Supersymmetry

Supersymmetry is the boldest of the symmetries physicists have proposed and the only one still unconfirmed. Every symmetry in the Standard Model relates like to like — one electron to another, one colour to another, a field to its rotated self. Supersymmetry (SUSY) does something no ordinary symmetry can: it relates particles of different spin, pairing every boson with a fermion and every fermion with a boson. The generator that does this, the supercharge $Q$ , carries half a unit of spin, so acting with it turns a spin-0 Higgs into a spin-½ higgsino and a spin-½ quark into a spin-0 squark. In the language of the algebra, $Q$ is a square root of spacetime translation itself: two supersymmetry transformations compose into an ordinary shift in space and time. The motivation is not aesthetic tidiness but a concrete disease of the Standard Model. The Higgs is the one elementary scalar we know, and scalar masses are quadratically sensitive to every heavy scale a theory contains — the Higgs mass-squared drags in a correction of order the cutoff squared, which for a cutoff at the Planck scale is thirty-odd orders of magnitude too large. This is the hierarchy problem. Supersymmetry cures it automatically: because bosons and fermions contribute loop corrections of opposite sign, a Higgs whose every partner is supersymmetric receives corrections that cancel the runaway piece exactly, leaving only a mild logarithm. The same superpartners, as a bonus, make the three gauge couplings meet at a single high scale and supply a stable, weakly-interacting dark-matter candidate. This article builds SUSY from its algebra, proves the cancellation, and works the numbers that made it the dominant paradigm beyond the Standard Model for forty years.

Key takeaways

Supersymmetry is a spacetime symmetry whose generator $Q$ has spin ½: it maps every boson to a fermion and back, and $\{Q,Q^\dagger\}=2\sigma^\mu P_\mu$ makes it a square root of translations.
Its headline job is the hierarchy problem: a fermion loop and its partner scalar loop contribute to $m_H^2$ with opposite sign, so the quadratically divergent $\Lambda^2$ term cancels exactly.
Unbroken SUSY forces each particle and its superpartner to have equal mass; since no such partners are seen, SUSY must be softly broken, with partner masses set by a soft scale $m_{\text{soft}}\lesssim$ TeV for naturalness.
In the MSSM the extra superpartners change the running of the couplings so that $\alpha_1,\alpha_2,\alpha_3$ meet at $M_{\text{GUT}}\approx2\times10^{16}$ GeV with $\alpha_{\text{GUT}}^{-1}\approx24$ — unification the Standard Model narrowly misses.
With R-parity conserved the lightest superpartner (LSP, often a neutralino) is stable and is a leading WIMP dark-matter candidate.
Key idea: supersymmetry pairs each boson with a fermion so their opposite-sign loop corrections cancel the Higgs mass's quadratic divergence — taming the hierarchy problem while unifying the couplings and supplying a dark-matter candidate.

The core insight

Supersymmetry is a symmetry relating bosons and fermions whose generator squares to spacetime translation; by pairing each particle with an opposite-statistics partner it cancels the quadratic divergence in the Higgs mass, stabilising the electroweak scale against enormous ultraviolet corrections.

Where this lives today

Supersymmetry is simultaneously the most-searched-for and the most-constrained idea in particle physics. At the LHC, ATLAS and CMS have devoted hundreds of analyses to it — missing-transverse-energy searches for squark and gluino cascades, dedicated stop hunts, electroweak-ino and slepton searches, and long-lived-particle programmes for compressed or split spectra. The null results push the coloured superpartners above roughly $2$ TeV and stops above about $1$ TeV in minimal models, which fuels the "little hierarchy" tension and has driven theorists toward natural-SUSY spectra (light stops and higgsinos), R-parity violation, and split or high-scale supersymmetry. On the astrophysical side, the neutralino remains a benchmark WIMP for direct-detection experiments such as LZ and XENONnT, whose tightening cross-section limits carve directly into MSSM parameter space. Precision probes matter too: the muon $g-2$ measurement at Fermilab and rare B-meson decays at LHCb constrain the spectrum. And the framework's deepest legacy is structural — local supersymmetry is supergravity, and it is built into every viable superstring construction, so even if TeV-scale SUSY is not realised, the symmetry remains central to how theorists think about quantum gravity.

