---
title: Supersymmetry
module: Beyond the Standard Model
moduleNumber: 12
lessonNumber: 3
order: 1203
summary: >
  Supersymmetry relates fermions and bosons, pairing every Standard Model particle
  with a superpartner whose spin differs by one half. The pairing makes the scalar
  and fermion loop corrections to the Higgs mass cancel, removing the quadratic
  sensitivity to high scales; it sharpens the meeting of the three gauge couplings;
  and, when R-parity is conserved, it leaves the lightest superpartner stable and
  neutral, a natural dark-matter candidate. The LHC has excluded gluinos and light
  squarks below roughly two TeV.
topics: [Beyond the Standard Model]
draft: false
sources:
  - book: Griffiths
    ref: "Ch. 12 — Afterword (supersymmetry)"
  - book: Perkins
    ref: "Ch. 12 — Physics beyond the Standard Model (supersymmetry, the MSSM)"
  - book: Thomson
    ref: "Ch. 18 — The Standard Model and beyond (supersymmetry)"
---

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 $\tfrac12$ 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,

$$
Q\,|\text{boson}\rangle = |\text{fermion}\rangle,
\qquad
Q\,|\text{fermion}\rangle = |\text{boson}\rangle,
$$

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

$$
\{Q_a,\, \bar Q_{\dot b}\} = 2\,\sigma^\mu_{a\dot b}\, P_\mu,
$$

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-$\tfrac12$ 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-$\tfrac12$ 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-$\tfrac12$ 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).

$$
% caption: The Standard Model particles and their superpartners. Each fermion gains
% a scalar partner and each gauge or Higgs boson gains a spin-one-half partner, so
% every partner differs from its Standard Model particle by half a unit of spin.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0,
  sm/.style={draw=black, minimum width=22mm, minimum height=6.5mm, font=\scriptsize},
  sp/.style={draw, minimum width=22mm, minimum height=6.5mm, font=\scriptsize}]
  \definecolor{acc}{HTML}{4A6FA5}
  \node[font=\scriptsize] at (0,3.6) {Standard Model};
  \node[font=\scriptsize] at (3.6,3.6) {superpartner};
  \node[sm] (q) at (0,2.8) {quark (spin 1/2)};   \node[sp] (sq) at (3.6,2.8) {squark (spin 0)};
  \node[sm] (l) at (0,2.0) {lepton (spin 1/2)};  \node[sp] (sl) at (3.6,2.0) {slepton (spin 0)};
  \node[sm] (g) at (0,1.2) {gluon (spin 1)};     \node[sp] (gg) at (3.6,1.2) {gluino (spin 1/2)};
  \node[sm] (w) at (0,0.4) {W, Z, photon};       \node[sp] (wg) at (3.6,0.4) {wino, bino};
  \node[sm] (h) at (0,-0.4) {Higgs (spin 0)};    \node[sp] (hg) at (3.6,-0.4) {higgsino (spin 1/2)};
  \foreach \x/\y in {q/sq, l/sl, g/gg, w/wg, h/hg} \draw[->] (\x) -- (\y);
\end{tikzpicture}
$$

## 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

$$
\delta m_H^2\big|_{\text{fermion}} = -\frac{\lambda_f^2}{8\pi^2}\,\Lambda^2 + \ldots,
$$

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

$$
\delta m_H^2\big|_{\text{scalar}} = +\frac{\lambda_s}{16\pi^2}\,\Lambda^2 + \ldots .
$$

Supersymmetry relates the couplings so that each fermion is accompanied by two
scalars with $\lambda_s = \lambda_f^2$. The quadratic pieces cancel exactly:

$$
\delta m_H^2 = \frac{1}{16\pi^2}\big(\lambda_s - 2\lambda_f^2\big)\Lambda^2 + \ldots = 0
\quad\text{(quadratic part)} .
$$

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

$$
\delta m_H^2 \sim \frac{\lambda_f^2}{16\pi^2}\,\big(m_{\tilde f}^2 - m_f^2\big)\ln\frac{\Lambda}{m_{\tilde f}},
$$

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.

$$
% caption: The top-quark loop and the stop-squark loop contribute to the Higgs mass
% squared with opposite signs. Supersymmetry fixes their couplings so the quadratic
% cutoff dependence cancels, leaving only a mild logarithmic term.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % fermion loop (left)
  \draw[thick, dashed] (-0.6,0) -- (0.4,0);
  \draw[thick] (1.3,0) circle (0.55);
  \draw[thick, dashed] (2.2,0) -- (3.2,0);
  \node[above, font=\scriptsize] at (1.3,0.65) {top loop};
  \node[below, font=\scriptsize] at (1.3,-0.75) {sign: lowers};
  \node[font=\scriptsize] at (-1.1,0) {$H$};
  % plus
  \node[font=\normalsize] at (3.9,0) {$+$};
  % scalar loop (right)
  \draw[thick, dashed] (4.6,0) -- (5.6,0);
  \draw[thick, densely dotted] (6.5,0) circle (0.55);
  \draw[thick, dashed] (7.4,0) -- (8.4,0);
  \node[above, font=\scriptsize] at (6.5,0.65) {stop loop};
  \node[below, font=\scriptsize] at (6.5,-0.75) {sign: raises};
  % equals cancel
  \node[font=\scriptsize, align=center] at (9.4,0) {$=$ quadratic\\part cancels};
\end{tikzpicture}
$$

