# A question about the Dirac mass and Majorana mass

I am sorry if my question seems to be naive. For the free Dirac field, the Lagrangian is $$\mathcal{L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m_D)\psi$$ or expressed in the Weyl spinor, the mass term is $\eta^\dagger m_D\chi$. Then my question is that must the parameter $m_D$ be non-negative and real?

Furthermore, for the Majorana mass term $\eta^\dagger m_M\eta^c$, must the parameter $m_M$ be non-negative and real?

Then my question is that must the parameter $m_{D}$ non-negative and real?

It may have different sign, but it has to be real valued. Really, Dirac equation is nothing but a projector of the direct sum $\left(\frac{1}{2}, 0 \right) \oplus \left( 0 , \frac{1}{2}\right)$ of irreducible representations of the Poincare group on the parity-invariant subspace. Projector may be $i\gamma_{\mu}\partial^{\mu} + m$ as well as $i\gamma^{\mu}\partial_{\mu} - m$.

More formal explanation is that both $i\gamma_{\mu}\partial^{\mu} \pm m$ operators lead to correct dispersion relation after multiplying on $i\gamma_{\mu}\partial^{\mu} \mp m$.

But this parameter has to be real since lagrangian has to be real.

must the parameter $m_{M}$ non-negative and real?

It may have arbitrary sign and, in principle, it can be complex (if we don't have specific interaction terms). The reason is that for Majorana case we don't have $U(1)$ global symmetry, so that by redefining the Weyl spinor, namely $\eta$, to $\eta \to e^{i\theta}\eta$, we obtain $e^{2i\theta}$ multiplier, which can absorb complexity of the mass. In the end, however, lagrangian has to be real.