The physical picture

The Higgs mass is unstable in a way no other Standard-Model mass is. A fermion's mass is protected by chiral symmetry and a gauge boson's by gauge symmetry, so their corrections grow only like the logarithm of the cutoff. A scalar has no such shield: virtual particles in a loop attached to the Higgs propagator drag its mass-squared up by an amount proportional to the square of the heaviest scale in the theory. Supersymmetry supplies the missing shield by brute pairing: for every fermion whose loop pushes $m_H^2$ down, a partner scalar pushes it up by an equal and opposite amount — the minus sign being the statistics factor that every closed fermion loop carries. The first figure is that cancellation, drawn as the two diagrams whose sum is finite.

Figure

The naturalness cancellation of the Higgs mass. A top-quark loop lowers the Higgs mass-squared by a term −λ_t²Λ²/8π², while the two stop-scalar loops raise it by +λ_t²Λ²/8π²; the quadratic Λ² dependence cancels exactly and only a mild logarithm of the superpartner mass survives

Cancelling the divergence is only worth doing if the partners are close in mass to the particles they protect — and here the algebra is unforgiving. Because $Q$ commutes with the four-momentum, it commutes with $P^2$ , the mass-squared operator, so a particle and its superpartner in an unbroken theory are exactly degenerate. We do not see a scalar electron at $0.511$ MeV, so SUSY, if real, is broken, lifting the partners to a soft scale $m_{\text{soft}}$ . Naturalness then reads off the surviving logarithm: the residual Higgs correction is of order $m_{\text{soft}}^2$ , so keeping the electroweak scale natural wants superpartners not far above a TeV. The second figure shows the reward the same superpartners deliver elsewhere — the running of the three gauge couplings, which in the supersymmetric spectrum converge on a single point.

Figure

One-loop running of the three gauge couplings from M_Z = 91.19 GeV. In the MSSM (solid) the inverse couplings α₁⁻¹, α₂⁻¹, α₃⁻¹ meet at M_GUT ≈ 2×10¹⁶ GeV with α_GUT⁻¹ ≈ 24, whereas in the Standard Model (dashed) the three lines splay apart and never cross at one point

The core equations

Supersymmetry is defined not by a Lagrangian but by an algebra — the way its generator combines with the generators of spacetime. The Coleman–Mandula theorem forbids any ordinary (bosonic) symmetry from mixing spin with internal quantum numbers, but it assumes commutators only. Supersymmetry evades it by using an anticommuting spinor charge, and the defining relation is the anticommutator below.

\{Q_\alpha,\, Q^\dagger_{\dot\beta}\} = 2\,(\sigma^\mu)_{\alpha\dot\beta}\,P_\mu,\qquad \{Q_\alpha,\,Q_\beta\}=0,\qquad [\,Q_\alpha,\,P^\mu\,]=0 \qquad (1)

where: $Q_\alpha$ — the supercharge, a two-component Weyl spinor ( $\alpha=1,2$ ) carrying spin ½; $Q^\dagger_{\dot\beta}$ — its conjugate; $\sigma^\mu=(\mathbb{1},\vec\sigma)$ — the Pauli four-vector; $P_\mu$ — the energy–momentum operator. Reading: two SUSY transformations compose into a spacetime translation, so $Q$ is literally a square root of $P_\mu$ — the reason SUSY is a spacetime symmetry, not an internal one. Because $Q$ shifts spin by half a unit and commutes with $P^\mu$ , it moves between boson and fermion states of the same momentum, hence the same mass — organising the spectrum into supermultiplets and forcing the mass degeneracy.

