Why do the $1/2$ factor appear in the Majorana mass Lagrangian? In case of Dirac neutrino there is no $1/2$ factor in the mass Lagrangian but for Majorana type neutrino there is a half factor in the mass Lagrangian.
 A: The real reason is in following.
Let's assume Majorana field:
$$
\Psi_{M} = \Psi_{L} + \hat{C}\bar{\Psi}^{T}_{L}, \quad \hat{C} = i\gamma_{2}\gamma_{0}, \quad \Psi_{L} = \begin{pmatrix} \psi_{L} \\ 0 \end{pmatrix}.
$$
By using this notation it's not hard to see that kinetic term is equal to
$$
\bar{\Psi}_{M}\gamma^{\mu}\partial_{\mu}\Psi_{M} = 2\bar{\Psi}_{L}\gamma^{\mu}\partial_{\mu}\Psi_{L},
$$
while
$$
\bar{\Psi}_{M}\Psi_{M} = \Psi^{T}_{L}\hat{C}\Psi_{L} + h.c.
$$
So if we want to start from $\Psi_{L}$, not from $\Psi_{M}$, we need to write lagrangian in a form
$$
L = 2\bar{\Psi}_{L}\gamma^{\mu}\partial_{\mu}\Psi_{L} - m (\Psi^{T}_{L}\hat{C}\Psi_{L} + h.c.),
$$
or in form
$$
L = \bar{\Psi}_{L}\gamma^{\mu}\partial_{\mu}\Psi_{L} - \frac{m}{2} (\Psi^{T}_{L}\hat{C}\Psi_{L} + h.c.).
$$
A: The short asnwer to your question is that the overall factor $\frac{1}{2}$ from the Lagrangian of a Majorana field (in the 4-component notation)
$$\mathcal{L}=\frac{1}{2}(\bar{\psi}i\gamma^{\mu}\partial_{\mu}\psi -m\bar{\psi}\psi)$$
compared to the general Dirac Lagrangian is usual for self-conjugate fields and it is introduced to ensure a consistent normalization of the field operators in QFT. 
