satisfied by uniquely [ JCAP07(2005)012 9 is I (1) ) null (2.8) (2.6) (2.7) U 9 . n 2.6 ) × n , duce a new particle, ]. The lesson is I ), with the upper − 66 ntro , p . ( while in equation ( 5 65 IO ψ 7 O ) µ uation has four linearly x µ = p i 5 ... here represents a nsection ∓ tains the configuration space . ass and opposite charge to an electron. O Dirac, Nobel Lecture, 1933 Spin-half fermions with mass dimension one nitial hesitation to identify the associated ntion i ,γ ssical framework. But in a quantum lity being apparent from the context 2identitymatrices, ]. In brief: reluctant to i stacks.iop.org/JCAP/2005/i=07/a=012 × i ):=exp( y, the Dirac eq 62 PAM σ x O ( − ψ nacla ariant manner would have assured that no ving the same m unless prohibited, by a conservation law, or i l, induce transitions between the classically al intuition may ask for avoiding the doubling O σ and detailed atte stand for 2 and ], not to say antiparticles [ I µ 07 (2005) 012 ( itio = ∂ 64 i mentum–space Dirac equation , . . ), the =i tency of the notation, 63 ab in =2).Obviousl 2.3 µ n p have their standard Weyl-representation form: )=0 ign for antiparticles, one ob )=0 ,γ µ x p ( γ ( )and( ψ ψ ) 2.2 ) I I m .Forconsis m .The 3 IO OI 8 − γ ∓ 2 µ ]. Furthermore, it may be noted that Dirac’s i γ µ ]: one should not impose mathematical constraints on a representation ∂ identity matrices, their dimensiona 1 p ssed Lamb shift [ = 61 µ γ µ particle an anti-electron’. The name ‘positron’ was suggested to Anderson by Watson Davis 68 γ n 0 0 , γ γ ( γ (i i × 67 − n like to discuss, a feature which led to the prediction of the positron’. ample, in equations ( := One would thus be inclined to introduce, as a new assumption of the theory, are llowing insistence on ‘only two degrees of freedom for a spin one half-particle’, Dirac 5 I ]). In the 1933 Nobel lecture Dirac unambiguously writes: ‘There is one feature of these equations which γ hat only one of two kinds of motion occurs in practice 4identitymatrix. le with a new particle is well documented by Schweber in [ t Fo 62 lectron, and opposite charge. By 1931 Dirac was to write so himself: ‘A hole, if there were one, would be a uld now × ee The quote is from [ This result will be derived and also given So, for ex in which they appear matrix (in the aboveindependent equation, solutions. Letting allowed and the classically forbidden sectors one, or hardly any one, raised an objection.inescapable Had [ Dirac taken that path, a local of the degreesSuch of constraints may freedom have orframework, a limited a the validity folklore interactions i may will, demand in a genera definite spin for particles, etc. space to obtainintuitions, an or prevalent interpretation folklore. which The physic satisfies certain empirically untested physical could have proposed aThe constraint fact which projected that out he two could of have the done four so degrees of in freedom. a cov 9 7 8 2.2. Dirac’s insight: not projecting out antiparticles The implied wave equation is the mo Here, Isho Dirac initially identified theimmediately new saw particle that with suchth the an proton. identification was Heisenberg, not Oppenheimer, tenable Pauli, andWe Tamm, the may and new Weyl call particle(see must [ such carry the a same mass as partic a4 gauge theory based onIt such would a have covariant mi framework would have lacked physical viability. Journal of Cosmology and Astroparticle Physics new kind of particle, unknown to experimental physics, ha Dirac equation: sign for particles, and lower s with JCAP07(2005)012 ]. as 10 68 3 2 )is . 2. Elko (2.9) / 10 (2.12) (2.10) (2.11) ) equals 2.11 ) ,j =1 K (0 j ⊕ 0) ,itmakesthe j, 11 ] .Equation( 71 ∗ , ces. The existence of a J 70 − nded Weyl space for which ’, as is done implicitly in all om we shall encounter 2 = σ 1 i − − ]—because such an identification Spin-half fermions with mass dimension one Θ ces may carry a relative phase, in J 69 )left-ha 1 2 as a single physical field which carries spin- , µ stacks.iop.org/JCAP/2005/i=07/a=012 ψ (2)s, with the following generators of lined here carries a quantum-mechanical follow Dirac’s insight and not project out ly a relativistic element via the Lorentz orm. The chosen form is justified on the SU )representationspa ]. Weyl spa ,j 72 rf 0) right-handed Weyl space, where allows for the introduction of ( , , . (0 1 2 ) ) j ⊕ K K 07 (2005) 012 ( i elow)—degrees of freed 0) g–Feynman interpretation of antiparticles ceases to be equivalent +i − ersal operator. We use the representation j, J J 2, we have the (0 ], recent work of Kaloshin and Lomov confirms our interpretation [ ( om identifying Θ with ‘ ( K. / 1 2 1 2 67 σ uckelber . = = = O iΘ A B J 1 2timerev In this latter context the suggestion is to consider as unphysical the practice of 2, we have the ( − / / ]. iΘ O complex conjugates any Weyl spinor that appears on its right, and σ : : 67 − 10 0 ,and A B 2werefrainfr 0 = K / (2) (2) J = omb of this structure emerges a new symmetry, i.e., that of charge = =1 represent the generators of rotations and boosts, respectively, for any of er’s spin-1 A SU C SU Θ= j K trary spin it is defined by the property Θ components. Apart from [ 2). uckelberg–Feynman interpretation of antiparticles [ ,and / 1 2 0 σ and = 2). For spin- / J ection rule, for some reason. Here, we shall does not exist for higher-spin ( From the w considerations on the subject—see, for example, [ Wigner time-reversal operator for all the St¨ representation spaces. In this paper, however, our attention is focused on The derivation of the Dirac equation as out connection betweenmanifest. particle–antiparticle symmetry and time reversal operator B σ is +i( It may be worth noting that the St¨ This seemingly logical position encounters an element of opposition when one applies it to a related problem i( 1. Even for 2. This form readily generalizes to higher spins. Furthermore, as required by to the standard interpretation in cosmological context [ 11 10 asel following grounds, and invites the remarks: of Rarita–Schwinger field [ K conjugation. The operator associated with this symmetry is For an arbi Journal of Cosmology and Astroparticle Physics The the relevant finite-dimensional representationFor spaces which may be under consideration. similar—i.e., anti-self-conjugate (see b projecting out the lower-spin components; and to, instead, treat deliberately written in a slightly unfamilia − well as ΘistheWign Here, the operator transformation: aspect in allowingthe for the sense fact madetransformation that explicit properties the above, two of anddepends the concurrent Weyl on spinors. the In existence turn of the two very spacetime existence of the latter JCAP07(2005)012 ) . 11 13 2.3 (3.2) (3.1) is an (2.13) ]. ), ( λ ) 74 ζ )below. ’s particle 2.2 4.12 .Theboost transforms as ) K 2 is an unspecified p ,Dirac ( γ λ 14 ∗ R ζ − φ ). This allows us to transforms as a left- . ∗ . † = p ) ( 12 Θ) p R 2) ρ ( C / φ ζ L 1 , transforms as a left-handed φ (0 ) κ p ( L Spin-half fermions with mass dimension one = φ 1 − 2)-space charge conjugation operator, stacks.iop.org/JCAP/2005/i=07/a=012 space—and which become eigenspinors / 0) , 1 2 ly indirect but physically insightful path. , / This brief review makes it transparent (1 (0 κ ⊕ ersal operator, has the property ost operators written in equations ( 0) , . 07 (2005) 012 ( 2 . , / nors and vice versa—see equation ( ∗ † =0 2] transforms as a right-handed spinor—where / 0) , 2) representation 2timerev 2 2) ) σ is an unspecified phase. These results are in agreement [ / / / / p 1 1 , ρ (1 ( − , ζ ∗ L κ (0 φ κ (0 the origin and form of the charge conjugation operator. We = ,andthe(1 ⊕ = ⊕ transforms as a right-handed spinor—where 1 takes, up to a spinor-dependent global phase Θ) ily seen to yield the standard form, 2) 1 − λ ) ’s spin-1 0) / is given some specific values. 0) , − ζ 1 , C p , 2) spinors which are different from that of Dirac—which, of course, 2 λ 2 ( / / (0 ζ ∗ L / 2) (1 2] Θ 1 transforms as a right-handed spinor, then ( κ / φ , / 1 , ) lm of spacetime symmetries. ⊕ σ (0 (0 p Θ) ( )isread κ C, κ 0) tion of λ ⊕ , summarized R Θ[ ζ 2 φ / 0) licity eigenspinors of charge conjugation operator, or Eigenspinoren des (1 ]. 2.11 , ollow the latter, and will shortly argue that if ,theWigner κ 2 73 ormal treatment of this result can be found in the classic work of Streater and Wightman [ operator if / erg’s classic on gravitation and cosmology, and since we cannot do a better job than that the reader is nded spinor—where nor dependence may be removed by appropriate redefinitions without changing the physical content of C So, particles and antiparticles are offsprings of a fine interplay between the quantum The opera rther, Θ Amoref The spi However, in general, boosts do not leave the time-order of events unchanged. This leads to interesting paradoxes, ,commute: 14 13 12 The details are asare follows: Hermitian and because inverse the to each bo other, we have 3.1. Formal structure of Elko Equation ( and again this necessitates the2ofWeinb existence of antiparticles.referred This to has been [ discussed elegantly in section 13 of chapter Journal of Cosmology and Astroparticle Physics the theory. This makes the notion of particle/antiparticle frame independent realm and the rea spinor, then ( aleft-ha phase; (b) if operator, We have just Ladungskonjugationsoperators (Elko) 3. Dual-he Keeping with our pedagogicthe style, charge we note: conjugation operator. the Dirac spinors are thus not eigenspinors of spinors into Dirac’s antiparticle spi When combined, these observations imply that: (a) if define (1 Fu unspecified phase—with a similar assertion holding true for handed spinor, then ( now proceed topurely obtain mathematical approach, its or eigenspinors. adopt a slight Towards this task one may take a direct and We shall f C also belong to the (1 of the JCAP07(2005)012 ) λ ζ 12 p ( λ (3.6) (3.8) (3.4) (3.7) (3.3) (3.5) )are p 0) and ( , ) llowing: λ 2 / phases, nthe )isthefo . 1, if the ,the 3.8 . ± ) rite , p ) same ( 0 ) ∗ R ( ) ) φ p ∗ L 0 ( ). The minus sign results in p φ ( ∗ ) ( R p L ∗ R ( p φ φ ( Θ) S iΘ φ ρ Spin-half fermions with mass dimension one ρ R we may w 2 ζ − φ commutes with the boost operator ( σ 2 σ ∓ p . stacks.iop.org/JCAP/2005/i=07/a=012 · ) )and p )= emand of observability), these become σ ( entical to the standard Majorana spinors )= p ). To obtain explicit expressions for 0 . ρ ( p ( p )= ∗ S ( ( ± A ed λ p A ) ( in appendix A.2, while the explicit forms of ρ 0 ( )= ± L etailed analysis of Dirac spinors (see section 5.5 p ). Since φ 07 (2005) 012 ( ( ,ρ 2) components to be the 0 ,λ ( / , )and . ,ρ ) ± L 1 Θ i licity eigenstates p ) , 0 φ ( ) ∓ ± ( p ]. As a consequence, the following spinors belong to the ) A 0 ( ± L )isgiven . ( ) ) λ p ∗ L 69 φ ∗ L = = ) p ( nberg’s d 0 p φ ,Cρ φ ( ∗ L ( 3.8 p ± ∗ ρ ) ( L L φ p ) L φ φ 2 Θ) ( ). We find that this choice has important physical consequences 0 φ λ σ )tobehe 0) and (0 λ ( p ζ +i Θ )= entation space: , 0 ( ± L ± ( ndix A.1. The physical content of the result ( ± ( 0 2 λ φ ,ζ ( L / i self-conjugate spinors: ± L φ ± φ )= Θ )= p )= )= p p 0 ( nappe = ( p p · · 2) repres S ( ( λ he rest spinors. These are / ζ Cλ σ λ σ λ λ 1 , restricted to the values has opposite helicity of (0 ∗ ⊕ )] this result holds for all ,are 0 ρ 0) 2) components of ( )aregiveni 0) ]). In particular, as we shall confirm, that the choice affects the parity and locality ζ , , acterized by a single-helicity and become id / anti-self-conjugate spinors: ± L 0 2 confine to this subsection only—that since iΘ = 8 2 1 / ( φ / , ± L (1 φ With this restriction imposed, we have Journal of Cosmology and Astroparticle Physics So as not towe obscure the physics by notational differences, it is helpful to note—a choice 3.2. Distinction between Elko and Majorana spinors (1 and with Ramond’s observation in [ Θ[ κ Next, we choose the the The derivation of equation ( The plus sign yields and concurrently note that where the upperself-conjugate sign spinors. is We now for have self-conjugate a spinors, choice in and selecting the the helicity lower of sign the yields (1 the anti- we first write t Confining ourselves toeigenspinors of real the eigenvalues charge (th conjugation operator with eigenvalues, (0 for reasons which parallelof [ Wei properties of the constructedthe field. helicity For for the the moment it (1 suffices to note that if one chooses char JCAP07(2005)012 ). ). 0) 13 0 , 13) ( 2 3.8 (3.9) S / )and (3.10) (3.12) (3.11) λ ). This ), while ) we start 3.8 p p ). After ( ( L spinors are λ φ 3.8 Elko ]. 76 oosted . , quation ( ∗ ∗ ) ) 0 )), we take the same original 0 er are dual-helicity objects. , ( ( ) ) p ) )(3. − L − L ( 0 0 0 0 exploit e ( ( ∗ L φ φ ( ( . − L − L } φ ) Spin-half fermions with mass dimension one φ φ + −} 2 , , 0 σ ( + iΘ } S {− S { ile in the same limit ± h − +i Θ λ λ we consid stacks.iop.org/JCAP/2005/i=07/a=012 − ther helicity. Then, when constructing 2) component. The b h, / ,wh { it is now useful to familiarize oneself by 1 λ m m , Elko ressions as the boost operator that appears ,andthen )= )= p p alently + + p 0 0 ( ( 2) · Elko 0) transforming component of the E E / ), which incidentally do not constitute an osite helicity to that of the original , 1 −} −} , , p , O 2 σ − ( (0 + / + 07 (2005) 012 ( p ρ A { S { 1+ 1 κ ), the boost operator, and the structure of the by 0) )(or,equiv )inoneortheo m m , lds 3.8 2 p p p m m / ( O ( + + ,λ ,λ 2 2 · ∗ L (1 L κ φ )yie E E φ identically vanishes σ )intheaboveexp ∗ ∗ ). For this reason 0 iΘ ) ) ( . 3.11 3.8 S 0 0 ± 15 ( ) ( ) )= λ )= )= 0 0 + L + L p ( ( p p ( φ φ ( ( try refers to the (1 + L + L } } Elko h φ φ helicity, i.e., we do not flip its helicity by hand. This causes the (1 −} + − , , ). For one thing, there is only one (not two) boost operator(s) in the iΘ + h, ]). This choice violates the spirit of the result contained in equation ( dded to the manuscript as an answer to remarks by E C G Sudarshan [ 2) representation space. The simplification that appears here is due to a S { { S {− 2) component / +i Θ − λ λ λ 75 sa / 1 3.11 , , 1 wa , 30 (0 0) component, 3.2 )= )= , ⊕ 0 0 2 ( ( lly respect the spirit and the content of the result contained in equation ( / 0) the (0 } } )inthesame , The results of the above discussion lead to four rest spinors. Two of them are self- Similar remarks apply to the + + , , p 2 ( Section / L A {− S {− simplification, equation ( transforming component to carry the opp 15 λ conjugate, constructing them in their fully explicit form. 3.3. Explicit form of Elko Having thus seen the formal structure of (see, e.g., [ We fu The first helicity en the (1 which, in the massless limit, independent set of in equation ( (1 In the boosts, we replace φ does not. We hastendifferent to pre-factors warn to the reader that one should not be tempted to read the two Journal of Cosmology and Astroparticle Physics with therein lies our point of departure from Majorana spinors. That is, for our is dictated by equation ( λ and the other two are anti-self-conjugate, the second entry encodes the helicity of the (0 now obtained via the operation fine interplay between equation ( JCAP07(2005)012 14 ,is (3.22) (3.18) (3.16) (3.19) (3.21) (3.14) (3.17) (3.20) (3.15) ) Elko dual fined as ile the second does elf-conjugate as well and not to imply the lement of asymmetry. } ,wh + )isde , p upper and lower position is the Kronecker symbol. negative-definite norm for ( ξ h h , . −}{− , ) ) δ + 0 0 { ( ( )holdsfors ε ,and .The } Spin-half fermions with mass dimension one } , + −} , , , −} 0 + 0 , , , , + γ 3.19 0 0 + γ A {− A { { {− ) γ γ ) Elkos λ λ stacks.iop.org/JCAP/2005/i=07/a=012 ε p † † p ( ( − ) † β ) is a hindrance to physical interpretation † h λ p m m p identically vanishes , , ( ψ ( have a real norm. The new dual must have β α 0 0 p p ± + + ε 1= , h h γ γ troduce an unjustified e S ∓± real definite norm, and (b) in addition, it must A ∓ δ Elko † † − have an imaginary bi-orthogonal norm as was E E ρ ρ ) ) Elko − p := p − 07 (2005) 012 ( ):=i ( ( ):= 1 1+ } } p ), the Dirac dual spinor Elko p ( + −} + −} our Dirac spinors and ( , , p , , A A h ( α / / + + m m ξ {− { S {− ):= S { ¬ ψ λ ):=+ ε λ λ m m p p + + ( ( 2 2 i lds . ppendix B.2. ∓ E E ∓ , 0 , − A ± γ S ± ) ¬ λ ¬ λ norm for two of the four ). The Dirac dual, for comparison, may then be re-expressed in )yie p ]. For the sake of a ready reference, this is recorded explicitly ( 2) spinor p † )= +i )= ( / )= )= ξ λ 41 sanyofthef 1 p p 3.19 p p ( ( , , ( ( } } ): ): (0 40 + −} + −} p ):= , , , , p A A ( ( ⊕ / / + + p S S S A {− { A {− A { ( ¬ ¬ )aregivenina λ λ ξ λ Dirac Dual: λ λ Elko Dual: λ dual can also be expressed in the following equivalent, but very useful, form: 0) , p ( 2 )represent ρ / p Elko ( ψ the antisymmetric symbol The not. where the Journal of Cosmology and Astroparticle Physics For any (1 3.4. A new dual for Elko Similarly, the anti-self-conjugate set of the boosted spinors reads the following equivalent form: as anti-self-conjugate With respect to the Dirac dual, the the property that: (a) it yields an invariant already noted inin [ appendix B.1.and The quantization. imaginary Enormous norm simplificationdefine of of a interpretation new and dual calculation with occurs respect if to we which secure a positive-definite with Explicitly, equation ( where In the massless limit, the first of these spinors of indices has been chosen only to avoid expressions like use of a metric to raise and lower indices. Equation ( Up to a relative sign, a unique definition of such a dual, which we call an the remaining two. Any other choice will in JCAP07(2005)012 15 ]: 77 (4.1) (3.26) (3.27) (3.23) (3.24) (3.28) (3.25) )have ) 3.25 ness relation )–( ]. In that pure 3.23 20 .Thecomplete spinors under the operation } + , , . I I Spin-half fermions with mass dimension one Elko {− = = , stacks.iop.org/JCAP/2005/i=07/a=012 ) ) −} p p , ( ( h + A α v { ) ¬ λ jugate spinors. Equations ( p ) ( p h ( v . , A α , 07 (2005) 012 ( , λ − αα αα self-con parity asymmetry ) hh − hh p ) ( mδ mδ h p 2 for Elko mδ mδ ( u ) 2 ,and I − S α p − P ¬ λ es and antiparticles carry opposite, rather than the same, − ( h ) ). We have laboured this point as different expressions are rticle and antiparticle have same parity u )= = )= +2 p ( and )= +2 p p )= 2 S α ( ( 2) p 3.18 ) p λ / ( C ( A α S α (1 h λ λ h ± u v ) ) α = ) ) p p h CPT ( p ( p bosons: pa fermions: particle and antiparticle have opposite parity. ranges over two possibilities: ge the only textbook which tells a more intricate story is that by ( ( ( )and( m m S α A α h 1 1 h dual thus defined, we now have (by construction) α 2 2 ¬ ¬ u v λ λ ]. The only known explicit construct of a theory which challenges the 3.17 8 stage for this section we begin by quoting the unedited textbook wisdom [ Elko eets the expectations of Wigner. tional wisdom was reported only about a decade ago in 1993 [ en ,m nv and Journal of Cosmology and Astroparticle Physics 4.1. Commutativity of To set the useful in various contexts. With the 3.5. Orthonormality and completeness relations for Elko which, ondefinitions use ( of results given in appendix B.2, shows these to be equivalent to The subscript of spatial parity.of This charge prepares conjugation, spatial usElko parity, to and show time-reversal that operators, the when acting square upon of the the combined operation To ourWeinberg knowled [ 4. Establishing In this section we present the detailed properties of clearly shows the necessity of the anti- co their direct counterpart in Dirac’s construct: spin-1 bosonic theoryrelative particl intrinsic parity. This manifests itself through the anticommutativity, as opposed JCAP07(2005)012 ) 16 )we (4.7) (4.6) (4.2) (4.4) (4.8) (4.9) (4.5) (4.3) 4.10 (4.11) (4.10) p ( ) h nge sign v the charge ], and that 6 )and( 2cha / 4.9 )and p Elko p · ( 1 2 h σ u . ) p . ( h ction can be summarized u ive. The parity operator is iΦ ie − ing on the Spin-half fermions with mass dimension one reads )= helicity operator . R p stacks.iop.org/JCAP/2005/i=07/a=012 } − p prise that the parity operation is slightly ( )tobeposit h [cf equation (4.18)] − ,whenact → 4.7 u } ,ofthe )), the 0 θ h , γ I π, p iΦ C, P cos( = { on these spinors. This a attention the classic work of Wigner [ + 07 (2005) 012 ( 1)-space’s charge conjugation and parity operators. , 2 to be real. This fixes the phase ) , φ ator commute, rather than anticommute (as they do C φ )=e P or, (0 P p → ( ⊕ ained in the sign, still remains. This ambiguity does not . h , . ) ) u ) 0) )sin( p : h. p , θ p cuss its relevance to the cosmological ‘horizon problem’. In ( ty oper R ( θ, φ ( h − 0 h h mmutat v γ u − v alues of sin( eigenspinors of the parity operator. Equations ( iΦ . iΦ , = : − π )  y, as cont R . R φ . 0 h i → R γ 0 eigenv ± )= + )=e )= )=ie θ iΦ → 2) representation space it reads γ p p ]. p p / ( ( ( ( )cos( h 7 = 1 h h h h θ : =i =e , etheantico iΦ .Webrieflydis (0 P e R≡{ Pu Pv Dirac Spinors Pv Pu R P (sin( ⊕ p Elko 0) , := 2 / p To calculat Given these remarks it does not come as a sur rthermore, while acting on the Dirac spinors, ect the physical consequences. It is fixed by recourse to text-book convention by taking The remaining ambiguit therefore fixed to be aff the sign on the right-hand side of equation ( Because for the theorywe based upon must Dirac require spinors relative the intrinsic parity is an observable, Journal of Cosmology and Astroparticle Physics to the commutativity, of the (1 Fu now need, in addition, the action of subtle for of Lee and Wick [ conjugation operator and pari for the Diracsection case). where we We bring shall to have our more reader’s to say about these matters in the concluding In a somewhat parallel fashion we shall now show that for the spin- That is, Dirac spinorsimply are Similarly, With Thus the (1 under the operation of This has the consequence that eigenvalues, JCAP07(2005)012 - 17 P and )—to I (4.12) (4.18) (4.14) (4.15) (4.17) (4.13) (4.16) ection − ) eal that 2.11 -refl = find that it ). P dual-helicity 2 p )rev ), ( P p 2 , . / ( ) ) pinors under 2 )reads 4.15 .We / +1 p p ( ( u )s +1 } } . p v + + 4.14 , , ( 0)-Weyl and positive- Elkos 0)-Weyl and negative- ,inbeing , A {− S {− n, in the massless limit → −} , →− 2 , )and( λ λ 2 )requires ) / ) + operators is an important i / A { spect, our results coincide p p Elko ]. Despite similarities, our four ( λ − ( P 2 4.14 6 2 4.15 / / 1 1 − . − )= +i nors. For each case it is found )= and . fthe Spin-half fermions with mass dimension one p p ( ( C )and( −} −} )spi , , .Inthisa For this reaso stacks.iop.org/JCAP/2005/i=07/a=012 p + + [cf equation (4.16)] ( A { S { v 4.14 Elko , ,u ) ]oneacho to the charged-particle spinors. The origin of ,v p =0 ) helicities. However, in the massless limit, the ( ), one can readily obtain the action of the )and )—taking a special note of equation ( p 2 } . / ( ,Pλ p ,Pλ ) C, P 1 2 ) ( ) 3.5 / 07 (2005) 012 ( p − [cf equation (4.13)] u p p 1 4.12 ( v ( [cf equation (4.11)] ( C, P } − ), in the massless/high-energy limit the { + rthermore, an apparently paradoxical asymmetry is u −} −} , , , , nstance, the second equation in ( . opposite I four + + ifference does not seem to affect many of the general force only positive-helicity (1 A {− ), carrying negative-helicity (1 →− S { 3.14 A { → − λ p ), and ( entically vanish. λ λ ) ) i ( 2) .Fu : ]=0 i p / } p ), and ( 1 ( − ( P = − , + , 2 ection resides in the fact that the 2 4.15 / 2 (0 / yields the result A {− C, P κ )and( P 4.10 +1 +1 λ ,onthe ), ( )= )= )= +i -refl u v gh a full formal generalization of the Wigner–Weinberg analysis } p p p P ( ( ( 3.13 ), ( Elko and eigenspinors of the parity operator. Following the same procedure as } } lly vanishes. The same happens to the 4.14 : :[ −} + + .Yet,thisd , , , C, P 4.9 0) )and , tivity and anticommutativity of the + { 2 S { A {− S {− not p / ( : 12.1 } (1 C Pλ Elko Pλ Dirac Spinors Elko Pλ + κ are ach of them: 2)-Weyl spinors to be non-vanishing. 