Right-handed neutrino

A right-handed neutrino transforms as $(1,1{)}_0$ under $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$, so it is neutral under every Standard Model gauge interaction.

A right-handed neutrino is a singlet of the entire Standard Model gauge group. Its same-chirality bilinear with the field obtained by charge conjugation therefore admits the displayed renormalizable Majorana mass term, in addition to a Higgs-induced Dirac mass term. It violates lepton number by two units. Analogous terms for electrically charged Standard Model fermions carry nonzero electric charge; quarks also have nontrivial color charge. The presence of a right-handed field permits mass but does not force a nonzero mass parameter.

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 $M$.

In the type-I regime $M_L\simeq 0$, $M_R\gg m_D$, the light mass is $m_D^2/M_R$. Specifying only the right-handed scale does not fix a numerical light mass. A choice of Dirac mass is an extra assumption, not a result of matrix diagonalization.

In an all-left-handed Weyl basis $({\nu }_L,{\nu }_R^c)$, the symmetric matrix has left and right Majorana entries $M_L,M_R$ and off-diagonal Dirac entry $m$. Real eigenvalues can have signs; physical Majorana masses are nonnegative after appropriate field rephasings.

The Weinberg operator is the dimension-five Standard Model operator $(LH)(LH)/\Lambda$. After the Higgs field condenses, it gives a left-handed neutrino Majorana mass of order $v^2/\Lambda$.

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.

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.