A Pause for Reflection — Question 3

Question 3

Lifetime and width identify a scale without uniquely naming an interaction

Two neutral resonances have measured widths 20 MeV and $2.0\times10^{-6}$ MeV. Use $\hbar=6.58\times10^{-22}$ MeV s.

1. Find both lifetimes and their ratio.

2. Explain mathematically why the narrow width alone cannot prove a weak decay if a strong amplitude is allowed but phase space is small.

Question 3 - Solution

Lifetime and width identify a scale without uniquely naming an interaction

(a) Method and calculation

$$\boxed{\tau_A=\hbar/\Gamma_A=3.29\times10^{-23}\ {\mathrm{s}}}.$$$$\boxed{\tau_B=3.29\times10^{-16}\ {\mathrm{s}},\quad\tau_B/\tau_A=10^7}.$$

(b) Interpretation and further calculation

$$\Gamma\propto|\mathcal M|^2\Phi,\quad\Phi\to0\ \text{near threshold}.$$$$\boxed{\Gamma\ll1\ \text{on a chosen energy scale does not uniquely fix }\mathcal M}.$$

Physical check

Quantum-number selection rules, available channels and threshold suppression must accompany lifetime-based classification.