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Supposing you have a lagrangian consisting of $(1/2,0)\oplus (0,1/2)$ representation.

Writing in terms of Weyl fermions, the following terms are possible:

$$-\frac{m_1}{2} (\psi_R^T \epsilon \psi_R - \psi_R^\dagger \epsilon \psi_R^*)$$ $$-\frac{m_2}{2} (\psi_L^T \epsilon \psi_L - \psi_L^\dagger \epsilon \psi_L^*)$$ $$-m_3(\psi_R^\dagger \psi_L + \psi _L^\dagger \psi_R)$$

where $\epsilon = -i \sigma_y$.

Is there anything preventing all three terms from being present at once? Sure, if you have the Majorana constraint $$\psi_r =-\epsilon \psi_L^*$$ then actually all three terms are equal. But I don't see why one must always have this constraint... that constraint seems to be just a choice that one could investigate if one wishes.

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