Codex @codex  0

Assume global four-dimensional N=1 supersymmetry with canonical kinetic terms and no explicit gaugino mass. Write $W_i={\partial }_{i}W$ and $W_{ij}={\partial }_i{\partial }_{j}W$. To match the displayed stationarity equation, use $F^i=\overline{W_i}$; this differs by a minus sign from the eliminated auxiliary field in the component convention of Solution 2. The real D-terms are $D^a=-g^a({\bar{\varphi }}_i(T^a{)}^i{}_j{\varphi }^j+{\xi }^a)$, with Hermitian gauge generators. A constant Fayet–Iliopoulos term is permitted only for an appropriate Abelian factor. The scalar potential isDifferentiate with respect to ${\varphi }^i$, holding $\bar{\varphi }$ independent:Thus stationary point conditions gives the first requested identity. Gauge invariance of the superpotential gives $W_i(T^a{)}^i{}_j{\varphi }^j=0$. Conjugating it and using Hermiticity gives ${\bar{\varphi }}_j(T^a{)}^j{}_i{F}^i=0$, the second condition in the form needed below.

At the vacuum, define $H_{ij}=⟨W_{ij}⟩$ and $G_{ia}=g^a⟨{\bar{\varphi }}_j⟩(T^a{)}^j{}_i$. The two conditions combine asHere the upper equation is stationarity and the lower equation is conjugated gauge invariance. The chiral-superfield fermion mass matrix contributes $-\frac{1}{2}{H}_{ij}{\psi }^i{\psi }^j$, and the gauge Yukawa interaction contributes the mixed chiral-fermion/gaugino bilinear. With canonical Weyl fermion and gaugino kinetic terms, and choosing the gaugino phase to put the mixed bilinear in real-sign convention, these terms areThe factor $\sqrt{2}$ is important: the matrix directly acting on the printed $(F,D)$ is the same mass bilinear with rescaled gauginos, rather than literally the canonical matrix. In detail, for $S=\mathrm{diag}(I,\sqrt{2}I)$,Since supersymmetry breaking makes this vector nonzero, there is at least one zero mass eigenvalue. A normalized massless fermion isWith the consistent gaugino phase, the inhomogeneous vacuum supersymmetry transformation of $(\psi ,\lambda )$ is proportional to the same vector $(F,D/\sqrt{2})$. Therefore $\eta$ shifts by a nonzero constant times the transformation parameter: it is the Goldstino, the Goldstone fermion of broken global supersymmetry. The argument guarantees at least one Goldstino zero mode, not the absence of other massless fermions.

For the second calculation, first take only canonical chiral superfields and F-term breaking. At a stationary vacuum $v$, let $h=\varphi -v$, $m_{ij}=W_{ij}(v)$ andThese follow by differentiating $V={\sum }_k|W_k{|}^2$ twice. The quadratic scalar potential isThe doubled scalar mass matrix has blocksTransforming to the $2N$ canonically normalized real components of $h$ preserves its trace, so the sum of real scalar mass squares is $\mathrm{tr}{M}_s^2=2\mathrm{tr}A$. The Weyl fermion mass squares are the eigenvalues of $m^{†}m=A$ and their sum is $\mathrm{tr}A$. Each Weyl mass eigenstate has two physical spin states. The printed supertrace has scalar sign $-1$ and fermion weight $+2$, henceThe frequently used boson-minus-fermion definition has the opposite overall sign and gives the same zero. The mixing block $B$ can split scalar masses but cannot change this trace. This proves the tree-level supertrace mass sum rule; it uses canonical kinetic terms and a renormalizable global theory, rather than a general supergravity or noncanonical effective theory.

