Past exam of the particle and nuclear physics course of the University of Cambridge / 2013 / Paper 1 / 2 / c

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- Solution c

Solution

The electroweak gauge group is

S U (2)_{L} \times U (1)_{Y}

. Left-handed fermions form weak-isospin doublets and right-handed charged fermions are singlets under

S U (2)_{L}

. Before symmetry breaking the gauge fields are

W^{1}, W^{2}, W^{3}

and

B

, with couplings

g, g^{'}

. A Higgs-doublet vacuum expectation value breaks the symmetry to electromagnetic

U (1)

.

In units

ℏ = c = 1

, the physical combinations and tree-level relations are

W^{\pm} = \frac{W ^{1} \mp i W ^{2}}{2}, A = B cos θ_{W} + W^{3} sin θ_{W}, Z = - B sin θ_{W} + W^{3} cos θ_{W},

e = g sin θ_{W} = g^{'} cos θ_{W}, M_{W} = gv /2, M_{Z} = \frac{v}{2} g^{2} + g^{'2}, M_{W} = M_{Z} cos θ_{W} .

The photon remains massless. charged currents change members of weak doublets, whereas the

Z

mediates weak neutral currents with parity-dependent couplings. At low energies

W

exchange reduces to the Fermi contact interaction,

G_{F} / 2 = g^{2} / (8 M_{W}^{2})

; the apparent weak suppression reflects the massive mediator. This unifies the gauge description, not the numerical equality of all low-energy couplings.^[ref]

Neutral-current neutrino interactions and subsequent observation of massive weak vector bosons tested this structure. In a neutral-current neutrino experiment, an isolated hadronic interaction without an accompanying charged lepton is a candidate signal; entering neutral hadrons and missed charged-current leptons are important backgrounds. Gargamelle's neutral-current evidence supplied a direct experimental check beyond ordinary beta decay.^[ref]

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