Majorana mass matrix seesaw and renormalizable interactions

Question

Sometimes the general seesaw matrix:

$$\begin{pmatrix} M_L & M_1\\ M_2 & M_R \end{pmatrix}$$

and its just said that in order to get renormalizable interactions one must impose the condition $M_L=0$. Why is this so?

Answer 1

Your question is virtually answered in this one.

The SU(2)×U(1) gauge-invariant fermion bilinears corresponding to each term of the matrix have their origins in the terms of dimension [d], $$ M_L: ~~~~~ \frac{1}{v}\left( L ^T i \sigma _2 \cdot \tilde\phi^* ~~\tilde \phi \cdot L^c \right) \qquad [5],\\ M_1: \qquad\qquad \qquad y_1 \overline{L}\cdot \tilde{\phi}~ R \qquad [4],\\ M_2: \qquad\qquad \qquad y_2 \overline{R}~ \tilde{\phi}^* \cdot L \qquad [4],\\ M_R: \qquad \qquad M R ^T i \sigma _2 C R \qquad [3]. $$

All fermion bilinears contribute dimension [3], and each Higgs, providing the v.e.v., contributes [1]. The non-diagonal ones are SM-like Dirac masses. The $M_R$ is a huge non-SM lepton number (doubly)violating Majorana one, and

• the $M_L$ is the famous dimension [5], so nonrenormalizable, Weinberg operator. It too is a Majorana mass violating lepton number by 2.