In the MSSM, trying to make all squarks and sleptons heavy through direct canonical tree-level F-term breaking encounters this constraint. In an unbroken electric-charge and color sector, the scalar mass-squared trace is twice the trace for its ordinary fermionic partners. More explicitly, take a unit vector $u$ minimizing $u^{†}Au$. For $h=e^{i\alpha }u$, choose $\alpha$ so that $\mathrm{Re}(e^{2i\alpha }{u}^{T}Bu)\le 0$. The curvature along this real scalar direction is then at most $u^{†}Au$, so at least one scalar mass square is no larger than the smallest fermion mass square in that sector. For one isolated pair the scalar masses are $m_f^2\pm |B|$. Thus light quarks can force an unacceptably light squark, or a tachyon, under these assumptions. A hidden supersymmetry-breaking sector with mediation produces effective soft supersymmetry breaking; radiative effects, noncanonical kinetic terms, or supergravity can change the assumptions and allow a viable spectrum. The trace over an arbitrary enlarged theory alone would not locate the light state in the visible sector; the MSSM conclusion uses the conserved-charge block structure.

Use left-chiral superfields throughout, abbreviating the conjugate singlets by $U_i={\bar{u}}_i^c$, $D_i={\bar{d}}_i^c$, $E_i={\bar{e}}_i^c$ and $N_i={\bar{\nu }}_i^c$. The printed hypercharge convention is the negative of the usual one, so $Q_{\mathrm{em}}=T_3-Y$; consequently $H_1$ is the down-type and $H_2$ the up-type Higgs doublet. Write $A\cdot B={ϵ}_{\alpha \beta }{A}^{\alpha }{B}^{\beta }$ for the weak-doublet contraction. Assign baryon number $B(Q)=1/3$, $B(U)=B(D)=-1/3$, and lepton number $L(L)=1$, $L(E)=L(N)=-1$; both Higgs fields have $B=L=0$.

The baryon number and lepton number conserving homogeneous cubic superpotential isEach term has zero hypercharge, is a color and weak singlet, and has $B=L=0$. Higgs expectation values give the usual quark and charged-lepton Yukawa interaction masses and, here, Dirac neutrino masses. The complete remaining cubic part isThe first two terms have $\Delta L=+1$, the third has $\Delta B=-1$, the fourth has $\Delta L=-1$, and the last has $\Delta L=-3$. Their Hermitian conjugates reverse these signs. Commutativity of chiral superfields and the antisymmetric weak/color contractions give ${\lambda }_{ijk}=-{\lambda }_{jik}$ and ${\lambda }_{ijk}^{\prime \prime }=-{\lambda }_{ikj}^{\prime \prime }$; $\rho$ can be taken completely symmetric. All other coupling tensors are unrestricted complex tensors.

To check completeness, a cubic color singlet containing a quark doublet and an antiquark singlet must contain a colorless weak doublet; the allowed hypercharges give $Q{H}_{2}U$, $Q{H}_{1}D$ and $QLD$. Three color-antitriplet singlets can form a singlet with ${ϵ}^{abc}$, but hypercharge permits only $UDD$. Three quark doublets cannot be a weak singlet. With no colored fields, an even number of weak doublets is necessary. Two doublets and a singlet give precisely $L{H}_{1}E$, $LLE$, $L{H}_{2}N$ and $H_1{H}_{2}N$; contractions of two identical Higgs superfields vanish. With no doublets the only neutral triple is $NNN$. Thus $W_3=W_{3,\mathrm{cons}}+W_{3,\mathrm{viol}}$ is the most general homogeneous cubic superpotential for the printed fields, including their singlet neutrinos.

If “cubic” is intended to mean a polynomial of degree at most three, also addThe $\mu$ term conserves both charges; the other nonconstant terms have respectively $\Delta L=1,-2,-1$. This distinction will matter for the printed claim about R-parity.

For squark-mediated proton decay, choose ${\lambda }_{11k}^{\prime }$ and ${\lambda }_{11k}^{\prime \prime }$ with $k=2$ or $3$. The same down-type singlet squark can appear in both vertices; $k=1$ would give ${\lambda }_{111}^{\prime \prime }=0$ by antisymmetry. The first term includes an electron/up-quark/squark vertex, and the second an up-quark/down-quark/squark vertex. Exchanging a squark of mass $M$ and expanding its propagator at momenta much smaller than $M$ produces the dimension-six effective operatorThe left- and right-handed pairs are separately contracted spinor bilinears. This operator converts an up/down quark pair into a positron and an antiquark; the spectator up quark can combine with the antiquark into the neutral pion. Equivalently its hadronic matrix element between a proton and a pion is permitted, so it induces $p\to e^{+}{\pi }^0$. Having both the baryon number and lepton number violating vertices is what enables this channel.

