Physics Letters B 760 (2016) 164–169 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb A Lagrangian for mass dimension one fermionic dark matter Cheng-Yang Lee Institute of Mathematics, Statistics and Scientific Computation, Unicamp, 13083-859 Campinas, São Paulo, Brazil a r t i c l e i n f o Article history: Received 16 January 2016 Received in revised form 25 June 2016 Accepted 26 June 2016 Available online 30 June 2016 Editor: S Dodelson Keywords: Elko Fermionic dark matter Mass dimension one fermions a b s t r a c t The mass dimension one fermionic field associated with Elko satisfies the Klein–Gordon but not the Dirac equation However, its propagator is not a Green’s function of the Klein–Gordon operator We propose an infinitesimal deformation to the propagator such that it admits an operator in which the deformed propagator is a Green’s function The field is still of mass dimension one, but the resulting Lagrangian is modified in accordance with the operator © 2016 The Author Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Funded by SCOAP3 Introduction The theoretical discovery of Elko and the associated mass dimension one fermions by [2,3] is a radical departure from the Standard Model (SM) These fermions have renormalizable selfinteractions and only interact with the SM particles through gravity and the Higgs boson These properties make them natural dark matter candidates Since their conceptions, Elko and its fermionic fields have been studied in many disciplines The graviational interactions of Elko have received much attention [8,9,11–18,20,23,29,36,46,47,49] while its mathematical properties have been investigated by da Rocha and collaborators [10,22,24–28,38] These works established Elko as an inflaton candidate and that it is a flagpole spinor of the Lounesto classification [42] thus making them fundamentally different from the Dirac spinor In particle physics, the signatures of these mass dimension one fermions at the Large Hardon Collider have been studied [7,30] In quantum field theory, much of the attention is focused on the foundations of the construction [5,6, 19,32–34,41,43–45] Their supersymmetric and higher-spin extensions have also been carried out by [40,50] An important result is that the fermionic field and its higher-spin generalization violate Lorentz symmetry due to the existence of a preferred direction This led Ahluwalia and Horvath to suggest that the fermionic field satisfies the symmetry of very special relativity [4,21] One question remains unanswered in the literature What is the correct Lagrangian of the mass dimension one fermion? Since the field is constructed using Elko as expansion coefficients which satisfy the Klein–Gordon equation, the naive answer would be the Klein–Gordon Lagrangian But this has two unsatisfactory aspects Firstly, the resulting field-momentum anti-commutator is not given by the Dirac-delta function Secondly, the propagator is not a Green’s function of the Klein–Gordon operator We propose an infinitesimally deformed propagator such that it is an Green’s function to an operator The resulting Lagrangian determined from the operator does not have the above mentioned problems and is still of mass dimension one The Elko construct We briefly review the construction of Elko and its fermionic field For more details, please refer to the review article [1] Elko is a German acronym for Eigenspinoren des Ladungskonjugationsoperators They are a complete set of eigenspinors of the chargeconjugation operator of the ( 12 , 0) ⊕ (0, 12 ) representation of the Lorentz group The charge-conjugation operator is defined as C= O −i −1 −i O K where K complex conjugates anything to its right and spin-half Wigner time-reversal matrix = −1 σ −1 = −σ ∗ is the (2) Its action on the Pauli matrices E-mail address: cylee@ime.unicamp.br (1) σ = (σ1 , σ2 , σ3 ) is (3) http://dx.doi.org/10.1016/j.physletb.2016.06.064 0370-2693/© 2016 The Author Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Funded by SCOAP3 C.