Q\,|\text{boson}\rangle=|\text{fermion}\rangle,\quad Q\,|\text{fermion}\rangle=|\text{boson}\rangle,\qquad [\,Q,\,P^2\,]=0\ \Rightarrow\ m_{\text{boson}}=m_{\text{fermion}} \qquad (2)

where: each supermultiplet contains equal numbers of bosonic and fermionic degrees of freedom; a chiral multiplet holds one complex scalar (2 real bosonic states) and one Weyl fermion (2 fermionic states). Reading: superpartners come in boson–fermion pairs degenerate in mass whenever the symmetry is exact. The physical payoff lives in the Higgs self-energy. A Dirac fermion coupled by a Yukawa term $-\lambda_f H\bar f f$ contributes a one-loop correction to $m_H^2$ that diverges quadratically with the cutoff $\Lambda_{\text{UV}}$ , and — crucially — with a minus sign from closed-fermion-loop statistics.

\Delta m_H^2\big|_{\text{fermion}} = -\,\frac{|\lambda_f|^2}{8\pi^2}\,\Lambda_{\text{UV}}^2 \;+\; (\text{logarithms}) \qquad (3)

where: $\lambda_f$ — the Yukawa coupling; $\Lambda_{\text{UV}}$ — the scale at which new physics enters (as large as $M_{\text{Pl}}\sim10^{19}$ GeV). Reading: with the top quark ( $\lambda_t\approx1$ ) this term alone is astronomically larger than the observed $m_H^2=(125\ \text{GeV})^2$ unless something cancels it. Supersymmetry supplies exactly that. Each Dirac fermion carries two complex scalars whose quartic coupling is fixed by SUSY to $\lambda_S=|\lambda_f|^2$ ; a scalar loop carries a plus sign, so the two contributions cancel the $\Lambda_{\text{UV}}^2$ piece and leave only a logarithm set by the superpartner mass.

\Delta m_H^2\big|_{\text{scalar}} = +\,\frac{\lambda_S}{16\pi^2}\,\Lambda_{\text{UV}}^2+\dots \;\;\xrightarrow{\ \lambda_S=|\lambda_f|^2,\ \times 2\ }\;\; \Delta m_H^2\big|_{\text{total}} \sim \frac{|\lambda_f|^2}{4\pi^2}\,(m_S^2-m_f^2)\,\ln\frac{\Lambda_{\text{UV}}}{m_S} \qquad (4)

where: $m_S$ — the scalar-superpartner mass; $m_f$ — the fermion mass. Reading: when SUSY is exact ( $m_S=m_f$ ) the correction vanishes; when it is softly broken the leftover is set by the mass splitting, not by $\Lambda_{\text{UV}}$ — the hierarchy problem is solved provided $m_S-m_f\lesssim$ TeV. The same superpartners rewrite the running of the gauge couplings. Each inverse coupling runs linearly in $\ln\mu$ at one loop, and the extra fields change the slope coefficients $b_i$ so all three lines meet.

\alpha_i^{-1}(\mu) = \alpha_i^{-1}(M_Z)-\frac{b_i}{2\pi}\ln\frac{\mu}{M_Z},\qquad b_i^{\text{MSSM}}=\Big(\tfrac{33}{5},\ 1,\ -3\Big) \qquad (5)

where: $i=1,2,3$ labels $U(1)_Y$ (GUT-normalised), $SU(2)_L$ , $SU(3)_c$ ; $b_i^{\text{SM}}=(41/10,-19/6,-7)$ for comparison. Reading: with the MSSM coefficients the three couplings converge on $\alpha_{\text{GUT}}^{-1}\approx24$ at $\mu\approx2\times10^{16}$ GeV — a coincidence the Standard-Model coefficients miss.

Derivation ACancellation of the quadratic divergence in the Higgs mass

The goal is to show explicitly that the leading ultraviolet sensitivity of the Higgs mass-squared cancels between a fermion and its scalar superpartners, and to identify what controls the finite remainder. We compute the two one-loop self-energy contributions, impose the supersymmetric relation between the Yukawa and quartic couplings, add them, and read off both the exact cancellation of the $\Lambda_{\text{UV}}^2$ term and the logarithmic residual that survives once the symmetry is broken — the single most important calculation behind the whole SUSY programme.