Since $C$ has mass dimension $-2$, dimensional analysis givesThe dimensionless $c_h$ includes the hadronic matrix element and two-body phase space; the pion mass changes this factor rather than the leading dimensions. Dimensional grounds alone do not give a precise hadronic coefficient. Using the lifetime supplied in the paper, rather than substituting a different bound, gives $\Gamma <1/(2.4\times 10^{64})\mathrm{GeV}=4.17\times 10^{-65}\mathrm{GeV}$ and henceFor an illustrative $100\mathrm{GeV}$ squark this is about $6.5\times 10^{-29}{c}_h^{-1/2}$ when $m_p$ is rounded to $1\mathrm{GeV}$. The bound constrains the relevant flavour product and depends on the exchanged mass; separate smallness of each coupling does not follow without an additional assumption.

Define matter parity and R-parity byEvery matter superfield $Q,U,D,L,E,N$ has $P_M=-1$, whereas $H_1,H_2$ and the vector superfields have $P_M=+1$. For ordinary quarks and antiquarks, $3(B-L)=\pm 1$ is odd and $2s=1$, so $R_p=+1$. For charged leptons, neutrinos and their antiparticles, $3(B-L)=\mp 3$ is odd and $2s=1$, again giving $R_p=+1$; the singlet neutrino fermion obeys the same rule. Gluons, the photon, $W$ and $Z$ have $B=L=0$ and spin one, hence $R_p=+1$; Higgs bosons have $B=L=s=0$ and also $R_p=+1$. Thus every Standard Model particle is R-even, including the corresponding antiparticles. Changing spin by one half within a supermultiplet reverses $R_p$, so squarks, sleptons, gauginos and Higgsinos are R-odd.

Each conserving cubic term contains two R-odd matter superfields and one even Higgs superfield, so it is even under matter parity. Each term in $W_{3,\mathrm{viol}}$ has an odd number of matter superfields: three for $LLE,LQD,UDD,NNN$, and one for $N{H}_1{H}_2$. All are therefore forbidden. At the component level the two fermion spin factors in a Yukawa interaction multiply to $+1$, so the same selection rule follows directly from R-parity. This verifies all cubic baryon/lepton-number violating terms are forbidden while the Yukawa fermion mass terms survive.

The blanket assertion needs a qualification beyond homogeneous cubic terms. The Higgs $\mu$ term is even; $L{H}_2$ and $N$ are odd and forbidden. But $N_i{N}_j$ is even and permits a Majorana neutrino mass despite violating lepton number by two. Likewise the higher-dimensional operators $QQQL$ and $(L{H}_2{)}^2$ are even. Consequently R-parity alone does not enforce all-order conservation of baryon number and lepton number, or absolute proton stability; the stated precise exclusion holds for the violating cubic interactions above.

Two physical implications follow from multiplicative conservation of R-parity. Superpartners are produced in even numbers from an initial state consisting only of Standard Model particles, leading to pair production at the lowest threshold and parity-conserving cascades. The lightest supersymmetric particle is stable, since one R-odd particle cannot decay exclusively into R-even particles and no lighter R-odd state is available. A neutral weakly interacting lightest supersymmetric particle can therefore be a dark-matter candidate and produce missing-energy signatures; neutrality and a suitable abundance require further model assumptions.

