Quantum Field Theory I

14 CHAPTER 1. INTRODUCTIONS
• ν µ -capture 25 ν µ µ − → ν e
¡
e −
the diagram
the amplitude M fi ≈ g2 m 2 2 µ
8 k 2 −MW
2
the cross-section σ CMS = g4 2 |⃗p e|
1024π |⃗p µ|
• µ-decay 26 µ − → ν µ ν e
¡
e −
the diagram
≈ g2 2 m2 µ
8M 2 W
m 2 µ
MW
4
the amplitude M fi ≈ g2 2
8
m 2 µ
k 2 −M 2 W
≈ g2 2 m2 µ
8M 2 W
the decay rate dΓ = 1 1 g2 4 m4 µ
(2π) 5 2E µ
δ 4 (P
64MW
4 f −P i ) d3 p e d 3 p νe d 3 p νµ
2E e 2E νe 2E νµ
Γ = 1
2 11 1
(2π) 3 g 4 2 m5 µ
M 4 W
25 k 2 ≈ m 2 µ can be neglected when compared to M 2 W
26 The diagram is the same as in the previous case, but the muon-neutrino leg is now
understood to be in the final state.
Since there are three particles in the final state, we cannot use the ready-made formulae
for the two-particle final state. The relevant steps when going from dΓ to Γ are: the identity
δ 4( P f −P i
) d
3 pν
e
2E νe
d 3 p νµ
2E νµ
= 1 8 dΩνµ , the standard trick d 3 p e = p 2 edp edΩ e = p eE edE edΩ e and
finally p e = E e for negligible m e, leading to ∫ max
min EedEe = ∫ m µ/2
0 E edE e = m 2 µ /8. The
correct result is 8 times our result.
3