Supersymmetry · study.

Every symmetry considered so far commutes with the spin operator: gauge transformations, flavor rotations, and spacetime translations all map fermions to fermions and bosons to bosons. Supersymmetry is the one remaining possibility — a symmetry whose generator carries spin $\frac{1}{2}$ and therefore turns a fermion into a boson and back. It doubles the particle spectrum, and the doubling addresses three separate problems at once: the instability of the Higgs mass under radiative corrections, the imperfect meeting of the gauge couplings, and the absence of a Standard Model dark-matter candidate.

The superalgebra and superpartners

A supersymmetry generator $Q$ transforms a bosonic state into a fermionic one,

Q ∣ boson ⟩ = ∣ fermion ⟩, Q ∣ fermion ⟩ = ∣ boson ⟩,

so $Q$ raises or lowers the spin by half a unit. The anticommutator of two supersymmetry generators is a spacetime translation,

{Q_{a}, \overset{ˉ}{Q}_{\dot{b}}} = 2 σ_{a \dot{b}}^{μ} P_{μ},

which ties supersymmetry to the Poincaré group: it is a spacetime symmetry, not an internal one, and its square is the momentum operator. Because $Q$ commutes with $P^{2} = m^{2}$ , exact supersymmetry would force each particle and its superpartner to have the same mass.

The particles pair as follows.

Sfermions. Each spin- $\frac{1}{2}$ quark and lepton gains a spin- $0$ partner, the squark and slepton. A left- and a right-handed fermion each get their own scalar, so the electron has two scalar partners.
Gauginos. Each spin- $1$ gauge boson gains a spin- $\frac{1}{2}$ partner: the gluino for the gluon, the winos and bino for the electroweak bosons.
Higgsinos. Two Higgs doublets are required (one to give mass to up-type quarks, one to down-type and leptons, and to cancel anomalies), and each has a spin- $\frac{1}{2}$ higgsino partner.

The electroweak gauginos and higgsinos mix into mass eigenstates: four neutral neutralinos and two charged charginos. This minimal spectrum is the Minimal Supersymmetric Standard Model (MSSM).

The Standard Model particles and their superpartners. Each fermion gains a scalar partner and each gauge or Higgs boson gains a spin-one-half partner, so every partner differs from its Standard Model particle by half a unit of spin.

Cancelling the Higgs-mass divergence

The strongest theoretical motivation for supersymmetry is the behavior of the Higgs mass under quantum corrections. A scalar mass receives loop corrections from every particle it couples to. A fermion loop, such as the top quark, contributes

δ m_{H}^{2}_{fermion} = - \frac{λ _{f}^{2}}{8 π ^{2}} Λ^{2} + \dots,

quadratic in the ultraviolet cutoff $Λ$ . If $Λ$ is the GUT or Planck scale, this correction is thirty orders of magnitude larger than the physical $m_{H}^{2} \approx (125 GeV)^{2}$ , and an enormous fine-tuning of the bare mass is needed to leave the small remainder. A scalar loop contributes with the opposite sign,

δ m_{H}^{2}_{scalar} = + \frac{λ _{s}}{16 π ^{2}} Λ^{2} + \dots .

Supersymmetry relates the couplings so that each fermion is accompanied by two scalars with $λ_{s} = λ_{f}^{2}$ . The quadratic pieces cancel exactly:

δ m_{H}^{2} = \frac{1}{16 π ^{2}} (λ_{s} - 2 λ_{f}^{2}) Λ^{2} + \dots = 0 (quadratic part) .

What remains is a logarithmic dependence proportional to the superpartner-fermion mass splitting,

δ m_{H}^{2} \sim \frac{λ _{f}^{2}}{16 π ^{2}} (m_{\tilde{f}}^{2} - m_{f}^{2}) ln \frac{Λ}{m _{\tilde{f}}},

which stays small provided the superpartner masses $m_{\tilde{f}}$ are not far above the electroweak scale. Naturalness of the Higgs mass thus predicts superpartners near a TeV.

The top-quark loop and the stop-squark loop contribute to the Higgs mass squared with opposite signs. Supersymmetry fixes their couplings so the quadratic cutoff dependence cancels, leaving only a mild logarithmic term.

