invariant-mass reconstruction
Most interesting particles never reach a detector. A heavy particle made in a collision lives for a vanishingly short time and decays into lighter, longer-lived fragments before any instrument can register it directly. It is like a snowflake that melts the instant it lands: you never see the flake, only the droplets. Invariant-mass reconstruction is the technique that runs this backwards — measuring the droplets to figure out the snowflake.
The method rests on conservation of four-momentum. Detectors measure the energy and momentum of each stable decay product. Add those four-momenta together, then compute the invariant mass of the sum using mass squared equals total-energy squared minus total-momentum squared. Because invariant mass is conserved and frame-independent, the result is precisely the mass of the parent that decayed, even though the parent is long gone. Do this for one event and you get one number; do it for millions and the parent reveals itself as a sharp bump rising above a smooth background at its mass.
This is how new particles are discovered and known ones are measured. The Z boson, the J/psi, and the Higgs boson were all found as peaks in reconstructed invariant-mass spectra — the Higgs famously as a bump near 125 GeV in the mass of photon pairs and of four-lepton combinations. The width of the bump even tells you about the particle's lifetime through the energy-time uncertainty relation. The honest caveat is that decays into neutrinos, which escape undetected, leave the reconstruction incomplete, forcing physicists to fall back on transverse quantities.
Reconstructed mass^2 = (E1 + E2)^2 - (p1 + p2)^2 c^2. Two muons from a decay reconstructing to 91 GeV/c^2 signal that their parent was a Z boson.
Adding up decay products and computing their invariant mass resurrects the mass of a particle that already vanished.
Reconstruction only works for decay products you can fully measure; neutrinos slip out undetected, so decays involving them cannot be reconstructed into a clean mass peak and require missing-energy methods instead.