= Solution
<supersymmetry> relates <bosons> and <fermions> through fermionic symmetry generators. A main motivation is the <Lorentz scalar> <hierarchy problem>: related <boson> and <fermion> loop contributions cancel particular ultraviolet sensitivities in an exactly supersymmetric theory. <soft supersymmetry breaking> leaves residual sensitivity to the <superpartner> scale, so a very heavy spectrum does not automatically solve naturalness. Supersymmetric matter content can also improve gauge-coupling unification, and a stable neutral lightest <superpartner> can be a dark-matter candidate. None of these motivations is itself experimental evidence; <supersymmetry> must be broken if ordinary particles and their partners have different observed <masses>.
If <R-parity> is conserved, <superpartners> are produced in pairs and cascades end in a stable <lightest supersymmetric particle>. A <neutralino> LSP can escape the detector, giving <missing transverse momentum>. Colored <squarks> and <gluinos> can generate energetic <jets>, sometimes b-tagged <jets> and <leptons>, plus that imbalance. An <inclusive supersymmetry search> classifies broad event samples by <jet> and <lepton> multiplicities, <transverse momentum> sums and missing <momentum>, without requiring a fully reconstructed decay chain.
Background controls are part of the physics argument. Lost-lepton W and top events can be constrained using <lepton> <collider control regions>; invisible-Z production can be related to visible-Z or photon samples with appropriate transfer factors. Instrumental imbalance in multijet events needs detector-quality checks and data sidebands. An excess in a <collider signal region> should survive uncertainty in those extrapolations, efficiencies and <luminosity>, with independently defined validation regions and attention to the number of searches performed.
An <exclusive supersymmetry search> targets a more specific production mode and decay pattern. Electroweak chargino-neutralino production can yield three <leptons> plus missing <momentum>; a cascade through a <slepton> can yield a dilepton invariant-mass edge that constrains <mass> combinations. Such searches often have cleaner backgrounds but depend more strongly on branching fractions and intermediate <masses>. An endpoint does not necessarily determine every absolute <mass> without additional observables.
Different spectra require different strategies. For a <compressed supersymmetric spectrum>, small parent-daughter <mass> differences make the visible decay products soft. <initial-state radiation> can recoil against the pair and provide a trigger, supplemented by soft-lepton reconstruction. The ATLAS compressed-spectrum study explicitly uses small <neutralino> <mass> differences and <initial-state radiation> to define a dedicated search rather than applying a hard-lepton selection indiscriminately.<Search for electroweak production of supersymmetric states in scenarios with compressed mass spectra at 13 TeV with the ATLAS detector by the ATLAS Collaboration>[arXiv:1712.08119, abstract and compressed-spectrum search strategy; methodological illustration published after this exam]{ref} If <R-parity> is violated, the LSP can decay visibly, so high-multiplicity <jet> or <lepton> channels may matter more than large missing <momentum>. CMS separately studied hadronic and leptonic final states for this reason.<Searches for R-parity-violating supersymmetry in pp collisions at 8 TeV in final states with 0-4 leptons by the CMS Collaboration>[CMS-SUS-14-003, abstract and independent lepton-multiplicity analyses]{ref} Long-lived particles instead motivate displaced vertices, disappearing tracks, abnormal <ionization> or time-of-flight measurements. Thus \b[there is no single model-independent missing-energy cut that covers every supersymmetric spectrum].
For the hypothetical relic-density calculation, use the supplied distribution and retain its prefactor exactly. Put $M=mc^2$, $\Theta=k_BT$ and $\Theta_0=k_BT_0$. Then
$$
n_f=6\left[\frac{M\Theta_f}{(\hbar c)^2}\right]^{3/2}e^{-P},\qquad
\Theta_f=\frac{M}{P}.
$$
<number density>, denoted $N$ in the paper, is written $n$ here to distinguish it from a total particle count. The <cosmological particle freeze-out> <temperature> in <energy> units is
$$
\boxed{\Theta_f=\frac{38}{36}\,\mathrm{GeV}=1.0556\,\mathrm{GeV}.}
$$
Assume a stable nonrelativistic relic, negligible further <annihilation> and $aT=\mathrm{constant}$ after <cosmological particle freeze-out>, using <sharp-freeze-out relic dilution>. Its comoving particle number is conserved, so
$$
n_0=n_f\left(\frac{a_f}{a_0}\right)^3
=n_f\left(\frac{\Theta_0}{\Theta_f}\right)^3
=\boxed{6\left(\frac{\Theta_0}{\hbar c}\right)^3P^{3/2}e^{-P}.}
$$
The particle <mass> cancels from this present <number density> under the stated approximation. Its present rest-energy density is $Mn_0$, not merely $n_0$:
$$
n_f=4.4193\times10^{34}\,\mathrm{m}^{-3},\qquad
n_0=4.8766\times10^{-4}\,\mathrm{m}^{-3},
$$
$$
\boxed{\rho_0=38n_0=1.8531\times10^{-2}\,\mathrm{GeV\,m}^{-3},\qquad
\frac{\rho_0}{\rho_c}=3.7062\times10^{-3}.}
$$
This is only \b[0.371% of the <critical density>], not a universe-closing component. The correct further comparison is with the measured dark-matter fraction, not an assumption that all <critical density> must be <dark matter>. In standard <cosmology> this abundance would be a small subcomponent, despite a possible collider discovery.
The <exponential freeze-out sensitivity> makes the result highly sensitive to $P$. Also, thermal <entropy> release means $aT$ is not exactly constant through the whole cosmic history. If total comoving <entropy> is conserved, the above <number density> acquires a factor $g_{*s,0}/g_{*s,f}$, involving <entropy degrees of freedom>. Those degrees of freedom and a full <annihilation> history are not given, so no precise correction is justified from the supplied data.
\Image[/past-exam-of-the-particle-astrophysics-course-of-the-university-of-cambridge/2016/paper-1-relic-sensitivity.png]
{title=The supplied <cosmological particle freeze-out> model is exponentially sensitive to the decoupling parameter}
{description=Original calculation using only the stated <mass>, present <temperature>, $\hbar c$ and <critical density>. The marked point is $P=36$; the curve omits changes in <entropy degrees of freedom> and residual <annihilation>.}
Establishing the nature of <dark matter> would require more than measuring a missing particle's collider <mass>. Measure its spin, charges, interactions, decay lifetime or stability limits, and <annihilation> or coannihilation processes; compare the resulting relic calculation with cosmological observations. Seek compatible direct nuclear-recoil signals and indirect <annihilation> or decay products, while controlling astrophysical and detector backgrounds. A collider-invisible lifetime need not be cosmologically long, and nonthermal production could change the abundance. \b[A consistent particle identity, cosmological abundance and independent non-collider evidence are needed to connect the hypothetical discovery to the <dark matter>.]
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