The Hadrons — Question 5

Question 5

A resonance peak shape determines its full width at half maximum

An isolated hadronic resonance follows $\sigma(E)=\sigma_0(\Gamma/2)^2/[(E-E_R)^2+(\Gamma/2)^2]$, with $\Gamma=100$ MeV. Ignore instrumental broadening.

1. Find the energies where the cross section is half its peak value.

2. Find the cross-section fraction at $|E-E_R|=100$ MeV and the integrated area in terms of $\sigma_0$ and Gamma.

Question 5 - Solution

A resonance peak shape determines its full width at half maximum

(a) Method and calculation

$$\sigma/\sigma_0=1/2\Rightarrow(E-E_R)^2=(\Gamma/2)^2.$$$$\boxed{E=E_R\pm50\,\mathrm{MeV},\quad\mathrm{FWHM}=100\,\mathrm{MeV}}.$$

(b) Interpretation and further calculation

$$\boxed{\sigma(E_R+100)/\sigma_0=50^2/(100^2+50^2)=.200}.$$$$\boxed{\int_{-\infty}^{\infty}\sigma(E)dE=\pi\sigma_0\Gamma/2}.$$

Physical check

The infinite integration limits are a line-shape idealization. Thresholds, background interference and detector resolution can distort an actual hadronic resonance.