Step 1. Couple the Higgs to a Dirac fermion by $\mathcal{L}\supset-\lambda_f H\bar f f$ and evaluate the one-loop correction to $m_H^2$ with a hard cutoff $\Lambda_{\text{UV}}$ . The closed fermion loop carries an overall minus sign from Fermi statistics, giving $\Delta m_H^2|_f=-\dfrac{|\lambda_f|^2}{8\pi^2}\Lambda_{\text{UV}}^2+\dots$ Why: a scalar mass is unprotected, so the correction grows as the cutoff squared; the sign is the fingerprint of fermionic statistics and is what makes the cure possible.

Step 2. Couple the Higgs to a complex scalar $S$ by the quartic $\mathcal{L}\supset-\lambda_S|H|^2|S|^2$ and evaluate its one-loop correction. A bosonic loop carries no statistics sign, so it comes out positive: $\Delta m_H^2|_S=+\dfrac{\lambda_S}{16\pi^2}\Lambda_{\text{UV}}^2+\dots$ Why: the boson and fermion loops are quadratically divergent with the same magnitude structure but opposite sign — the raw material for a cancellation, if the couplings are related.

Step 3. Impose supersymmetry. A single Dirac fermion is two Weyl fermions, and SUSY assigns each Weyl fermion a complex scalar partner and fixes the quartic coupling to $\lambda_S=|\lambda_f|^2$ . Two scalars therefore contribute $2\times\dfrac{|\lambda_f|^2}{16\pi^2}\Lambda_{\text{UV}}^2=\dfrac{|\lambda_f|^2}{8\pi^2}\Lambda_{\text{UV}}^2$ . Why: the cancellation is not a numerical accident to be tuned — supersymmetry forces the exact coupling and multiplicity that make it happen at every order.

Step 4. Add the two contributions: $\Delta m_H^2=-\dfrac{|\lambda_f|^2}{8\pi^2}\Lambda_{\text{UV}}^2+\dfrac{|\lambda_f|^2}{8\pi^2}\Lambda_{\text{UV}}^2=0$ at leading order. The dangerous $\Lambda_{\text{UV}}^2$ term is gone identically. Why: this is the precise sense in which SUSY solves the hierarchy problem — the Higgs mass no longer knows about the Planck scale.

Step 5. Restore reality by breaking SUSY softly, $m_S\neq m_f$ . Keeping the subleading pieces, the leftover is logarithmic, $\Delta m_H^2\sim\dfrac{|\lambda_f|^2}{4\pi^2}(m_S^2-m_f^2)\ln(\Lambda_{\text{UV}}/m_S)$ . Why: the residual is controlled by the superpartner mass splitting, not the cutoff, so naturalness of the electroweak scale demands $m_S-m_f\lesssim$ TeV — the prediction that put superpartners within LHC reach. *The $\Lambda^2$ coefficients $-|\lambda_f|^2/8\pi^2$ (fermion) and $+2\times|\lambda_f|^2/16\pi^2$ (two scalars) sum to zero identically, and the residual correction is $\propto(m_S^2-m_f^2)\ln\Lambda$ .*

Go deeper · Derivation B · 1 minBoson-fermion mass degeneracy from the superalgebra

Derivation BBoson-fermion mass degeneracy from the superalgebra

The aim is to prove, straight from the algebra of equation (1), two structural facts that make supersymmetry both powerful and experimentally constrained: that every particle is exactly degenerate in mass with its superpartner when the symmetry is unbroken, and that the vacuum energy of an unbroken theory is exactly zero, so a nonzero vacuum energy is the order parameter for SUSY breaking. Both follow from traces and expectation values of the anticommutator, with no dynamical input.

Step 1. Since $[Q_\alpha,P^\mu]=0$ , the supercharge commutes with $P^2=P^\mu P_\mu$ , the mass-squared Casimir. A state and the state obtained by acting with $Q$ therefore share the same eigenvalue of $P^2$ . Why: mass is a label that $Q$ cannot change, so boson and fermion members of a multiplet are forced to be degenerate.

Step 2. Apply this to any supermultiplet: acting with $Q$ maps a boson of mass $m$ to a fermion of the same mass $m$ , so $m_{\text{boson}}=m_{\text{fermion}}$ exactly in the unbroken theory. Why: this is why SUSY cannot be exact in nature — a $0.511$ MeV scalar "selectron" would have been found long ago; the observed absence forces SUSY breaking.