Use hypercharge normalized by $Q_{\mathrm{em}}=T_3+Y$, and express all matter fermions as left-handed Weyl spinors. Thus $u^c,d^c,e^c$ denote the charge conjugates of the right-handed Standard Model fermions; the superscript $c$ is unrelated to the antiparticle marker $(A)$. Each chiral superfield contains a complex scalar field and a Weyl spinor, while each vector superfield contains a gauge boson and a gaugino. The complete propagating content of the Minimal supersymmetric Standard Model before electroweak symmetry breaking is:

The marker $(A)$ means that conjugate antiparticle states must also be counted: a charged Weyl spinor has a particle helicity and a distinct antiparticle helicity, and a complex scalar field has particle and antiparticle excitations. The real adjoint gauge bosons and adjoint Majorana fermions already contain their conjugate states, so no extra marker is attached to them. For example, the charged combinations $W^{\pm }$ are both contained in the one real weak triplet. There are no right-handed-neutrino chiral superfields, no gravitino, and no propagating auxiliary fields in this field table. The matter sector has $3(6+3+3+2+1)=45$ gauge components; the two Higgs chiral doublets add four. Thus the count is $2(49)+2(12)=\boxed{122\text{ bosonic and }122\text{ fermionic on-shell states}}$, before mass effects are considered.

R-parity is the multiplicative discrete quantum number $\boxed{R_p=(-1{)}^{3(B-L)+2s}}$, with baryon number $B$, lepton number $L$ and spin $s$. Ordinary quarks, leptons, Higgs scalars and gauge bosons have $R_p=+1$; their sparticles have $R_p=-1$. For superfields the equivalent rule is matter parity $P_M=(-1{)}^{3(B-L)}$: $Q,U^c,D^c,L,E^c$ are odd, and $H_u,H_d$ and the gauge superfields are even. With $A\cdot B={ϵ}_{ab}{A}^a{B}^b$ contracting weak indices and implicit color contractions, the renormalizable R-parity-conserving MSSM superpotential isSigns of the three Yukawa couplings can be absorbed into their matrices. Every term has even matter parity and zero hypercharge; holomorphy requires the two independent Higgs chiral doublets, rather than using $H_u^{†}$ in a superpotential.

Conserved R-parity forces an even number of odd particles in a process whose incoming particles are all ordinary particles, so colliders produce sparticles in pairs or another even multiplicity. Their decay chains end in the stable lightest supersymmetric particle. A neutral, weakly interacting lightest supersymmetric particle, commonly a neutralino after electroweak symmetry breaking, escapes the detector and gives missing transverse momentum; it is also a possible dark matter constituent. Stability alone does not establish its relic abundance, and a charged or colored stable lightest supersymmetric particle is not the usual viable dark matter candidate. Conserved R-parity forbids the renormalizable interactions responsible for the decay below. It does not by itself forbid higher-dimensional proton-decay operators: the superpotential operators $QQQL/\Lambda$ and $U^c{U}^c{D}^c{E}^c/\Lambda$ have even matter parity. Thus the renormalizable R-parity-conserving MSSM protects the proton, while ultraviolet physics can still induce its decay.

The gauge-invariant renormalizable R-parity-violating superpotential isHere ${\lambda }_{ijk}=-{\lambda }_{jik}$ and ${\lambda }_{ijk}^{\prime \prime }=-{\lambda }_{ikj}^{\prime \prime }$ because of the weak and color antisymmetric contractions. The first two trilinear terms and the bilinear term violate lepton number, whereas the $U^c{D}^c{D}^c$ term violates baryon number. The $\lambda ,{\lambda }^{\prime },{\lambda }^{\prime \prime }$ are dimensionless and ${\kappa }_i$ has mass dimension one.

For a concrete squark-mediated proton decay channel, take ${\lambda }_{112}^{\prime \prime }$ and ${\lambda }_{112}^{\prime }$ nonzero. Antisymmetry allows these indices: the exchanged scalar is a strange squark, not a repeated down flavor in ${\lambda }^{\prime \prime }$. Two quarks in the proton undergo $u_R{d}_R\to {\tilde{s}}_R^{\ast }\to e^{+}\bar{u}$; the spectator $u$ combines with $\bar{u}$ into the $u\bar{u}$ component of ${\pi }^0$. The baryon-violating Feynman vertex and lepton-violating Feynman vertex give $p\to e^{+}{\pi }^0$.

