Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 2 / f

Table of contents
- Solution f

Solution

The leptons consist of

L_{i L} = (ν_{i L}, e_{i L})^{T} \sim (1, 2)_{- 1/2}, e_{i R} \sim (1, 1)_{- 1},

with charged-lepton Yukawa coupling

- \overline{L}_{L} Y_{e} H e_{R} + h.c.

Adding gauge-singlet right-handed neutrinos

ν_{R} \sim (1, 1)_{0}

permits the Dirac Yukawa coupling

- \overline{L}_{L} Y_{ν} H ν_{R} + h.c.,

which gives

M_{D} = v Y_{ν} / 2

. Because

ν_{R}

carries no Standard Model gauge charge, a Majorana mass term

ν_{R}^{T} C M_{R} ν_{R} /2

is also allowed. For

M_{R} ≫ M_{D}

, the seesaw mechanism produces light neutrino masses of order

M_{D} M_{R}^{- 1} M_{D}^{T}

.

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