Majorana field Lagrangian in curved spacetime

Question

In this paper(An Exact Cosmological Solution of the Coupled Einstein-Majorana Fermion-Scalar Field Equations), the Majorana field Lagrangian has been stated as

$\mathcal{L}_M = i \bar{\psi} \left(\gamma \wedge *\nabla\right) \psi - i m\bar{\psi}\psi*1$

where $\bar{\psi}$ is the charge conjugated spinor field $\bar{\psi}=\psi^{\dagger} \mathcal{C}$ with charge conjugation matrix, $\mathcal{C}=\gamma_0$.

In contrast to this, the same Majorana field Lagrangian has been written in this paper (Variational Field Equations of a Majorana Neutrino Coupled to Einstein’s Theory of General Relativity) as,

$\mathcal{L}_M = \dfrac{i}{2} \bar{\psi} *\gamma \wedge \nabla \psi + \dfrac{i}{2} m\bar{\psi}\psi*1$

Now even if the mass term is neglected, the rest parts of two Lagrangians aren't seem to be same in both the cases!

What am I missing and what is the correct Lagrangian for a general Majorana field in curved spacetime? Are these two Lagrangians similar (or somehow same) in some sense?

Disclaimer: I am not very much habituated in the exterior calculus, maybe that's why such confusion is arising.

Answer 1

I would write the action (in $n=2,3,4$ mod 8, spacetime dimensions where Majorana's exist) in a more conventional form as
$$ S= \int d^n x \sqrt{-g}\frac 12 \bar \psi {\mathcal C}\gamma^a e_a^\mu \left(\partial_\mu + {\textstyle \frac 12} {\omega^{bc}}_\mu \sigma_{bc}+ m\right)\psi, $$ where $\mathcal C$ is defined independently of the gamma-matrix representation by ${\mathcal C} \gamma^a {\mathcal C}^{-1} =- \gamma^{aT}$. I see no advantage to the differential form language, but it seems popular in some areas. I do think that the factor of two is correct. The $i$'s are metric-signature dependent.

I have a longer answer here: Do I need Gamma matrices in Majorana representation in the Lagrangian of a Majorana fermion?