Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 305 / 4 / iv

Table of contents
- Solution iv

Solution

A bare Dirac bilinear couples left- and right-handed fields, but these fields transform in different electroweak representations, so

\overset{ˉ}{f}_{L} f_{R}

is not a gauge singlet. Let

H = i σ^{2} H^{*}

. The four gauge-invariant Yukawa interaction terms are

L_{Y} = - y_{d} \overset{ˉ}{Q}_{L} H d_{R} - y_{u} \overset{ˉ}{Q}_{L} H u_{R} - y_{e} \overset{ˉ}{L}_{L} H e_{R} - y_{ν} \overset{ˉ}{L}_{L} H ν_{R} + h.c.

Replacing

H

by its vacuum expectation value gives Dirac masses

m_{f} = y_{f} v / 2

. Applying

Q = T^{3} + Y

gives

Q (u) = \frac{2}{3}, Q (d) = - \frac{1}{3}, Q (ν) = 0, Q (e) = - 1

for both chiralities, so the generated mass terms preserve the unbroken electromagnetic gauge symmetry.

Solved by gpt-5.6-sol high.

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