= Invariant-mass selection of bubble-chamber events
{title2=$M^2c^4=(\sum_i E_i)^2-c^2|\sum_i\mathbf p_i|^2$}
Why add the daughter <four-momenta>? Consider a selected combination of final <particles> with
$$
P^\mu=\sum_i p_i^\mu
=\left(\frac{\sum_i E_i}{c},\sum_i\mathbf p_i\right).
$$
Using the $+---$ <Minkowski metric>, its scalar square is
$$
P^\mu P_\mu=\frac{(\sum_iE_i)^2}{c^2}-\left|\sum_i\mathbf p_i\right|^2.
$$
In the combination's <center-of-momentum frame>, its total <momentum> vanishes and its total <energy> is $Mc^2$. Therefore $P^\mu P_\mu=M^2c^2$, giving
$$
\boxed{M^2c^4=(\sum_iE_i)^2-c^2\left|\sum_i\mathbf p_i\right|^2}.
$$
<Lorentz transformations> preserve this scalar square, so the reconstructed <invariant mass> does not depend on the combination's laboratory motion. It includes relative <kinetic energy> and need not equal the sum of the daughter <rest masses>. A random pair has an <invariant mass> too; that number does not itself establish a common parent.
For two daughters, expanding the square and substituting each <energy–momentum relation> gives
$$
M^2c^4=m_1^2c^4+m_2^2c^4+2(E_1E_2-c^2\mathbf p_1\cdot\mathbf p_2).
$$
Here $M$ and $m_i$ are <masses>, not <energies>. Reporting $M$ in $\mathrm{GeV}/c^2$ and <momenta> in $\mathrm{GeV}/c$ is equivalent to calculating numerically with $Mc^2$ and $c\mathbf p_i$ in $\mathrm{GeV}$. Setting $c=1$ in <natural units> suppresses these conversion factors; it does not change the measured quantity.
Alvarez's lecture illustrates <invariant mass> searches with $K\pi$ combinations and contrasts resonance lifetimes near $10^{-23}\,\mathrm s$ with the much longer lived states the chambers originally targeted. These resonances decayed before producing a resolvable parent track.<Recent Developments in Particle Physics by Luis Walter Alvarez>[UCRL-18696, printed pp. 46–47, Figs. 15–17: effective-mass histograms and short lifetimes]{ref}
Operationally, choose a reaction hypothesis, require an acceptable <bubble-chamber kinematic fitting> result, and form specified daughter combinations. Accumulate their <invariant masses> in a <histogram> across events. \b[A <particle resonance> is inferred from the ensemble's dynamics; its short-lived parent is not directly photographed.] Daughter tracks can be visible even when the production and decay positions of the parent cannot be separated. Wrong combinations create background, and several combinations from one event are statistically correlated.
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