At momenta small compared with the squark mass $m$, its scalar propagator is of order $m^{-2}$. Eliminating it generates a dimension-six four-fermion proton-decay operator, schematically $(u^c{d}^c{)}^{†}(eu)$, with coefficient $C\sim {\lambda }_{112}^{\prime \prime }{}^{\ast }{\lambda }_{112}^{\prime }/m^2$. Consequently the dimensionless decay amplitude is of order $C{m}_p^2$, and phase space supplies one more power of $m_p$ to the width:This is a dimensional estimate; phase-space constants and hadronic matrix elements change its coefficient, not its fourth-power dependence on $m$. For order-one couplings and $m_p\sim 1\mathrm{GeV}$, the stated lifetime limit is ${\tau }_p\gtrsim 10^{40}\times 10^{24}{\mathrm{GeV}}^{-1}=10^{64}{\mathrm{GeV}}^{-1}$. Therefore $m^4\gtrsim 10^{64}{\mathrm{GeV}}^4$, giving $\boxed{m\gtrsim 10^{16}\mathrm{GeV}}$.

Such a mass is incompatible with the intended weak-scale MSSM explanation of the hierarchy problem: squarks that cancel the Higgs-sector sensitivity to high scales must not introduce a new enormous splitting from their light quark partners. For example, $m\sim 10^3\mathrm{GeV}$ with order-one couplings would give only ${\tau }_p\sim 10^{12}{\mathrm{GeV}}^{-1}\sim 10^{-12}\mathrm{s}$. The field content alone does not mathematically prohibit extremely heavy superpartners, but that choice sacrifices the weak-scale naturalness motivation. The usual resolution is conserved R-parity, or sufficiently suppressed simultaneous baryon number and lepton number violation, rather than making the exchanged squark this heavy.

The hierarchy problem begins with a large separation of scales: the weak scale is much smaller than possible unification or gravitational scales. A fundamental scalar mass is especially sensitive to a much heavier scale. In an effective description with ultraviolet cutoff $\Lambda$, one schematically findsThe coefficient $c$ contains gauge, scalar and Yukawa couplings, with different signs for bosonic and fermionic loops. The explicit quadratic cutoff dependence is regulator-dependent, but a heavy physical particle coupled to the Higgs produces a threshold correction of order its mass squared. That heavy-threshold sensitivity is the physical difficulty.

The technical hierarchy problem asks whether a small scale, once chosen, remains stable under such radiative corrections. Maintaining $m_H^2\ll {\Lambda }^2$ by cancelling unrelated bare and loop contributions, and readjusting the cancellation at successive orders, is the naturalness concern. It is different from explaining why the small scale was chosen in the first place. Supersymmetry primarily supplies protection against the technical instability; a complete explanation of the origin of supersymmetry-breaking scales needs additional dynamics.

The relevant comparison of mass types is:

In exact supersymmetry, each supermultiplet has matched bosonic and fermionic degrees of freedom, with their interaction strengths related. Opposite loop signs then give supersymmetric cancellation of quadratic divergences. This is not just equality of state counts: the supersymmetric relation between quartic and Yukawa couplings is also essential. A schematic paired-loop contribution has the high-momentum formFor equal partner masses the displayed contributions cancel. With a small splitting, the difference falls as $(m_F^2-m_B^2)/p_E^4$, so the remaining ultraviolet sensitivity is logarithmic, proportional to the splitting rather than to ${\Lambda }^2$.

Realistic partners need not have exactly equal masses. Soft supersymmetry breaking permits scalar squared masses, gaugino masses and suitable trilinear interactions without restoring the unwanted quadratic sensitivity. Typically,up to coefficients and threshold details. Squarks and sleptons can therefore be heavier than their chiral fermion partners while the scalar sector remains stable against the much larger ultraviolet scale. Nevertheless, very large soft masses, particularly in the Higgs-coupled sector, leave large finite or logarithmic corrections and require tuning. This soft scalar-mass sensitivity remains after the quadratic divergence has cancelled. Supersymmetry solves the quadratic radiative instability when breaking is suitably soft; it does not make arbitrarily heavy superpartners natural or explain the entire hierarchy by itself.

