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The identities $S_L^{†}{\bar{\sigma }}^{\mu }{S}_L={\Lambda }^{\mu }{}_{\nu }{\bar{\sigma }}^{\nu }$ and $S_L^T{\sigma }^2{S}_L={\sigma }^2$ show respectively that the kinetic term and both mass bilinears are Lorentz invariant. Infinitesimally, the first identity makes $u_L^{†}{\bar{\sigma }}^{\mu }{u}_L$ transform by the vector law stated in the question, while the second makes $u_L^T{\sigma }^2{u}_L$ a scalar.

Varying the anticommuting components of $u_L^{†}$ givesThe factor two from varying the antisymmetric quadratic form cancels the $1/2$ in the mass term. This is a Majorana mass term: under $u_L\mapsto e^{iq\alpha }{u}_L$ it carries charge $2q$, so a nonzero mass is compatible with an unbroken electric charge only for $q=0$. A charged massive fermion instead needs an independent Weyl field of opposite chirality to form a Dirac mass.

In the convention $(SU(3{)}_c,SU(2{)}_L{)}_Y$, one Standard Model fermion generation including a right-handed neutrino isA bare Dirac mass term pairs left- and right-handed fields in the same gauge representation, but every charged left-handed Standard Model fermion is an $SU(2{)}_L$ doublet while its right-handed partner is a singlet. The Higgs field $H:(1,2{)}_{1/2}$ and $\tilde{H}=i{\sigma }^2{H}^{\ast }:(1,2{)}_{-1/2}$ permit the gauge-invariant Yukawa interactionsThe neutral Higgs vacuum expectation value turns them into masses after electroweak symmetry breaking.

Because ${\nu }_R$ is a gauge singlet, it may also have a large Majorana mass term $M$. Together with its Dirac mass $m_D$, the seesaw mechanism gives a light neutrino mass $m_{\nu }\simeq m_D^2/M$. If no right-handed neutrino is retained, the same low-energy physics is encoded by the dimension-five Weinberg operator $(L_{L}H)(L_{L}H)/\Lambda$.

Now let $\varphi$ be an electroweak scalar triplet with $Y=1$. Its $T^3$ weights are $1,0,-1$, and electric charge is $Q=T^3+Y$, so its components have chargesIn the given matrix convention,Thus the electromagnetic-neutral direction satisfies ${\varphi }_3=0$ and ${\varphi }_1-i{\varphi }_2=0$.

The gauge-invariant type-II seesaw mechanism Yukawa interaction isThe two lepton doublets have total hypercharge $-1$, which is cancelled by $Y(\varphi )=1$, and the displayed $SU(2)$ contraction is a singlet. Expanding it containsTherefore a neutral condensate $⟨{\varphi }^0⟩\ne 0$ preserves electromagnetism and generates a left-handed-neutrino Majorana mass term proportional to $y_{\varphi }⟨{\varphi }^0⟩$.

The type-I seesaw mechanism combines a Dirac neutrino mass $m_D$ with a much larger right-handed-neutrino Majorana mass term $M$. Diagonalizing the mass matrix gives a light eigenvalue of order $m_D^2/M$ and a heavy eigenvalue of order $M$.

The type-II seesaw mechanism couples two lepton doublets to an electroweak scalar triplet. A small neutral-triplet vacuum expectation value directly generates a left-handed neutrino Majorana mass term.