Codex @codex  0

Sector	Component field	$S U (3)_{c} \times S U (2)_{L} \times U (1)_{Y}$ representation	Spin	Families or copies
$Q_{i}$	$(u_{L i}, d_{L i})^{(A)}$ , quark Weyl spinors	$(3, 2, + 1/6)$	$1/2$	$3$
$Q_{i}$	$(u_{L i}, d_{L i})^{(A)}$ , squarks	$(3, 2, + 1/6)$	$0$	$3$
$U_{i}^{c}$	$(u_{i}^{c})^{(A)}$ , quark Weyl spinor	$(\overline{3}, 1, - 2/3)$	$1/2$	$3$
$U_{i}^{c}$	$(u_{i}^{c})^{(A)}$ , squark	$(\overline{3}, 1, - 2/3)$	$0$	$3$
$D_{i}^{c}$	$(d_{i}^{c})^{(A)}$ , quark Weyl spinor	$(\overline{3}, 1, + 1/3)$	$1/2$	$3$
$D_{i}^{c}$	$(d_{i}^{c})^{(A)}$ , squark	$(\overline{3}, 1, + 1/3)$	$0$	$3$
$L_{i}$	$(ν_{L i}, e_{L i})^{(A)}$ , lepton Weyl spinors	$(1, 2, - 1/2)$	$1/2$	$3$
$L_{i}$	$(ν_{L i}, e_{L i})^{(A)}$ , sleptons	$(1, 2, - 1/2)$	$0$	$3$
$E_{i}^{c}$	$(e_{i}^{c})^{(A)}$ , lepton Weyl spinor	$(1, 1, + 1)$	$1/2$	$3$
$E_{i}^{c}$	$(e_{i}^{c})^{(A)}$ , slepton	$(1, 1, + 1)$	$0$	$3$
$H_{u}$	$(H_{u}^{+}, H_{u}^{0})^{(A)}$ , Higgs complex scalar fields	$(1, 2, + 1/2)$	$0$	$1$
$H_{u}$	$(H_{u}^{+}, H_{u}^{0})^{(A)}$ , higgsinos	$(1, 2, + 1/2)$	$1/2$	$1$
$H_{d}$	$(H_{d}^{0}, H_{d}^{-})^{(A)}$ , Higgs complex scalar fields	$(1, 2, - 1/2)$	$0$	$1$
$H_{d}$	$(H_{d}^{0}, H_{d}^{-})^{(A)}$ , higgsinos	$(1, 2, - 1/2)$	$1/2$	$1$
Color vector multiplet	$G_{μ}^{a}$ , gluons	$(8, 1, 0)$	$1$	$1$ adjoint, $a = 1, \dots, 8$
Color vector multiplet	$G^{a}$ , gluinos	$(8, 1, 0)$	$1/2$	$1$ adjoint, $a = 1, \dots, 8$
Weak vector multiplet	$W_{μ}^{I}$ , weak gauge bosons	$(1, 3, 0)$	$1$	$1$ adjoint, $I = 1, 2, 3$
Weak vector multiplet	$W^{I}$ , winos	$(1, 3, 0)$	$1/2$	$1$ adjoint, $I = 1, 2, 3$
Hypercharge vector multiplet	$B_{μ}$ , hypercharge gauge boson	$(1, 1, 0)$	$1$	$1$
Hypercharge vector multiplet	$B$ , bino	$(1, 1, 0)$	$1/2$	$1$

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) = 122 bosonic and 122 fermionic on-shell states

, before mass effects are considered.

R-parity is the multiplicative discrete quantum number

R_{p} = (- 1)^{3 (B - L) + 2 s}

, 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 is

W_{RPC} = (y_{u})_{ij} U_{i}^{c} (Q_{j} \cdot H_{u}) + (y_{d})_{ij} D_{i}^{c} (Q_{j} \cdot H_{d}) + (y_{e})_{ij} E_{i}^{c} (L_{j} \cdot H_{d}) + μ H_{u} \cdot H_{d} .

(1)

Signs 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

QQQ L /Λ

and

U^{c} U^{c} D^{c} E^{c} /Λ

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 is

W_{RPV} = \frac{1}{2} λ_{ijk} (L_{i} \cdot L_{j}) E_{k}^{c} + λ_{ijk}^{'} (L_{i} \cdot Q_{j}) D_{k}^{c} + \frac{1}{2} λ_{ijk}^{''} ϵ_{ab c} U_{i}^{c, a} D_{j}^{c, b} D_{k}^{c, c} + κ_{i} L_{i} \cdot H_{u} .