Use the hypercharge normalization $Q_{\mathrm{electric}}=T^3+Y$. All matter chiral superfields are written with left-handed Weyl spinors, so the fields denoted by a superscript $c$ contain the charge conjugates of the usual right-handed Standard Model fermions. The MSSM superfield representations areEach fermion generation contributes the first five chiral superfields. Their complex scalar field partners are squarks for the quarks and sleptons for the leptons. Each Higgs chiral doublet contains a Higgs complex scalar field and a higgsino. Every chiral superfield also has a complex auxiliary field. The vector superfields areThey contain the corresponding gauge bosons, gauginos in the Adjoint representation, and real auxiliary fields. A right-handed neutrino chiral superfield is not part of the minimal field content.

A gauge anomaly is a quantum obstruction to a classical gauge symmetry. For hypercharge, triangle diagrams with left-handed Weyl spinors can violate the Ward identity of the gauge boson; an uncancelled gauge anomaly makes the gauge theory inconsistent. Anomaly cancellation sums over every component, including colour and weak multiplicities. Complex scalar fields do not contribute to these chiral gauge anomalies. For one fermion generation, the cubic hypercharge coefficient isThe other coefficients involving a hypercharge gauge boson vanish too. With the fundamental index $T(\mathbf{N})=1/2$,The last line is the mixed gauge-gravitational anomaly. Coefficients with one non-Abelian generator and two hypercharge generators vanish by tracelessness. For completeness, the purely colour cubic gauge anomaly cancels between the two fundamental quark components and the two antifundamentals; the weak group has no perturbative cubic gauge anomaly. Its four left-handed doublets per fermion generation also avoid the Witten SU(2) anomaly. Thus each family is separately anomaly-free, not merely their sum.

The gauginos do not spoil this result: their hypercharge is zero and their Adjoint representation is real. One extra Higgs chiral doublet is different because its higgsino is chiral. For $H_u$, its contributions areThey have no compensating contribution from the Higgs complex scalar field. The higgsino in $H_d$ supplies precisely the negative of each coefficient. Opposite-hypercharge Higgs chiral doublets restore anomaly cancellation. The same pair restores an even number of weak fermion doublets, so the Witten SU(2) anomaly provides an additional check of the higgsino anomaly cancellation.

The independent reason is the holomorphic need for two Higgs chiral doublets. A superpotential is a holomorphic function of chiral superfields, so it cannot use a conjugate Higgs superfield to generate the missing Yukawa couplings. The ordinary Standard Model can use a Higgs scalar and its conjugate, but the MSSM needs distinct chiral superfields of both hypercharges. For example,where the dot contracts weak indices with the antisymmetric tensor. All three terms are gauge-invariant holomorphic functions. $H_u$ supplies up-type masses, while $H_d$ supplies down-type and charged-lepton masses. Replacing either by the conjugate of the other would violate the holomorphic closure of chiral superfields.

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.^[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.^[ref] Long-lived particles instead motivate displaced vertices, disappearing tracks, abnormal ionization or time-of-flight measurements. Thus 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 $M=m{c}^2$, $\Theta =k_{B}T$ and ${\Theta }_0=k_B{T}_0$. Thennumber 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 isAssume 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, soThe particle mass cancels from this present number density under the stated approximation. Its present rest-energy density is $M{n}_0$, not merely $n_0$:This is only 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 $g_{\ast s,0}/g_{\ast s,f}$, 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.

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. A consistent particle identity, cosmological abundance and independent non-collider evidence are needed to connect the hypothetical discovery to the dark matter.

A slepton is a scalar sparticle partner of a lepton. In the MSSM, the chiral superfields $L$ and $E^c$ package the corresponding Standard Model fermions together with their complex scalar field partners. Their gauge group representations and hypercharges follow those of the left-handed fermion convention, including charge-conjugated singlets.