= Solution
In the <Breit-Wigner resonance cross section>, $E$ is the total <centre-of-momentum energy>, $E_0$ the resonance <energy>, and $p$ either incident particle's center-of-momentum <momentum> magnitude. The total <decay width> is $\Gamma=\hbar/\tau$, while $\Gamma_i$ and $\Gamma_f$ are partial widths into the entrance and specified exit channels. Their ratios to $\Gamma$ are the corresponding <branching fractions>. The factor $g$ averages over incident <spin> states and sums over the resonance's magnetic substates.
In <natural units>, $\hbar=c=1$, the resonant partial-wave cross-sectional scale is an area proportional to the squared reduced de Broglie wavelength, $1/p^2$. Equivalently a partial-wave scattering amplitude carries one factor $1/p$, and its squared modulus supplies $1/p^2$. In SI area units the same scale is $(\hbar/p)^2$; if $p$ is expressed in <energy> units, multiply the final inverse-energy-squared answer by $(\hbar c)^2$.
For unpolarized spin-half incident <muons>, the <unpolarized resonance spin factor> is
$$
g=\frac{2J+1}{(2s_{\mu^-}+1)(2s_{\mu^+}+1)}
=\frac{2J+1}{4}.
$$
Hence \b[$g_Z=3/4$ and $g_H=1/4$]. The outgoing tau spins do not supply another independent multiplicity here: their allowed final states are already included in $\Gamma_f$.
Near a narrow resonance, treat the smooth prefactor $p^{-2}$ and the partial widths as constant across the line. At the peak the denominator is $\Gamma^2/4$. Half its height occurs when $(E-E_0)^2+\Gamma^2/4=\Gamma^2/2$, namely at $E=E_0\pm\Gamma/2$. Thus \b[the full width at half maximum is $\Gamma$] within the Breit-Wigner narrow-line approximation. A varying prefactor or convolution with the beam-energy distribution can modify an observed line shape.
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2015/paper-1-higgs-diagrams.png]
{title=Tree production and decay, fermion triangles, and the W-loop topologies for Higgs diphoton decay}
The production amplitude has a <muon> pair joined to a Higgs propagator through their <Yukawa coupling>, followed by the bottom-quark pair at a second Yukawa vertex. The separate $H\to b\bar b$ graph is a tree-level Higgs–<fermion> vertex. There are no tree-level $Hgg$ or $H\gamma\gamma$ vertices in the renormalizable <Standard Model>: their lowest nonzero graphs are loops. A <quark> triangle, dominated by the top contribution, produces two <gluons>. Charged-fermion triangles contribute to two <photons>; charged-W loops also contribute and are dominant in the Standard Model amplitude. <Photon> permutations and the W contact topology belong to the full amplitude; the diagram shows these topologies rather than claiming one triangle alone is the entire gauge-invariant W contribution.<Review of Particle Physics by Particle Data Group (2022)>[Status of Higgs <Boson> Physics, §11.2.4 and discussion on physical page 5 of loop-induced <gluon> and <photon> couplings]{ref}
Back to article page