Step 3. Now take the vacuum expectation value of the algebra. Setting $\mu=0$ and summing over the spinor index, the anticommutator gives $\sum_\alpha\{Q_\alpha,Q_\alpha^\dagger\}=2\,\text{Tr}(\sigma^0)\,P^0=4H$ , so the Hamiltonian is $H=\tfrac14\sum_\alpha(Q_\alpha Q_\alpha^\dagger+Q_\alpha^\dagger Q_\alpha)$ . Why: the energy operator is a sum of squares of the supercharges, a manifestly non-negative object — this is special to SUSY.

Step 4. Evaluate in the vacuum $|0\rangle$ . The expectation value $\langle0|H|0\rangle=\tfrac14\sum_\alpha(\lVert Q_\alpha^\dagger|0\rangle\rVert^2+\lVert Q_\alpha|0\rangle\rVert^2)\ge0$ , with equality iff $Q_\alpha|0\rangle=0$ for all $\alpha$ . Why: an unbroken vacuum is annihilated by $Q$ and has exactly zero energy; a positive vacuum energy signals spontaneously broken supersymmetry.

Step 5. Read off the order parameter: SUSY is unbroken exactly when $\langle0|H|0\rangle=0$ , and broken when $\langle0|H|0\rangle>0$ . Why: this reframes SUSY breaking as vacuum-energy generation and underlies the F-term and D-term breaking mechanisms of realistic models. *These two results follow directly from the algebra: $[Q,P^2]=0$ forces equal masses within a supermultiplet, and $H=\tfrac14\sum\{Q,Q^\dagger\}\ge0$ with $E_{\text{vac}}=0\Leftrightarrow Q|0\rangle=0$ .*

Limiting cases

Case	Result	Meaning
Exact (unbroken) SUSY	$m_{\text{boson}}=m_{\text{fermion}}$ , $E_{\text{vac}}=0$	perfect degeneracy; ruled out by experiment, so SUSY must be broken
Softly broken SUSY	$\Delta m_H^2\sim m_{\text{soft}}^2\ln\Lambda$	$\Lambda^2$ cancellation survives; natural if $m_{\text{soft}}\lesssim$ TeV
Heavy superpartners ( $m_{\text{soft}}\to$ large)	Standard Model recovered	hierarchy problem returns — the "little hierarchy" tension
R-parity conserved	lightest superpartner is stable	neutralino LSP is a WIMP dark-matter candidate
Local (gauged) SUSY	supergravity: graviton + gravitino	SUSY promoted to a theory including gravity

Common trapthe misconception that the LHC has ruled supersymmetry out

It has not, and the claim confuses a framework with its most economical incarnation. What the LHC has done is exclude gluinos and first-generation squarks below roughly $2$ TeV and stops below about $1$ TeV in the simplest scenarios — enough to squeeze the most natural version of the MSSM and revive a mild "little hierarchy" tension, but nowhere near a refutation of supersymmetry itself. SUSY is a colossal parameter space: spectra can be compressed, R-parity can be violated, or the breaking scale can be split or pushed far above the TeV at the cost of naturalness. A related trap is to expect superpartners at the same mass as their Standard-Model partners — that is the unbroken theory, already excluded; realistic SUSY is always softly broken, with partners at the soft scale. "No superpartners yet" bounds the breaking scale; it does not close the framework.

Worked examples

1 · How supersymmetry defuses the hierarchy problem (fully worked)

Estimate the radiative correction to the Higgs mass-squared from the top sector, first with a Standard-Model cutoff at the Planck scale and then in supersymmetry with 1 TeV stops. Use $\lambda_t^2\approx0.9$ , $M_{\text{Pl}}=1.2\times10^{19}$ GeV, $m_H=125$ GeV.

Step 1. In the Standard Model the top loop alone gives $\Delta m_H^2\approx\dfrac{3\lambda_t^2}{8\pi^2}\Lambda_{\text{UV}}^2$ (the colour factor 3 counts the quark colours). With $\Lambda_{\text{UV}}=M_{\text{Pl}}$ : $\dfrac{3(0.9)}{8\pi^2}=0.034$ , so $\Delta m_H^2\approx0.034\times(1.2\times10^{19})^2=\mathbf{5\times10^{36}\ \text{GeV}^2}$ .

