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

Use the same leading-order assumptions as in part a. Let $K=m_H/(8\pi v^2)$, so the partial widths are $K$ times their mass-squared and colour weights. The stated three-channel branching sum fixesThe Higgs branching-fraction normalization from mass-weighted channels therefore givesNo numerical value of $v$ is needed. At resonance, $g=1/4$ and $p\simeq m_H/2$, neglecting the incident muon mass in the kinematic factor. The peak resonance cross section and branching fractions relation givesUse the supplied rounded $\hslash c=0.197\mathrm{GeV}\mathrm{fm}$, rather than silently substituting a different conversion:This is the ideal unsmeared peak for the simplified inputs, not a beam-averaged collider event rate. Finite energy spread, radiation, polarization and radiative corrections require additional treatment.