2)-Weyl spinors id , / / S {− )identica 1 1 , , λ p ( , Elko acting on the rthermore, the consistency of equations ( −} , are not eigenstates of + P Fu The commuta Unlike the Dirac spinors, as already noted, equations ( S { Elko ects, combine Weyl spinors of λ structures of obj of Journal of Cosmology and Astroparticle Physics as follows: Using equations ( anticommutator, reflection. This situation isthe in sharp asymmetry contrast under That is, As a consequence of ( in the process showsset, that do the not remaining contain two, additional i.e., physical the content: first and the third equation in that to vanish: helicity (0 helicity (0 before, we now use ( The the evaluate the action of the commutator [ vanishes for e distinction between the Dirac spinors and the with the possibilities offered by Wigner’s general analysis [ construct differs fromthis the in Wigner–Weinberg section analysis in a crucial aspect. We outline may be desirable, our specific construct does not require it. conclusions. Even thou contained in these equations. For i Elko JCAP07(2005)012 18 . 4 our e Elko (5.1) (5.2) (5.3) . .The (4.21) (4.19) (4.20) 4 → d ) ) ]. But in Elko p → 8 ( ) now known, wn from p −} , ( + − A { Elko ,v ,λ on 3 itten as 3 d e T → → ) ) sson to be dra ]andWeinberg[ and p p ( ( , 32 } + ) cise not only gives a sharper + , p ( C, P Spin-half fermions with mass dimension one A {− S α ), we have ,v λ 2 can now be wr d , stacks.iop.org/JCAP/2005/i=07/a=012 as follows: 4.16 ,λ 2 → =0 e ]‘themainle Elko ) discussion of differences with Weinberg’s } )= +i p Elko 78 do not satisfy the Dirac equation. The next . atically satisfied by writing equations which , ( → p I 4 ( − ) , A α − P, T p 3 ], the 12.1 ( { , Elko 2 79 ,u = −} 07 (2005) 012 ( , , 1 acts on 2 + d eflects on the connection between ‘spin sums’, wave S { ) -covariant representation space. Yet, in the , C =1 P 5 → ,Tλ γ ) ) tion of all three of the ,λ p p CPT 1 ( ]=0 but it also sheds new light on the well known Dirac spinors. ( ( =i e A α + , λ ,i I T u i ,brieflyr j → and Dirac spinors. This exer effort. C, T − 2) is a d eflection asymmetry. This result has a similar precedence in − ) Elko / ij 5.2 p -r 1 ( = Ω , } : P 2 Elko + )= (0 , ) 4 :[ =1 j p .Withtheac ( S {− ⊕ I ab initio S α λ = establishes that massive hand it is helpful to make the following local change in notation: − i CPT 0) Tλ Elko e ( rsal operator , Spinors eevolution = 2 5.1 / 2 : T eement with Wigner: trast between Section The (1 subsection, i.e., section operators, and propagators. Theacon remaining three subsectionsindependent are existence devoted to to establishing thus confirming Wigner’s expectation. For a the end we hadin to large develop part much our of the formalism ourselves. So, what follows constitutes treatment, we refer the reader to section 5. Spacetim The existing techniques to specify spacetime evolution do not fully suffice for path we take carries its inspiration from works of Ryder [ Journal of Cosmology and Astroparticle Physics 5.1. Massive Elko do not satisfyFor the the Dirac equation task at For Dirac 4.2. Agr The time-reve one can immediately deduce that, in addition to ( implying and that at the same time, For Elko Adopting the procedure introduced in [ transform covariantly’. analysis is that special relativity is not autom the Velo–Zwanziger observation, who noted [ formalism, it carries JCAP07(2005)012 ) 19 5.3 nors (5.5) (5.4) (5.7) (5.9) (5.6) (5.8) (5.11) (5.10) ]) with ) 8 )spi p ( v 16 ,weobtain entation spaces e 1 − =Ω spinors. In momentum d ile that for 2) repres / operator with eigenvalues ). That is, they cannot be 1 I , µ )wh m . p µ µ + x ecast into the form γ lt more precise below. Moreover, µ µ , Spin-half fermions with mass dimension one p ntiparticle ) p i )=0 I µ p − 4. 2, γ pplying from the left the operator Γ ( m 0) and (0 , , v stacks.iop.org/JCAP/2005/i=07/a=012 )=0 , fore, on using ) ± 2 p .A I ( / =3 =1 µ d )canber u p m j j . ) µ I rties under spatial reflection. Since the mass 5.5 + γ ,    =Ω .There m µ for for I ), nor by ( I I d    p e i I particle and a I nspinors of the i µ − I 1 2 3 4 − − − γ d d d d m µ 07 (2005) 012 ( ccurs via exp( ⊗ p . , , −    µ 5 I I e. I I I I i i i I )o 1 γ µ i e e − − ⊗ − − p p ecomes := j j ( µ mγ Ω d d ), equation ( u γ 2 ). )areeige ) I ) )b ∗ = i I I σ p i m 5.3 m d ( − 5.3 ⊗ −B 2 2 + . v / / rticles: ( 5 µ mmediately tell us that each of the spinors in the set defined by I ,d II p i (1 II ∗ µ )i I I operator. We shall make this resu i    B−B γ − mγ rticles: ( d. B − +(1 := ( 1 2 3 4 µ ), one cannot naively go from the momentum–space expression ( e e e e 5.5 p )and µ ⊗    B µ mplied by ( x p )implyΓ I ectively—a fact emphasized by Weinberg (see page 225 of [ 1 2 1 2 plification, we introduce    γ ( µ = p u =Ω =ΩΓ 5.7 For pa For antipa := ij e e )and( Ω= Ω= Γ e Ω Γ Γ:= Ω] = 0 yields ,resp , ’partasi 5.3 m ij − follows the second entry in the direct product always refers to the spinorial part, while the first one cannot be annihilated by ( is a linear combination of the Dirac and For formal sim In what m ccurs via exp(+i With the definition In matrix form, Ω reads eigenspinors of the refers to the ‘ Journal of Cosmology and Astroparticle Physics to its configuration space counterpart. 16 where That is, Dirac’s In this language, equationand ( using [Γ But, equations ( Equations ( Elko space, the Dirac spinors are annihilated by ( + since the time evolution of the and o the observation that it is a result of how the (1 have been put together toterms carry carry simple opposite prope signs,Elko and hence are different for the particle and antiparticle, the JCAP07(2005)012 , µ 20 ∂ i .On uples while to be (5.12) (5.16) (5.14) (5.17) (5.15) (5.13) 17 ). It is → ) x )co µ ( p term, and h Elko v µ 5.17 =+1, p µ A γ  This is true for , )and )asksforaeight- x ( ]. h =0 1, the above equation 81 u 5.17 1and in the −       − ) ) β α ) ) δ p p := ]and[ p p ( ( ( ( = } } 80 or in equation ( } our-component Dirac spinor. −} + + −} S , , , , + −} nner as the coupled equations , ,  nors—each of which is a two- + + + S { A {− {− { S {− A { erat Spin-half fermions with mass dimension one ε λ λ λ λ − − ema .Equation( ,andviceversa. .       stacks.iop.org/JCAP/2005/i=07/a=012 ), has several formal similarities and −} µ I ermines , Elko x nforthef m 2.8 µ i + reveals dWeylspi p { i 1 − )det . − annihilate the neutral particle spinors ×       . ). The presence of , A )Ω .Thewaveop ) ) ) ) / 5.15 I S p p p p )=0 nnihilates each of the four 5.7 07 (2005) 012 (  ( ( ( ( on, we make the standard substitution ⊗ x } } . ( Elko )=0 I 5 )=0 ,a + −} + −} , , , , A I p p / + + )donot ( ⊗ ( S β S {− S { A {− A { I m )and( A β S β ]inexactlythesam mγ λ λ λ λ λ )exp λ λ m − uation confirms the earlier result of [ 2 83 p       β α , σ ( µ 5.14 β α ± β α ε O ∂ A ε ε I −    µ / 82 := Ω( µ I I S p µ m γ µ µ m m λ p operator, thus making precise the observation made above. To in the mass term, now make it impossible for the 2 i O µ γ O σ +i µ γ β α +i − p β α ε µ β β α α ):= δ ’s antisymmetric symbol defined as γ δ δ µ x µ µ µ m ( p ∂ OO p p µ A µ 3.4 degree of freedom with the µ µ / γ = γ terparts of equations ( S i γ γ } µ λ gate as well as anti-self-conjugate µ + , p OOO µ γ {− µ p not so for the wave operator for the component formalism [ self-conju for the right-handedcomponent Weyl spinor—yields and the left-hande As wave we equatio proceed, weit shall introduces eight see independent thatrepresentation degrees the of space. eight-component freedom for formalism an is inherently not four-dimensional called for as O OOO OO µ us, making the direct product explicit again, finally we reach the result γ The result contained in the above eq 1. The Dirac operator, i    Th eigenspinors of the 17 An explicit evaluation of Journal of Cosmology and Astroparticle Physics the existence of the Consistency with equations ( obtain the configuration-space evoluti yielding Its counterpart for the Dirac case, equation ( differences: reduces to These are coun and define recalling section which establishes that ( JCAP07(2005)012 ). 2) 21 / 1 reads , 5.15 (5.20) (5.19) (5.18) S ) th Dirac A , )) )and( θ 0) and (0 cos( 5.14 he mass term has , ture remains true 2 ,p / .Thus,bo ) β α φ δ I hrootoftheKronecker ) 2 )sin( m ,soonlyt θ .Thisfea ent from a phenomenological to (anti-)self-dual spinors in how to take the ‘square-root’ β α I δ − ) 2 µ µ sin( lds the Dirac equation. Taking p p m Spin-half fermions with mass dimension one µ µ γ ,p p − ) ,yie rtible, the wave equation can always )isdiffer φ µ β α stacks.iop.org/JCAP/2005/i=07/a=012 p to choose whic δ µ . nve , p )=( ± 5.17 ation on the nature of wave operator for β α )cos( taken that these spin sums reproduce the ε i.e., the trace-free symmetric ones, are also θ I S I S spin sums, and further develop the formalism I , − A A 3 m .Itiswellknown )=( i I β σ α sin( nors. Its explicit form will be obtained in the δ I S m ± − S I I 07 (2005) 012 ( ) − A β α A 2 + δ E, p )spi µ µ m and p p p ( of ‘helicities’ (or referring m 1 µ − µ m λ := ( γ µ σ γ δ p µ )( )( ± , µ p )= I β α )= p ε p p m I ( .Thus,weonlyhave ), and that in the representation in which the four energy– ( ∗ µ 0 der different square roots of the Klein–Gordon operator times a m λ A α S α − p ¬ λ µ ¬ λ trices, µ 3.25 γ ) +i p ) 0 λ µ p p β α ( γ ( δ ima lds erated. S α A α µ defines the phase relationship between the (1 instead produces the wave equations derived above, ( λ λ p = der’s attention to the fact that, using the results of appendix B.4, the µ β α S S γ α α ε . i A particles have to fulfill the Klein–Gordon equation. Turning this argument A φ nsional Kronecker i ± :( :ityie ght into a form where the first term reads I µ = just enum p Elko trix := ie β α µ Elko ) p ssible roots, but they are not considered here. λ 2 σ The other Paul to be considered. Taking the trivial root, and around, it is possiblewe to would provide like a totwo-dime ‘quick consi and dirty derivation’the neutral of case), the i.e., wave of equation: ( operator’—often, in introductorythe lectures, sense that it ( is even constructed in that way—in off-diagonal Majorana mass term whichequation. is often introduced This in is theElko context so of because the Dirac of the observ ( po for of be brou symbol we take. Since its roots are always i 2. The off-diagonal nature of the mass term in ( 3. The Dirac operator can be considered as a ‘square root of the Klein–Gordon where Journal of Cosmology and Astroparticle Physics 5.2. When wave operators and spinWe sums draw do not our coincide: a rea pivotal observation spin sum over self-conjugate spinors reads The ma and the one for the anti-self-conjugate spinors is given by Before attending totogether quantum some essential field new details theoretic about the structure of the theory we need to collect at the representation space level. transforming components of the momentum vector is given by, completeness relation ( next section. For the moment, note may be JCAP07(2005)012 ]. K 22 72 , ], (5.22) (5.21) (5.23) (5.24) O 60 ). This ) , 84 , 59 —where , 72 5.15 ( , K 20 60 ). The meaning , and other; and that it 59 ) , 5.19 32 5.14 trix times ( )and( it becomes identical, up to 2ma )withthe properties of the spinor. It is p 5.18 × ( T Elko ). Such an operator is required λ Spin-half fermions with mass dimension one 5.15 ehaviours, the reader may recall that ,and stacks.iop.org/JCAP/2005/i=07/a=012 P ]withcorrectionsnotedin[ , additional operator which annihilates the lity and statistics structure of the theory. ators for the Dirac spinors. C 32 , , ). For the I I )and( I m m m + − 5.14 pin sums and wave operators: introducing ± ve oper 2) spinor, µ µ . 07 (2005) 012 ( µ / p ) p p 1 µ µ 0 µ , γ ( . γ γ encodes the 2) (0 / 1 rthermore, such an operator is expected to shed light on ) ) , )= A )= ⊕ p p (0 p p ( ( ( ( χ 0) h 0) 2) h , , / trix u 2 u 2 1 A / , ) ) / (1 (0 p p ( ( ntum–space wave operators in equations χ χ h h )= 2ma u v 0 )singly.Fu hat is one of the pivotal observations for the theory. It affects the ( × 2) 2) p 0) / / ( , 2 (1 (1 )= λ / ± ± p (1 ( = = χ h h ξ different from the ones in ( The right-hand sides of the spin sums are not proportional, or unitarily but equally well to the Dirac framework. The method is a generalization of the lize the importance of these contrasting b ant multiplicative factor, to the relevant spin sums. Our walk in search of this Elko troduce a general (1 To rea We now make w This section is devoted to establishing the existence of such operators, and to reveal )andis these statements will become more clear as we proceed. p tire particle interpretation, the realized statistics, the propagator, and the locality ( tructure contrasts sharply with the Dirac case where the spin sum over the particle aconst operator is leisurely, andthese we operators do may not carry. refrain In from a stopping particle’s to rest look frame, at by other definition aspects [ which correspond to the momentum–space wa spin sums enter atSo, a with these profound observation level intheir mind, in it relation the is to important loca to waveinterpretation. decipher operators. the origins This of we spin sums do and next on our way to developing the particle and the one for the antiparticle spinors left unspecified at theform moment may except be that written, we require if it required, to as be a invertible. general Its invertible most 2 general en structure: connected, to the mome Journal of Cosmology and Astroparticle Physics The question which is now posed is: is there an 5.3. Non-trivial connection between the s λ Our task is tothat obtain this the operator operator(s) is defined nothing above. but For ( the Dirac case, it will be found Here, the complex 2 s spinors to have the property that it does not couple one of the annihilates these the structure of the spin sums which appear in equations ( textbook procedure to obtain ‘wave operators’ [ their origin and associated properties.to the We present a unified method which applies not only of We in JCAP07(2005)012 23 .31) (5.34) (5.30) (5.33) (5.27) (5.25) (5.26) (5.29) (5.28) (5.32) ) ecified the )aresp 0 ( 2) / 1 , ), and re-arranging, gives (0 χ 5.25 tor we are searching for. We )and Spin-half fermions with mass dimension one 0 ( )resultsin . 0) ) , . 2 stacks.iop.org/JCAP/2005/i=07/a=012 p ) / ( p 5.26 (1 ( 0) , space opera χ 2 . 2) lds / )inequation( / 1 (1 , , . χ (0 ) ) ation ( )yie 5.29 )=0 χ 0 0 )=0; (5 1 ( ( , p , − p ) 1 ( 0) 2) 07 (2005) 012 ( 1 ( , 0 − / 2 2) 5.25 − ( 1 2) 0) / , / , / . 1 2 0) 1 2) , (1 (0 ined into a matrix form, result in , , / 2) / 2 )inequ (0 / χ χ 1 / (0 (1 , , 1 , χ κ (1 χ tion ( (0 0) 2) , (0 )=0 / χ κ 2 5.28 1 κ I 1 1 D − p / , ( − − D (1 (0 − ξ ) llow from κ κ A A A p A ( 1 )+ )fromequation( 0) I I , )fo 0) − 0 p , 2 )= )= D )= )= )= ( − p 2 − ( / se of equa D / ( 0 0 p p 0 0) (1 0) ( ( ( ( ( , , lies (1 2) 2 κ 2 1 / 2) 0) 0) 2) 2) / χ / I , , 1 / / / − , 2 2 1 1 1 (1 (1 1 , / / , , − := := (0 D χ − χ (0 (1 (1 (0 (0 χ )imp O χ χ − χ χ D χ D jugates a spinor to its right. Once 5.24 )and rator p ( tuting for 0) , 2 / (1 which, as wenow will study soon its see, various properties. is the momentum– substi Journal of Cosmology and Astroparticle Physics complex con χ The ope which on immediate u Equation ( With the useful definition while similar use of equation ( Similarly The last two equations, when comb JCAP07(2005)012 ). ϕ 24 will 5.38 (5.36) (5.40) (5.39) (5.38) (5.42) (5.35) (5.41) (5.37) is due in the ), and ) p µ Elko p − E, )results equation ( =( ]. ), along with the 8 µ 5.40 )theform p 5.34 —see I 5.37 can be read off from the As a result, for the Dirac = A )and( 2 ) p · )and( part of this result for 5.39 of these operators in ), recalling σ , Spin-half fermions with mass dimension one . .The 5.36 A ) ) 5.37 )inequation( stacks.iop.org/JCAP/2005/i=07/a=012 ϕ ϕ ations ( · · linearity I 5.35 I σ σ − , . lds The counter ) ]; and that (b) in a quantum field theoretic ) p p )and( ( ( 72 v using the definition of the boost parameter u exp ( exp ( , ,yie O 8 nors O nors. . , . encodes the spin sums. 2) 5.36 / ) ) p ) )withequ ) 07 (2005) 012 ( m 1 m I I , · O ϕ ϕ ,and − (0 )spi )spi · m · m I σ κ p 5.22 ), gives equations ( p m I ( ( σ σ − + he well known momentum–space wave operators for the = ± I v u − µ − µ − )= 2.7 I )= + 2 γ γ ) and p p E µ µ ( ( for for p p h p 0) h roperty of Pauli matrices, ( · , ( ( )and( exp ( exp ( ycovariance[ u v 2 ) / ) )= σ 1 1 m m p (1 p ϕ ( ( , , not only determines the wave operator but it also determines the κ 5.21 ), the exponentials that appear in the above equations take the ection. − − h I I h · u v ,thesearet spinors + − O σ transparent that 2.4 =+ =+ 2) 2) = = / / t ± ) ) /m ) ) (1 (1 p p p p ,andthep ( ( ( ( = ± ± u v u v as in equations ( enexts A = = nts: (a) parit O O O h h A O exp ( µ nors quations ( γ for Dirac O tspi tically, we remind the reader that the writing down of the Dirac rest spinors, Using information contained in equation ( Up to a factorrepresentation of 1 space under consideration. The Using these expansions in equations ( This result makes i Journal of Cosmology and Astroparticle Physics Dirac res 5.4. The The Dirac representation space is specified by giving as shown bytwo Weinberg and requireme also by our independent studies, follows from the following Parenthe explicit expressions for framework (with locality imposed), the Dirac field describes fermions [ Comparing e given in equations ( form to the form of introducing and Majorana fields structure of the Feynman–Dyson propagator. be proved in th following spin sum: Exploiting the fact that ( JCAP07(2005)012 ) A 25 p ( λ (5.44) (5.45) (5.50) (5.49) (5.43) (5.46) (5.47) (5.48) (5.51) ) .Useofthe Elko ent that the for D ng use of the information in , ). Consequently, . S reads A 12.5 m p .Therequirem A Spin-half fermions with mass dimension one · , A + m p σ E · + stacks.iop.org/JCAP/2005/i=07/a=012 Elko σ + E I (anti-) self-conjugate case. Since the )andbymaki ugation operator completely determines , − S S I , 3.10 A ±A is given by in the explicit form of ) Dirac representation space gives . = 07 (2005) 012 ( , m m φ A p p i lf-conjugate S , lightly non-trivial but can be extracted from explicit · · )and( + + S A I − S σ σ I 2 E E A − A 3.9 p ) is specified by giving Elko )iss is Hermitian and a square root of the unity matrix, i.e., + + m − , S I S I I S I 2 + vanishes, we have A ) 5.43 − A A Elko )0 ∗ we must obta } m E 0 φ S , . λ m m ( 2 S + − O m m A 0exp( p β, m + + , 2 2 2 0 λ E p .Fortheanti-se Θ = = −A E ( E =+ exp (i 1 · λ in equations ( 2 ) ) − ± ± ± ζ ) p p ) ( σ = ( = p S S A { = = = = · = S A λ λ A .Notethat φ ). The result ( A D O O D β D A A σ i )given for Elko =( 0 † 5.20 ( ) O := ie λ S is the angle defined by the 4-momentum—cf the end of the paragraph above A λ φ In order to obtain =( dual-helicity eigenspinors of the charge conj S and forms of equation ( appendix A.1. TheExplicitly, given for representation the accounts self-conjugate for the complex conjugation involved. where where the last identity is due to the dispersion relation ( where Journal of Cosmology and Astroparticle Physics 5.5. The Similarly to the above, the A general procedure implemented for the where the plus (minus) sign refers to the anticommutator But because ( to be be JCAP07(2005)012 ). 26 p and − )has θ ( (5.57) (5.54) (5.52) (5.53) (5.58) (5.56) (5.59) (5.55) (5.60) )with to zero ) 5.56 −G − 5.19 P + )= P p )and( ( encodes the spin G )and( implies the identity determine the wave O ndependent of 5.55 G ei If one does not employ 5.18 i.e., to , p does not − ,andar O φ , to , ) ) Spin-half fermions with mass dimension one p tting the product p p ( ermore, it is observed that although ( A α S α ¬ λ stacks.iop.org/JCAP/2005/i=07/a=012 ¬ λ endent matrix ) ) p p ( ( . A α . , S α , ) -dep ) ) λ ) λ φ p p ( ( G depend only on S α A α −G λ λ + I ecall that arriving at results ( in going from + O I ( 07 (2005) 012 ( O ( m m 1 Elko 1 G er m m 2 2 m m ), which is nothing but the Klein–Gordon operator in )itfollowsthatse − − − . the structure of the Feynman–Dyson propagator. The ) . 12.5 5.48 = =+ π lds nd, the comparison of equations ( )= )= + ) ) )= + )= , p p + p p ) S p p ( ( ( ( ( ( S O A φ A λ )yie λ ( A α A α S α S α e, it is transparent that in this case as well ±G ¬ ¬ O λ λ ¬ ¬ λ λ O ,from( S I 1 2 )inmi ) ) 1 2 ) ) −G ( 5.51 O A p p 1 2 p p − − ( ( ( ( S α A α S α A α 5.46 := = λ λ λ λ )= = φ still determines := S A ± ( )and( α α α α the spin sums read G G P P P convenience we introduce the tter is considered in appendix B.6. Furth G 5.50 the dispersion relation ( rresponds to the behaviour of For later .Thisma p eeping relation ( of the spin sums do‘square roots not of coincide the with Klein–Gordonany the operator’ dispersion in wave the operator relation following they sense. can be still be considered as This co It is curious that the spin sums for the form a complete set of projection operators. Second, the definition of Some remarks are in order. First, it should be noted that the operators Journal of Cosmology and Astroparticle Physics K In terms of implies momentum space. More explicitly,already w invoked the dispersion relation. One may, going a step backward, define Thus, as in thesums. Dirac As cas a result, for the two new fields based upon Elko, operator but it part of the assertion which refers to the propagator will be proved below. equations ( JCAP07(2005)012 ), 27 is (6.6) (6.4) (6.3) (6.1) (6.5) (6.2) 5.49 ) Elko .From identical rated over m A which is often , P . | µ ) , x x and µ ( p † i µ S η ). This requirement x − ) µ P R p x )e ,wheninteg ( p +i η x ∈ ( ive-energy self-conjugate → )e γ x | A β p lies the dispersion relation. ¬ λ ( Q A β ) λ ), makes p ) 1posit ( ac case. β p ), with Spin-half fermions with mass dimension one is the corresponding probability. ( c γ 5.49 † β , =0imp . c + A A µ the second equality in equation ( ββ ∗ + stacks.iop.org/JCAP/2005/i=07/a=012 18 x nguish it from δ =0 P lds the amplitude for the particle to be µ µ A is ) S p x  } lf-conjugate particle of mass  µ ) P +i ]fortheDir he quantum field associated with p  tational structure for such amplitudes: the p x i )—the only difference being that now no ,yie p 85 )e − − ( p to A )e β ( p 5.48 . ( p x S β ( 3 ,c ¬ λ S β δ ) nthatt represents the physical vacuum. Therefore, the 07 (2005) 012 ( litude, to disti | λ ) 3 p ) ) ) ( p x p β ( π ( ( c | † β β ¬ η { c nt amp c )  fixed by the requirement that =(2 = x β ( β dual given by   η ) ) 19   ) ) (or, to be more precise, exp(i p p ,where p . ( ( | p ( )is