Step 2. Compare to the physical value $m_H^2=(125)^2=1.6\times10^4\ \text{GeV}^2$ . The bare mass must cancel the correction to a fractional precision $m_H^2/\Delta m_H^2=1.6\times10^4/5\times10^{36}\approx\mathbf{3\times10^{-33}}$ — a tuning of one part in $10^{32}$ , obscene and unexplained. This is the hierarchy problem.

Step 3. Now switch on supersymmetry. The stop loops cancel the $\Lambda_{\text{UV}}^2$ term (Derivation A), replacing $\Lambda_{\text{UV}}$ by the stop mass $m_{\tilde t}$ : $\Delta m_H^2\approx\dfrac{3\lambda_t^2}{8\pi^2}m_{\tilde t}^2$ . With $m_{\tilde t}=1$ TeV $=10^3$ GeV: $\Delta m_H^2\approx0.034\times10^6=3.4\times10^4\ \text{GeV}^2$ , i.e. $\Delta m_H\approx\mathbf{185\ \text{GeV}}$ .

Step 4. Compare again to $m_H=125$ GeV. The correction is now comparable to the mass it corrects, so no fine cancellation is needed — the electroweak scale is natural. SUSY converts a $10^{32}$ tuning into an order-one one, provided superpartners live near a TeV. (Restoring the logarithm and pushing $m_{\tilde t}$ above current limits worsens this to the few-percent "little hierarchy" tension — the price of the LHC's null results.)

2 · The unification scale and the GUT coupling

Using the MSSM one-loop running of equation (5), find the energy at which $\alpha_1$ and $\alpha_2$ meet and the common coupling there, then check whether $\alpha_3$ agrees. Inputs: $\alpha_1^{-1}(M_Z)=59.0$ , $\alpha_2^{-1}(M_Z)=29.6$ , $\alpha_3^{-1}(M_Z)=8.5$ , $M_Z=91.19$ GeV.

Step 1. Set $\alpha_1^{-1}(\mu)=\alpha_2^{-1}(\mu)$ with $b_1=33/5$ , $b_2=1$ and $L=\ln(\mu/M_Z)$ : $59.0-\dfrac{33/5}{2\pi}L=29.6-\dfrac{1}{2\pi}L$ . Collecting, $29.4=\dfrac{1}{2\pi}\Big(\dfrac{33}{5}-1\Big)L=\dfrac{28/5}{2\pi}L=0.891\,L$ .

Step 2. Solve: $L=29.4/0.891=\mathbf{33.0}$ , so $\mu=M_Z\,e^{33.0}=91.19\times2.2\times10^{14}=\mathbf{2.0\times10^{16}\ \text{GeV}}$ . This is $M_{\text{GUT}}$ .

Step 3. The common coupling: $\alpha_{\text{GUT}}^{-1}=\alpha_2^{-1}-\dfrac{1}{2\pi}L=29.6-\dfrac{33.0}{2\pi}=29.6-5.25=\mathbf{24.3}$ , so $\alpha_{\text{GUT}}\approx1/24$ .

Step 4. Check the strong coupling at the same $L$ : $\alpha_3^{-1}=8.5-\dfrac{-3}{2\pi}(33.0)=8.5+15.8=\mathbf{24.3}$ . All three inverse couplings agree at $24.3\pm0.1$ . The three lines pass through one point at $2\times10^{16}$ GeV — the quantitative case for supersymmetric grand unification. (With the SM coefficients $(41/10,-19/6,-7)$ the same exercise gives three different crossing scales — no unification.)

Physics of the portrait

Supersymmetry has no single discoverer; it was found independently on both sides of the Iron Curtain and only later recognised as the same structure. Its portrait is the idea itself and the handful of theorists who gave it its algebra, its first field theory, and its uniqueness theorem.

