Fermionic degeneracy and non-local contributions in flag-dipole spinors and mass dimension one fermions

Abstract

We construct a mass dimension one fermionic field associated with flag-dipole spinors. These spinors are related to Elko (flag-pole spinors) by a one-parameter matrix transformation $${\mathcal {Z}}(z)$$ Z ( z ) where z is a complex number. The theory is non-local and non-covariant. While it is possible to obtain a Lorentz-invariant theory via $$\tau $$ τ -deformation, we choose to study the effects of non-locality and non-covariance. Our motivation for doing so is explained. We show that a fermionic field with $$|z|\ne 1$$ | z | ≠ 1 and $$|z|=1$$ | z | = 1 are physically equivalent. But for fermionic fields with more than one value of z, their interactions are z-dependent thus introducing an additional fermionic degeneracy that is absent in the Lorentz-invariant theory. We study the fermionic self-interaction and the local U(1) interaction. In the process, we obtained non-local contributions for fermionic self-interaction that have previously been neglected. For the local U(1) theory, the interactions contain time derivatives that renders the interacting density non-commutative at space-like separation. We show that this problem can be resolved by working in the temporal gauge. This issue is also discussed in the context of gravity.

Chiral phases for massive fermions and spinor classifications

“…Taking the Dirac spinors from [1] with ψ = ψ † γ 0 , explicit calculations show that they belong to the 2nd class. The 4th class is a complete set of Majorana spinors also known as ELKO and the 5th class is obtained by a one parameter transformation of ELKO [5,6]. The fermionic field constructed from ELKO yields a mass dimension one field.…”

“…The fermionic field constructed from ELKO yields a mass dimension one field. For more details see [3][4][5][6][7][8][9][10] and references therein. In [2], Lounesto actually identified the 1st-3rd class to be the Dirac spinors.…”

“…Secondly, as we will show, for certain interactions, the observables are functions of α and β. 6 We start with the Yukawa interaction of the form…”

“…The Dirac spinors belongs to the 2nd class. ELKO (a complete set of Majorana spinors) and its generalisation belongs to the 4th-5th class [3][4][5][6]. The Weyl spinors belongs to the 6th class.…”

“…The fermionic fields for Dirac and Weyl spinors describe the massive and massless fermions in the SM. The fermionic fields for ELKO describe a mass dimension one fermion with renormalisable quartic self-interaction making it a potential dark matter candidate [3][4][5][6][7][8][9][10]. In light of existing works, a natural question arises -What are the physics of the 1st and 3rd class?…”