the covaria ( Elko † β † β )  | x ,shallbe 1 ) 6.5 ( 1 ,c ,c = | mE x ectively. This exercise may, in addition, also be viewed as a simple unity C s ) ) ) mE | ( ecipher that the state which contains 2 x ) p p x 2 are now written without invoking (  ∈ contains 1 positive-energy se ( ( ¬ η ) x → β † β ¬ η (

p x c c  ( 3 ) ) s ,resp 3 ) is A p Q    x ) , p x − 3 )wed π S ( ( 3 π λ d x s η P d | (2 (2 6.2 O le interpretation and mass dimensionality actor, | and integrated over all spacetime, call it cetime, yields tate the corresponding + x Invoking the dispersion relation, as in equation ( )= )= P → x We call This argument follows closely Hatfield’s discussion in [ x x lf-conjugate particle to propagate from ( ( ¬ The pre-f η covariant amplitude ( found anywhere in the universe; consequently, 19 18 6.1. The Elko propagator Once these fieldsse and anticommutators are given, the amplitude for a positive-energy η where the The s particle at dispersion relation is invoked explicitly. Then, but instead are defined using equation ( all spa is imposed by the quantum-mechanicalQ interpre 6. Partic consistency check. It will be established in the next sectio Journal of Cosmology and Astroparticle Physics referred to as an amplitude. to with equation ( Here JCAP07(2005)012 . , ,  28 x 20 x Elko → (6.9) (6.7) (6.8) x at article by one Q )  creation; m x reserved under as its (open) sp x ti is reabsorbed into .I )andthep . R )  ebeforeits ∈ t 6.1 α − ); rticles; so the covariant t x ( ( θ λ follows from the fact that the ,whereit  )]  x x | ( ) α  x Spin-half fermions with mass dimension one ( η exp[i ) lling destroys the time ordering of events. See gative-energy self-conjugate particles x → ( stacks.iop.org/JCAP/2005/i=07/a=012 physically equivalent process that we ) x this may be recast as ¬ η ( .Fromequation( appropriate Green function provides the .Ne λ  x T i-self-conjugate particle of mass x | er, the integration over the entire spacetime ).  nd propagates to Such a tunne x anti-self-conjugate pa ( . − cetime, carrying the lightcone of λ ,a x ] one cannot destroy a particl ) with an 5 x 07 (2005) 012 ( t ince fermionic amplitudes are antisymmetric under γ ) 21 x x − . ( espa .S then the process above lowers this charge by one unit  →  α t ng. The pseudo-scalar nature of x ( | 22 θ . Q )creates ishi )) 0) exp[i x t>t ( ed by the requirement that it coincides with the appropriate x | | ( η → ) )  ) ¬ η x destroy it, then we are also raising the new charge at > x x ( ( ( )  ,( λ η ¬ η x  ) ( .Ifwecreateanant ) he total covariant amplitude for the process under consideration is  x  − η ( x ( ,t (  ¬ η η x ,wherewe | | |T ↔ . x    rticle this interpretation is free from interpretational problems. For the x hich is pseudo-scalar under the operation of parity, is distinct from the Dirac charge—irrespective >t  ,w t ge ollowing this first-principle derivation so as to avoid using full quantum field theoretic formalism which nman–Dyson propagator is then nothing but a numerical constant times ar so take into account. If we consider positive-energy self-conjugate particles to gauge transformation ey ch important to make this reference to an inertial frame because quantum mechanically a particle has a finite and subsequently raises it by one unit at is As in the Dirac case, there is a distinct but quantum field. their self-/anti-self-conjugacy under the transformation eF x undary, as the classically forbidden region remains accessible quantum mechanically We are f It is Th se must al carry negative charge implicitly contains the assumptionElko of locality. The latter property, asprobability we of shall ‘tunnelling’ see, beyond is the not light cone fully of respected by the also footnote 12. 22 21 20 interacting case, the relation of For a free pa interpretation, we seeamplitude that for this process is of the fact whether it is zero, or non-van lo Journal of Cosmology and Astroparticle Physics the exchange propagating backwards in time, thatthe is, positive-energy opposite anti-self-conjugate particles, charge, carry Once again, wephysically meaningful cannot only for destroy a particle before we create it; hence this process is therefore, at additional interpretational structure. Moreov is required, and is not limited to th the local the vacuum. In a given inertial frame unit and lowering it by the same amount at Invoking the fermionic time-ordering operator Th where the constant is determin Green function. The quantum-mechanicalthe propagation one of in the caseparticle self-conjugate of is particle, the created like Dirac out of particle, the is vacuum the at process where a positive-energy self-conjugate transport it to bo JCAP07(2005)012 29 en in (6.15) (6.13) (6.14) (6.12) (6.10) (6.11) ) ), may be on of the )giv p 6.3 ( A λ . ) . t )  eexpansi t − . uation ( µ  ) t t − x ) ), we get ( )and ) µ µ t θ x p − x p ( ( −  θ − +i t ) S µ 5.58 x ( µ ( λ )e x · x θ  (     ) p − µ p µ p µ ,usingeq ) ( µ x i µ x (with the already made implicit x )+i − − . A β ( Spin-half fermions with mass dimension one x t µ µ µ p | λ using ( β − − )e ) p x ) ) µ t  ( x +i β p p +i µ x )( δ ( − p ( ( p )e ) p stacks.iop.org/JCAP/2005/i=07/a=012 i ( µ )e x ( † β . S β ( p − p p c † β i · E p ¬ λ ( ,and i ) ) c p ( −  )e . µ p − ) − A β x p p A β  + p ) ( ¬ λ ( ) )+ i − ( ¬ µ  λ )) e t p µ µ )) e β x ) S β ( S β es from the terms in th p φ − ) x c x 3 φ − →− ( λ ( p ¬ λ t ( µ ( p δ µ ) ( µ p G )( ( 3 t i ) β p p G x ) p | 1 ( A β − c p ( 07 (2005) 012 ( term by term. The first term may be written as µ + +i π − ( mE λ + E p )e  S β I +  2 )e I t +i +i λ ( β µ p )( p | )( ( x t θ ( +(2 t µ identically vanishes, this then yields the first term in β )) S β 3 )) e )) e ) p  A β − ) x λ − φ p φ p  ¬ λ +i ( β ) ( π ) ( 3  ( t |  t ) ( ) d | p G ¬ )e 1 η  ( p (2 θ ) ( ) p (  p θ ) p (  1 + ( −G mE p ( p ): β ) A β ( x 2 ) ( I I c S β β ( p mE λ 1 ) p c ¬ β ( λ )( ) η )( 3 2 1 (  )   c p 1 ( ) p t 5.56 t t ) E mE ) ( β p 3 3 E π to be 2 2 ( p p ) 2 d p − − − ( † β ( 3 (2 π 1 3 3 3 † β t c t t † β d ) ) ) ( c |T mE ( ( p p p β c | (2 3 3 3 θ θ π π π θ ). Under the integral, 2 )) d d d )and( p (2 (2 (2 | x ( + + − × × − ( † β |− 3 c ¬ ) η p 5.55 )  3 π )     p d ( (2 x ( rther simplification we invoke the spin sums for = = = β ilar evaluation of the second term gives η c ( x x x     → → → x x x Journal of Cosmology and Astroparticle Physics We now evaluate For fu  Next focus on the second term. Letting Q Combining both of these evaluations leads to the result replaced by assumption that thethe vacuum fact state that is normalized to unity). Taking further note of |T The only non-vanishingform contribution com Q equations ( Q Asim JCAP07(2005)012 ) 30 ,x  x (6.21) (6.20) (6.22) (6.18) (6.17) (6.23) (6.19) (6.16) ( . over all ) . FD  S x endence in i ) → x x ) − − φ 0 Q -dep ( )below). x p ( φ · G p i + e )+ 6.29 . ) I p ) t  ( t i referred direction − )), while in the second )–( . . − , . t E ) p t R t  direction. In this special − ( ( i ))( − ω ∈ + i z   p 6.27 E t ω stent with the observation ( e ( − γ  i ω i E − i +i +i ω ω )) π − e . )to d ω 2 2 φ 2 ) − ω i ( ( i( ω φ 0 π m m .Thus,theonly ( I G − d − ), with p Spin-half fermions with mass dimension one 2 =1 orm by the use of an integral in such a manner that an integral φ 6.18 γ G − − + + − = φ 0  +e µ µ I + ) 0 p p ( → ) lim + stacks.iop.org/JCAP/2005/i=07/a=012 I  p p 0 x µ µ +i ( lies along the p p − → lim 2  .Thisisconsi E → ned by the requirement that x  x   ( )= variant f · m  x i i I ) ω − t p i µ  )= but not on − e x − or in the absence of a p : t x − ) ) , ) − µ t µ µ x 23 θ t µ p x x − − ( µ x  − − θ t π ω ( t p 07 (2005) 012 ( µ µ lds µ 2 d ( ))( p x x and i θ ( (  p ( − µ µ i p p p e which depends on E i i but on )yie ) ) − − ) − , reen function (see equations ( G ω p p e e x µ x ( ( i( 1 1 p 4 4 ly on e 6.15 ( → E E ) ) term: p p 4 x 2 2 4 4 π π δ ). This substitution alters ( 4 3 d d 4 )) ) ) (2 (2 ) p p p p  φ 4 3 4 ( π π uld demand the amplitude to be exp(i π i ∝Q ( d d is now understood. The covariant amplitude is directly related to d E ) ends on (2 (2 G (2 − first term: second + = = − ,x 0 + ω  onality constant is determi I x ω x x 4 + + ( ( )dep → ) 0 0 → → e, one sho x = π x depends not on x  → → lim lim FD and set the result to unity   0 −  Q S For the Q × (2 For the  x p x   x → i i ( x · → eriod vanishes. With this result at hand, the covariant amplitude reduces to − − − p Q x ω = = x x The → → To be more precis and write the result as ssible x x 2 over one p the whole integrand comes from where the proporti where the limit Now we change variables: in the first integral the the Feynman–Dyson propagator: Inserting these integrals into ( 23 If there is nofree to preferred choose direction, a and coordinate system since in we which are integrating over all momenta, we are 6.2. Mass dimension one: the Elko propagat representation of the Heaviside step function: The above equation can be cast into a co case, the Journal of Cosmology and Astroparticle Physics that we have no preferred spacetime point. For this reason we integrate coincides with the appropriate G integral Q po Q Because of the indicated limit on the above integrals, we can drop the terms of the order JCAP07(2005)012 31 (6.26) (6.30) (6.28) (6.29) (6.25) (6.27) (6.32) (6.24) (6.31) ) ected mass )withmass x ( η . , . 2), the exp /    1 , I +i +i +i field, which also transforms ]. For the circumstance at m 2 2 ), the mass dimension of the 2 8 I m m + I m . µ m − Elko − ) p − A, B Spin-half fermions with mass dimension one µ µ x µ µ 0) and (0 p p γ p , µ µ − µ 2 p  p p stacks.iop.org/JCAP/2005/i=07/a=012 2, but 1, lies in the non-locality of the / x / ( ) ) ), constitutes a set of coupled equations. 4 )reads ) µ µ δ . µ , x x imensional arguments—which appear after massive fields, combining his dimensional . x − − − mensionality of fields and their relation with 8 − µ µ 5.17 6.28 µ x x x ( ( )=0 )=0 ( µ µ µ )= p p p p i i p ( i ( x 07 (2005) 012 ( − − } ,x − e e  → + −} e , , x x 4 4 + ( 4 reads ) ) p S {− p S { Q ) p 4 4 π π 2 4 π d d 1 Elko d FD (2 (2 m mλ mλ (2 i . S 2. The reason that for the Elko I − / − I = )+i ) 2 lds = m p p  m )= ( ( ):= − } + . +i + −} ,x , , ]—are more robust than the arguments that precede it. The reason  ,x 2 0yie I I  2 8 = + x S {− S { x m ( µ m ( → λ λ x 2), the mass dimension is not 3 ∂ .Soforthefieldofthetypes(1 lf-conjugate µ µ − /  → µ =i p p B 1 x Elko Dirac FD FD ∂ µ µ  , ), or its equivalent equation ( i S Q γ S  γ + (0 limit A lein–Gordon field. It is this circumstance that endows the 5.13 ⊕ 12.1.9) of [ tter, the counterpart of equation ( e euristic understanding of this crucial result may be gained as follows. now fixed we have ), a property we establish in section 0) , x ore, the propagator for the new theory is ( 2  η / ef In the absence of a preferred direction, the Elko propagator is identical to that of Ah er us, it is again clear that the new theory is different from that based upon Dirac spinors. dimension one. General discussionpropagators on mass can di behand found, it for is to example, be in noted that section Weinberg’s purely 12.1 d of [ equation ( Th ascalarK This is valid for the Majorana as well as the Dirac field. lies in the assumption of locality. For local field is 1 + arguments with those followingthat his for monograph’s a equation massive (12.1.2), field one of immediately Lorentz transformation sees type ( Journal of Cosmology and Astroparticle Physics That is, and it satisfies Th For the la With Taking th dimensionality follows to be 3 as (1 field Equation ( The set for the se JCAP07(2005)012 . . . 32 ∓} ±} ±} , , , (6.33) (6.36) (6.34) (6.35) is also {± {∓ {∓ ) ±} , ], a fixed ). Thus, we {∓ 86 c p ( λ )=0,whichon p ( −} , + S { λ ) ecomes 2 fine the Lagrangian density ). This has led some authors m )b − xamined if a preferred direction 6.32 Spin-half fermions with mass dimension one 6.34 ν pared to the Dirac case is that . p the two four-component degrees of µ p stacks.iop.org/JCAP/2005/i=07/a=012 ν ]. However, the spacetime evolution )=0 γ )and( µ p 83 ( γ hat the Klein–Gordon propagation is an , fluid, or some other external field such couples . 6.31 )istakentode −} 82 , rt makes the Elko field to carry mass dimension , ,equation( + nnihilates the remaining three I S { ads lf-conjugate particles with the reversed helicity λ 6.36 . µν )a )=0 ) η I 07 (2005) 012 ( p )gives( p )=0 2 I ( =0,it ( 2 p m ( =2 −} −} m , , m A 6.31 equations ( − } / + + ]=0,re ν S { S − I S { yanti-se ν ν λ λ λ ν ,γ p ,p p µ field, and the Klein–Gordon operator is the kinetic operator µ µ I I µ µ p p 2 2 γ —see m p p µ i { } γ µ µν m m γ + η Elko ν , − − γ I I )= ν ν {− 2 p + p p ( creates a positive-energy hole with the reversed dual helicity ν µ µ } field the crucial difference as com γ p p + , µ and ±} µν µν , γ ndependence of the theory must be re-e creates a positive-energy self-conjugate particle with dual helicity S {− η η destroys a negative-energy self-conjugate particle with dual helicity λ {∓ ), but rather, unless Elko −} )i c ±} ±} , , p , p . ( ( + † {∓ A {∓ { G property of the / c c ∓} S α , λ the creator of positive-energ {± For the That is, The The The holes carry the interpretation of anti-self-conjugate particles. Thus The • • eedom, the ‘square root’single of the Klein–Gordon operator is not a valid wave operator for a Journal of Cosmology and Astroparticle Physics The second of these equations may be re-written as have fr Substitution of this in equation ( which is nothing but the Klein–Gordon equation in momentum space. exploiting the commutator [ Using the anticommutator Similarly, it is seen that ( considered above makes it abundantly clea intrinsic for this field. Inpath other integral words, a approach conflict unlessfor arises equation the between Elko ( the framework. canonicalone, This quantization rather and circumstance than the three halves. background, a thermal bath, a reference as ainterpretational magnetic/Thirring–Lense structure is field worth taking of note of, a explicitly. neutron star. In addition, the following to suggest an eight-component formalism [ exists. This may come about because of a cosmic preferred direction [ JCAP07(2005)012 33 (7.4) (7.3) (7.6) (7.7) (7.8) (7.1) (7.5) (7.9) (7.2) . ) µ llow from x µ µ p x µ +i llows from the )fo p i p − ( ))e .  ), or equivalently )fo † β c p ))e ( ) 7.2 , p x E ( 5.17 ( ), and to establish the η µ E i x ) p µ ( x )(+i − p  )and † β ( c )( +i p p ¬ η ( ( p )and( )e β ( 2 A β c p λ ( A β m ) Spin-half fermions with mass dimension one  ¬ λ A β 5.16 )and − p λ ) ( p ) ) p ( p † β x ( β stacks.iop.org/JCAP/2005/i=07/a=012 ) ( ( c β c x † β η c ( c + µ ) ∂ + µ ) ,η + p ) x ) µ µ ( µ µ x x p x p x ( i µ ( ( E µ p ¬ η a − p ¬ 2 η ), we note that equation ( i µ +i . x 3 − ∂ ))e ( ) p 07 (2005) 012 (  )e )is L η 3 ))e π p x , p d p x x ( ( ) ( (2 ( 4 4 )=0 η S β x . E d d x i ( E ) λ ( )  ) x ) − ¬ η η ( µ p p  . )( η ( (  p I )(+i ∂ β = = ∂t ( 2 E ∂ p p c ∂t a ( ( gate to 2 m −L ) ) as contained in equations ( = S β 3 S β  ) x x ) λ β + ¬ ( ( λ · p ill be fixed below. Our aim now is to settle the statistics, i.e., x η ) π L 3  ( ν ) ¬ η )satisfies ,η ∂ d ∂ ) p ∂ · (2 p η ) ) x ( µ ( p ( µ x ∂ ( ∂t † β ∂ β π mconju ( η p c c ( )= E .Itw ¬ µν η a = = x 2 η R ( tor I tive/anticommutative properties of the 3 π H S × × H eevolution ) p ∈ /anticommutative properties of the 3 π β,β H d ) pera the momenta conjugate to (2 x x µ 3 3 p ( d d ), allows us to introduce the field operator: a 6.36 )= = = x ( commutative On-shell it becomes The field momentu The commuta action Journal of Cosmology and Astroparticle Physics The spacetim 7. Energy of vacuum and establishing the fermionic statistics H considering the field energy: result with which theThe previous field section o opened. Towards this task we proceed as follows. where To obtain in ( η while the Hamiltonian density reads JCAP07(2005)012 ) µ 34 p ( . ) a (7.15) (7.18) (7.19) (7.12) (7.10) (7.14) (7.11) (7.17) (7.16) (7.13) p ( ) β c ) . . ) p ( p menting this ) ( † β c β p c 2 ( ) A β p β λ ( ) † β . p c ( .Imple 2 m ) ) A β H µ p β ¬ 2 λ ββ ( p δ † β ) ( ) ) 2 c  p ) p a ( p ( p † β 3 . ( ) c p E − β ) 3 π 3 Spin-half fermions with mass dimension one c p ) ) d p p ( (2 3 ( π p − β ( d 3 c . ) ssociated statistics must be fermionic: (2 δ E stacks.iop.org/JCAP/2005/i=07/a=012 ) ) p ( 1 2 p p + ( )+ β ( 1 c p E ) E ββ ( 2 2 2 δ p nthea )+ S β ( 3 ) ), resulting in λ p β is consistent with fermionic fields, we fix  † β ) ) ( p c p π ( p )= E ( µ p E 2 − 3 p ), formally 07 (2005) 012 ( S β 2 ( β d )(2 x ¬ p ccur because both the self-conjugate as well as the λ β µ ( · o β ) 3 p ( δ m p p β p ), we get 3 b x ( ) x, . ) 3 ) 3 β 3 µ µ

