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Are both types of Majorana masses $$\mathcal{L}^L_M=-\frac{m_L}{2}[\overline{(\psi_L)^c}\psi_L + \overline{\psi_L}(\psi_L)^c]$$ and $$\mathcal{L}^R_M=-\frac{m_R}{2}[\overline{(\psi_R)^c}\psi_R + \overline{\psi_R}(\psi_R)^c]$$ $SU(2)\times U(1)$ invariant? If not, then both the masses must arise by $SU(2)\times U(1)$ symmetry breaking and both the mass scales are fixed by electroweak symmetry breaking scale. Right? Then why does one take $m_R>>m_L$ in see-saw mechanism? Why should we assume $m_R$ is around the GUT scale?

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In this context, the right-handed neutrino is a singlet under the Standard Model gauge groups. Only the right-handed neutrino is allowed a Majorana mass. The left-handed term is not gauge invariant. If the SM and right-handed neutrino fields were embedded into a Grand Unified gauge group, the right-hand term would break that symmetry. It is expected therefore that $m_R\sim m_{GUT}$.

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  • $\begingroup$ Why do you say that the left-handed term is not gauge invariant but the right-handed is? @innisfree $\endgroup$ – SRS Apr 5 '14 at 15:14
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    $\begingroup$ The left-hand is an SU(2) doublet with nonzero U(1) hypercharge. The right hand is a singlet. The Major an a masses don't contain $\sim\psi^*\psi$, but $\sim\psi\psi$, so they are not invariant. $\endgroup$ – innisfree Apr 5 '14 at 15:26

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