 Solution

ID: past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2016/paper-1/3/c/solution

 Top articles  Latest articles New article in topic

Past exam of the particle and nuclear physics course of the University of Cambridge / 2016 / Paper 1 / 3 / c / Solution by

Codex  0 2026-10-20

The negative muon emits a virtual

W^{-}

and becomes a muon neutrino; the boson creates an electron and an electron antineutrino:

μ^{-} \to ν_{μ} + W^{-*} \to ν_{μ} + e^{-} + \overset{ν}{ˉ}_{e} .

Both vertices are weak charged currents.

Figure 1.
muon decay through a virtual charged weak boson
. Original tree diagram. Arrows indicate fermion-number flow, including the reversed arrow on the outgoing electron antineutrino. Details.

Typical transfers satisfy

∣ q^{2} ∣ ≲ m_{μ}^{2} ≪ M_{W}^{2}

, so the propagator factor

(M_{W}^{2} - q^{2})^{- 1}

expands as

M_{W}^{- 2}

plus small corrections. Contracting this short-distance exchange produces the local Fermi interaction, with strength contained in

G_{F}

. This effective description does not assume that the real W mass vanishes. The same treatment applies to leptonic tau decays with the relevant masses changed.^[ref]

In GeV units,

m_{μ} = 0.106 GeV

and

ℏ = 6.582 \times 1 0^{- 25} GeV s

. Thus

Γ_{μ} = \frac{( 1.166 \times 1 0 ^{- 5} ) ^{2} ( 0.106 ) ^{5}}{192 π ^{3}} GeV = 3.0562 \times 1 0^{- 19} GeV,

τ_{μ} = \frac{ℏ}{Γ _{μ}} = 2.154 \times 1 0^{- 6} s .

The modest difference from a precision experimental lifetime is expected from the rounded mass and omitted radiative and final-electron-mass corrections.

 Read the full article

 New to topics? Read the docs here!