1 2 d 2 c π ,b d = , p d p ) ) ( ) ( 3.24 exp(i = p 2  p p to be ( ) ( ( a x  ) 1 2 3 † β m mb 3 3 2 3 ill yield a field energy which vanishes for a general configuration µ ) E p c E ) ) ) ) H d p ( p 2 1 π 1 µ π 3 0 π ( = † β ! ( p )w d results in (2 )and( ] (2 ( (2 3 p β irement 3 2 δ ( ,c m mb − ) 2 a ) will be dictated by the interpretation of H † β ) ) π − ) c p 3.23 p µ µ )= )= µ 2 ( 3 = p p µ (2 R 0 p = = β ( ( d 4 p / ( ( 0 c 2 2 ( 3 2 a a a H δ H  H a ∈ )and 3 ) integration gives (2 p ) ) p 0 ( 3 )=[1 µ ( π ors/anticommutators have been assumed. It is clear that commutative relations rs of 2 in each of the β d p p 3 c )bytherequ ( (2 ( δ µ b 3 p δ − ( een b = = That is, Journal of Cosmology and Astroparticle Physics H The spatial showing the first term in where Using the results ( after the usual normal ordering. For this reaso anticommutator on H While wecommutat have beenbetw careful about ordering of the various operators, no specific anti-self-conjugate parts of theorder to field obtain the contribute. zero-point energy With which this observation in mind, and in Since Thus, we have The facto and JCAP07(2005)012 ) 35 8.8 .21) (7.20) (7.22) (7.23) ) )shows amiltonian field for a 7.17 in the sense . p ,andanti-self- 3 µ , x d µ −} x p )tothe i vity of the H µ , 3 x − p + µ ( )d p )e { idea does not seem to work. 3 E , ac case. p +i } ( /h )e of equation ( + A β not touch the locality result ( p , ), the field operator and its ¬ λ ( H A β ver, this ) {− energy λ p ch do ) 7.14 ( urns out to be non-vanishing. Thus, it is β ]fortheDir p Spin-half fermions with mass dimension one ( c represents an energy assignment 87 † β . c + 0 he derivation of positi µ ββ H + stacks.iop.org/JCAP/2005/i=07/a=012 x tators do not vanish. δ =0 (7 µ µ , ) p x  } h µ al anticommutators required to state the ) +i )withitselft p  p ythesame i x p ration and remark on the massless limit. ( order. That is, while the field–momentum )e econd term in − = − ( η )isemployed.Howe p )e rder to obtain the standard fermionic harmonic oscillator β ( p π ( p 6.3 S β ( 3 no ,c .Notethatint ¬ λ S β δ ) } 07 (2005) 012 ( ) )i λ ) 3 p ) ) ( x p .Thes manticommu lies 2 ( p β ( π 7.21 ( c 24 † β ,η β to each unit-size phase cell (1 ) { c c x ( = =(2 η { idea of introducing extra terms whi β β   ) )   ) =1imp ) —contributes exactl p p p ( ( p ( ( † β † β −} 25 , 1 )withthe 1 ,c ,c + mE ) ) mE nics. The new zero-point energy, just as in the Dirac case, comes { . 2 p p , 2 7.21 p ). ( ( } p † β β ( c c + 3 † β 3 , ) p   ) p ,c have the result 3 π ) 3 π d p {− d (2 tablish the results with which the previous section opened. ( ), for each helicity of the self-conjugate and anti-self-conjugate degrees of (2 ssion of the new zero-point energy is similar to Zee’s in [ β c p ( lly, as we will see below, the anticommutator of ble to modify ( E but which cancel the terms in 1 2 We thus )= )= − x Our discu x Actua ( ( ¬ conceiva below, anticommutator exhibits thefield–field usual and form momentum–momentu expected of a24 local quantum25 field theory, the The emergent non-locality is at the second 8.1. Fundamental anticommutators for the Elko quantum field η Since in natural units algebra for and in the evaluationThus, of it the appears propagator below to only be ( reasonable to keep ( Journal of Cosmology and Astroparticle Physics freedom (hence the factor of 2), of statistical mecha with a minus sign, and is infinite These require imposing locality structure of the theory under conside In this section we present the three fundament 8. Locality structure We thus es of that each of the four degrees of freedom—self-conjugate: to obtain correct particle interpretation. Using equation ( η Elko dual become conjugate: given momentum JCAP07(2005)012 ) 36 )to 6.3 (8.2) (8.6) (8.5) (8.1) (8.3) (8.4) µ 8.5 x . .Next, ) ) µ t p i p ( = −  A t / ))e .  S β ticommutator ¬ λ p . ( ) ) p E µ p i ( x − ( µ − A )to p / i .Thismakes( S β A β . )( −  p 8.3 λ ¬ λ µ p x − ) ( µ = µ p p x † A β µ equal-time an to µ +i − p . ¬ λ x ( ) µ µ )e p +i ) A β µ p  x p e p λ x ( µ . p ( † +i Spin-half fermions with mass dimension one p µ ( p A − i A β )e ) +i β / ) ) ¬ − λ µ µ S β c p p ) µ x ( (  p ¬ x λ ,t ) µ x stacks.iop.org/JCAP/2005/i=07/a=012 µ  ( µ A β + A β x p p µ p ) p i x ( µ p λ λ ( S β µ ( p p − ) i +i ¬ A β λ ( E +i x e i ¬ p η ) − µ e λ A ) (  p / p ))e  ) † β integration reduces ( p  S β (  ∂ ) − +i ))e c ( ∂t ββ    † β λ p p S β µ c ( , ( p p p We begin with the ) ) + λ ) ))e (  x  (  E † β µ + µ p p β ,t p E p = x 07 (2005) 012 ( ( ( i µ ,c ( µ x +i β † ,c x ) p (+i ( − i ) µ µ E p 1 1 ( η p x † − i ) ) p ( µ mE mE ( † β − p p x )e i † β c 2 2 e ) ( −  ming the c − p )(+i † = ) A  x  (  p ( / )  · )e S β  ( p } S β ) ) p  p ( ) ( λ p i 3 3 p λ   p ) ( p ( S β ) ) e ( p p ,t ( S β  ¬ A β p λ π π S β 3 3 S β † 3 ( ¬ S β λ ¬ x λ d d λ ) ¬ λ p A β β (2 (2 ( ¬ λ ) 3 π c ) ¬  λ ) )  ) ) d ) p p p )  ,π (2 p p p ( ( ( ) ( ( p )andperfor ( p ) ) S β ( † β ( β S β A † β ,t λ p p c c / A β c λ β ( ( ββ x 6.4 S β λ c ( ¬ m λ i η β 1 1 2 − + { × × + × × mE mE 2 2 m i )and( 2 ), the only terms which contribute are 6.3 3 3 3 ) ) ) p p p 6.4 3 3 3 π π π d d d (2 (2 (2 Journal of Cosmology and Astroparticle Physics 8.1.1. Field–momentum anticommutator. Due to the anticommutatorsand imposed ( by the particle interpretation, i.e., equations ( Now first note that we are calculating an equal-time anticommutator. So, set The anticommutator on the right-hand side of the above equation expands to Here, we have used the fact that in the second terms change the integration variable from be Using ( JCAP07(2005)012 37 . (8.8) (8.9) (8.7) (8.10) (8.12) (8.14) (8.13) (8.11) T ecomes ) . b ) p p ) . p herefore, we − ( 5 − A β γ ( 2 λ A β γ 2 ¬ ) λ p ) ) m  p − By inspection of the p ( + µ ( − )to S β 2 x ( , λ µ p ceases to be valid in the A β 1 p 8.6 λ µ ) mE + +i od vanishes. T x anishes identically, giving µ 2 µ T x p + 8.8 µ )+ inner product with ) 1 p . +i i p ) µ p γ ( 3 −  Spin-half fermions with mass dimension one ( 0 φ x ) e p ( S β A β µ π 3 p T ¬ θγ λ G λ i d ) (2 − ) ) stacks.iop.org/JCAP/2005/i=07/a=012  x e ) p sin − p ( p T eequationv x ( p ( S β ( ) · S β ) A β λ )overoneperi p ) p λ . i p λ θ ( φ e ( ) (  A β 3 ) )). This simplifies ( β G x ) cos( β λ p 1 φ p 3 π ( mE ( ) pr − i ) d S β ) G 2  x (2 07 (2005) 012 ( integration yields the expression -axis, and thus the λ x x − p z ( )e + − x (  φ . ( 3 θ x ) · I (  S β δ · 3 ( p . i ) p p λ p i e ( =0 3 m +i π sin( e  ) =i d † β ) ) =0 )  (2 θ p x p } ) p 1 ( d } 1 ( ) ,c ( − ) ,t ) π † β x E  ming the E ,t ( 0  ,t · p x  3 3 ( p = x ( ,c i ) ) x p p β ( ) e 2 ( 3 3  c π π ¬ η ow show, d d 3 p m , ile an integration of ,η ) ,π (2 (2 p ( )  ) ) 3 π 2 + β }| }| ,t d ,t ,t p c ) ) 2 (2 x ββ x x ,wh  p (  ,t ,t ( (   φ m m ) η 1 1 η η )andperfor x x to be aligned with the 2 2 { + i { ×  ,t ( (   p we shall n x x = = d ,η ,η ( ) ) int it is emphasized that the ‘standard result’ ( B.20 ∞ ¬ − η ,t ,t 0 , x x x ) ( ( 2 η η ,t x 1 ( mπ η 4 Journal of Cosmology and Astroparticle Physics 8.1.2. Field–field, and momentum–momentum, anticommutators. The indicated spin sum equals 2 However, as |{ which exhibits only anticommutators and thus yields obtain At this po presence of a preferred direction, in general. The spin sum on the right-hand side of the abov Inserting ( |{ It turns out thatof not creation only and is annihilation theits result operators, vacuum non-vanishing, and expectation but also it value: on depends commutators. on We, anticommutators therefore, evaluate above calculation one readily obtains The second integral vanishes becausechoose in the absence of a preferred direction we are free to independent of JCAP07(2005)012 38 (8.21) (8.24) (8.15) (8.22) (8.17) (8.16) (8.19) (8.20) (8.23) (8.18) ) . 5 . γ 2 γ ) #$ 2 5 r γ ( 2 δ gular part just gives γ 2 ) 1 pr mπr 2 sin( .Thean  r, + x d 1 pr ) γ | − 2 π Spin-half fermions with mass dimension one r 0 r ( x | γ , δ + ) 0 1 mr , ∞ stacks.iop.org/JCAP/2005/i=07/a=012 ) γ − −∞ 0 e mr , , γ 3 ( ) mr ) 4 1 2 O mr ) pr pr ) xponential tail does contribute to the finite ]. In order to interpret the result it is useful ( mπr pr 1 (1 + )= pr )+ mr 88 1+ pr 4 2 0 ( r mr πJ 1 stributional sense, cf appendix B.5. ( 2sin( , = +( − J 07 (2005) 012 ( mr ) 2 = 2 2 rδ ) = ) r ) d =e ( θ ln ( ) 3 θ )—see [ pr pr x , ∞ ( }| ( πδ − cos( d −∞ ) cos( , γ 1 2 = ,t 2.12 pr )  1 2 1 i − Θ O mπr pr m p rated over all separations x i ) 4 ( ) = mr x )e = )d ( )e θ ,η Θ O )= = 1 pr ( θ ) 2 2 +( mr J pr mr r r 2 d( ,t − 2 ( − x e x e x sin sin( = ∞ ( sin( r r rδ 0 }| η 5 θ θ d d d p ) γ d d " 2 ,t ∞ ∞ ∞ ∞ ∞ π π 0  γ ninequation( 0 r 0 0 0 0 0 in the measure. The relevant radial integrals are integration the following two integrals have to be evaluated: x i 1 × |{ ( 2 p ). Note that we had to regularize one of the integrals. Thus, in order to obtain ,η πr ) integration the following two integrals are needed: is the Bessel function of the first kind. This implies is the Euler–Mascheroni constant. Obviously, the main (singular) contribution 8.8 ,t θ final 1 x γ J ( η actually is related to Wigner’s time reversal operator: Therefore, the final result is established: 5 γ 2 where Θ is give where comes from the coincidence limit, but the e part.locality This ( singular behaviour is to be contrasted with the regularity of ordinary Journal of Cosmology and Astroparticle Physics For the |{ For the where where the last result is only true in a di to consider non-locality integ The first term hasγ a Yukawa-like behaviour while the second one is localized. Note that the usual 4 and JCAP07(2005)012 ) ), 39 ,t local after 8.25 x limit, (8.26) (8.25) ( π ) m onjugate &    he quantity ,dropout 0 ) r )) with itself that x ,t ].  ( )andthec ding to some extent. x π 90 x .Thequantity( ( ( η ¬ m η . 0 )(or ∂ γ ∂t 1 x , ( γ ) ,weconsidert η 0 1 m ,t r x . Spin-half fermions with mass dimension one ay be mislea ( = he behaviour expected from a 26 ¬ η ect consequence of the non-triviality he mass dimensionality of the field. & stacks.iop.org/JCAP/2005/i=07/a=012 ∂ ∂t nt    ), and the associated momentum }    ei ontained in the underlying representation ) er to the question is non-trivial is already on-locality and containing a distributional ,t ous case. Again it is found that the vacuum yisadir ,t disagrees with this expected wisdom is quite tc x )displayst  % ( um fields, independent of spin, enjoy a smooth x η ( = 8.8 non-locality carries its signatures in the physical ,η & 07 (2005) 012 ( ) field ,t    )quant x Elko } ( ,j ) η { ,t (0 Elko     spin sums which underlie non-locality end up modifying the x ⊕ ). In the absence of a preferred direction, this modification ( 0) % ,π j, ) x Elko Elko ned in equation ( − re, the massless-limit decoupling of the right and left transforming ,t x and taking the derivative with respect to x events carrying spacelike separation. In the context of the physics ( ]all( at the representation space level m m π m { 89    dded to the manuscript as an answer to a question by Yong Liu [ Elko m heory. It is only in the anticommutators of )asobtai Elko d .Therefo is singular. Specifically, this can be seen from the anticommutators studied d esult which is independent from the cutoff x % ( was a π inberg [ 8.1.2 by the fact that Elko 8.2 spin sums. the non-local anticommutators suggest that there may be non-vanishing physical iplication with The evaluation of It is also our duty to bring to our reader’s attention that according to a pioneering The question now arises if the Elko Section non-locality becomes negligible. 26 It is worthwhile recalling that the field anticommutator between 8.3. Signatures of Elko non-locality in the physical amplitudesmomentum and cross-sections in section It is clear frombased the on above discussion that the massless limit of the quantum field theory 8.2. Massless limit and non-locality quantum field t aconvergentr Journal of Cosmology and Astroparticle Physics can be performed inexpectation full value analogy to is the non-trivial, previ exhibiting n part. It is non-trivial that both the divergent contribution and the regulator, mult may be used to estimate the sensitivity of non-locality to mass. In the large- components of hinted at amplitudes and cross-sections. That the answ fermionic propagator (of to the amplitudenot for as propagation a non-local from propagation one but spacetime as point a chang to another exhibits itself correlations for Yet, work of We space alone. This is due totogether the carry fact additional that information the to tha non-locality emerges. Technically, non-localit of non-singular massless limit. Thatsimple: our Weinberg’s result analysis explicitly assumes locality JCAP07(2005)012 40 30) tions (8.28) (8.32) (8.27) (8.31) (8.29) ) ar. But this )itmaybe n )therela 3.24 unusual factors )) acquire an unwanted p ( 7.22 E 2and2disappe / )would .The uation ( +2 0 t 8.8 0 H ors: for an ordinary Klein– )( -matrix framework, may have p one may take for example the )andeq ( S † α Spin-half fermions with mass dimension one into free and interaction parts, pressions on the right-hand side. c int hat the factors 1 3.23 , H reliminary investigation of the effect 2 H / , replaced by )= stacks.iop.org/JCAP/2005/i=07/a=012 ) unitarity holds also for the interaction 0 p ap t t ( ; 2 − )sucht 0 † α t / pens with fact )(8. ( c ) 0 0 H ˜ t n ). For ss-sections. We find that non-locality affects ( . H 7.14 ) )intheex i nonical anticommutator ( 2 0 = t, t / − H p int ( ) in ( ( )with † H 0 0 7.11 )e t ( U a er space and with a relative minus sign, because E ) H x − ˜ tH . t 07 (2005) 012 ( , ( x d ) 7.22 0 ± t H x t t, i t 0 ( ( ( − , η η H e H 0 ) 2 i n han the Lagrangian density). One may now employ the 2 / ated ov η x ) / − ) and thus the ca ) ( )) 0 0 t 0 ecause from equation ( t π the free Heisenberg fields, p ¬ η − ( t − ( t t, t coincides with the free-field expression ( is given by ( ( )b 0 E ( = equation ( 0 and † exp 2 H 0 † i H H 0 η i U 9.3 )(integr η H − T ) nrathert plit the total Hamiltonian 0 x 9.3 ( )= )=e H )itisevidentthat η )=e )= . nents should be noted; they are needed, as will be shown below. )or( x x ) 0  )( x 27 t, t, ( p ( ( t, t t, t 9.2 )or( 7.11 ( ,where ( ( ¬ η H H 0 α c η U U η int 9.2 )isgivenby H x rs of 2 are a consequence of the presence of two Elko-modes: self-dual and 2): , tion ( / )= 0 + 2. t p / ( ( 0 address this possibility we here undertake η ) α H c 2intheexpo the time-evolution operator N een spacelike separated regions of the very early universe. n eed the Hamiltonia To this end we s To The full Heisenberg field may now be defined as Standard methods lead to the well known result (again with a somewhat unusual / ) = ∈ 0 Of course it is possible to change the definition of n H H of interactions on physical amplitudesthese, and cro and hencein we a establish non-trivial that manner.role it In for is particular, higher-order endowed correlation it with functions. will observable be physical shown consequences that non-locality plays a decisive Journal of Cosmology and Astroparticle Physics significant bearing onbetw the horizon problem. The latter asks for causal thermal contact will change the normalizationfactor of of 1 ( ( 27 of very early universe this, if borne out by the relevant In order to verify that Hamiltonian in ( deduced that where factor of 1 are very helpful. AgainGordon something field unusual one hap obtains instead with From equa of 1 last term in ( we n free Hamiltonian to construct These facto anti-self-dual ones JCAP07(2005)012 41 ), the (8.39) (8.38) (8.36) (8.37) (8.34) (8.41) (8.35) (8.40) (8.33) )appear ) 5.55 5.56 . ), ( & 5.55    ) n sums ( ), as expected. x  ( , } H 0 t, t ) ( η , y ) U ( , n px i x px − )thespin i ( = )e ¬ η operators are on the right-hand † , )e p H 0 . ) ( Spin-half fermions with mass dimension one 3.24 . β )) p ¬ η 2 ) x 0 c ( / S β y ( ) A β ), ( ¬ 0 λ − ,t + t  ... λ x t η ( stacks.iop.org/JCAP/2005/i=07/a=012 † β − † β ) ( p { c 3.23 i t c 2 ( β β U − x 0 + I ( e I H )( i ) denotes time ordering. The time-evolution H 0 0 G − ) p η := ct ourselves to the bilinear case and address := e . ( ) T y normal-ordered product. We will not attempt , 2 t, t − + 2 ( olution is given by / ( − E − ¬ η ) x I β 2 ¬ . ¬ η η ( t U ) 3  ) − 07 (2005) 012 ( t ) H 0 p + ) ( x 3 ¬ π η ( p t, t H + d i ( , ( η ) (2 ¬ η ,η − 1 )and U px e  i x 1 = px ) 2 i ( − ¬ t mE η / N )=1,itss − ) ( t, t 2 H 0 0 )e ( 2 t = η )e = H int t, t orthonormality relations ( p ) − 28 U ( alogy to Wick’s theorem, to study the relation between the t rturbative quantum field theory one may express correlation p } ( 1 ( 3 H ( ) 0 x ) ) U , p A β S β y ( y H 3 ( π − ¬ i λ λ )= ( d  η β odinger equation H 0 raction Hamiltonian written as a Heisenberg operator with respect )= − (2 β ¬ c η  ¬ η ,t c ¬ η β + 0 ) β , res of the quartic case. The notation t I ) t, t )=e x I + (  ( ( x T η ( η U    := U := ) t, t := + = 0 ( ∂ β + + η ∂t operators on the left-hand side in each expression. By virtue of ( hy instead of the ¬ T η η { η I i % U t, t † β one obtains ( )istheinte c ˜ t 0 U ( denotes (fermionic) normal ordering, i.e., all H int ssential featu >y N H 0 the abbreviation Thus, as in standard pe The next step is, by an x As usual all anticommutators are to be understood as being matrix-valued in spinor-space, rather than scalar- ubsequently. Journal of Cosmology and Astroparticle Physics s For allows the decomposition with 28 where valued. That is w to the free fieldsoperator (‘interaction fulfils picture’) the and Schr¨ functions (with respect tofields the ground in state terms ofstate of the interacting of (a theory) the Taylor betweencorrelators series free interacting of of) theory) the correlation of form functions free (with Heisenberg respect fields. to the Therefore ground it is sufficient to consider Note that time-ordered product of correlators andto the do this in full generality but rather restri below e anticommutator evaluates to where With the initial condition side and all JCAP07(2005)012 42 ). ]). ) ], 5 +]). w ++], ( − (8.44) (8.46) (8.43) (8.42) (8.45) 8.13 − + ) + ¬ η , −−−− − ) } [ z ) ±±±± , − y ], [ x ( spin sum, the ( +] − p ) i + times the Elko- + ¬ − η w no longer will be i e , ( ) . ) − G T z + . . same w −−− ( ¬ [ η (  ) , − ) p + η . ( z ¬ +i η )) { ( } A β ) ) 2 φ ) − )) is sufficient ([+ ++] λ x ( y y η ( ( m ( ) G ) tractions + y + + − η p ( 8.43 −− + ¬ η ( [ ¬ η . µ I , , S β + ) p ( ) Spin-half fermions with mass dimension one } y ¬ η λ )+ µ ) x − +con ) ( p z lity one has to study higher-order x w ( ( β x ++] ( − nary contraction )(or( p ( i − ) η ) stacks.iop.org/JCAP/2005/i=07/a=012 e + + + µ ) { w ) y ¬ ,η η η -loca ( p ) − − 8.41 p + ( ) [ µ x ( ¬ η , y ( x −G ( ( ) E 1 ) + N )+ordi µ I z 2 mE y p η + i ( w ( 2 3 ¬ η ( η − ) p ¬ ) ). Incidentally, this is the η e )= ) 3 π −{ + y z ) 4 d w 3 07 (2005) 012 ( ( ¬ η ( ) ) (2 p ( x ) p − 4 ( ) ¬ π η w 3 π B.20 + η ( d η y d ) ) ( (2 ¬ η (2 where the latter procedure leads to x + x ( − + ( ¬ η η ¬ N η + ) lishes a standard result: time-ordering decomposes into + η y } = 0 = = ( ) part) − } N → z lim . }  ) components yields 16 terms which we denote by [ + ( ) ) y ¬ η − = y z ( )= ± ( ( ) lds )estab ,η )= − + x w ¬ ) η ) ( ( ¬ η )(prop x ) ), which will be recalled for convenience: w rk contrast to Dirac fermions. A simple calculation yields ,η , + ( ( x ) ) + x, y x 6.14 η )yie ( + ( ( ¬ η x x ) ¬ η + ( ( η η z x, y ) ) ( ( η + − z Elko z FD 5.56 − η η ( ( T S { { + −{ η ,insta − +], in order to make this peculiarity explicit and to establish the connection η Elko FD η ) analogous steps yield − ) S = y 0 y ( ( on with ( ]) and in two terms two contractions of that type appear ([+ ¬ η + ) −

