 Bubble-chamber kinematic fitting

Alvarez describes a progression from measured coordinates to momenta and angles, then conservation-constrained fits testing alternative track identities. Computer programs made the repeated tests practical; an insufficiently constrained event could retain several hypotheses.^[ref]

A mathematical formulation explains what such a fit can establish. Let $y$ collect measured momenta and angles, with positive-definite covariance matrix $V$. For a chosen reaction hypothesis, adjust them to $x$ by minimizing a generalized least squares objective,The constraints $g$ express four-momentum conservation and any independently known daughter masses. For each proposed identity, calculate energy from the energy–momentum relation, $E_i^2=m_i^2{c}^4+c^2|{\mathbf{p}}_i{|}^2$. A missing neutral particle adds unknowns; a measured decay vertex can add geometric constraints. Compatibility depends on the measurement errors and the independent constraints, not just the smallest raw ${\chi }^2$ among unrelated fits.

A simple independent check uses a single momentum conservation constraint. Suppose two measured one-dimensional momenta are $0.32$ and $-0.28\mathrm{GeV}/c$, each with independent standard deviation $0.02\mathrm{GeV}/c$, and a stipulated parent at rest requires their sum to vanish. Minimizing the two equal-weight squared changes gives fitted values $0.30$ and $-0.30\mathrm{GeV}/c$, with ${\chi }^2=2$. A fit repairs a mismatch according to a hypothesis; it does not prove that hypothesis. This one-component illustration is not a complete reaction fit.