-Y Lee / Physics Letters B 760 (2016) 164–169 The complete set of Elko is constructed from a four-component spinor of the form ϑ φ ∗ (p, σ ) χ (p, α ) = φ(p, σ ) (4) where φ( , σ ) is a left-handed Weyl spinor in the helicity basis with = lim|p|→0 pˆ and σ · pˆ φ( , σ ) = σ φ( , σ ) (5) so that σ = ± 12 denotes the helicity Here α = ∓σ denotes the dual-helicity nature of the spinor with the top and bottom signs denoting the helicity of the right- and left-handed Weyl spinors respectively The spinor χ (p, α ) becomes the eigenspinor of the charge-conjugation operator C with the following choice of phases C χ (p, α )|ϑ=±i = ±χ (p, α )|ϑ=±i (6) thus giving us four Elkos Spinors with the positive and negative eigenvalues are called the self-conjugate and anti-self-conjugate spinors They are denoted as C ξ(p, α ) = ξ(p, α ), (7a) C ζ (p, α ) = −ζ (p, α ) (7b) There are subtleties involved in choosing the labellings and phases for the self-conjugate and anti-self-conjugate spinors The details, including the solutions of the spinors can be found in [6, sec II.A] The Elko dual which yields the invariant inner-product is defined as [1,48] ¬ ξ (p, α ) = [ (p)ξ(p, α )]† , (8a) ¬ ζ (p, α ) = [ (p)ζ (p, α )]† (8b) where † represents Hermitian conjugation and is a block-offdiagonal matrix comprised of × identity matrix = O I I O The matrix (p) = (9) 2m α ¯ p, α ) − ζ (p, α )ζ¯ (p, α ) ξ(p, α )ξ( (10) The bar over the spinors denotes the Dirac dual The dual ensures that the Elko norms are orthonormal ¬ ¬ ξ (p, α )ξ(p, α ) = −ζ (p, α )ζ (p, α ) = 2mδαα (11) and their spin-sums read ¬ ξ(p, α ) ξ (p, α ) = m[G (φ) + I ], (12a) α ¬ ζ (p, α ) ζ (p, α ) = m[G (φ) − I ] (12b) α where G (φ) is an off-diagonal matrix ⎛ ⎜ ⎜ G (φ) = i ⎜ ⎜ ⎝ eiφ p = |p|(sin θ cos φ, sin θ sin φ, cos θ) (14) where ≤ θ ≤ π and ≤ φ < 2π Multiply eqs (12a) and (12b) with ξ(p, α ) and ζ (p, α ) from the right and apply the orthonormal relations, we obtain [G (φ) − I] ξ(p, α ) = 0, (15a) [G (φ) + I] ζ (p, α ) = (15b) Since these identities have no explicit energy dependence, the corresponding equation in the configuration space has no dynamics and therefore cannot be the field equation for the mass dimension one fermions Nevertheless, writing the above identities in the configuration space for λ(x) is non-trivial and is a task that must be accomplished in order to derive the Hamiltonian This issue is addressed in the next section Identifying the self-conjugate and anti-self-conjugate spinors with the expansion coefficients for particles and anti-particles, the two mass dimension one fermionic fields and their adjoints, with the appropriate normalization are d3 p (x) = (2π )−3/2 [e −ip ·x ξ(p, α )a(p, α ) 2mE p α + e ip ·x ζ (p, α )b‡ (p, α )], d p (x) = (2π )−3/2 ¬ (16a) [e ip ·x ξ (p, α )a‡ (p, α ) ¬ 2mE p α + e −ip ·x ζ (p, α )b(p, α )], ¬ λ(x) = ¬ λ(x) = (x)|b‡ =a‡ , ¬ 0 −e −i φ eiφ −e −i φ 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ (13) The angle φ is defined via the following parametrization of the momentum (16b) (16c) (x)|b‡ =a‡ (16d) ‡ Here a(p, α ) and b (p, α ) are the annihilation and creation operators for particles and anti-particles They satisfy the standard anti-commutation relations {a(p , α ), a‡ (p, α )} = {b(p , α ), b‡ (p, α )} = δα α δ (p − p) (p) is defined as 165 (17) Note that for the creation operators, we have introduced a new operator ‡ in place of the usual Hermitian conjugation † This follows from the observation that since the Dirac and Elko dual are different, it suggests that the corresponding adjoints for the respective particle states may be different Assuming they are different, it may then become necessary to develop a new formalism for particles states with the new ‡ adjoint in parallel to [31] This is an important issue that deserves further study but since it does not affect our objective of deriving the Lagrangian, we shall leave it for future investigation The Lagrangian: defining the problem There are two reasons why the Klein–Gordon Lagrangian are unsatisfactory for the mass dimension one fermions Firstly, the field does not satisfy the canonical anti-commutation relations (CARs) since the field-momentum anti-commutator is not equal to i δ (x − y) I Instead, it is given by1 {λ(t , x), πkg (t , y)} = i d3 p (2π )3 e −ip·(x−y) [ I + G (φ)] (18) For the rest of the paper, we will be working with λ(x), but the results hold for (x) also 166 C.