¬ η -based framework via the η ) k erties, our first task is simply n x Elko ( Spin-half fermions with mass dimension one for d ) ...η stacks.iop.org/JCAP/2005/i=07/a=012 U ) ]). 1 8 ssociated canonical momentum x interactions or non-polynomial potentials ( k/M le terms are not strictly forbidden, in the he physical amplitudes and cross-sections ¬ η non-locality anticommutator ( ) referee. hing is a novel feature of Elko as compared to y. To examine t 1 x we must make. ( η cribes. We suggest the symbols 07 (2005) 012 ( othe JCAP )andthea inction from the Dirac framework. x ( N es is what interactions can η )des x = ( is the mass dimension of the interaction, and η like separated events. Such a calculation is far from trivial, ) d n appears to prohibit a naive application of power-counting x ‘non-local contractions’ between ( having studied its various prop ¬ η ) Elko lko with dark matter n x working assumption ( )and x ( ,andinto η ...η ) Elko FD 1 x S ( ¬ η ) Non-locality of 1 x limited number of terms appear in the action. Simple arguments of power counting ( Thus, Wick’s theorem in abbreviated form reads arguments. Nevertheless, we takecould them well as be good that starting non-localitytime point is anticommutator in a between higher-order the effect hope because, that after it all, the equal- Having addressed the relevance of non-local contributions for The next question which aris Equipped with the tools of Wick’s theorem (which may be derived in a standard With these results the primary task of this subsection has been established, i.e., we η the reasons for the success of the Standard Model stems from the fact that only a rturbation theory in the previous section, and to proceed further, we explicitly state an This section was added on suggestion from a T introduction of to name the particles the field 29 Journal of Cosmology and Astroparticle Physics low-energy scale), where 4 + for the Higgs.low-energy While regime these thatrole non-renormalizab we because are they able are to suppressed explore by experimentally factors they ( play essentially no very renormalizability prohibit for example 4-Fermi Standard Model fields.of Towards the goal of answering this question we note that one GUT or the Planck scale (cf e.g. section 12.3 in [ 9. Identification of E Having established the kinematic structure of an particles—with obvious symbolic dist fashion by complete induction) one mayfeatures now are derive the found Feynman in rules this for Elko; way, besides no unusual the remarkable behaviour encoded in the have shown that for an interacting theory t itself. That such contractionsthe are Dirac non-vanis case, and is directly linked t the Elko-propagator and the non-triviality of contractions of Elko with itself. carry non-trivial signatures of non-localit the cosmological horizon problemscattering amplitudes now for requires space calculationbut the of analysis correlations of between this section makes it, in principle, well defined propagator i where ‘contractions’ decompose into ‘ordinary contractions’ between pe observation and the JCAP07(2005)012 ) ς 44 9.1 self- (9.1) (9.4) (9.2) (9.3) ) coupling tral with Elko particles—as , 2 )isneu fields or two .This ) Elko x . x ( ( the Standard Model η counting arguments. 30 η ) Elko ) Elko dimensionless x ( 9.3 with ssible. The fact that ( ¬ η are ¬ ς E E α ,α and gauge field with associated field E )andpower- )arepo Spin-half fermions with mass dimension one ς λ x )+ tandard Model Lagrangian density ( x 9.3 η ( ion value. rpart of the quartic self-interaction his regard. η , stacks.iop.org/JCAP/2005/i=07/a=012 ) ) x x are distinguished by their charges—it is ( ( ss, and , LHC. Clearly, a thorough analysis of this photon and the latter has been severely a field is present mixings between them are ¬ η ) µν in sharp contrast to x hese two subsections involves a conventional ther Standard Model (SM) fermions explains 2 F 2 ( efortheS ) ) η m , x ) U ) ( Elko x − x η aquestionint ( ( 07 (2005) 012 ( ) ] ¬ η ν x int k/M ( ) esameexpectat L the Elko m η ,γ field. Of course, if more than one scalar field should be x µ µ ( ∂ γ m φ ) )+ ) )[ x x x ( Elko x ( ( ( † ¬ η ¬ φ η µ Elko E ∂ E L  λ uppressed as ( )= )= )= )= x ss x may carry a coupling to an Abelian ( x ( ]. The indicated smallness is not unexpected as (1) gauge transformations. Thus, the dominant interaction between x ima Vassilevich for raising ( Higgs doublet, ( ection with a conclusion that dawns on us unexpectedly: the fact that ¬ ς )i U 92 int ηF new Elko int φη , ), for example of the form 9.2 L L L L x 91 and ( ς µν )isthe has not been detected yet. However, since it does interact with the Higgs there F x ( φ almost do not interact with the matter content of the Standard Model makes them and particles of the Standard Model is expected to be via ( Elko ¬ ς While With this approximation made, the interactions of is local. So, non-locality manifests itself when at least two momenta appear together in th We end this s It is also worth noting that the SM-counte ssible. However, as there seems to be no way to distinguish different We are grateful to D trength is a possibility that it might be discovered at issue would be desirable. s contains no interactions with gaugewhy fields or o sensible to introduce only one present in Nature additional interactions of the form ( opposed to the Standard Model fermions which exercise on relic density, the second subsection studies the gravitational collapse of a In the next two subsections wethese strive candidate to constrain particles. the mass While and the the first relevant cross-section of of t 10. Constraining the Elko mass and the relevant cross-section prime dark matter candidates. 30 interaction could be of significant physical consequence, as we shall remark below. fields are entirely governed by mass dimension one of As a result the following additional structur po constants. Obviously, if more than one Journal of Cosmology and Astroparticle Physics contained in ( where and constrained [ the coupling constant has tothey be very lead small. to Such terms an affect photon effective-mass propagation because term for the comes to exist: respect to local Elko JCAP07(2005)012 . H ]. H 45 φ φ ge- H akly φ Elko 101 m , (10.1) → ς ¬ elar ∼ number ) 100 ς ndication σ X m ]forwe Elko ). More detailed 101 ge the comoving 10.1 ,samei etermines the ,the unity. ς scenario chan which seed th 10.2 ]. ς . n 93 . is provided by the Higgs: not 31 . | l )thend are set to ø  B 2 ς M k 9.3 Elko n v erms in equation ( | )does H − ant Spin-half fermions with mass dimension one φ rtheWIMP 2 9.2 H φ rticles are considered to be tightly → EQ ς stacks.iop.org/JCAP/2005/i=07/a=012 ς ¬ nconst n ς lution is determined by the Boltzmann σ  ry to an extent. The collapse analysis also ht pa of equation ( ], and Kolb and Turner [ & aged product of the cross-section | ): dates are a subset) may be useful, we shall The collapse, with eventual rebound, should l E ø ,whereasfo λ 100 H M ning of each of the t aver φ Elko v | 07 (2005) 012 ( m ant H ,andBoltzman φ H , given in equation (

c φ ς ill contribute to fluctuations in → ς ¬ m ς gconst te for cosmic gamma-ray bursts [ Elko )w σ % ], Dodelson [ 9.2 = 99 tree level. Instead, in conjunction with gravitational interactions 33 ) 3 ]forthephysicalmea a 99 t ς d n ere use units where the thermodynamic situation is as follows. The thermal contact of the The couplin particle as we wish to reserve that symbol for the Hubble rate. ,atthe d( mass. ς ]. At the same time, as will be seen in section 32 3 ggs . n i − H 98 of equation ( , Elko a φ he equilibrium value. This leaves one unknown density, i.e., that associated rH E ]–[ Elko fo α + is governed by ,t eric WIMP scenario for dark matter considers two heavy particles X which Elko 94 represents a Higgs particle and shall be assumed to have a mass of roughly 150 GeV. We do not use H ς H EQ rtic self-interaction of H n φ om and Goobar [ n φ tructure formation in the Cosmos. zero on. For the WIMP dark matter particle. Its evo
With these remarks in mind, and under the assumption that Fermi statistics can be In its essentials the analysis that follows is an adaptation of the one presented by In the considered scenario, Agen ¬ ς ee, e.g., section 9.2 of [ considered as a process that precedes virialization of the dark matter cloud. We thus S Here, For convenience, we h + trength [ density ignored for a zeroth-order understanding of the relic abundance of anon- cosmic plasma made of the Standard Model particles with equati ς with scale s Journal of Cosmology and Astroparticle Physics textbook derivations with a view towards various involved assumptions can be found, for instance, in [ 33 31 32 Bergstr¨ arises from an entirelyMeV-range different consideration. For these reasons we concentrate on the interacting massive particles (WIMPs). Since(of a which comparison with the apresent supersymmetry-implied generic the WIMP candi scenario analysis in such a manner as to make this apparent and the Møller velocityrelative (which velocity of in the the two cosmic annihilating comoving frame can be identified with the predicts an explosive eventshould in be the early considered history a of candida each galaxy containing dark matter. It consider both of these approaches complementa In a series of publicationss the case for a MeV-range dark matter particle has been gaining be 10.1. Relic densitycomparison of with Elko: WIMP constraining the relevant cross-section, mass, and a primordial Elko cloud.consistency It with is the encouraging relic that density the analysis. latter calculation finds an element of the symbol annihilate into two light particles. The lig coupled to the cosmic plasma,equals so that their number density in the cosmic comoving frame annihilation rate through the thermally The qua density of JCAP07(2005)012 , ς 1 46 n can − ,as 25. Elko I . a ):= Elko .At ,the t − =2), ς ¬ ( (10.2) (10.3) T ς γ CPT ) = H n g 2 ) ectrons and and side of hanexample CPT ack and forth today is of the ς framework the ρ , 4 slosh b ]. For such a scenario, (1–100 MeV) = 14 roughly scales as T Elko ∗ ,wehave( 30 2 g T π 102 =6),andel . ) Elko mass, and wit Elko ν ns: for the Standard Model fields T  g )ontheleft-h ( 2 ς ς 3 ∗ n g a ς energy density − Spin-half fermions with mass dimension one 75. If there is a relativistic n considering purely unusual Wigner classes. . := thermal distribution continues this 2 ergy density of all the matter at a $ i ubtle. If one is considering purely the Standard stacks.iop.org/JCAP/2005/i=07/a=012 EQ ς g Elko is used as an abbreviation for n Elko ς  n is respected by the . tence (for the quadratic appearances of Dirac fields, abits ¬ ς & trum of the cosmic microwave background =20MeVfor | l sign means that the summation is confined to ς =fermions ø )istheen i are connected via the usual definition m M T CPT and tical). So, at least for the process under consideration, i.e., 8 7 CPT ( v H ς | ρ en 07 (2005) 012 ( + H id with the following observatio i φ g (1–100 MeV) = 10 H as a serious dark matter candidate is established by is ∗ into the cosmic plasma of the early universe and on φ taking note of the fact that The symbol 2 g  ) → ons CPT ς ¬ 34 ς pb, freeze out occurs at a temperature of about 2.1 MeV . Elko σ ¬ ς ), where ng Elko =bos CPT i % T ndard Model particles and the " ( =1 4 = gardi and H outside the derivative. This yields es any statement on T | l 2 ς 3 3 ς ø ,Sta ,andthen ) 30 π 3 n H T M ent re φ T 3=: v aT | is respected. The general situation is likely to be more subtle and is expected to violate / m and the Hubble rate t )= H ) d d φ symmetry is fine. The same remains true if one is a T T has both the standard and unusual Wigner fields present, the differing actions of 3 .Thismak H ( ( =4).Thisyields ≈ 4 )by φ hile for the unusual Wigner class under consideration, i.e., T ρ CPT → , fields, the effect of ( ± ggs mass, the high-energy tails of the e ς ,w ¬ T H CPT I sone g ς πGρ φ 10.1 σ 8 represents helicity degrees of freedom associated with a specific component in the +  lify the statem Elko section is the cosmic scale factor. Since symmetry. i H =+ g φ a 2 ) = Before The analysis of the angular power spec Now, multiplying and dividing the factor of ( trons (
CPT ς ¬ We qua + CPT nduce a relative sign between the amplitude terms which involve only the Standard Model fields, and those ˙ Journal of Cosmology and Astroparticle Physics the temperatures of the order of 1–100 MeV, the contributions arise from photons ( those particles which may be considered relativistic at the relevant temperature, cosmic plasma, and the primed summation (see below). The viability of ς 34 Here, divides equally among component in the indicated point in the temperature range then where posi thermally. Metaphorically speaking,end Higgs it serves sucks as and a sprays two-headed the fountain: on its one value of given temperature, andnon-relativistic the matter overdot components indicates contribute negligibly, a it derivative is with given by respect to time. Since equation ( combined comoving number density of i involving an unusual Wignerwith class. a Dirac However,as as field, well long no as as such an relative amplitude does sign not comes contain into the exis contraction of same order as the critical energy density at the present epoch. the scale factor implies a spatially flat Friedmann–Robertson–Walker universe [ showing that, for this representative example, the the other it doesbelow the the Hi same forprocess the for Standard a Model while. particles. For Once a the representative temperature falls allows us to move ( But as soon a proven in Model physics three flavours of neutrinos and associated antineutrinos ( ( a/a JCAP07(2005)012 , f 47 x (i.e., (10.6) (10.7) (10.9) (10.8) (10.4) (10.5)  (where (10.10) deplete ∞ ) . x f ∞ = x & =1pbwe | x ubstitution 75. l . . x ø | l to