Origins · 1 minsupersymmetry and how this was discovered

supersymmetry

the boson–fermion symmetry with Q · B⟩=

Supersymmetry emerged from three nearly independent lines of work in the early 1970s. In Moscow in 1971, Yuri Golfand and Evgeny Likhtman wrote down the first extension of the Poincaré algebra by anticommuting spinor generators — the birth of the four-dimensional superalgebra — but publishing behind the Iron Curtain, their work went almost unnoticed in the West for years. The idea reached the wider community only in 1974, when Julius Wess and Bruno Zumino, motivated by two-dimensional string worldsheet symmetries, constructed the first renormalisable four-dimensional field theory of interacting scalars and fermions — the Wess–Zumino model — and made supersymmetry a tool physicists could compute with. The following year Rudolf Haag, Jan Łopuszański and Martin Sohnius proved the decisive structural result: given the Coleman–Mandula no-go theorem, a spinorial (graded) charge is the only way to nontrivially unite spacetime and internal symmetries, singling supersymmetry out as mathematically unique. Pierre Fayet built the first supersymmetric electroweak models, and in 1981 Savas Dimopoulos and Howard Georgi assembled the Minimal Supersymmetric Standard Model, whose gauge-coupling unification and hierarchy-problem solution made it, for four decades, the leading candidate for what lies beyond the Standard Model.

History

1967

— Coleman and Mandula prove spacetime and internal symmetries cannot be nontrivially combined with bosonic generators alone

1971

— Golfand and Likhtman construct the first four-dimensional supersymmetry algebra

1974

— Wess and Zumino write the first renormalisable interacting theory of a scalar and a fermion

1975

— Haag, Łopuszański and Sohnius show a spinor charge is the unique way around Coleman–Mandula

1981

— Dimopoulos and Georgi formulate the Minimal Supersymmetric Standard Model

Going further

Texts. Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 20 (radiative corrections and the hierarchy problem) and §22.3 (running couplings and unification) · Griffiths, Introduction to Elementary Particles, 2nd ed., Ch. 12 (beyond the Standard Model) · Halzen & Martin, Quarks and Leptons, Ch. 15 · Thomson, Modern Particle Physics, Ch. 18 · Schwartz, Quantum Field Theory and the Standard Model, Ch. 22 (naturalness). Specialist. S. P. Martin, A Supersymmetry Primer, arXiv:hep-ph/9709356 (the standard pedagogical reference) · Wess & Bagger, Supersymmetry and Supergravity. Key numerical checks: the $\Lambda^2$ coefficients $-|\lambda_f|^2/8\pi^2$ and $+2\times|\lambda_f|^2/16\pi^2$ cancel exactly; $[Q,P^2]=0$ implies mass degeneracy and $H\ge0$ ; the MSSM slopes $b_i=(33/5,1,-3)$ give $M_{\text{GUT}}=2.0\times10^{16}$ GeV and $\alpha_{\text{GUT}}^{-1}=24.3$ ; and the stop-loop estimate yields $\Delta m_H\approx185$ GeV for $m_{\tilde t}=1$ TeV, compared with a $10^{32}$ tuning at a Planck cutoff. The boundary data $\alpha_{\text{em}}^{-1}(M_Z)=127.9$ , $\sin^2\theta_W=0.231$ , $\alpha_s(M_Z)=0.118$ are standard PDG values.

Prerequisites. Standard Model Summary · Higgs Mechanism · Spontaneous Symmetry Breaking · Electroweak Gauge Group · Local Gauge Principle. Leads to. The Hierarchy Problem · Dark Matter · Grand Unification · Open Questions.

Ask the tutor: Ask me to walk through the boson–fermion loop cancellation line by line, or to show how the MSSM slopes steer the three couplings onto one point at $10^{16}$ GeV.

What exactly is a superpartner, and how many new particles does SUSY predict?

A superpartner is the particle you get by acting with the supercharge $Q$ on a known one: same charges and (in the unbroken theory) same mass, but spin differing by half a unit. Every Standard-Model fermion gains a spin-0 partner — squarks for quarks, sleptons for leptons — and every gauge boson gains a spin-½ partner, the gauginos (gluino, winos, bino). The Higgs sector, which needs two doublets in SUSY, gains higgsinos; after electroweak mixing the neutral gauginos and higgsinos combine into four neutralinos and the charged ones into two charginos. In total the MSSM roughly doubles the particle count and gives five physical Higgs states.

