 Majorana neutrino mass from the Weinberg operator

For the gauge-singlet composite $\chi ={\tilde{\varphi }}^{†}L$, the interaction $G{\chi }^T{C}^{-1}\chi -G^{\ast }\bar{\chi }C{\bar{\chi }}^T$ is electroweak gauge-invariant. At the Higgs field vacuum, $\chi =v{\nu }_L/\sqrt{2}$, giving $\frac{1}{2}(G{v}^2{\nu }_L^T{C}^{-1}{\nu }_L-G^{\ast }{v}^2{\bar{\nu }}_{L}C{\bar{\nu }}_L^T)$. The Grassmann variation of a chiral Majorana mass term therefore identifies the mass parameter as $G{v}^2$, with physical mass $|G|v^2$. The dimension-five interaction has a coupling of mass dimension minus one. Under lepton number phase rotation, its two terms acquire factors $e^{\pm 2i\alpha }$, so it violates continuous lepton number by two units. It is an effective extension of the minimal renormalizable Standard Model.