= 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 fixes
$$
\frac K\Gamma=\frac{0.67}{m_\tau^2+3m_c^2+3m_b^2}
=\frac{0.67}{84.99\ \mathrm{GeV^2}}.
$$
The <Higgs branching-fraction normalization from mass-weighted channels> therefore gives
$$
B_{\mu\mu}=\frac{0.67(0.106)^2}{84.99}=8.858\times10^{-5},\qquad
B_{b\bar b}=\frac{0.67(75)}{84.99}=0.59125.
$$
No 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 gives
$$
\sigma_{\rm peak}=\frac{4\pi g}{p^2}B_{\mu\mu}B_{b\bar b}
\simeq\frac{4\pi}{m_H^2}B_{\mu\mu}B_{b\bar b}
=\boxed{4.21\times10^{-8}\ \mathrm{GeV^{-2}}.}
$$
Use the supplied rounded $\hbar c=0.197\ \mathrm{GeV\,fm}$, rather than silently substituting a different conversion:
$$
\boxed{\sigma_{\rm peak}=1.63\times10^{-39}\ \mathrm{m^2}
=1.63\times10^{-11}\ \mathrm b\simeq16.3\ \mathrm{pb}.}
$$
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.
Back to article page
