Past exam of the particle physics course of the University of Oxford / 2022 / 5 / c

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- Solution c

Solution

Write

f = (v + h) / 2

. The Pauli matrices give the weak isospin operators

τ^{+} = (0010), τ^{-} = (0100), τ_{3} = \frac{1}{2} (10 0 - 1) .

For the Higgs doublet

Φ = (0, f)^{T}

with weak hypercharge

Y = 1

,

τ^{+} Φ = (f 0), τ^{-} Φ = 0, τ_{3} Φ = (0 - f /2), Q Φ = (τ_{3} + \frac{1}{2} I) Φ = 0.

The Higgs field direction, including its vacuum expectation value, is electrically neutral. Consequently the electromagnetic U(1) leaves the vacuum invariant: electroweak symmetry breaking preserves electromagnetism and the photon remains massless. The result concerns this neutral vacuum direction; it does not say that both components of the full Higgs doublet have zero electric charge.

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