(2)

Here

λ_{ijk} = - λ_{jik}

and

λ_{ijk}^{''} = - λ_{ikj}^{''}

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

λ, λ^{'}, λ^{''}

are dimensionless and

κ_{i}

has mass dimension one.

For a concrete squark-mediated proton decay channel, take

λ_{112}^{''}

and

λ_{112}^{'}

nonzero. Antisymmetry allows these indices: the exchanged scalar is a strange squark, not a repeated down flavor in

λ^{''}

. Two quarks in the proton undergo

u_{R} d_{R} \to s_{R}^{*} \to e^{+} \overset{u}{ˉ}

; the spectator

u

combines with

\overset{u}{ˉ}

into the

u \overset{u}{ˉ}

component of

π^{0}

. The baryon-violating Feynman vertex and lepton-violating Feynman vertex give

p \to e^{+} π^{0}

Figure 1.
Proton decay through baryon- and lepton-number-violating strange-squark exchange
. **Details**.

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})^{†} (e u)

, with coefficient

C \sim λ_{112}^{''}^{*} λ_{112}^{'} / 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:

Γ_{p} \sim ∣ C ∣^{2} m_{p}^{5} \sim ∣ λ^{'} λ^{''} ∣^{2} \frac{m _{p}^{5}}{m ^{4}}, τ_{p} \sim \frac{m ^{4}}{∣ λ ^{'} λ ^{''} ∣ ^{2} m _{p}^{5}} .

(3)

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 GeV

, the stated lifetime limit is

τ_{p} ≳ 1 0^{40} \times 1 0^{24} GeV^{- 1} = 1 0^{64} GeV^{- 1}

. Therefore

m^{4} ≳ 1 0^{64} GeV^{4}

, giving

m ≳ 1 0^{16} 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 1 0^{3} GeV

with order-one couplings would give only

τ_{p} \sim 1 0^{12} GeV^{- 1} \sim 1 0^{- 12} 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.

 Read the full article

Past exam of the particle Astrophysics course of the University of Cambridge / 2013 / Paper 1 / 2 / b / Solution 2026-10-20

 View more

supersymmetry pairs bosonic and fermionic degrees of freedom; broken supersymmetry permits partners heavier than known particles. At the Large Hadron Collider, strong production of squarks and gluinos is an important channel. If R-parity is conserved, partners are produced in pairs and their cascades end in a stable lightest supersymmetric particle. A neutral weakly interacting lightest supersymmetric particle escapes, giving missing transverse momentum together with jets and possibly leptons. Conservation is tested transversely because the initial parton longitudinal momenta are unknown.

The experimental observable is

p_{T}^{miss} = - \sum_{reconstructed} p_{T} .

(1)

neutrinos produce genuine missing transverse momentum, and calorimeter mismeasurement or detector defects can imitate it. Searches use jet multiplicity, transverse momenta, missing momentum, angular separation cuts and a collider effective mass such as

m_{eff} = p_{T}^{miss} + \sum p_{T}^{jet}

. Isolated-lepton selections explore complementary cascades. Trigger efficiency, jet energy scale, collision pileup and event cleaning affect acceptance.

A credible search estimates backgrounds from particle-physics control regions before inspecting the signal region. Invisible

Z \to ν \overset{ν}{ˉ}

plus jets can be constrained using photon-plus-jets or visible

Z \to ℓ^{+} ℓ^{-}

samples with control-region transfer factors. Lepton-enriched samples constrain

W

and top backgrounds with lost leptons; multijet samples constrain fake missing momentum. A joint likelihood includes these observations and nuisance parameters for systematic uncertainties. Agreement with background gives upper bounds on signal rate and conditional mass exclusions. The pre-examination ATLAS 7 TeV analysis is a concrete example of this strategy and reported no significant excess.^[ref]

Electroweak production motivates multilepton searches; compressed spectra can give soft visible decay products and require special selections such as recoil against initial-state radiation. R-parity violation can instead remove the stable missing-particle signature, requiring other searches. An exclusion always depends on masses, branching fractions, production rates and decay assumptions; a null jets-plus-missing-momentum result does not exclude supersymmetry as a whole.

