= Missing four-momentum in electron-positron collisions
{title2=$P_{\rm miss}=P_{\rm initial}-P_{\rm visible}$}
Use <natural units> $c=1$ and the <Minkowski metric> with signs $(+,-,-,-)$. In the ideal <centre-of-momentum frame>, assume a known initial <four-momentum> $P=(\sqrt{s},\mathbf0)$, no unaccounted radiation and exact visible reconstruction. Define
$$
P_{\rm miss}=P-p_e-p_\mu=(E_{\rm miss},\mathbf p_{\rm miss}),\qquad
M_{\rm miss}^2=P_{\rm miss}^2=E_{\rm miss}^2-|\mathbf p_{\rm miss}|^2.
$$
The <missing four-momentum in electron-positron collisions> is a subtraction of measured quantities. It is not itself a <particle detector> hit. With $M_{e\mu}^2=(p_e+p_\mu)^2$, expansion gives the useful identity
$$
M_{\rm miss}^2=s+M_{e\mu}^2-2\sqrt{s}(E_e+E_\mu).
$$
For any collection of future-directed massless invisible <four-momenta> $q_i=E_i(1,\widehat{\mathbf n}_i)$,
$$
M_{\rm miss}^2=2\sum_{i<j}E_iE_j(1-\widehat{\mathbf n}_i\cdot\widehat{\mathbf n}_j)\geq0.
$$
Thus a positive <missing invariant mass> excludes exactly one massless invisible <particle> under the stated assumptions. It does not exclude one massive invisible <particle>, identify a <neutrino flavor>, or determine the number of massless invisible <particles> beyond that lower bound. Several collinear massless <particles> can even have zero combined <invariant mass>. Detector resolution and unaccounted radiation invalidate a literal event-by-event interpretation of these ideal inequalities.
For an original exact example in <natural units>, choose total <energy> $6\,\mathrm{GeV}$ and visible massless <four-momenta>, in $\mathrm{GeV}$,
$$
p_e=(1;1,0,0),\quad p_\mu=(1;0,1,0),\quad
P_{\rm miss}=(4;-1,-1,0),\quad M_{\rm miss}^2=14\,\mathrm{GeV}^2.
$$
Two possible invisible massless <four-momenta> are $(2;-1/2,-1/2,\sqrt{14}/2)$ and $(2;-1/2,-1/2,-\sqrt{14}/2)$. Splitting either into two positive collinear parts gives a three-particle interpretation with identical recoil. Alternatively, one invisible <particle> of <mass> $\sqrt{14}\,\mathrm{GeV}$ has that recoil. This example tests the ambiguity of the subtraction, without asserting any of these alternatives is a <tau lepton> decay.
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