= Simulated two-track invariant-mass example
Take two stipulated oppositely charged daughters with equal <rest mass> $m=0.14\,\mathrm{GeV}/c^2$ and exactly opposite <momenta> of magnitude $p=0.30\,\mathrm{GeV}/c$. Their total <momentum> is zero, so
$$
E_1=E_2=\sqrt{0.14^2+0.30^2}\,\mathrm{GeV}=0.33106\,\mathrm{GeV},\qquad
M=0.66212\,\mathrm{GeV}/c^2.
$$
This is a constructed event, not an identification of a historical <particle>. Boosting both <four-momenta> by $\beta=0.60$ along the first daughter's direction gives total <energy> $0.82765\,\mathrm{GeV}$ and total <momentum> $0.49659\,\mathrm{GeV}/c$. Substitution recovers the same <invariant mass>, although the laboratory <energy> has increased.
The following independent simulation stipulates 700 signal combinations from a <Cauchy distribution> centered at $0.80\,\mathrm{GeV}/c^2$ with scale $0.020\,\mathrm{GeV}/c^2$, truncated to $0.50<M<1.10\,\mathrm{GeV}/c^2$, plus 2400 background combinations whose density decreases linearly across that interval. Each combination is converted to a back-to-back two-daughter event with $m=0.14\,\mathrm{GeV}/c^2$. Independent isotropic directions are chosen, each Cartesian <momentum> component receives <normal distribution> noise with <standard deviation> $0.012\,\mathrm{GeV}/c$, and <invariant mass> is reconstructed from the perturbed <momenta>. All inputs and counts are invented. No historical experimental points or fitted historical parameters appear.
\Image[/hydrogen-bubble-chamber-simulated-invariant-mass.png]
{title=A stipulated resonance component produces an excess above a smooth simulated invariant-mass background}
{description=Original simulation of two-daughter events, with invented counts, a truncated Cauchy signal and a linearly falling background. The two panels share the same generated events. The right panel adds chosen momentum noise before reconstructing invariant mass; this is not an experimental discovery plot.}
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