-Y Lee / Physics Letters B 760 (2016) 164–169 ⎛ ¬ where πkg = ∂ λ/∂ t is the conjugate momentum of the Klein– Gordon Lagrangian Secondly, the propagator obtained from the fermionic time-order product is ¬ S (x, y ) = | T [λ(x)λ( y )]| =i d4 p (2π )4 e −ip ·(x− y ) I + G (φ) p · p − m2 + i d p (∂ μ ∂μ + m2 ) S (x, y ) = −i (2π )4 e −ip ·(x− y ) [ I + G (φ)] (19) (20) In eqs (18) and (20) the problem resides in the three and fourdimensional Fourier transform of G (φ) which are non-vanishing 3.1 The inverse of I + G (φ) In a consistent field theory, we expect the fields to satisfy the canonical anti-commutation relations and that the Lagrangian and propagator to have the same symmetry In this respect, it is evident that a Klein–Gordon Lagrangian is inadequate to describe the mass dimension one fermions Here we derive the Lagrangian that provides a complete description of λ(x) We start by determining the operator O (x) in which the propagator given by eq (19) is a Green’s function O(x) S (x, y ) = −i δ (x − y ) (21) According to eq (20), this can be achieved by determining the inverse of I + G (φ) and the corresponding operator in the configuration space However this matrix is non-invertible since det[ I + G (φ)] = (22) This problem can be bypassed by considering a more general matrix I + τ G (φ) where τ is a real constant Its inverse is [ I + τ G (φ)]−1 = I − τ G (φ) − τ2 (23) The singularities can be avoided by taking the limit τ → ±1 This is possible since a simple calculation shows that the inverse is well-defined for all values of τ [ I + τ G (φ)]−1 [ I + τ G (φ)] = − τ2 1−τ I=I (25) The inverse is then given by eq (23) Details on the mathematical foundation of the proposed deformation and inverse can be found in [44] To determine the form of [ I + τ G (φ)]−1 in the configuration space, we first need to define an operator G which corresponds to the matrix G (φ) in the configuration space For this purpose, fractional derivatives must be introduced since the matrix elements of G (φ) are proportional to e ±i φ and that it√can be expressed in terms of complex momenta p ± = ( p ± ip )/ as e ±i φ ± ∓ −1 1/2 = [p (p ) ] (26) There are many definitions of fractional derivatives The fractional derivative that is appropriate for our task is the Fourier fractional derivative [35, pg 562] [37] The general properties of fractional derivatives, including Fourier’s definition, are given in Appendix A The operator G is defined as 1/2 −1/2 ⎞ 1/2 −1/2 −1/2 1/2 ∂− −∂+ ∂− ∂+ −∂+ ∂− ⎟ ⎟ ⎟ ⎟ ⎠ 0 0 √ where ∂± act on the complex coordinates x± = (x1 ± ix2 )/ (27) An interesting property of the Fourier fractional derivative, which turns out to be important for our construct is its non1/2 −1/2 f (x), uniqueness To see what this means, let us consider i ∂+ ∂− an element of G f (x) where f (x) is a test function with the Fourier transform d3 p (2π )3 e −ip ·x F (p) (28) Using eq (A.6), we obtain −1/2 ∂− d3 p f (x) = −1/2 (2π )3 e −ip ·x e −i π /4 e −in− π p + F (p) (29) where we have used the fact that i −1/2 = e −i π /4 e −in− π has two 1/ roots with n− = 0, Acting on the expression again with i ∂+ we obtain 1/2 −1/2 i ∂+ ∂− f (x) = d3 p (2π )3 (i ω)e −ip ·x e −i φ F (p) (30) where ω = e i (n+ −n− )π with n± = 0, Due to the ambiguity of the phases, in order to obtain G (φ) in the momentum space, all the elements of G f (x) must have the same phases so that d3 p G f (x) = (2π )3 ωe−ip·x G (φ) F (p) (31) For functions such as f (x) comprised of a single Fourier transform, ω is a global phase and its value is unimportant