Past exam of the gauge field theory course of the University of Cambridge / 2012 / Paper 1 / 3 / a

A bare Dirac mass couples left and right chirality, so the distinct electroweak representations prevent writing it as a gauge-invariant term before symmetry breaking. Instead the Yukawa interactions areWeak indices are contracted to singlets. Substituting the vacuum value gives mass matrices $M_f=Y_{f}v/\sqrt{2}$ and a physical Higgs coupling $-{\bar{f}}_L{M}_f{f}_R\sigma /v$ plus its conjugate. Biunitary diagonalization defines mass eigenstates; a mismatch between the up- and down-quark left rotations appears in the CKM matrix. Gauge-boson masses arise from the Higgs kinetic term, whereas these fermion masses arise from the Yukawa term.

With only the minimal field content there is no right-handed neutrino, so no renormalizable neutrino Yukawa term. Adding a gauge-singlet $N_R$ permits both a Dirac scale $m_D=y_{\nu }v/\sqrt{2}$ and a Majorana scale $M_R$. For one family the symmetric mass matrix in a left-chiral basis isIts signed eigenvalues are $(M_R\pm \sqrt{M_R^2+4m_D^2})/2$. Rephasing gives positive physical masses, and for $|M_R|\gg |m_D|$ the seesaw mechanism yieldsFor $m_D\sim 100\mathrm{GeV}$ and $M_R\sim 10^{14}\mathrm{GeV}$, the light scale is about $0.1\mathrm{eV}$. Integrating out the heavy state generates the dimension-five Weinberg operator, whose Higgs vacuum insertion gives the same suppression. Alternatively a Dirac neutrino can have a very small Yukawa coupling with lepton number conserved. A tiny observed neutrino mass does not by itself distinguish these mechanisms.