Improved coupling unification

The one-loop running coefficients change when the superpartners are added above their mass threshold, because each new particle contributes to the beta functions. With the MSSM spectrum the coefficients become

(b_{1}, b_{2}, b_{3}) = (\frac{33}{5}, 1, - 3),

replacing the Standard Model values. The altered slopes make the three inverse couplings meet at a single point rather than merely passing close, and they raise the unification scale to

M_{X} \approx 2 \times 1 0^{16} GeV,

which also lengthens the predicted proton lifetime beyond the Super-Kamiokande bound. The precision of the meeting under the MSSM is often cited as indirect evidence for TeV-scale supersymmetry, though it depends on the assumed superpartner thresholds.

With the supersymmetric spectrum switched on above a TeV, the three inverse couplings change slope and meet at a single point, in contrast to the near-miss of the Standard Model running.

R-parity and the dark-matter candidate

The most general supersymmetric Lagrangian contains terms that violate baryon and lepton number and would drive proton decay at an unacceptable rate. They are forbidden by imposing a discrete symmetry, R-parity, defined by

R = (- 1)^{3 (B - L) + 2 s},

which is $+ 1$ for every Standard Model particle and $- 1$ for every superpartner. Conservation of $R$ has two consequences.

Superpartners are produced in pairs.
The lightest supersymmetric particle (LSP) is absolutely stable, because a single superpartner cannot decay to Standard Model particles alone without changing $R$ .

If the LSP is the lightest neutralino — an electrically neutral, weakly interacting mixture of the bino, wino, and higgsinos — it is a natural candidate for dark matter. Its interactions have roughly the weak-scale strength, and a stable relic of weak-scale mass freezes out of the early universe with about the observed dark matter abundance, the WIMP miracle argument developed in the dark-matter lesson.

Collider signatures and current limits

R-parity conservation shapes the experimental search. At a hadron collider, strongly interacting superpartners — squarks and gluinos — are produced in pairs and cascade down to the LSP, which leaves the detector unseen. The signature is jets or leptons recoiling against large missing transverse momentum, carried off by the two invisible LSPs.

A supersymmetric cascade at a collider. A produced gluino decays through squarks and lighter neutralinos down to the stable LSP, which escapes and leaves an imbalance of transverse momentum, the search signature.

No superpartner has been seen. The LHC excludes gluinos below about $2.2$ TeV and first- and second-generation squarks below roughly $1.8$ TeV in the simplest models with a light neutralino LSP.¹ These limits are already in tension with the naturalness argument, which prefers superpartners near a few hundred GeV; the gap between the predicted and excluded mass ranges is the current form of the hierarchy problem. Supersymmetry may still exist at higher masses, with more compressed or more elaborate spectra that soften the collider signature, but the simplest natural version is disfavored.

The LHC excludes a region of the gluino-squark mass plane. Points below and to the left, where the superpartners are light, are ruled out; heavier spectra remain allowed but are less natural.

Summary

Supersymmetry pairs each Standard Model particle with a superpartner whose spin differs by $\frac{1}{2}$ : sfermions for the quarks and leptons, gauginos and higgsinos for the gauge and Higgs bosons, mixing into neutralinos and charginos. Its generator satisfies ${Q, \overset{ˉ}{Q}} \sim P$ , tying it to spacetime. The pairing makes the quadratically divergent fermion and scalar loop corrections to the Higgs mass cancel, leaving only a logarithm and predicting superpartners near a TeV; it sharpens the three-coupling unification to a single point at $M_{X} \approx 2 \times 1 0^{16}$ GeV; and, with R-parity conserved, it leaves a stable neutral LSP as a dark-matter candidate. The LHC has excluded gluinos and light squarks below about $2$ TeV, placing the natural version under pressure.²³

Current LHC mass limits on gluinos and squarks (roughly $2.2$ TeV and $1.8$ TeV in simplified models with a light neutralino) are compiled in the Particle Data Group supersymmetry review, pdg.lbl.gov. ↩
Griffiths, Ch. 12 (Afterword), introduces supersymmetry, the superpartner naming, and the role of the LSP. ↩
Perkins, Ch. 12, and Thomson, Ch. 18, develop the MSSM, the Higgs-mass cancellation, the improved coupling unification with the MSSM beta-function coefficients, R-parity, and the collider missing-momentum signature. ↩