= Solution
\b[Two mechanisms are a <tau lepton> pair with two leptonic decays, and a <W boson> pair with two leptonic decays.] With equal beam <energies>, the laboratory is the <centre-of-mass frame>, and <four-momentum conservation> gives $E_{\rm miss}=\sqrt{s}-E_{e^-}-E_{\mu^+}$ in <natural units>.
For the first mechanism,
$$
e^+e^-\to\gamma^*,Z^*\to\tau^+\tau^-,\qquad
\tau^+\to\mu^+\nu_\mu\bar\nu_\tau,\qquad
\tau^-\to e^-\bar\nu_e\nu_\tau.
$$
There are four escaping <neutrinos>. Each <tau lepton> decay proceeds through a virtual <W boson>. The fixed-charge final state has <branching fraction> $B_\mu B_e$; the factor two for the inclusive $e^\pm\mu^\mp$ sample does not apply when the charges are specified.
For the second mechanism,
$$
e^+e^-\to W^+W^-,\qquad
W^+\to\mu^+\nu_\mu,\qquad W^-\to e^-\bar\nu_e.
$$
There are two escaping <neutrinos>. The leading <Feynman diagrams> comprise both $s$-channel <photon>/<Z boson> exchange and $t$-channel <electron neutrino> exchange. Their <scattering amplitudes> interfere; the complete set is needed for a consistent high-<energy> prediction. In particular, the neutrino-exchange diagram alone does not define an independently observable production rate.
\Image[/past-exam-of-the-nuclear-and-particle-physics-course-of-the-university-of-oxford/2021-leptonic-production-diagrams.png]
{title=Tau-pair decay and the three leading W-pair production diagrams}
A <tau lepton> pair has enhanced production at the <Z boson> <particle resonance>, $\sqrt{s}\simeq91\,\mathrm{GeV}$. The four-<neutrino> <missing energy> has a broad continuous distribution. To derive a useful schematic, neglect daughter <masses> and parent <spin> correlations. In an unpolarized <tau lepton> rest frame, the <Michel spectrum> for $x=2E_\ell^*/m_\tau$ is
$$
f(x)=2x^2(3-2x),\qquad0\leq x\leq1,\qquad \langle x\rangle=\frac7{10}.
$$
A <Lorentz boost> gives $E_\ell/E_\tau=x(1+\beta_\tau u)/2$, with $u=\cos\theta^*$ uniform on $[-1,1]$ in this approximation. For $\beta_\tau\simeq1$, the charged-daughter <probability density function> in $z=E_\ell/E_\tau$ is
$$
g(z)=\int_z^1\frac{f(x)}{x}\,dx
=\frac53-3z^2+\frac43z^3,\qquad0\leq z\leq1.
$$
For two independent decays, $y=E_{\rm miss}/\sqrt{s}=1-(z_1+z_2)/2$ has the <convolution>
$$
p_\tau(y)=2\int_{\max(0,1-2y)}^{\min(1,2-2y)}
g(z)g(2-2y-z)\,dz,\qquad0\leq y\leq1,
$$
and $\langle y\rangle=0.65$. It vanishes at both endpoints and peaks on the larger-missing-<energy> side. This is a shape illustration, not the exact polarized <Z boson> prediction; <spin> correlations, acceptance, and radiative effects alter it.
The <W boson> process turns on near $\sqrt{s}=2m_W\simeq161\,\mathrm{GeV}$. At exact on-shell <production threshold>, each <neutrino> has <energy> $m_W/2$, so
$$
\boxed{E_{\rm miss}=m_W=\frac{\sqrt{s}}2\quad\text{at threshold}.}
$$
The <cross section> vanishes at the ideal two-body threshold, so this narrow limiting spectrum must not be called the maximum-rate spectrum. The large <W boson>-pair rate occurs above threshold, around $200\,\mathrm{GeV}$. There $\beta_W=\sqrt{1-4m_W^2/s}$ and each <neutrino> has
$$
E_\nu=\frac{\sqrt{s}}4(1+\beta_W\cos\theta^*),\qquad
\frac{\sqrt{s}}4(1-\beta_W)\leq E_\nu\leq\frac{\sqrt{s}}4(1+\beta_W).
$$
Thus
$$
\boxed{\frac{1-\beta_W}{2}\leq\frac{E_{\rm miss}}{\sqrt{s}}\leq\frac{1+\beta_W}{2}.}
$$
An isotropic independent-decay schematic is triangular, centered at $E_{\rm miss}/\sqrt{s}=1/2$, with endpoints approximately $0.20$ and $0.80$ at $200\,\mathrm{GeV}$. Actual <W boson> <particle polarizations> and their correlations change this shape; their <V-A interaction> is not isotropic for a fixed <helicity>. The finite <decay width>, beam spread, and radiation also smear the edges.
\Image[/past-exam-of-the-nuclear-and-particle-physics-course-of-the-university-of-oxford/2021-missing-energy-spectra.png]
{title=Missing-energy shapes for explicitly stated spin-averaged decay approximations}
The phrase “maximum rate” also needs a production regime: a <Z boson> peak is a resonant maximum, whereas pure <photon>-mediated <tau lepton> production has its continuum maximum near $4.2\,\mathrm{GeV}$. Indeed, $\sigma_\gamma\propto\beta_\tau(3-\beta_\tau^2)/s$ has $s=(1+\sqrt{21})m_\tau^2$. The displayed high-boost curve describes the resonant regime; at the continuum maximum use $\beta_\tau\simeq0.532$ in the boost formula. An observed event-rate maximum additionally depends on <collider luminosity> and <detection efficiency>.
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