I am glad to hear back from the inventor @Dharam Vir Ahluwalia of ELKO! And I really appreciate the chance to have a in-depth discussion with the true experts of the ELKO theory.
There is a growing body of literature on the ELKO spinors (see references here), which are alleged to be mass dimension one fermions and can be a dark matter candidate.
But is the ELKO spinor a red herring? Is the mass dimension one fermion term irrelevant at the standard model energy scale?
A Dirac type fermion Lagrangian can be written as $$ L = i\bar{\psi}\not D\psi - m\bar{\psi}\psi $$ while the Lagrangian for ELKO type fermion is $$ L = \bar{\psi}\partial^\mu\partial_\mu\psi - m'^2\bar{\psi}\psi $$ Actually the most general fermion Lagrangian should read $$ L = i\bar{\psi}\not D\psi + M^{-1}\bar{\psi}\partial^\mu\partial_\mu\psi- m\bar{\psi}\psi \tag{1} $$ (or equivalently: $$ L = iM\bar{\psi}\not D\psi + \bar{\psi}\partial^\mu\partial_\mu\psi- m'^2\bar{\psi}\psi \tag{2} $$ where $m'^2 = Mm$. it's just a matter of re-scaling the fermion field.)
The ELKO kinetic term $M^{-1}\bar{\psi}\partial^\mu\partial_\mu\psi$ observes the Lorentz symmetry, so ideally it should be included in the modern effective quantum field theory framework.
The key question here is the magnitude of the ELKO term $M^{-1}\bar{\psi}\partial^\mu\partial_\mu\psi$. How large should $M^{-1}$ be? The naturalness principle tells us that $M$ should be of the Planck scale $$ M \sim M_{Planck} $$ so that the ELKO term is drastically suppressed by the order of $$ \frac{p}{M_{Planck}} $$ where $p$ is the momentum/energy scale of the physics process in concern.
Additionally, the ELKO term $M^{-1}\bar{\psi}\partial^\mu\partial_\mu\psi$ breaks the axial symmetry $$ \psi \rightarrow e^{\theta i\gamma_5}\psi $$ Hence this term is further suppressed due to t' Hooft's technical naturalness argument, analogous to the suppression of the axial-symmetry-breaking fermion mass term $m\bar{\psi}\psi$.
With that, shall we regard the ELKO term as irrelevant, unless you are dealing with Planck scale quantum processes where all bets are off.
Reply to @Dharam Vir Ahluwalia comment: "the presented Lagrangian has dimensionality mismatch between various terms".
As to "dimensionality mismatch", that is why I included the mass parameter $M$ in equation (1) and (2). This parameter $M$ plays the central role in my argument that the ELKO term is diminishingly small compared with the normal Dirac spinor term. And therefore, the ELKO term can be regarded as virtually nonexistent at sub-Planck energy scales.