But since λ(x) is a sum of the Fourier transform of the self-conjugate and anti-self-conjugate spinors, the action of G on the field yields2 G λ(x) = (2π )−3 d3 p 2mE p α [e −ip ·x ωξ ξ(p, α )a(p, α ) − e ip ·x ωζ ζ (p, α )a‡ (p, α )] (24) where we have used the identity G (φ) = I Therefore, to obtain the inverse of I + G (φ), we must first perform a τ -deformation I + G (φ) → I + τ G (φ) 0 f (x) = −1/2 1/2 ∂+ ∂− This is not a Green’s function of the Klein–Gordon operator ⎜ ⎜ G =i⎜ ⎜ ⎝ (32) Since the phases can take the values of ±1, G λ(x) has four possible solutions Out of the four possibilities, as we will show in the next section, the only solution which yields a positive-definite free Hamiltonian is ωξ = −ωζ = (33) This then gives us the equation G λ(x) = λ(x), (34) which is the counterpart of eqs (15a) and (15b) in the configuration space Having defined G , the inverse of I + τ G (φ) in the configuration space is given by A= I −τ G (35) − τ2 Now we apply A to eq (20) The τ -deformed propagator is ¬ In this paper, we assume that the fermionic field λ(x) and its dual λ(x), both of which are a sum of the Fourier transform of the self-conjugate and anti-selfconjugate spinors and its dual, are well-defined C.-Y Lee / Physics Letters B 760 (2016) 164–169 d4 p S (τ ) (x, y ) = i (2π )4 e −ip ·(x− y ) I + τ G (φ) p·p − m2 When A acts on the S (τ ) (x, y ), we must take limit τ → 1, we obtain +i (36) ω = so that in the O(x) = A(∂ μ ∂μ + m2 ) = ¬ H= (39) where we sum over the repeated indices The operator A is dimensionless so λ(x) remains a mass dimension one field The field equation is Aab (∂ μ ∂μ + m2 )λb = (40) The operator A does not affect the solutions of the field equation but as we will show in the subsequent section, it ensures that the field satisfies the canonical anti-commutation relations 3.2 Canonical anti-commutation relations The CARs of λ(x) are determined by {λ(t , x), λ(t , y)}, {π (t , x), π (t , y)} and {λ(t , x), π (t , y)} The first anti-commutator identically vanishes [6, sec III.B] When the Lagrangian is Klein–Gordon, the second anti-commutator identically vanishes and the third is given by eq (18) We show that the field associated with eq (39) satisfies the CARs by computing the relevant anti-commutators The conjugate momentum is ¬ ∂ λa πb ( y ) = Aab ( y ) ( y) (41) ∂t which differs from πkg ( y ) by a factor of A( y ) Since fractional derivatives commute, the anti-commutator between the conjugate momentum is {π (t , x), π (t , y)} = O (42) τ -deformed field-momentum anti- lim {λ(t , x), π (t , y)}(τ ) = i δ (x − y) I (43) τ →1 We now show that the free Hamiltonian is positive-definite This is achieved by showing that the free Hamiltonian, as a func¬ tion of λ(x) and λ(x) is identical to the one given in [40, eq (97)] In order to obtain the correct Hamiltonian, we must take the con¬ jugate momentum associated with both λ(x) and λ(x) into account where the later is defined as ∂L ∂λb = −Aab ¬ ∂t ∂ λa /∂ t (46) ¬ (44) ¬ ¬ d3 x (∂ μ λa ∂μ λa − m2 λa λa ), (47) the Hamiltonian simplifies to (38) L = Aab (∂ μ λa ∂μ λb − m2 λa λb ) ¬ λa (x) Using the identity (37) Based on the form of O (x), the Lagrangian for λ(x) is πa = − d3 x Aab (∂ μ λa ∂μ λb − m2 λa λb ) The operator O (x), in which the propagator is a Green’s function of is therefore A direct evaluation of the commutator now yields ¬ lim A(∂ μ ∂μ + m2 ) S (τ ) (x, y ) = −i δ (x − y ) I τ →1 ¬ lim Aab λb (x) = τ →1 167 ¬ d3 x − ∂λa ∂ λa ¬ ¬ − ∂i λa ∂ i λa + m2 λa λa ∂t ∂t (48) This is identical to [40, eq (89)] and is therefore positive-definite Conclusions The propagator and Lagrangian proposed in this paper addressed the outstanding problems of the mass dimension one fermions They preserve the mass dimensionality and renormalizable self-interaction The field satisfies the CARs and the propagator is a Green’s function to the operator given in eq (38) While the mass dimension one fermionic field violates Lorentz symmetry, in light of the new Lagrangian, it