= 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.<Recent Developments in Particle Physics by Luis Walter Alvarez>[UCRL-18696, printed pp. 30–33: HYDRO, PANG, GUTS, KICK and SUMX]{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,
$$
\chi^2=(x-y)^{\mathsf T}V^{-1}(x-y),\qquad g(x)=0.
$$
The constraints $g$ express <four-momentum conservation> and any independently known daughter <masses>. For each proposed identity, calculate <energy> from the <energy–momentum relation>, $E_i^2=m_i^2c^4+c^2|\mathbf p_i|^2$. 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.
Back to article page