M ø ), the left-hand f v ,aswellas ς M | x H v ghly 10 m | φ /T ( H ς H ∗

M 20 T . v | ependence on σv MeV ς  H

is φ Elko 5 m H =4 ∞ φ 10 or light dark matter [ T ,havealready particle, at freeze out the → ), equation ( day cr ς 1MeVisvery ¬ ± × . ρ =20MeV to ς e 2 σ ς 31 10.5  . ]. The emergent relevant cross- ≈ Elko 9 33 as the Spin-half fermions with mass dimension one f ,m . 98 lies that a rough estimate of relic T 0 . 3. To see this explicitly, the above 72, is . . ] ]–[ is the 0 } amuchweakerd ≤ ] stacks.iop.org/JCAP/2005/i=07/a=012 94 . ln [ ln ns, except around 20 MeV and ς 3 pb ≈ A 0 =1pb

=0 ς Ω T − ς cr 30 is obtained for λ & . ∞ | σv m h energy density l find ≤ )thisimp  /ρ Y ø ς ς by Fayet’s analysis f M ρ =0 MeV ς ln [3 may occur, and since 25 m we v . ς { 10.7 ,andonly | m ± Elko = def .Fora20MeV H ≈ e pb 07 (2005) 012 ( 8 φ given in equation ( pb ς

3 significant reheating of the cosmic plasma occurs. 0

H − 5ln ]) . ς φ T GeV) arks and lepto 0 en this result we envisage that the asymptotic value σv λ 10 σv →  ∞ no  ς ¬ 101 − Y 14 re, the ς × 5 )gives0 ς ] − σ 5 m A 10 . % contribution to dark matter energy density today is very 10 ς , 8 3 λ × 1 gnificant reheating of the cosmic plasma after 10.15 × ≈ for ≈ AL gamma-ray satellite [ ∞ 0 31 indicates the respective values in the epoch when . mass in MeV, while Elko T T 3 ς ∞ 100. The value Ω .Therefo 9 617 [ln [3 0 1MeV.Giv . . Y ∞ 1 a 30 ∞ 4 2 2 ≤ a 1 ∞ 0 is √ Elko nepochwhen ≈ T T tion (5.43) of [ 1MeVallthequ ≈ 2ln . 0 ∞ GeV f 2 a MeV ς ≈ = 2 a T f ς − m 47 ≈ Elko x ρ 3 WIMP not only agrees with the considerations presented in section − f is the pb 10 ) ≤

T ς 0 =2,equation( 10 presentative case considered above, for the present epoch characterized by × T ssumption of no si m 0 σv pb MeV ς × 

89 725 K (1 K = 8 7. In conjunction with equation ( /a . 2 m . . . 1 9 ∞ σv  is reached in a T carries a strong dependence on = =2 ≈ =2 undary of where a slight reheating by ∞ ς ς 0 ∞ f ς a x Journal of Cosmology and Astroparticle Physics This gives (cf equa ρ That is, by temperature is abundance of and heated the photons in the cosmic plasma. Since For the re T For this reason also supported by observationsSpace on the Agency’s 0.511 INTEGR MeVsection gamma-ray line is made also by similar the to European the one suggested the range 1 around 1 pb one can readily obtain Ω bo where Using the representative value of Since the critical density today, assuming working a Y value for the obtained resultencouraging. for This means that by choosing For where the subscript its asymptotic value,dramatically and with the a subscript heavy( 0 WIMP represents scenario, the where present significant epoch. reheating takes This place, and contrasts Ω Ω formalism immediately yields JCAP07(2005)012 ¬ ς a 49 (see Elko non- and as the 2 E Elko (10.16) ς λ ) mass as H Elko m Elko evaluated using ,with his circumstance. B /k 2 interaction producing c are severely suppressed. H m should be Elko – ∼ ς Elkos ρ H . T 37 Elko Spin-half fermions with mass dimension one Such an evaluation may provide ged from the observation that for justification for t rtant distinction from the WIMP the Standard Model cosmic plasma rtic self-interaction (requiring cloud . 64 for the same range of stacks.iop.org/JCAP/2005/i=07/a=012 . ange, the collapse of a primordial eric WIMP scenario the characteristic qua 8.3 0 ccurs due to some quantum gravity effect with o timpo ≤ apriori ς ives rise to the virialized dark matter clouds ualitative transformations: Ω referee. Elko Elko’s fq xies, and cosmic gamma-ray bursts independent constraint on the Elko mass, ≤ ), or (b) by the tter core at the centre of galaxies. 3, in a gen 07 (2005) 012 ( to ssed mass r B . 47 JCAP . cross-section will depend quadratically on can be further gau cloud. Its spatial extent is assumed to be a few times /k 2 H ¬ ς kma pb φ –

mc H ς tandard-model, galaxies. The overlap is assumed purely . φ ς ¬ σv ∼ Schematically, we study a galactic-mass primordial ς cloud which emerges after the above-indicated rebound and )gives0  V ς → T is roughly 0 ¬ ς on → ,forthediscu 100. ς Elko 10.15 ς ¬ ς ς ≤ R 10.2 WIMP that appears in the above estimate of → MeV ς H ς ¬ φ m ς H φ P )) is evident. What is not clear is what precise contribution the ≤ → ς ¬ ς 9.3 σ =1,equation( dark matter central cores for gala is a rebound caused (a) either by represents a primordial is the virialized ς ¬ ς ¬ ς ¬ pb ς ς ς

thermalization-inducing process. that of a typical galaxy. R temperature of about P The pair of Higgs (requiring cloud temperature of about Higgs mass). The possibilityis that rebounds also considered. V The sensitivity of Ω In order that Ω first give a brief run-through, and then proceed with the details. M σv 6 This section was added on suggestion from a • • •  first-principle candidate fordictates dark the matter, above-mentioned but characteristic temperature it also carries with it the property which is severely restricted bytemperature the below which mass the dimension productionThis and one annihilation temperature aspect. of is This determined introduces a by characteristic the Higgs mass. Therefore, not only is that temperature for production andfew annihilations hundred of WIMPs GeV. is The put thermal in contact by of hand to be around a 37 10 We 10.2.1. A brief run-through. Here we consider a possible scenario which g 10.2. Collapse of a primordial Elko cloud: which overlap with luminous,on observational s grounds, and we provide no As shown in section Journal of Cosmology and Astroparticle Physics a the perturbative procedureadditional insights outlined and novel features in that areof section not apparent from this the presented section. calculations That above, i.e., 1 Here, clouds also leaves a significant dar cloud which undergoes the following se equation ( locality will make.scenario, and it This remains is to be perhaps studied the in detail. mos JCAP07(2005)012 50 Elko dhas (10.17) ) clou bly due to en here has been analogue of a Elko envelope. erizes either of the Elko ς particles undergoing cloud is greater than ς age the ς .Itisproba ed before the cloud radius ,of 38 tst R ’charact dark matter central cores for ∗ ]—the approach tak core, the 103 cloud is induced by the process M Spin-half fermions with mass dimension one nduced structure of the Standard .Attha 6 Elko , -i nent dark matter configuration we 39 C Elko ,where‘ λ )) stacks.iop.org/JCAP/2005/i=07/a=012 B this to happen a substantial fraction of Elko 9.3 P /k ]. 2 m tter the lower mass bounds on the lightest m c rcumstances, the newly available radiation )–( 104 ∗ hould yield if the presented scenario is to be m 9.1 lly and/or to emit neutrinos, and to cool down ≈ of the present work [ ∼ Ch 07 (2005) 012 ( v1 ∗ T th the Planck scale) is reach ndition which will be examined below (and roughly The cloud continues to collapse, and its temperature ), (b) it suggests 10 If this requirement is not met, then one ends up with —see explicitly. 10.1 —a co m, R ,G arXiv 3 B Ch licity we assume a spherical distribution, characterized by mass R P , m associated wi ,c,k m , in section QG rries three basic results: (a) it sets a lower bound of about 1 MeV for ≈ core, or a black hole. These may, or may not, have association with T hich appear in the additional m For simp ecome a black hole or a degenerate Ch ng assumption that non-locality plays an insignificant role. Then, the ,ca (or H M reaches a temperature Elko φ is reached when the spatial extent of the collapsing H rmine the mass and size of such a rema T ∗ + T ction we exhibit H agrangian density (see equations ( or galaxies, and (c) it predicts an explosive event in the early life history of galaxy φ tationally induced collapse. , as a dark matter candidate. In essence for ς handrasekhar limit enerate T
Some elements of our exercise are textbook like. Yet, the requirement just enunciated The physical criterion that provides the above-enumerated results is the requirement To dete ¬ (called simply ς (of the order of a typical galactic mass), and initial radius After this work was submitted to In this se ς andard arguments of balancing the Fermi pressure against the gravitational potential + hat ndependently adopted by Vanderveld and Wasserman [ soars until it ς agravi M m 39 38 10.2.2. Details. The scenario in which the rebound of a primordial i the C yields a rich setencountered of these results. considerations As in a parenthetical literature remark, on it dark is emphasized matter that we have not crosses the Chandrasekhar limit. adeg luminous galaxies. t typical formation. Journal of Cosmology and Astroparticle Physics the possibility to radiate electromagnetica provided the fact that thefor lower example, mass bound in derived the below context is of often SUSY superseded dark by ma stricter bounds; SUSY particle are wellMeV-range within Elko mass. the 10–100 GeV range. These, however, do not apply to signals black hole formation).pressure may Under cause thesefated part ci to of either the b cloud to explode, while leaving an imploding remanent two mass scales w Model L neutron star. If suchenvelope a must, scenario at is present, toprediction, explain carry but the dimensions what dark of detailed matterviable the calculations problem order the s the exploding galactic size. This is not a the released explosive energy must be re-deposited to the expanding make the worki st energy yield the following critical Chandrasekhar values: JCAP07(2005)012 G is 11 al 51 M 10 mass ecast c/G ≈ ς (10.19) (10.20) (10.21) (10.18) oes not mtot ) we r star is a typical particles: α ,where the coupling 2 2 ς represent the := − p − p cloud d ∗ P m )asasu m R as input a rough m 3 P 3 P equals the energy Ch the absolute lower is characterized by 3 is either the m .Forthis m ς Elko , M ∗ ∗ ∗ Ch /m R pc R − rvationally star ∗ m .Let R ,and , 2 expressed in keV. α ∗ -induced new Lagrangian ssume that m cloud when it reaches the − alculations in this section carry Ch M R m ς ≈ R 10 /mc and G Elko  × Ch M 3 is approximately the number of . R mass Spin-half fermions with mass dimension one M 6 ergy burst—predominantly made ≈ .Sinceobse 2 ς 3 1 star )asallourc . x M α , H Elko stacks.iop.org/JCAP/2005/i=07/a=012 /m .Weshalla ≈ ile releasing ( ote of the fact that m m ∗ 2 10.19 is reached when the spatial extent of the c m ∗ scales which = = Ch ecture that the collapse physics of such a T ,wh ∗ ∗ γ,ν eed—over the timescale of its collapse—the ,andanen m m Ch M ς M M for for ), we write ,R 07 (2005) 012 ( .Takingn sets the boundary between stability and instability H particles in the cloud, while } represents the M M ς m lds ς Ch 3 9 x . / 12 1 1 patial extent of the collapsing ,R erplay with gravity, are such that the 10 )yie envelope, ≈ Ch H ς × . is the proton mass, and inserting for (the only two mass , g M ∗ m 6 2 ς star . p { c 2 p p 3 10.19 star m 1 33 α ∗ ater than the Chandrasekhar limit, is the associated Compton length, m α 2 p m m m 2 ς m 10 1 C √ x m λ = he Higgs mass × ≈        nder the assumption that the initial = 2 ∗ is the number of ome knowledge of plausible values of 77 : ,andtheirint = . Ch mine the condition for which E star .U 40 NR ∗ ss. The set in the following form: GM λ mass, α M m ,equation( a T , cloud is gre as input, one may obtain the ratio ς M/m E Ch Ch is the solar mass, and ς rkable equation can be read in two ways: with the α 11 is R R configuration when the average kinetic energy gained per = not include a factor of 2 on the rhs of equation ( 10 trepresentst m N M lso marks the complete onset of general relativistic effects. To proceed further it is and ≈ particles, or a black hole with mass We now exa With these observations in mind, we conj Elko ς und of da We do Ch ,ori star develop significant densityformation fluctuations of to smaller s structures. Journal of Cosmology and Astroparticle Physics order-of-magnitude estimates. where 40 is known, this immediately implies the following results for the mass of the represents the luminous mass of a typical galaxy, where of gamma rays, and neutrinos—carrying cloud is similar toof that of a supernova explosion. It leaves behind a degenerate structure an constants m radius which characterizes thetemperature s stellar mass (=3 collapsing stars in the same, and This rema M bo the estimate for the number of starsα in a typical galaxy may be derived; on the other hand, with for the mass of rebounding essential to gain s where the Planck m associated with a Higgs or density carries) JCAP07(2005)012 ) = 52 the H ]and 10.24 m m (10.22) (10.24) (10.23) ) itself, and 106 ,thesum  , M cloud, and ,themass 41 ∗ ∗ is very finely identification 105 m m he observation 2 cloud m Elko ≈ /c For .But,aslongas central degenerate is due to the order- ¬ Elko m ς 42 . m 2keV . M . and = ,the 3 P ]. Kusenko [ H ∗ ς m 2 p intrinsic parity asymmetry. m m ing the Higgs mass 107 m star ≈ ial mass of the α ∗ Spin-half fermions with mass dimension one while rebounding . ill of the order of unity and thus m 2 H , sar kicks and that the same particle mass, we now make t particles, m m m 2 p to occur, the round bracket in ( stacks.iop.org/JCAP/2005/i=07/a=012 itself. That one cannot make a more M m 6 = = ,andwrit star )isst Elko ∗ ∗ M 11 α Elko aexplosions[ der of the init tes) or as a virtue (the rebounding cloud is cases where the fermions of the Standard m m tmustcarryan − 10.24 1 obtained by saturation of this inequality. for for =10 hat a sterile component with a mass of about .Forthis one is tempted to note that, apart from the dark 07 (2005) 012 ( M. = M he bracket, we get ∗ star cloud the viability criterion cannot be met 2 )was Ch α ∗ m arries a mass of the order of m /c m M ≈ becomes of the or ]andsupernov 2, the above equation yields − Elko − 10.20 m Ch 1 2 )insidet ≤ M 106 cloud c M fraction of , 2MeV /c alue of H ile ( . x = := ]havearguedt the 10.20 105 ≤ 0–1 ood m Elko ,wh . γ,ν γ,ν g 108 3keV 1 ≈ Ch M M logical dark matter. However, for the argument to work for pulsar ∗ are to be considered as lower bounds because the requirement one has R m ntative v + + = ς ς ≥ m ,with1 of the solution ∗ M m M 2 provides a good candidate to explain pul R /c ecome too small. For the solution with 2 must be a ive to Elko details and thus may probe Elko physics). Unless ords, for /c ecide between these two values of the γ,ν favour to be a dark matter candidate. M To d In + and using equation ( 100 GeV Elko That is, observationally, the mass of the dark matter cloud should be of the order of a typical galactic mass ς In other w H that to impose is must not b again the rebounding and its spatial extentfor must extend beyond luminous extent of galaxies. This we call the minimal viability criteria 42 41 Taking a represe These values of Such an asymmetry is naturally built in the may also bekicks cosmo it is essential that the sterile componen Since without a rebound of the matter problem, there are two outstanding Model of particle physicsThese fail to are provide pulsar the kicks astrophysical [ consequences expected of them. of-magnitude nature ofno the prediction calculation from and order-of-magnitudevery may estima sensit be considered as a drawback (there is tuned the quantity in round brackets in ( x core (or, a black hole) carries a mass of about 10 Journal of Cosmology and Astroparticle Physics hence its difference from the latter ceases to have a reliable meaning. one confines oneself to the set of assumptions we have used, the 4 On the extreme rhsM of the above equation, identifying the factor outside the bracket as the associated burst of energy, carries almost the entire mass. For of the remanent core becomes of the order of M order-of-magnitude estimate is more robust. 2–20 keV Dolgov and Hansen [ precise statement for the mass of the rebounding cloud near JCAP07(2005)012 1 53 − are m ¬ ς ) and ], followed ς 95 , 94 [ cloud, if et al Elko cal and non-local field on the ensor theories or quintessence may be characterized by two ∗ . T Spin-half fermions with mass dimension one ], Boehm ς cloud, the neutronic core is replaced 109 more fundamental problem with such stacks.iop.org/JCAP/2005/i=07/a=012 cenarios with critical temperatures of sionless coupling constant, much like mma-ray line seen by the European eutrino radiation may carry a mass of ga ed. Moreover, for stability reasons the ], purport to read a dark matter particle Elko ea have additional support from two recent wo s 2 ly unknown cosmic phenomenon, it is not ust not be overestimated. There are several 96 dimension one component (Elko) to a mass of fermionic matter and are accompanied by [ ch mixing of lo ob /c he problem appears to be non-trivial and has ]doesinscalart we st H field, which is a scalar of mass dimension two, 111 07 (2005) 012 ( et al , m is to explain dark matter. It is desirable to improve (or, a black hole)—indicated here to be of the order ≈ 110 sses. For an Elko 511 MeV . ς m ionally induced collapse of an Elko if analysis can be presented at this stage of our work. In this AL gamma-ray satellite [ 2 /c al solar ma ]. identification meets the minimal viability criteria well. Yet, the rate core of H 115 m ]–[ ectively. observations on 0 of the solution masses: the Higgs mass and the mass of = to 20 MeV 112 ∗ 2 ,resp m ς /c ssive degene T favour solar masses—while the gamma-ray and n in the range of 1–20 MeV. 6 nd reason that the following sequence of multiple rebounds is realized. In that dditional observations of Beacom models [ Therefore, our simple order-of-magnitude considerations above cannot be used to Quartic Elko self-interactionsessential have way been incoupling the neglected. constant energy in rangeemerges. These front consider As of may a this result contribute term thisNon-locality in may has contribute an to to has the be rebound been such of the that collapsing neglected, a Elko cloud. repulsive but interaction because it scales essentially with The may couple tothe the Jordan–Brans–Dicke field Ricci [ scalar with a dimen contributions from it could beNon-standard relevant gravitational here. interactions couldinstance, be the of importance ‘square’ in of this the context—for In conclusion, the gravitat In Given that we are confronted with a tru acteristic and ss yo • • • H determine uniquely the massof of 1 Elko, keV but it appears to be likely that it lies in the range by a ma of 10 strength of this viabilityopen and points identification regarding m these considerations. Journal of Cosmology and Astroparticle Physics an identification: how is one to add a mass noted here gives rise toviability too criteria. high In a value addition, for there the seem galactic core. It thus violates the minimal dimension three-halves field (neutrinos)?no T clear answer due to subtle questions whi Space Agency’s INTEGR by a ma the order of galactic mass and the burst temperature these limits further, butmore to detailed this study, end possibly one has combining to the address t the caveats mentioned above in a T to serve as darkstellar matter, objects must which be leave(a) a qualitatively degenerate an similar core to expanding type-IIradiation envelope carrying supernova of sever explosions matter, of and (b) an intense electromagnetic and neutrino works. From one hand, and mass dimension three-halves and one on the other, raises. char be event one can noTherefore, longer no rely quantitative on the one-mass scale dominance scenario outlined above. JCAP07(2005)012 54 itself ), and 10.26) (10.25) ) .Inthis 9.4 2 Elko /c ectromagnetic )and( atter candidate, 5TeV 9.3 . no postulate has to )( x ( ]. Over the years it  µν )istheel Elko F x ) ( 118 x µν ( ]–[ F µν 116 F mass being 0 Spin-half fermions with mass dimension one M  level. Latest measurements by two tion and Higgs-mediated explosions Elko σ stacks.iop.org/JCAP/2005/i=07/a=012 )+ x tfrombeingadarkm (  opositronium life time puzzle which had φ nant interactions with the SM fields: ) x ( ak and GUT fields do not carry any significant † self-interac  φ cloud successively soars to this temperature one 07 (2005) 012 ( ) esults in the x ( Elko φ ) Elko x . ( 3 † ,thisr cloud rebounds without leaving a remanent core. / 2 φ 1 /c M P λ Elko ]. llapsing m star GeV 2 p Standard Model Higgs doublet, and α )= 120 m 19 , x ( 10 L ]. Yet, we suspect these claims for the simple reason that the evidence then only one additional mass scale comes into the picture, and it is 119 field. Yet, apart from the assumption that standard power-counting apart from an explosion induced by quartic self-interaction of = × are dimensionless coupling constants. 122 2 )isthe m proposal for dark matter has some noteworthy similarities to and differences . , M x een overwhelming already because mirror stars would have already been .Astheco  Elko ( restrict the interactions with the Standard Model matter to the specific form. Elko P rror world carries same gauge group as that of the Standard Model, with all φ 121 =1 m Elko almost all of the and P rticle masses in the mirror world are set degenerate to the masses of the Standard m M mass to be ecall that mirror matter, which restores parity symmetry by introducing a parallel left-handed fields replaced by right-handedThe fields mirror and world vice has versa. the following domi Mirror symmetry predicts mirror stars, and claims an indirect support for their The Model particles of the worldThe in mi which we reside. Pa field strength tensor—with primed fieldsλ representing their mirror counterparts, while where • • • be made to must have b existence [ the mirror counterpart just enumerated,one is obvious, of and carries the itsrenormalizability roots arguments in provide mass a dimension good guide, in the case of from the mirror matter framework. The similarity between equations ( independent groups, however, have discoveredlatest a systematic experiments source provide ofelectrodynamics a errors and [ decay now rate these which agrees well with predictions of quantum acquired a statistical significance well above the 5- had been suspectedcould that also mirror provide matter, a apar resolution to the orth Journal of Cosmology and Astroparticle Physics universe, is postulated to be endowed with the following properties. We r 10.3. Similarities and differences from the mirror matter proposal given by coupling to scenario, if one assumes that the Electrowe It is a viable and attractive dark matter candidate [ another explosive phasegravity occurs induced at effect theexplosive occurs Higgs element, at temperature, andnot the and should succeed Planck an the in temperature. causing unknownElko a quantum- significant Should rebound, then the a latter one-mass carry scale scenario an predicts the may expect that With scenario JCAP07(2005)012 ] . 55 43 one 127 fermion ) unless Elko rror neutrino. fermions, it is more be a mi hree-halves. In contrast, re, in this section, we now Elko rror matter postulates the may matter provide most natural Spin-half fermions with mass dimension one Elko stacks.iop.org/JCAP/2005/i=07/a=012 ]. The introduction of an matter carries with itself, and with the 125 atter–antimatter asymmetry. We have not , ]. ered in a Thirring–Lense gravitational field cle physics, cosmology, and experiments such ng any untested spacetime symmetries. For in the gravitational background of a rotating ]. While pursuing these studies, the following 123 proposal set forth here in this paper does not ]. 123 Elko een investigated, due to subtle questions which he identification of the described particles with neutrinos and of luminous SM-model stellar objects 21 07 (2005) 012 ( 124 Elko Elko was entirely overlooked. He tter carry mass dimension t rror matter and non-locality carries. The mi tyetb ¬ ς ssible interactions with the Standard Model fields. matter carry mass dimension one. alerted that t Elko both mi Elko ni and Mohapatra [ heory based on ]. The minimal troys the very motivation upon which the mirror symmetry is based. ]thereaderis framework does not have to invoke a parallel universe. It thus avoids framework carries a well defined non-locality, whose physics motivation ing mass dimensionalities of the mirror and 126 127 icles in a Thirring–Lense gravitational field Elko Elko . ς eloped a full t pping spatial regions and yet the indicated circumstance does not seem to arise. and utrinos made there, in view of the work presented here, is no longer tenable. The correct identification is The fermions ofthe the fermion(s) mirror of ma the The limited interactions which the Standard Model fields, arewhat dictated effects by the its intrinsic massabsence dimension. of a At large present, class isGiven of not differ po known natural that the suspected sterile component of neutrinos Such a suggestion has been made in [ non-locality and its mass dimensionMirror raises. mattermeteorites [ demandssuggest the any such objects. existence of mirror stars, planets, and even These remarks aside, as a sterile component has no On the other hand, the mirror–matter framework is local at a quantum field theoretic One of advantages of our proposal, which to some extent is a subjective perception, ς t, an interesting possibility with parity symmetry being spontaneously broken has been In the context of [ • • • • In the above discussion the role of 11. Elko part as those envisaged for mirror matter [ candidates for darkthis matter reason, without and invoki distinguish due their to implications their for similarities, astroparti it should be considered as an urgent task to observations should be kept in mind. have been discussed at some length in [ level. The suggested by Berezhia Ye the inevitable formation offramework mirror of stars, unbroken and mirrorawayoutasitdes mirror symmetry. planets—which Broken are mirror contained symmetry in does the not seem to be is that the invokes some arguments as to whyoverla the dark matter and luminous matter of galaxies occupy with antine 43 observed as numerous dark companions gravitational source. Yet, an important physical implication has already been studied [ find that ityet may carry dev significance for the m in which Elko single-particle states are consid Journal of Cosmology and Astroparticle Physics JCAP07(2005)012 56 ]. is the (11.5) (11.3) (11.4) (11.1) (11.2) 129 ) , specifies J represents r alification, 128 ω rry opposite exist and the which appears ,ca : ), and lf-conjugate state does G 45 .Thequ Elko r explicitly): is its radius, and R , Spin-half fermions with mass dimension one G is the usual speed of light, , ). Using phases which the coupling of stacks.iop.org/JCAP/2005/i=07/a=012 c p he-envelope’ calculation is the following: t t ( 2) transforming components of the state | | / which appears in a particle’s spin/helicity ∓} 1 b b , A , Thus if we introduce a se · · / S {± eralized flavour-oscillation clocks [ p p λ | | 2 2 is the mass of . 07 (2005) 012 ( are × × ), where M ) t A . i i 0) and (0 r / , S cancels out. This happens because the osc + − 2 / , w ∓ ω. J· ( 3 A , 2 r ), refers to the ensemble-averaged time as measured by a ,where 3( |± neutron star. ∓ ω t 2) Weyl components, for each of the R , 2 − 2 / 11.2 c 1 GM |± ), J , field induces, we find the following oscillations is studied in the polar region. MR  2 5 ar mass 5 4 b 11.2 2 S G 2) : exp 0) : exp c )and( = 2 , / 4sol . . ∓ 2 1 J≈ , volution = , / osc 11.1 b (0 |± ω (1 )and( tion has been made under the assumption that we prepared the test particle in such a way that 2fora1 44 . la 0) and the (0 0 , 11.1 in the gravitational environment of a rotating gravitational source, such as a is the Newtonian gravitational constant, cu ≈ 2 representing the Thirring–Lense field of the source star. In the weak field limit it ssociates particle nature to self-conjugate sector, and provides an antiparticle al / S R a G is given by b 2 ec ,andthatitse cancel out. In ( stationary clock situatedensemble-averaged, in is the necessary for gravitational gen environment at In phases ( in a generic quantum evolution and the The circumstance which allows for a ‘back-of-t r − J This is a valid approximation all the way to a neutron star as the dimensionless gravitational potential , Th • • = + The spinors associated with Since in theElko presence of a preferred direction, the study of the wave equation for 45 44 This example is an important physical case where a preferred direction the (1 more general structure of the spacetime evolution must be invoked. Here angular momentum of thethe radial rotating coordinate gravitational distance of sourceof the (denoted region by where the particle is observed. The magnitude neutron star, then each of the (1 with oscillation probability sin GM/c p Journal of Cosmology and Astroparticle Physics these states with the the associated angular frequency. A couple of observations are immediately warranted. with is given by picks up equal and opposite phases (where we now display helicities. Based onparticle the states particle interpretation to developed| carry in the this same paper property. we expect single- JCAP07(2005)012 57 ], 8 and (12.1) (12.3) (12.2) C ) egroup ummary represent ¬ ς onclude that and ς ,isnotas matter–antimatter ea composed entirely 12.6 Elko resented construct and ,weareledtoc )field.As 12.5 x Spin-half fermions with mass dimension one ( Elko η -based quantum field theory carries, oscillations, where ¬ ς stacks.iop.org/JCAP/2005/i=07/a=012 Elko  articles of the Standard Model, if ς ]. Specifically, in the context of Weinberg’s ncluding remarks with additional details of 8 nd parity operators, respectively, then

