General mass term for Majorana fermions

Question

I have a question regarding the most general mass terms for Majorana fermions.

In the literature it is often stated that the most general Majorana mass term is

$$ \mathcal{L}^M_{\rm mass} = -m_D\overline{\psi}_L\psi_R - \frac{1}{2}m_L\overline{\psi^c_L}\psi_L - \frac{1}{2}m_L\overline{\psi^c_R}\psi_R $$ where $\psi^c_L = (\psi_L)^c$ and $\psi^c = C\overline{\psi}^T$ is the standard charge conjugation. By using the standard properties of $C$ one can find that $$ \psi^c_{L/R} = (\psi_{L/R})^c = C\overline{P_{L/R}\psi}^T = C\gamma^0P_{L/R}\psi^* = P_{R/L}C\overline{\psi}^T = (\psi^c)_{R/L} $$ However, since Majorana-fermions obey the Majorana condition $\psi^c = \psi$, we would have $\psi^c_{L/R} = \psi_{R/L}$. Thus we could re-write both the Majorana mass terms into the same form as the Dirac mass term, so it is ambiguous to write the Majorana mass terms.

Obviously I am wrong somewhere in the argument here. I think it has to to with that $\psi^c_L$ isn't actually the same object as $\psi_R$, but that we form the Majorana $$ \Psi_L = \begin{pmatrix}\psi_L \\ -i\sigma_2\psi_L^*\end{pmatrix} = \begin{pmatrix}\psi_L \\ \psi_L^c\end{pmatrix}, \,\Psi_R = \begin{pmatrix}i\sigma_2\psi^*_R \\ \psi_R\end{pmatrix} = \begin{pmatrix}\psi^c_R \\ \psi_R\end{pmatrix} $$ but I am not sure, so I seek the knowledge of this forum. Thank you in advance for any contributions!

Answer 1

When it comes to "Majorana fermion", one has to be very careful, since there are two conflicting definitions:

• The old-fashioned definition according to Majorana: Majorana-fermions obey the Majorana condition $\psi^c = \psi$, which means that the right-handed and left-handed portions of the fermion are related via charge conjugation $\psi^c_{L/R} = \psi_{R/L}$.
• The modern definition: the fermion is endowed with Majorana mass $m_R\overline{\psi_R^c}\psi_R$ which implies lepton-number violation and can be verified experimentally by neutrinoless double-beta decay. In this case, the right-handed and left-handed portions of the fermion could be independent which allows for a separate Dirac mass $m\overline{\psi_L}\psi_R \neq m\overline{\psi_R^c}\psi_R$ since $\psi_{L} \neq \psi^c_{R}$.

You are confused because you are simultaneously invoking the two contradicting definitions of "Majorana fermion". To avoid trouble, you are strongly encouraged to pick only one definition and stick with it. See more explanations here.