= Narrow resonance and instrumental resolution
{title2=$\tau=\hbar/\Gamma$}
A simple isolated <particle resonance> model starts with a survival amplitude proportional to
$$
a(t)=e^{-iE_0t/\hbar}e^{-\Gamma t/(2\hbar)},\qquad t\geq0.
$$
Its survival <probability> is $|a(t)|^2=e^{-\Gamma t/\hbar}$, giving mean rest-frame lifetime $\tau=\hbar/\Gamma$. The <Fourier transform> of this one-sided amplitude is proportional to $(\Gamma/2-i(E-E_0))^{-1}$; its squared modulus gives a <Lorentzian line shape>. Write its normalized density as
$$
L(E)=\frac1\pi\frac{\Gamma/2}{(E-E_0)^2+(\Gamma/2)^2},\qquad
\int_{-\infty}^{\infty}L(E)\,dE=1.
$$
At $E-E_0=\pm\Gamma/2$ the density is half its peak, so its intrinsic <full width at half maximum> is $\Gamma$. This is a local isolated-line approximation, extending the offset coordinate over the real line; physical production thresholds, interference and energy-dependent <decay widths> can require a richer <quantum scattering> model.
Now stipulate a centered <normal distribution> for the measurement response, with <standard deviation> $\sigma$. A measured coordinate $x$ has density
$$
G_\sigma(x-E)=\frac{e^{-(x-E)^2/(2\sigma^2)}}{\sqrt{2\pi}\sigma},\qquad
O(x)=\int_{-\infty}^{\infty}L(E)G_\sigma(x-E)\,dE.
$$
This <convolution> describes the smearing of an underlying line, with constant acceptance and no background. For a pair spectrometer the response summarizes <invariant-mass resolution>; for an idealized collider scan it summarizes incoming <center-of-mass energy> spread. These are different experimental sources of uncertainty even when the same mathematical response is useful.
\Image[/narrow-resonance-resolution-and-lifetime.png]
{title=An arbitrary narrow line broadened by two stipulated Gaussian responses}
{description=Original numerical convolution using a Lorentzian with intrinsic width 0.20 times an arbitrary energy scale and Gaussian standard deviations 0.40 and 1.20 times that scale. Left: densities with unit area on the whole line. Right: each profile divided by its own peak. No historical points, cross-section table or published curve were copied or fitted.}
Choose an arbitrary scale $E_*$, intrinsic <decay width> $\Gamma=0.20E_*$ and response <standard deviations> $0.40E_*$ and $1.20E_*$. Numerical integration gives apparent <full width at half maximum> values of approximately $1.05E_*$ and $2.93E_*$, respectively. The state and its rest-frame lifetime are identical in both cases. The <normal distribution>'s width alone is $2\sqrt{2\ln2}\,\sigma$; adding that width to $\Gamma$ in quadrature is not an exact rule for this <convolution>.
For a purely illustrative $E_*=1\,\mathrm{MeV}$, taking $\hbar\simeq6.5821\times10^{-22}\,\mathrm{MeV\,s}$ gives $\tau\simeq3.2911\times10^{-21}\,\mathrm s$. These are invented line parameters, not <J psi meson> measurements. Substituting the larger apparent width into $\hbar/\Gamma$ would incorrectly shorten the inferred lifetime by nearly a factor of fifteen. An unresolved width can support an upper bound on intrinsic <decay width>, and consequently a lower bound on lifetime, only after the instrumental response and model assumptions have been assessed.
The discovery peak and its later interpretation answer different questions. The <J psi meson> is understood as <charmonium>, a bound <charm quark>–<charm antiquark> system. Richter's 1976 lecture distinguishes the initial discovery from subsequent tests of competing explanations, including the related level structure and later evidence for particles carrying uncancelled charm. A narrow peak alone did not establish all these properties in 1974.<From the Psi to Charm – The Experiments of 1975 and 1976 by Burton Richter>[From the Psi to Charm – The Experiments of 1975 and 1976, introduction and sections 7.4–8: charmonium interpretation and subsequent charm searches]{ref}
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