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ID: past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2018/section-a/a1/a/solution

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Past exam of the particle and nuclear physics course of the University of Cambridge / 2018 / Section A / A1 / a / Solution by

Codex  0 2026-10-18

Treat the quoted maximum as the resonant Z boson contribution described by the Breit-Wigner resonance cross section. The Electron and Positron each have spin

1/2

, while the Z boson has spin one, so

g = \frac{2 ( 1 ) + 1}{[ 2 ( 1/2 ) + 1 ] ^{2}} = \frac{3}{4} .

Neglect the electron mass at this centre-of-mass energy. Each incident momentum magnitude is then

p_{i} = E_{0} /2

in natural units. Write

B_{e} = Γ_{ee} /Γ

. At the resonance peak the denominator is

Γ^{2} /4

, and the same partial decay width appears for production and decay:

σ_{peak} = \frac{π g}{p _{i}^{2}} \frac{Γ _{ee}^{2}}{Γ ^{2} /4} = \frac{12 π}{E _{0}^{2}} B_{e}^{2} .

The peak resonance cross section and branching fractions relation contains

B_{e}^{2}

, because the electronic channel occurs at both ends of the resonant process.

For an area in ordinary units, restore

(ℏ c)^{2}

. The original PDF prints

ℏ c = 197.5 MeV

, which is dimensionally incomplete: the intended standard conversion is

197.5 MeV fm = 0.1975 GeV fm

. Also,

2.0 nb = 2.0 \times 1 0^{- 37} m^{2} = 2.0 \times 1 0^{- 7} f m^{2} .

Consequently

B_{e} = \frac{σ _{peak} E _{0}^{2}}{12 π ( ℏ c ) ^{2}} = \frac{( 2.0 \times 1 0 ^{- 7} ) ( 91.2 ) ^{2}}{12 π ( 0.1975 ) ^{2}}, B (Z^{0} \to e^{+} e^{-}) = 0.03363 ≃ 3.4%.

This estimate uses the supplied resonance model. The full physical elastic Electron-Positron cross section also contains nonresonant photon exchange and interference; an experimental use of the formula must isolate the resonant contribution.

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