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The important point is the fact that such a mass term breaks the gauge symmetry (Edit: I am assming that you want to build the Majorana mass term using SM available fields -- no extension considered -- of which there is only $\nu_L$). Namely, the desired term is (one generation suffices): $$\frac{1}{2}\,M\, \nu_L^T \,\mathcal{C}^\dagger\,\nu_L\, +\, ... 4 The SU(2) \times U(1) electroweak gauge theory has 4 symmetry generators, of which the vacuum breaks only three -- corresponding to W^+, W^- and Z. The vacuum is symmetric under the generator of electromagnetism, hence the photon cannot interact with the vacuum. Think of the vacuum as a medium -- if it was not symmetric under EM, then it could ... 3 Lets analyse the Majorana condition and the Majorana mass term. A massive Majorana neutrino \chi_j (a Majorana spin 1/2 fermion) having mass m_j>0 can be described in a local quantum field theory (eg. the standard model) by a four component spin 1/2 field \chi_j(x) which satisfies the Dirac equation and the Majorana condition which reads:$$ ...
The easiest way to form $SU(2)$ singlets in the most general way is to use the techniques of Young Tableau. The method is discussed from a physicists perspective in many lecture notes online. One such example is given here. Using such method its easy to show that 2 lepton doublets make a singlet and a triplet under $SU(2)$, 2 \otimes 2 = 3 ...