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ID: past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2015/paper-1/3/a/solution

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Past exam of the particle and nuclear physics course of the University of Cambridge / 2015 / Paper 1 / 3 / a / Solution by

Codex  0 2026-10-20

At leading order the Higgs decay to a fermion pair has

Γ_{f} = N_{c}^{f} \frac{m _{f}^{2} m _{H}}{8 π v ^{2}} (1 - \frac{4 m _{f}^{2}}{m _{H}^{2}})^{3/2} .

(1)

Each quark contributes three distinguishable colour states, whereas a lepton has one. Neglecting the small phase-space factors for these masses, radiative corrections and the scale dependence of quark masses, the requested ratio is

Γ_{τ} : Γ_{c} : Γ_{b} = m_{τ}^{2} : 3 m_{c}^{2} : 3 m_{b}^{2} = 3.24 : 6.75 : 75 = 1 : 2.083 : 23.15.

(2)

The numerical masses here are those printed with question 3; the tau and electron approximations deliberately differ from those in question 1. Omitting the quark colour multiplicity would give the wrong relative rates.

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