Why does adding a boson loop cancel a fermion loop — where does the minus sign come from?

The minus sign is a theorem of quantum field theory, not an assumption. Every closed fermion loop carries an overall factor of $-1$ , a direct consequence of the anticommuting (Fermi–Dirac) statistics of the fields — the same statistics behind the Pauli exclusion principle. Boson loops carry no such factor. So a fermion and a boson coupled to the Higgs with the same strength contribute quadratic divergences of equal size and opposite sign. Supersymmetry's job is precisely to guarantee "the same strength": it locks the scalar's quartic coupling to the square of the fermion's Yukawa coupling, so the cancellation is exact and holds order by order rather than being a fragile numerical coincidence.

Is supersymmetry the same thing as the MSSM?

No — supersymmetry is the general framework, the MSSM is one specific model built on it. Supersymmetry itself only says that $Q$ exists and satisfies the superalgebra; it does not fix the particle content, the breaking, or the superpartner scale. The MSSM is the most economical realistic choice: take the Standard Model, give every field a partner, add a second Higgs doublet (needed for anomaly cancellation and to give mass to both up- and down-type quarks), and parametrise the breaking with "soft" mass terms. Many alternatives exist — the NMSSM adds a singlet, split SUSY lifts the scalars far above the gauginos, and R-parity-violating models change every collider signature. Ruling out the MSSM in some mass window does not rule out supersymmetry.

What is R-parity and why does it make the lightest superpartner a dark-matter candidate?

R-parity is a discrete symmetry defined by $R=(-1)^{3(B-L)+2s}$ , where $B$ , $L$ , and $s$ are baryon number, lepton number, and spin. Every Standard-Model particle has $R=+1$ and every superpartner has $R=-1$ . If $R$ is conserved, superpartners are produced and destroyed only in pairs, and — crucially — the lightest superpartner has nothing lighter to decay into while conserving $R$ , so it is absolutely stable. If that state is neutral and colourless, typically the lightest neutralino, it interacts only weakly and survives from the early universe as a relic — the profile of a WIMP. The abundance a TeV-scale neutralino freezes out with lands close to the observed dark-matter density, the "WIMP miracle".

Has the LHC's failure to find superpartners killed supersymmetry?

It has constrained it sharply but not killed it. The strongest bounds are on the coloured superpartners — gluinos and light-flavour squarks above roughly $2$ TeV, stops above about $1$ TeV — because those are copiously produced if they exist. But these limits assume simple spectra with distinctive missing-energy signatures. Compressed spectra hide the signal in soft, untriggerable decay products; R-parity violation removes the missing energy entirely; and split or high-scale SUSY places the superpartners beyond current energy. What the null results genuinely do is erode naturalness: the most motivated version, with superpartners near the electroweak scale, is now in tension, reviving a mild "little hierarchy" fine-tuning of a few percent. That constrains the scale, not the existence, of supersymmetry.

Why do the couplings unify in the MSSM but not in the Standard Model?

Because the superpartners change the slopes at which the couplings run. Each coupling's running rate is set by its beta coefficient $b_i$ , which counts the charged and coloured particles circulating in loops. Adding the full set of squarks, sleptons, gauginos, higgsinos, and a second Higgs doublet shifts the Standard-Model coefficients $(41/10,-19/6,-7)$ to the MSSM values $(33/5,1,-3)$ . The change is not uniform: it slows the running of the strong coupling and speeds the others just enough that all three inverse couplings, plotted against $\ln\mu$ , become straight lines through a single point at $\sim2\times10^{16}$ GeV. In the Standard Model the same three lines cross pairwise at three different scales spread over two orders of magnitude — close enough to be suggestive, but missing true unification by many standard deviations. Broken SUSY keeps this virtue because the breaking is soft: it enters only through mass terms, which shift the spectrum without spoiling the divergence cancellation, so unification and naturalness both survive as long as the soft scale is near a TeV.