is nevertheless a welldefined quantum field in the sense that it has a positive-definite free Hamiltonian, it satisfies the CARs and furnishes fermionic statistics These properties are highly non-trivial They require careful choices of expansion coefficients, adjoints These results strongly suggest that the mass dimension one fermions have a well-defined space-time symmetry The τ -deformed Lagrangian differs from the original Klein– Gordon Lagrangian proposed by Ahluwalia and Grumiller [2,3] On the one hand, it resolves the problem of the CARs But on the other hand, the new Lagrangian suggests that the associated interactions for the fermions must now be functions of A(x) Constructing well-defined interactions may be difficult as the operator has a pole at τ = If it is not removed or cancelled, it can make the scattering amplitudes divergent and thus non-physical Despite the difficulties, the proposed τ -deformation may still be of value to the theory Recently, it was suggested that by constructing a τ -deformed field adjoint, it is possible to obtain a fully Lorentzcovariant theory [45] If the mass dimension one fermionic fields are invariant under very special relativity as proposed by Ahluwalia and Horvath, the effects of Lorentz violation would be minimal since very special relativity is compatible with the null results of the Michelson– Morley experiments and other well-known relativistic effects [4, 39] We would expect discrete symmetry violations and scattering cross-sections involving mass dimension one fermions to have dependence on a preferred direction On a broader picture, this theoretical construct presents an interesting new paradigm Space-time symmetry may be a mere reflection of the symmetry of the rods and clocks comprised of the SM particles Space-time, according to the rods and clocks made of dark matter, may paint a completely different picture The Hamiltonian is then given by the Legendre transformation ¬ H= ∂ λa d x ∂t ∂λb Aab ∂t ∂λb − Aab ∂t Acknowledgements ¬ ∂ λa −L ∂t (45) In obtaining the first term, we have used the definition of A and the integration by parts rule for the Fourier fractional derivative The first two terms can be simplified further since in the limit τ → 1, we have a simple identity I would like to thank N Faustino and G.S de Souza for discussions at the early stage of this work I am grateful to R da Rocha for suggestions and reading the initial manuscript During the revision of the manuscript, I have benefited greatly from numerous discussions with D.V Ahluwalia This research is supported by the CNPq grant 313285/2013-6 168 C.-Y Lee / Physics Letters B 760 (2016) 164–169 Appendix A Fractional derivatives All fractional derivatives satisfy the following properties Let α be an arbitrary real number, in the limit α → n where n is a positive integer, dα dn f ( x ) = f (x) α →n dxα dxn lim (A.1) The operations are linear dα dxα dα dxα dα [c f (x)] = c dxα f (x), [ f (x) + g (x)] = dα dxα (A.2a) f (x) + dα dxα g (x) (A.2b) The Leibniz rule is ∞ dα ( f g) = α dx α j =0 j dα − j f dxα dj dx j g (A.3) where the binomial coefficient is generalized to arbitrary real numbers by the (α ) function α (α + 1) ( j + 1) (α − j + 1) = j (A.4) The Fourier fractional derivative is defined as follows Let f (x) and F (k) be two single-variable functions related by the Fourier transform f (x) = F (k) = ∞ 2π 2π dk e −ikx F (k), (A.5a) dx e ikx f (x) (A.5b) −∞ ∞ −∞ The definition of the fractional derivative on f (x) is a straightforward generalisation of the usual 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[7] A Alves, F de Campos, M Dias, J.M Hoff da Silva, Searching for Elko dark matter spinors at the CERN LHC, Int J Mod Phys A 30, no 01 (2015) 1550006 [8] A Basak, J.R Bhatt, S Shankaranarayanan,... (A. 7) References [1] D.V Ahluwalia, On a local mass dimension one Fermi field of spin one- half and the theoretical crevice that allows it, 2013 [2] D.V Ahluwalia, D Grumiller, Dark matter: a