ssured by Wigner’s work on the extended for bosons: gauge/Higgs particles. for fermions: leptons, quarks with the dark matter, these oscillations may for fermions: the p for bosons: construct of [20]. ¬ ς ,σ,m 07 (2005) 012 ( , , , p and .Forthep , =0 |− ς 46 =0 ]=0 component, thus inducing an nm C} ]=0 C , C} ℘ , C ¬ ς , , the exception of subsections P {P [ m P [ {P properties = oiding confusion we take note that the presented construct is a

T ,σ,n ]. In that work a general result is reached that the Poincar´ 6 p ,and turn is an extension of Wigner’s considerations, equation (2.C.10) of [ P interacts only very weakly with the matter content of the Standard Model. P| , asses: C mmetries. In fact, he showed that the following additional structure is allowed: Elko ner states ‘Much of the material which follows was taken from a rather old but unpublished manuscript esy egroup[ gmann, A S Wightman, and myself’. rigin of these observations in fact lies in an unpublished work of Bargmann, Wightman, and Wigner. our effort had a chance of providing us with a theoretically viable and ig will in time develop a ], Wigner argues that the above class does not exhaust all possibilities allowed by eo 6 ]W ς 6 generically represent charge conjugation a Th P In [ Poincar´ Standard Model: 46 Poincar´ extended with space,to time, and support space–time fourunderlying reflections type allows every of representation quantum space field theories. Each of these theories differs in the 12.1. Elko as a generalization ofThat Wigner–Weinberg classes phenomenologically novel theory seemed a interpretation for the anti-self-conjugate sector of the particles and antiparticles associated with the neutral of In [ by V Bar Journal of Cosmology and Astroparticle Physics our work. This section, with In this section we combine the customary co 12. A critique and concluding remarks have an important role tobaryonic play matter–antimatter in cosmology, asymmetry although is weafter hasten all, unlikely to to add be that the a observed consequence of this because, asymmetry. Since we identify the Wigner Cl work, which in Yet, forgeneralization the of Wigner–Weinberg sake context [ of av rotations in gravitational environments induce of our work but isreader’s intended attention as additional an structure integral whichand part the of in our addition exposition. tofield Its point theories purpose out at is the that tothe Planck locality bring scale. previous to is section, However, most giving the likely level us not of the rigour a luxury now stable of changes feature some and of speculative it quantum remarks. mirrors JCAP07(2005)012 , α 58 ulted). .This )which ) 4.2 8.20 e–Heisenberg rg algebras of represents one ]maybecons σ ], 133 , 8 132 ]. define a degeneracy, and is the momentum vector, 131 ation, we give the following p lism is a generalization of the eandHeisenbe its a non-locality ( n, m i.e., the dual-helicity index , Spin-half fermions with mass dimension one ]. The counter-example contained σ ime and non-commutative energy– 8 ,and )exhib 3 stacks.iop.org/JCAP/2005/i=07/a=012 x J ( cs are unstable algebraic structures, i.e., n-locality contained in the field based on η rences therein. In some simple models—for instance, rpreted as associated with self-conjugate kasourforma ]. etotheaboveequ 57 , ] ]andrefe ws for the result obtained in section 55 antum field theories. This is apparent from Lee– 07 (2005) 012 ( 23 , 130 , 22 terpreted as a contradiction, but as a generalization. 56 represent one-particle states,