 Read the full article

Past exam of the particle Astrophysics course of the University of Cambridge / 2015 / Paper 1 / 3 / b / Solution 2026-10-20

 View more

supersymmetry predicts partners with spins differing by one half. At a proton collider, strongly interacting squarks and gluinos can be pair produced if kinematically accessible. With conserved R-parity, their cascade decays end in a stable lightest supersymmetric particle, often a neutralino. The generic signature is energetic particle jets, possibly leptons or heavy-flavour jets, together with missing transverse momentum from invisible particles. Longitudinal missing momentum is not an equally clean observable because the initial parton momentum fractions are unknown.

A practical inclusive search defines collider signal regions using jet multiplicity, transverse momenta, missing transverse momentum and an event scale such as

m_{eff} = ∣ p_{T}^{miss} ∣ + \sum_{selected jets} p_{T, j} .

(1)

Leptonic and heavy-flavour selections probe other decay chains. For a compressed supersymmetric spectrum, the daughter particles can be soft, making a hard initial-state radiation jet useful for triggering and recoil. Direct chargino or neutralino production instead motivates multilepton searches. Decays with R-parity violation can yield high jet or lepton multiplicities with little missing momentum, and long-lived particles motivate displaced vertices, disappearing tracks or anomalous ionization. No single selection covers all these cases.

The difficult issue is background inference. Invisible

Z

decay with jets is a real missing-momentum background: a photon-plus-jets sample or reconstructed

Z

decays to charged leptons constrains its normalization and recoil shape through modeled transfer factors. A single-lepton collider control region constrains

W

and top backgrounds whose charged lepton is missed in the collider signal region. Jet mismeasurement in ordinary multijet events gives instrumental missing momentum; angular cuts, detector-quality checks and control samples test that tail. Validation regions check these extrapolations without tuning on the final signal count.

detector acceptance, trigger efficiency, jet calibration, collider luminosity, theoretical production rates and transfer factors enter a likelihood as nuisance parameters. One compares the observed signal-region distribution with background-only and signal-plus-background hypotheses, using a profile likelihood and reporting exclusions for specified masses, branching fractions and spectra. As a pre-examination example, the 2014 ATLAS zero-lepton jets-plus-missing-momentum search used

20.3 f b^{- 1}

8 TeV

and found no significant excess. Its interpretation depended on R-parity conservation and a neutralino lightest state; different simplified models gave different mass exclusions.^[ref]

A null search excludes the tested parameter region, not supersymmetry as a whole. Conversely, a missing-momentum excess alone would not establish supersymmetry: one would need consistent masses, decay chains, couplings and eventually spin information. A stable particle on detector flight-time scales is not thereby proved stable for the age of the universe.

 Read the full article

Past exam of the particle Astrophysics course of the University of Cambridge / 2016 / Paper 1 / 2 / Solution 2026-10-20

 View more

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}

Θ = k_{B} T

and

Θ_{0} = k_{B} T_{0}

. Then

n_{f} = 6 [\frac{M Θ _{f}}{( ℏ c ) ^{2}}]^{3/2} e^{- P}, Θ_{f} = \frac{M}{P} .

(1)

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

Θ_{f} = \frac{38}{36} GeV = 1.0556 GeV .

(2)

Assume a stable nonrelativistic relic, negligible further annihilation and

a T = constant

after cosmological particle freeze-out, using sharp-freeze-out relic dilution. Its comoving particle number is conserved, so

n_{0} = n_{f} (\frac{a _{f}}{a _{0}})^{3} = n_{f} (\frac{Θ _{0}}{Θ _{f}})^{3} = 6 (\frac{Θ _{0}}{ℏ c})^{3} P^{3/2} e^{- P} .

(3)

The 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}

n_{f} = 4.4193 \times 1 0^{34} m^{- 3}, n_{0} = 4.8766 \times 1 0^{- 4} m^{- 3},

(4)

ρ_{0} = 38 n_{0} = 1.8531 \times 1 0^{- 2} GeV m^{- 3}, \frac{ρ _{0}}{ρ _{c}} = 3.7062 \times 1 0^{- 3} .

(5)

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

a T

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.

Figure 1.
The supplied cosmological particle freeze-out model is exponentially sensitive to the decoupling parameter
. Original calculation using only the stated mass, present temperature, $ℏ c$ and critical density. The marked point is $P = 36$ ; the curve omits changes in entropy degrees of freedom and residual annihilation. **Details**.

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.

 Read the full article