Supersymmetry

Every symmetry considered so far — Poincaré, gauge, internal — relates bosons to bosons and fermions to fermions. Supersymmetry (SUSY) is the unique possible symmetry that relates bosons to fermions, extending the Poincaré algebra by fermionic generators. Though not yet observed, it is the most-studied extension of the Standard Model, motivated by the hierarchy problem, gauge unification, and dark matter, and it is a powerful theoretical laboratory because SUSY theories are unusually calculable.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

The unique extension of Poincaré

A famous no-go theorem — Coleman–Mandula — states that in a relativistic QFT with a mass gap, the only possible symmetries combining spacetime and internal transformations are the trivial product Poincaré $\times$ internal. There is exactly one loophole: the theorem assumes bosonic (commutator) symmetry generators. Allowing fermionic (anticommutator) generators $Q_{\alpha }$ evades it, and the Haag–Łopuszański–Sohnius theorem shows the unique such extension is supersymmetry, with generators satisfying

$$\boxed{\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}=2{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }.}$$

The anticommutator of two SUSY transformations is a spacetime translation — SUSY "square-roots" the energy–momentum operator. The generator $Q$ is a spinor, so it changes spin by $\frac{1}{2}$:

$$Q|\text{boson}⟩=|\text{fermion}⟩,Q|\text{fermion}⟩=|\text{boson}⟩.$$

Supermultiplets and superpartners

SUSY organizes particles into supermultiplets containing equal numbers of bosonic and fermionic degrees of freedom with the same mass and quantum numbers. Every known particle would have a superpartner differing in spin by $\frac{1}{2}$:

Since no superpartners are seen at the masses of their partners, SUSY must be spontaneously broken (the superpartners are heavier, beyond current reach). The formalism is most elegant in superspace, where ordinary spacetime is augmented by anticommuting Grassmann coordinates $\theta$, and fields become superfields — single objects packaging a supermultiplet.

The hierarchy problem motivation

SUSY's leading phenomenological motivation is the hierarchy problem: the Higgs mass receives quadratically divergent loop corrections $\delta m_H^2\sim {\Lambda }^2$, which — with $\Lambda$ near the Planck scale — require an absurd fine-tuning to keep $m_H\sim 125$ GeV. In a SUSY theory, the boson and fermion loops in a supermultiplet contribute with opposite signs and cancel:

$$\delta m_H^2{|}_{\text{boson loop}}+\delta m_H^2{|}_{\text{fermion loop}}=0.$$

The quadratic divergence is removed exactly (softly broken SUSY leaves only a mild logarithmic sensitivity $\sim m_{\text{SUSY}}^2\ln \Lambda$), stabilizing the electroweak scale — provided the superpartners are not too heavy.

Non-renormalization theorems

SUSY makes theories dramatically more calculable through non-renormalization theorems: certain quantities receive no perturbative corrections beyond a fixed order. The superpotential (holomorphic couplings) is not renormalized at all in perturbation theory; some quantities are one-loop exact. These theorems make possible exact nonperturbative results — most famously Seiberg–Witten theory, the exact low-energy solution of a 4D gauge theory including instanton effects to all orders. SUSY theories are the closest thing to exactly solvable interacting QFTs in four dimensions, making them an invaluable theoretical laboratory even if SUSY is not realized in nature.

Other motivations and status

• Gauge coupling unification: extrapolating the running couplings of the Standard Model, the three gauge couplings nearly meet at $\sim 10^{16}$ GeV, and with superpartners in the loops they meet much more precisely — suggestive of grand unification.
• Dark matter: if $R$-parity is conserved, the lightest supersymmetric particle (LSP) is stable and a natural weakly-interacting dark-matter candidate.
• Local SUSY = supergravity: gauging SUSY necessarily includes gravity (the graviton and gravitino), a step toward unifying gravity with the other forces, and a low-energy limit of string theory.
• Status: the LHC has found no superpartners up to the TeV scale, pushing simple SUSY models into tension with naturalness. SUSY remains theoretically central but experimentally unconfirmed.

Summary

• SUSY is the unique (Haag–Łopuszański–Sohnius) extension of Poincaré with fermionic generators; $\{Q,\bar{Q}\}\sim P$.
• Particles pair into supermultiplets with superpartners (squarks, gluinos, higgsinos, …); SUSY must be spontaneously broken.
• Boson–fermion loop cancellation solves the hierarchy problem; superspace/superfields are the natural formalism.
• Non-renormalization theorems make SUSY theories exactly solvable in ways ordinary QFTs are not.

Where this leads

• The theory SUSY would extend: the Standard Model.
• Gravity from local SUSY: QFT in curved spacetime.
• Exact/nonperturbative structure: solitons and instantons, conformal field theory.

References

• Wess & Bagger, Supersymmetry and Supergravity.
• Martin, A Supersymmetry Primer, arXiv:hep-ph/9709356.
• Weinberg, The Quantum Theory of Fields, Vol. 3.
• Seiberg & Witten, Nucl. Phys. B 426, 19 (1994).