432
22.1
22. The GSW Gauge Theory of Electroweak Interactions
Problems
(a) Using the representation for α, β and γ 5 introduced in section 20.2.2
(equation (20.14)), massless particles are described by spinors of the
form
(
)
E
1/2
φ +
†
u =
(normalized to u u = 2E)
φ −
where σ · p ˆφ ± = ±φ ± , p ˆ = p|p|. Find the explicit form of u for the
case p ˆ = (sin θ, 0, cos θ).
(b) Consider the process ν ¯ μ +μ
−
→ ν ¯ e +e
− , discussed in section 22.1, in the
limit in which all masses are neglected. The amplitude is proportional
to
G F v ¯(¯ ν μ , R)γ μ (1 − γ 5 )u(μ
− , L)¯ u(e
− , L)γ
μ (1 − γ 5 )v(¯ ν e , R)
where we have explicitly indicated the appropriate helicities R or L
(note that, as explained in section 20.2.2, (1 − γ 5 )/2 is the projection operator for a right-handed antineutrino). In the CM frame,
let the initial μ
− momentum be (0, 0, E) and the final e
− momentum be E(sin θ, 0, cos θ). Verify that the amplitude is proportional to
G F E
2 (1 + cos θ). (Hint: evaluate the ‘easy’ part v ¯(¯ ν μ )γ μ (1 − γ 5 )u(μ
− )
first; this will show that the components μ = 0, z vanish, so that only
the μ = x, y components of the dot product need to be calculated.)
22.2 Verify equation (22.20).
22.3 Check that when the polarization vector of each photon in figures 22.7(a)
and (b) is replaced by the corresponding photon momentum, the sum of these
two amplitudes vanishes.
22.4 By identifying the part of (22.45) which has the form (22.57), derive
(22.58).
22.5 Using the vertex (22.48), verify (22.79).
22.6 Insert (22.29) into (22.151) to derive (22.153).
22.7 Verify that the neutral current part of (22.159) is diagonal in the ‘mass’
basis.
22.8 Suppose that the Higgs field is a triplet of SU(2) L rather than a doublet;
and suppose that its vacuum value is
(
)
0
‹0| ˆ
(
)
φ|0› =
0
f
22.1
22. The GSW Gauge Theory of Electroweak Interactions
Problems
(a) Using the representation for α, β and γ 5 introduced in section 20.2.2
(equation (20.14)), massless particles are described by spinors of the
form
(
)
E
1/2
φ +
†
u =
(normalized to u u = 2E)
φ −
where σ · p ˆφ ± = ±φ ± , p ˆ = p|p|. Find the explicit form of u for the
case p ˆ = (sin θ, 0, cos θ).
(b) Consider the process ν ¯ μ +μ
−
→ ν ¯ e +e
− , discussed in section 22.1, in the
limit in which all masses are neglected. The amplitude is proportional
to
G F v ¯(¯ ν μ , R)γ μ (1 − γ 5 )u(μ
− , L)¯ u(e
− , L)γ
μ (1 − γ 5 )v(¯ ν e , R)
where we have explicitly indicated the appropriate helicities R or L
(note that, as explained in section 20.2.2, (1 − γ 5 )/2 is the projection operator for a right-handed antineutrino). In the CM frame,
let the initial μ
− momentum be (0, 0, E) and the final e
− momentum be E(sin θ, 0, cos θ). Verify that the amplitude is proportional to
G F E
2 (1 + cos θ). (Hint: evaluate the ‘easy’ part v ¯(¯ ν μ )γ μ (1 − γ 5 )u(μ
− )
first; this will show that the components μ = 0, z vanish, so that only
the μ = x, y components of the dot product need to be calculated.)
22.2 Verify equation (22.20).
22.3 Check that when the polarization vector of each photon in figures 22.7(a)
and (b) is replaced by the corresponding photon momentum, the sum of these
two amplitudes vanishes.
22.4 By identifying the part of (22.45) which has the form (22.57), derive
(22.58).
22.5 Using the vertex (22.48), verify (22.79).
22.6 Insert (22.29) into (22.151) to derive (22.153).
22.7 Verify that the neutral current part of (22.159) is diagonal in the ‘mass’
basis.
22.8 Suppose that the Higgs field is a triplet of SU(2) L rather than a doublet;
and suppose that its vacuum value is
(
)
0
‹0| ˆ
(
)
φ|0› =
0
f