nt preprint by Chryssomalakos and Okon [ coefficients. In relevance to the theory which we present e–Heisenberg algebra as presented by Mendes still respects formalism into a further generalization of the Wigner–Weinberg -based quantum field ections associated with ,σ,n ecovariantqu p | his introduces two stabilizing dimensionful constants which may be remains unknown at present. Elko Elko .T )isnottobein but calls for non-commutative spacet ttention a rece 48 , a 47 tivity and quantum mechanics. Each of the indicated algebras is stable by ]. The 4.21 7 al theory for the remaining matter degrees of freedom (for reviews [ +1)spinproj iled mathematical and interpretational examination of the stability-based framework of Mendes, we ased on incorporating gravitational effects in a quantum measurement process also suggest an element are superposition j ta de nm oourreader’s ℘ algebra, which is unstable. The stabilized Poincar´ in the mathematical sense they are not rigid. special rela itself. What, however, can be claimed is the following: locality cannot be considered a ra dual-helicity Elko is the parity operator and carries properties defined on p 76 of [ t, its direct connection with the underlying no Fo It should be pointed out that the idea that quantization of gravity leads to non-locality and/or non-commutative ilaton gravity in two dimensions—these ideas turn out to be realized straightforwardly, i.e., quantization of gravity (c) The algebra for relativistic quantum field theory is the combined Poincar´ (a) The Galilean relativity and classical mechani (b) Their deformation towards stability results in Poincar´ momentum space 47 48 is consistent withwhich this we expectation. discovered with that Yet, of we Lee cannot and Wic fully identify the non-local element Lorentz algebra Lee and Wick have arguednon-locality that [ the non-standard Wigner classes must carry an element of 12.2. On Lee–Wick non-locality, and Snyder–Yang–Mendes algebra robust feature of Poincar´ Wigner framework for which Lee–Wick considerations apply. is no longer sufficiently general. In referenc Wick paper as wellwhy as does the this explicit circumstanceand construct exist? observations which of we The Mendes present. answer, that we The [ think, question lies then in arises: the important reminder notational details: the Arguments b Journal of Cosmology and Astroparticle Physics of spacetime non-commutativity and non-locality [ spacetime has various roots; see, for example, [ framework, and while remainingto in Weinberg’s agreement result with contained Wigner in we equation provide (2.C.12) a of counter-example [ here, while theand degeneracy anti-self-conjugate index states, may the be counterpart inte of the bring t the d yields a non-loc observation casts the in the resultYe ( introduces additional structure and allo of the (2 P the JCAP07(2005)012 ) ). 59 3.1 2.3 (12.6) (12.7) (12.4) theory ) )and( Elko the dispersion 2.3 This possibility ]. . ), ( ded or the relation I − 137 2.2 , . ,andthat = )istheinverseof( ng–Mendes algebra, but 2 ) A 136 2 , 2 2.2 p E + )arediscar 2 )with( 131 − m 2 Spin-half fermions with mass dimension one p √ nyder–Ya 12.6 5.33 ± + m 2 2 to support two type of spinor: those stacks.iop.org/JCAP/2005/i=07/a=012 4 − in ( iplicity = 1. iplicity = 1 m rtant physical criterion for the viability )) ( = ndependent of E O 2 m    ) mult mult 2 0) + ]isi , ) 2 appears / O in quantum field theories based on stabilized E E ( (1 , + κ , It is conceivable that for example in the context of m O , 2 iplicity = 2 (12.5) 07 (2005) 012 ( 2) 2 p m / (2 p 1 , (2 + (0 + mult 2 κ − 2 m 2 m = p + 2 , stent with the starting point as ( p + is modified. 2 2 − + p 2 ], and that the algebra found on algebra-stability grounds by 2) m m / m ( m 2 1 + 2 , e–Heisenberg algebra, i.e., the S − √ 2 135 (0 − , ± ]= κ m rac theory will suffer modifications in any extension which respects m ld 2 O = 134 − = )isinconsi and 2 = )yie m–space wave operator, E E Det[ E 0) ,    2 12.7 ssociated with / 2) / a 12.6 liminary examination of the Snyder–Yang–Mendes algebra suggests that the (1 1 , κ (0 e–Heisenberg algebra. Following Mendes, that stability of algebra associated with κ sical theory should be considered as an impo 0) een , Apre Yet, just as Di 2 ndes is precisely the one that Yang had arrived at more than five decades ago. hy / (1 ssociated with the usual dispersion relation, relations ( κ However, it is now to be noted that the Det[ is again taken up in appendix B.6. non-commutative momentum space the latter scenario might be realized. and those Journal of Cosmology and Astroparticle Physics Evaluating the determinant of the operator 12.3. A hint for non-commutative momentum space Me identified with the Planckin length and 1947 cosmological Yang constant.algebra, had It also already is suggested worth arguedcarry taking earlier that note curvature in that the [ the lack same of year, translational is invariance remedied of if Snyder’s spacetime is allowed to Obviously, ( The momentu a usual expectation on localityas commutators/anticommutators ‘locality cannot is hold.completely not We lose a phrase its it stable conventional feature meaning of quantum field theories’, and suspect that it will ap Poincar´ establishes presented here. Itlocality. may serve as a low-energy model to explore the consequences of non- still remains a useful low-energy model, the same may be expected for the Thus, there are only two options: either the solutions ( of a theory has also been emphasized by Chryssomalakos [ the stabilized Poincar´ betw JCAP07(2005)012 , ) . ), 60 ab uld 7.5 5.56 (12.8) )sho exotica ) exotica operator operators 12.8 ). The spin such higher he resulting P , dual-helicity )and( I content. field to higher CPT − j all 6.13 ations, all arise and 5.55 = ). Equation ( ,equation( C Elko 3 2 2 7.4 .Their p ). The action and the 2) + / 2 j> ). Wigner’s expectation 1 6.2 m Elko , ). It is to be contrasted persion rel ). The propagator, in the rtant consequences for the √ (0 ± )and( structure, and t 3.25 rth considering. In such an m ⊕ 6.28 2 6.20 7.3 )and( )–( − 0) , Spin-half fermions with mass dimension one = Elko .Assuchfor 6.1 2 3 2 E / 3.23    ) ence from the Dirac construct. The ,j stacks.iop.org/JCAP/2005/i=07/a=012 (0 ), while the square of the κ eparture from the standard high-energy =1and 0) j, j smayhaveimpo The covariant amplitude which determines 4.16 ( κ anticommutativity, of the ]mayalsobewo tified when one realizes that various bsence of a preferred direction = spin sums is found in equation ( other differ 68 , 07 (2005) 012 ( ea 2 49 )onlyfor on spaces which carry a single-spin p 67 + Elko ). With respect to this dual, the orthonormality and 2 the formal structure of the ). Like the Dirac dual, there exists a new dual for 12.8 m Diracian propagator are presented in equations ( 3.8 √ 3.19 utative spacetime, modified dis ± )showsan m are contained in equations ( 2 Elko − 4.21 )provide = ation ( dtheresult( E Elko    3.5 ) ,j is contained in equation ( verifie (0 )–( ndicative of the fact that the above-mentioned non-commutativity is κ 0) ars to carry significant physical promise as to give such a task an element 3.3 ns is perhaps not too difficult a conceptual exercise. If such an exercise j, Elko ( lly relevant representations spaces is far from complete, and a serious representation spaces, such as the spinor-vector Rarita–Schwinger field, an terested in the main theoretical expressions we now provide a brief reference κ fined in equ .Itisi )spi e, we have 2 1 ,j analysis along the lines of [ propagator in terms of the = ting on (0 ered a conjecture. j study appe ⊕ the resulting covariant amplitude given in equation ( quantum field and its dual are given in equations ( .Itisde An explicit calculation shows that For other The purpose of these remarks is simply to emphasize that the physical content of Elko 0) To be precis j, 49 For a reader in nature follows fromElko the result ( 12.5. A reference guide to some of the key equations list. Equations ( It may be worth noting that a generalization of the (1 12.4. Generalization to higher spins probably representation-space dependent. unless Journal of Cosmology and Astroparticle Physics be consid for commutativity, as opposedwhile to ac completeness relations are enumerated in equations ( gives the canonically conjugate momentum. the as contained in equation ( Lagrangian density for Elko ( sums which finally give the spin fields will carry mass dimension one. Thi is undertaken it may reveal that inphysical th understanding of representati ab initio various physica analysis, the spinorpropagator sector will would almost have certainly to carry be new physical given content. an with initio of urgency. This becomes even more jus such as non-locality, non-comm in constructs whichprogramme go may far be beyond considered the as point-particle a context. minimal d As such, the suggested physics framework with a potential to allow a systematic study of the indicated absence of a preferred direction, is written in equation ( JCAP07(2005)012 . ) ). ). ), 61 ons Elko Elko 9.2 Elko 8.26 10.21 10.15 ) eracti fertile playing equation ( class of int oscillation frequency )) is discussed around w ). It defines the physical ( ). trophysical observations, ¬ η ) gation operator carries mass 12.8 z masses. The fractional 10.20 ( η ) is to be read at equation ( Spin-half fermions with mass dimension one y ,isgivenby ς ( titmayserveasa Elko )and( ¬ η ). The non-locality anticommutators ) stacks.iop.org/JCAP/2005/i=07/a=012 he possibility of Planck-scale physics. Elko x ( 12.7 6.30 antitative argument yields three different η ,andvariousas ( mt mstance forbids a large kground revealed the interesting possibility non-locality to its mass can be obtained theory carries no free parameters and is aper is: a fermionic quantum field based 2chargeconju T / njugate states, with an interaction. A remarkably simple equation Elko een at equation ( cal consequences of non-locality. framework with dark matter. Equations ( Elko Elko 07 (2005) 012 ( quations ( Elko ,iss mass and resides in the well established spacetime Elko ), and in a remark which encloses equation ( ggs– Elko particles. 8.20 of the spin-1 Elko sHi 50 ), ( Elko ntained in e 8.8 ). A hint for non-commutative momentum space, and its possible after their German name. refore, for a theoretical physicis lytical expressions for possible )provide 11.5 Elko 9.3 ). The quartic self-interaction of ude this long exposition with two remarks. rtic self-interaction. This suggests a first-principle identification of the new lled these )giveana 8.45 ). The non-locality as manifest in 8.25 10.25 gauge and matter fields of the Standard Model, while allowing for an interaction with governed entirely bysymmetries. the The ground for examining phenomenologi We concl th the Dirac propagator recorded in equation ( lues for the mass of the new dark matter candidate. These values are 3 keV, 1–1.2 MeV, We have ca (a) The non-locality that appears in 50 The unexpected theoretical result of this p 12.6. Summary on dual-helicity eigenspinors wi are found inAn equations estimate ( offrom the ( sensitivity of the of oscillations between self- and anti-self-co Although interactions withparticles gravity in a were Thirring–Lense not gravitational bac our maingiven concern, by equation a ( brief study of dependence on representationsuggested space by the (i.e., results spin co content of the probing test particle), is Journal of Cosmology and Astroparticle Physics equation ( dimension one, and not three-halves.with This circu the Higgs field. Inwith addition, owing a to qua massfield dimension one, with the dark introducedto field matter. is what endowed constitutes Thus, dark regarding matter, we the have question provided we a asked new possible in answer, the namely introduction as contribution to the total cosmic energy density, Ω and ( which connects theparticle number masses, of including stars that of in a typical galaxy with three relevant elementary while equation ( viability for the identification of the Considerations on thesuggest relic a abundance 20 of MeV mass for The question onindicated dark interaction energy calls will forcollapse perhaps of some a very be galactic-mass specific cloudlike the of explosion properties the subject for for new the of particles. a collapsingva cloud. gravitationally In a particular A induced subsequent it semi-qu asks paper.and for 0.5 a TeV supernova- The as loweris bounds. held The responsible firstthe of for these phenomena, the values arises and explosion, if the the the quartic last second self-interaction value value results results fro from Higgs being behind JCAP07(2005)012 ] 62 138 (A.1) (A.5) (A.2) (A.4) (A.3) ve the ) dusimpro referee for his/her detailed and aised also helpe Spin-half fermions with mass dimension one , ) θ JCAP tefully acknowledges financial support stacks.iop.org/JCAP/2005/i=07/a=012 . , , , r/his suggestions which, in particular, led cos( and Dima Vassilevich for correspondence, 2 , 2 2 2 2 Ggra ) 2 2 2 φ/ i φ/ φ/ φ φ/ φ/ i ot to carry any Standard Model interactions. i φ/ i i odinger fellowship J-2330-N08 of the Austrian φ/ i φ/ i i − − − − 2)e 2)e 2)e 2)e 2)e .Thequestionsr 2)e 2)e 07 (2005) 012 ( θ/ )sin( 2)e θ/ θ as θ/ θ/ θ/ θ/ θ/ θ/ p 10.1 cos( sin( sin( isin( sin( cos( − , to be zero. icos( ) icos( isin( − 2 and φ ϑ 2 1 ϑ ϑ i i . m m 8.3 e e √ √ )cos( and m m θ 1 10.2 √ √ ϑ plicit form )= )= sin( 0 0 ( ( )= )= 0 0 + L − L rd Arnowitt, Abhay Ashtekar, Chryssomalis Chryssomalakos, Naresh ( ( φ φ = uation of spin sums (cf appendix B.4) the following identities are useful: − L + L 2 2 φ σ p σ φ (0) ± L een supported by the Erwin Schr¨ φ )taketheex 0 A-K acknowledges CONACyT (Mexico) for funding this research through Project rts of this paper are based on Concluding Remarks and an Invited Talk [ ( ± L nd)’; and in that context we thank all the organizers, and in particular Hans Klapdor- The presented theory doesthe not sense made postulate precise this above. absence as an input but requires it in For the eval Pa DV φ dhich, Salvatore Esposito, Steen Hansen, Achim Kempf, Yong Liu, Terry Pilling, Lewis (b) Ordinarily, dark matter is postulated n the Ryder, Parampreet Singh, George Sudarshan, We thankDa Richa at times extended, and/or discussions. We thank a Acknowledgments In this paper we take Journal of Cosmology and Astroparticle Physics Representing the unit vector along A.1. The presentation of section constructive review of this long paper and for he presented at ‘PhysicsFinla beyond theKleingrothaus for Standard his early Model: encouragement.and enjoyable D hospitality beyond frompart CIU the at of Desert the this Universityorganizacion 2002 of de paper Zacatecas (Oulu, mi in was estancia autumn en conceived. 2003 Zacatecas. while Muchas gracias a Gema Mercado Sanchez por la 32067-E and IUCAA (India)work for has its b hospitality which supported part of this work. DG’s to the addition of sections Appendix A. Auxiliary details Science Foundation (FWF). JCAP07(2005)012 63 (B.2) (B.3) (B.4) (B.1) (A.8) (A.9) (A.6) (A.7) (A.11) (A.13) (A.10) (A.12) ) ]—we find the The associated . , 139 lly vanish. We take , m, m, 41 2i )=0 )=0 − p p ( ( imaginary. −} −} , , )= )= +2i )identica + + p p p S { A { ( ( ( λ λ A ) ) −} −} Spin-half fermions with mass dimension one , , λ p p ) + + ( ( A { S { p ( λ λ −} −} , , S ) ) stacks.iop.org/JCAP/2005/i=07/a=012 + + λ p p S { A { ( . . ( . ults in } } λ λ ∗ ∗ ∗ + + , , are intrinsically ) ) ) . 0 0 )and A {− S {− 0 ( ( ∗ ( λ λ p ± L ± L ( ± L ) Elko S φ φ spinors φ m, 0 m, λ ( 07 (2005) 012 ( . ) )thenres 2i i ± L , ∓ ∓ p , , ∓ (p) − φ ∗ ( ± 3.2 λ )gives A = = = , . =0 = λ ∗ ∗ ∗ ± ∗ ∗ (0)) 3.7 )=0 )= )=0 )= +2i ) ) ) ± L )) )) p p p p = 0 0 0 ( ( ( ( ( ( 0 0 φ ( ∗ lf-conjugate spinors reads } } } } ( ( )) = 1 )) = 0 relations for the self-conjugate spinors: ± L ± L ± L + + + + 0 0 ± L ∓ L ) , , , , φ φ Θ( ( ( φ φ φ 0 S {− S {− A {− A {− ± L ∓ L ( ( ( 2 2 λ λ λ λ φ φ p ± L p to be zero—a fact that we explicitly note [ p ) ) ) ) ). σ σ ( ( · · φ † † † † · 2 p p p p 1 ( ( ( ( from equation ( )) )) )) )) ϑ Θ 3.8 Θ − } } 0 0 0 0 p ∗ σ + −} + −} ( ( ( ( σ Θ , , , , · 1 σ ± L ± L ± L ± L + + Θ − σ ∗ S {− S { A {− A { and φ φ φ φ Θ. So σ λ λ λ λ ( ( − ( ( Θ Θ 1 jugating equation ( − ϑ ombinations of the type bi-orthonormality = licity properties of 1 − lds equation ( nally, left-multiplying both sides of the preceding equation by Θ, and moving Θ through ,yie Appendix B. Elkology details Journal of Cosmology and Astroparticle Physics B.1. Bi-orthonormality relations for On setting A.2. He Complex con as they imply following But Θ Fi p Their counterpart for anti-se Substituting for Additional helpful relations are Or, equivalently, while all c note that the bi-orthogonal norms of the JCAP07(2005)012 . } 64 ) p ( (B.9) (B.6) (B.7) (B.5) )and (B.10) (B.11) (B.12) A p ) ( ,ρ ) ). In the −} p , ( + W S S { ρ ρ { , . ) ) p p aginary. ( ( } } . . I + + I , , . )(B.8) ) = A {− S {− nors: = p p λ λ ( ( ]. i )areim } − p + −} ) , , ) )spi ( 139 p + Spin-half fermions with mass dimension one p p , ( A M ion space. Therefore, there cannot S {− S { ( ( } ρ ρ λ ρ )= +i )= + ) ) 41 , −} , , p p p p stacks.iop.org/JCAP/2005/i=07/a=012 ( ( + ( ( A {− S { } } } 39 λ ρ , + + + −} tent with this observation, we find that ) , ) , , , ) )and ) p p incorporate the self- as well as anti-self- + 38 p ( A {− S {− p A {− A { ( p ( } ( ( . λ λ + λ −} , −} nors through the following identities: , , −} , = = + nors. In the massless limit, nfirms that a physically complete theory of + S {− A { + A { ), these spinors are given by A { iΘ λ λ m m λ ) ρ 07 (2005) 012 ( ,ρ ,ρ )spi real, while the 2 2 ) ) ) M +iΘ − p − )co p )spi ( p p p − − ( I I rticularly simple orthonormality, as opposed to bi- ) ( ( ( )+ p λ p −} ( p , B.5 , ( −} −} −} ( )are ρ , , , ) + } S { p + )= )= + + −} p + , S { ( S { A { , λ ( p p + λ λ λ ( ( iΘ) A S M i A { , S {− } − .Apa S W λ λ ρ + −} − − ) , , ) ) λ +iΘ )+ I + p p p ( A {− A { I ( p S ( ( ( ρ ρ } } ion are in the spirit of [ } ) ) − )= )= +i −} + + , , , + p p p p , + 2) is a four-dimensional representat )= ( ( ( ( S { A {− / A {− } p 1 2 A {− ned above are in the Weyl realization (subscripted by λ 1 λ ( + −} λ −} −} ρ , , ) , , , A ) , = + + + p S S {− S { M p A { S { spinors ( (0 ( } λ λ λ ρ − S + ρ ), or with the physically and mathematically equivalent set, } ⊕ + } , + nors are related to the , )obtai p (p) 0) ( identically vanish S {− p , ρ S {− than four independent spinors. Consis ( A λ 2 ) λ A / )spi , p ,λ S p ( ) ( λ p The completeness relation ( ρ m −} ( 1 , m 1 S 2 2i + A { λ Journal of Cosmology and Astroparticle Physics B.3. Elko in the Majorana realization The B.2. The Now, (1 − completeness relation is Majorana realization (subscripted by orthonormality, relation exists between the One is also freethese to two choose sets. some appropriate combinations of neutral particle spinors from The results of this sect where Calculations show that the − An associated completeness relation also exists, and it reads Using thesecompleteness identities, relations for one the may immediately obtain the bi-orthonormality and ρ fundamentally neutral particle spinors must conjugate spinors.{ However, one has a choice. One may either work with the set be more the JCAP07(2005)012 65 (B.20) (B.13) (B.17) (B.19) (B.14) (B.15) (B.16) (B.18) nserted ) ,    ) y, the whole θ ) θ E cos( i 0 .Havingi sin( finally produced. p A − p φ lishes the same result A − +i )is ) 5.18 . )estab θ .Consequentl , , )e , A ) ) θ ) S 0 G 3.15 cos( 2 ( } p ) Spin-half fermions with mass dimension one E S −G sin( p + + , φ m A I − I i p S {− 2 − m )( + − ) )( i e ¯ stacks.iop.org/JCAP/2005/i=07/a=012 λ )and( p p ) − m − E 0 ), the result ( ( + lds ( − + 3.14 replaced by m = )i E −} E , 2 θ rward. We perform it here explicitly for the † E 12.5 has to be replaced by + ( S )yie ), ( S { )) )0 S =( =( ). λ θ 0 cos( † T ( † A E B.6 3.13 − i 0 + L p ) 07 (2005) 012 ( ) ) )=i ) sin( 5.19 φ φ p − p p p . ( 0 ( ( nary’ Dirac bars. By virtue of the identities )( p ( − +i A α S α erscript 0 ( e S α λ ( λ −} S A β ¬ , λ − I − L )and( λ + ) A ) ) φ S { S p p p ¯ ) λ y‘ordi ) ( ( ( − ) θ 3.12 S p S α A α S α † 0 I ) −A λ λ λ ( − A ogether with ( θ ). )) } ( ), ( = cos( S β 0 + −} −} −} )withsup , E )t , , , 0 ( 2 λ i p + + + sin( − L m σ 5.54 S {− B.7 { { { i , , , φ ∗ + φ λ p i } } } an overall sign and ) 3.18 − B.14 S T − )( + + + , , , e 0 A ) ( {− {− {− := = ( )containsonl p    + L 2 = = = ( α α α φ S σ S A β λ the result is displayed in ( )and( =2 B.14 S ) p as defined in ( ogether with ( ( B.13 −A S β G )t λ In a similar fashion the following spin sums may be evaluated: The definition ( The ‘twisted’ spin sums relevant to non-locality turns out as = A β 3.17 with Journal of Cosmology and Astroparticle Physics B.4. Spin sums The evaluation of the spin sums is straightfo self-dual spinors and( sketch briefly the result for the anti-self-dual ones. The definition with A In conjunction with the dispersion relation ( Note that ( and one obtains and expression acquires as in ( JCAP07(2005)012 66 .      (B.26) (B.21) (B.22) (B.23) (B.27) (B.28) (B.24) (B.25) ) ]inthe ) θ 2 1 +i ) θ m  k 140 cos( + − +i 1 p sin( 0 2 2  φ . p i p i . ) k i ) − k 1 2 − is nothing but the ( −  − k ie − k ( k  − i πδ 1 πδ − i − 1 2 − ) λ 2 0 = θ − 2 k = ) →  lim  ) m ) θ 2 k cos( ) d + 1 +i k d 2i k p ( 2 ( 0 k sin( φ p ( neral result = Spin-half fermions with mass dimension one → πδ p πδ 0  lim +i x → )  lim +i ie = − +i  llows from i stacks.iop.org/JCAP/2005/i=07/a=012 − − x k ) 2 , , 1  k 1 i( )fo ) ) i +1) k .Thege $ ) − p p − ) λ θ λ e k x p )0 2 i( − + ) aluated by methods described in [ P θ Γ( P − B.27 − − x m 2 ( cos( e ( E E x − ) − , , θ  + ) A β p sin( 0i λπ/ − ) i ¬ 07 (2005) 012 ( k λ 2 +i 2 φ x p ( i k p ) k † i  i( − ( )=0 )=0 )=2( )=2( e πδ − =ie +i ) ie p p p p k πδ ( ( ( ( p x i( )canbeev − } } +i ∞ e d − 0 } −} + + −} +i , , , , entation of ( ( x A A A A ) / / / / ) + + x  1 k S β 8.16 S S S S { {− {− { θ 1 2 2 η, η ¬ λ +i λ λ λ λ ∞ 2 1 0 k k { ) † † † † 0 i( → θ m P  lim cos( e ) ) ) ) λ + 1 k p p p p p 2i P 0 x d )+ ve repres ( ( ( ( sin( i d 2 2 φ 0 } } p p pplied to obtain ∞ 0 ( p − + −} + −} → +i 0 , , , ,  i 1 A A A A 2i → A β / / / / + +  lim ie 0 S S S S {− { {− { ¬ 0 λ := lim := k λ λ λ λ − denotes the principal value. In our case the quantity → † d →  lim d  lim x x     P d d ) = = p ( kx kx =2 S β .Analternati =1canbea ¬ r λ sin sin λ x x " ∞ ∞ 0 0 β Journal of Cosmology and Astroparticle Physics context of the Fourier transformation of B.5. Distributional part of An integral such as that in ( and and radius with Finally, the identities may be useful in various contexts. The symbol JCAP07(2005)012 = O as 67 p ). It Elko p σ × (B.29) (B.31) (B.30) (B.32) , n ) I trivially tandard particles ). Because A := =( l µ Elko σ he operator φ, B.32 ersion relations mplying further d lly there will be / p .Forthes . with particles may probe )d A µ ile for θ σ rtue of ( =0 µ particles may probe the } sin B Elko p / ,wh · by vi = )appears. A )ortwodisp l ) Elko is not independent from still hold. As t , B B µ =(1 φ σ nential representation has been 12.5 12.5 {A µ d n Spin-half fermions with mass dimension one 5.3 in general. A possibility for non- vanishes without i .Thus,generica , / and B properties which is proportional to A on the matrix p p O B µ we obtain σ )d stacks.iop.org/JCAP/2005/i=07/a=012 = µ , θ not )by and A B, general also a non-standard dispersion )exp( ˜ CPT B µ trast to the Dirac case. A different way of Elko sin but for specific cases other representations 2.3 = σ ]=0 / .Theexpo µ n A A, B A · , is 1, else it is 2. The first case is trivial and =exp . ich is orthogonal to the direction of propagation, =(1 A, B σ 2) ,incon 12 , O )and( / can be derived from 07 (2005) 012 ( 0 =0 wh n 1 , p A ˜ , [ B n (0 2.2 1 µ component, =exp( − respect to it. Note that in the Dirac case n encoding the addressed to what extent µ and p for Elko and non-standard dispersion relations γ | D A 5 , 2 is sensitive to it. Consequently, p ). Clearly, the explicit form of the dispersion relations will O γ | ,where with −DD y–momentum space or deformations of the Lorentz group O A, κ ± =0 A I the rank of σ ) ally has half rank because the lower block linearly depends on I 12.6 } µ ). Thus, the determinant of I he standard dispersion relation ( p σ ecified operators · µ tors which can be considered as ‘natural’ in the context of )= p ˜ 5.30 =exp B σ =det , D∝ concreteness we suppose 0) , ±G )maxim 2 O pera / )onlyt {A I . ( ). We emphasize that (1 )exp( dependence of φ 1 2 n det κ µ depends on a direction 5.33 2.3 , φ σ G µ , ), ( in ( A lly has a non-vanishing =(0 2.2 trix as defined in ( O and to decompose µ ]=0 n θ D 0 In the light of this angular dependence we consider the possibility of a boost operator Regarding the multiplicity of the dispersion relations it should be noted that the Finally, the question will be For the sake of d but it is independent from / typica =exp( ,σ osen in order to make invertibility manifest, , p A particles has been addressed in section standard boost o d commutes with all these projections,[ while for B has block form with mutually commuting non-singular entries, its determinant is given by writing the projection operators is Journal of Cosmology and Astroparticle Physics B.6. On the The ma D it depends on relation. To this end let us replace ( different from the standard one, implying in with with non-commutativity of energ different from the wayto Dirac the particles behaviour of do. the Such matrix a difference, if any, can be traced back the unit matrix onlybut for the Dirac spin particles. sum operator The dispersion relation is independent from it, with some asch yet uns might be more useful. All identities derived in section p is useful to introduce an adapted orthogonal Dreibein matrix restrictions. either one dispersion relationwith with multiplicity 1 multiplicity as 2 in as ( in ( aspects of non-standard dispersion relations in a way different from Dirac particles. the upper one. If depend on the choice of the boost operator, but may arise only for very special choices of choice ( Therefore, for Dirac particles JCAP07(2005)012 68 541 22 (B.33) (B.34) (B.35) .The 2002 ] ) } , 1964 ) 083509 Mod. Phys. , ,t as Lounesto’s  70 /0410119 x D 2001 ( th SPIRES Int. J. Mod. Phys. , Astrophys. J. )andthesame Elko p- ¬ η one obtains after , , he ) 0 astro-ph/0412059 110 [ 2002 1983 , γ gr-qc/0411131 B.33 laces ,t , 6 5 , } . on and Breach) p 37 x γ ) 5 ( ge: Cambridge University 2 Phys. Rev. Lectures of the Istanbul γ ( ,t γ 2  ¬ Class. Quantum Grav. Preprint η ) γ 2 Preprint x { /0304245 ) ( r Preprint 2004 2 th ( , d, which p r δ 2005 2004 ,η p- ( ) ] 2 2004 , (Cambrid , δ 2427 and he , 2004 ,t 2 tational bosons , } x 1 Spin-half fermions with mass dimension one 361 ) ( Helv. Phys. Acta mπr 1 η ,t ]appeare  2 { expectation values, mπr astro-ph/0401341 0 x 2 Preprint stacks.iop.org/JCAP/2005/i=07/a=012 undations 1933 141 ( − γ † astro-ph/0301505 , ursey (New York: Gord of non-locality discussion 1 Fo − ,η γ 1 2003 ) entary Particle Physics 0 ∼− ). By virtue of the identities γ , astro-ph/0204521 γ Preprint ] ,t 0 an dynamics as an alternative to dark matter vol 1 0 ][ γ x γ mr ( Preprint )edFG¨ ] an conjugation and thus provides 8.20 † − On the spin of gravi (erratum) ( mr 2004 ] e ). η ) , − 3 { Phil. Trans. R. Soc. econtext 07 (2005) 012 ( e ,t 2003 ds in Elem  3 SPIRES mr , [ of Fields x 8.20 ). SPIRES ( 069901 ] mr [ 2003 † mπr tian and anticommute with each other. It should be /0101009 , tueckelberg field 1+ 263 71 4.16 4 mπr ,η th (1/2,1/2) representation space: an ab initio construct ) 1+ astro-ph/0412378 4 D − [ 40 eS Modified Newtoni p- 1385 ,t )byHermiti ] he Th = = x um Theory ( 495 matrices, 148 ][ gravitation theory for the MOND paradigm of the Newtonian dynamics: implications for galaxies † antum gravity at astrophysical distances? & & ] he negative of ( 8.20 η γ and rotational curves in dark galactic halos oretical Physics (1962) Space inversion, time reversal, and other discrete symmetries in local field 433 Qu gr-qc/0205072 ' ing part as (       0 he ][ phasizes its differences and similarities with the Majorana spinors. The authors also Phys. Rev. (  γ ures on dark matter ) ) ] SPIRES ile this paper was being proof read, [ fT [ tter: early considerations tter: introduction ∼ ,t The Quant ,t  The mass of the missing baryons in the x-ray forest of the warm-hot intergalactic lo  Nature Phys. Rev. Relativistic Dark matter on galactic scales (or the lack thereof) ts the same behaviour of the distributional part as ( , x x  )isjustt oo SPIRES ( ( ) .Wh [ † Vector field 1377 Unitary representations of the inhomogeneous Lorentz group including reflections gr-qc/0404042 astro-ph/0403694 ch Amodification matrices are Hermi ¬ SPIRES η ,t 2005 1966 TASI lect nd Weyer H, [ et al  ][ ][ Dark ma Dark ma , ,η Spectral displacement of extra galactic nebulae 16 , Cosmography: cosmology without the Einstein equations , rS ) ) x B.33 γ ftheremain ( A 1621 ,t ,t 371 ¬ η x x means equivalence of corresponding vacuum mme 11 , ( ( ) † SPIRES SPIRES [ D [ medium Ann. Rev. Astron. Astrophys. Group Theoretical Concepts andSu Metho 270 theories Lett. 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Notes SPIRES 2000 D [ ed 263 , /9307118 522 SPIRES /9409134 [ 821 ] SPIRES th A th 977 SPIRES 520 p- [ p- SPIRES ] 106 (New York [ B he 73 BEYOND 2003: he 171 [ (erratum) ition and analysis of ][ ][ 063519 307 Ann. Math. SPIRES Phys. Rev. World Sci. L ett. 14 3082 [ 207 entation spaces: 679 2003 67 , 77 107 /0312171 33 D gapore: World Scientific) 378 ex/0408008 1939 Support of evidence for 1997 1991 , tt. 5854 SPIRES ph , , p- Phys. Rev. [ SPIRES repres ] Phys. L [ p- (Sin ) he 86 Data acquis .Le Evidence for neutrinoless double he ,j 439 oscillation from the LSND neutrino oscillations from LSND 1957 tt. Nuovo Cim. ][ (0 2001 , Rev. Mod. Phys. 9 1855 e e Nucl. Instrum. Meth. , ¯ Phys. Rev. ν ν ⊕ und. Phys. Phys. Rev. A Phys. Rep. .Le /0201231 ] 11 ] aDecay 0) → → 1937 Preprint Fo Spin-half fermions with mass dimension one 2001 ph A tt. 2004 , µ µ j, Phys. 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