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

Table of contents
- Solution d

Solution

Let

H = i τ^{2} H^{*}

. Gauge invariance permits the Yukawa interactions

L_{Y} = - \overline{Q}_{L} Y_{d} H d_{R} - \overline{Q}_{L} Y_{u} H u_{R} + h.c.,

where

Y_{u}, Y_{d}

are arbitrary complex

3

by

3

matrices in generation space. Their color and weak indices are contracted to singlets, and the hypercharges in each term sum to zero.

After symmetry breaking,

L_{Y} \supset - \overline{d}_{L} M_{d} d_{R} - \overline{u}_{L} M_{u} u_{R} + h.c., M_{d} = \frac{v}{2} Y_{d}, M_{u} = \frac{v}{2} Y_{u} .

The same terms couple the physical Higgs field proportionally to the quark mass matrices.

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