## 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

$$
(b_1, b_2, b_3) = \left(\frac{33}{5},\, 1,\, -3\right),
$$

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

$$
M_X \approx 2\times10^{16}\ \text{GeV},
$$

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.

$$
% caption: With the supersymmetric spectrum switched on above a TeV, the three
% inverse couplings change slope and meet at a single point, in contrast to the
% near-miss of the Standard Model running.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->] (0,0) -- (8.0,0) node[right, font=\scriptsize] {log energy};
  \draw[->] (0,0) -- (0,4.6) node[above, font=\scriptsize] {inverse coupling};
  % SUSY threshold marker
  \draw[dashed] (2.2,0) -- (2.2,4.3);
  \node[below, font=\scriptsize, align=center] at (2.2,0) {SUSY\\threshold};
  % three lines bending at threshold to meet at one point
  \draw[very thick] (0.3,4.2) -- (2.2,3.4) -- (6.8,1.7);
  \node[left, font=\scriptsize] at (0.3,4.2) {U(1)};
  \draw[very thick, dashed] (0.3,2.9) -- (2.2,2.55) -- (6.8,1.7);
  \node[left, font=\scriptsize] at (0.3,2.9) {SU(2)};
  \draw[very thick, densely dotted] (0.3,0.9) -- (2.2,1.15) -- (6.8,1.7);
  \node[left, font=\scriptsize] at (0.3,0.9) {SU(3)};
  \fill[acc] (6.8,1.7) circle (2.4pt);
  \node[acc, right, font=\scriptsize] at (6.9,1.7) {single point};
\end{tikzpicture}
$$

## 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

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

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](/particle-physics/beyond-standard-model/dark-matter-candidates).

> **Definition (Lightest supersymmetric particle).** The lowest-mass state with
> $R = -1$. Under conserved R-parity it cannot decay and is stable. A neutral,
> colorless LSP interacts only weakly, escaping detectors as missing momentum and
> serving as a dark-matter candidate.

## 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.

$$
% caption: A supersymmetric cascade at a collider. A produced gluino decays through
% squarks and lighter neutralinos down to the stable LSP, which escapes and leaves
% an imbalance of transverse momentum, the search signature.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0,
  bx/.style={draw, minimum width=16mm, minimum height=6mm, font=\scriptsize}]
  \definecolor{acc}{HTML}{4A6FA5}
  \node[bx] (g) at (0,2.4) {gluino};
  \node[bx] (sq) at (2.6,2.4) {squark};
  \node[bx] (n2) at (5.0,2.4) {neutralino};
  \node[bx, draw=acc, fill=acc!10] (lsp) at (5.0,0.6) {LSP (escapes)};
  \draw[->, thick] (g) -- (sq) node[midway, above, font=\scriptsize] {jet};
  \draw[->, thick] (sq) -- (n2) node[midway, above, font=\scriptsize] {jet};
  \draw[->, thick] (n2) -- (lsp) node[midway, right, font=\scriptsize] {leptons};
  \node[acc, below, font=\scriptsize, align=center] at (5.0,-0.1) {missing transverse\\momentum};
\end{tikzpicture}
$$

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.[^pdg-susy] 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](/particle-physics/beyond-standard-model/hierarchy-problem-naturalness).
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.

$$
% caption: The LHC excludes a region of the gluino-squark mass plane. Points below
% and to the left, where the superpartners are light, are ruled out; heavier spectra
% remain allowed but are less natural.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.2,0) node[right, font=\scriptsize] {squark mass};
  \draw[->, black] (0,0) -- (0,4.4) node[above, font=\scriptsize] {gluino mass};
  % excluded region shaded (lower-left)
  \fill[acc!12] (0,0) -- (4.2,0) -- (4.2,1.2) -- (1.2,1.2) -- (1.2,3.6) -- (0,3.6) -- cycle;
  \draw[acc, very thick] (4.2,0) -- (4.2,1.2) -- (1.2,1.2) -- (1.2,3.6);
  \node[acc, font=\scriptsize] at (1.4,0.7) {excluded};
  \node[black, font=\scriptsize, align=center] at (4.3,3.2) {allowed\\(heavier)};
  \node[black, below, font=\scriptsize] at (4.2,0) {2 TeV};
  \node[black, left, font=\scriptsize] at (0,3.6) {2 TeV};
\end{tikzpicture}
$$

## Summary

Supersymmetry pairs each Standard Model particle with a superpartner whose spin
differs by $\tfrac12$: sfermions for the quarks and leptons, gauginos and higgsinos
for the gauge and Higgs bosons, mixing into neutralinos and charginos. Its
generator satisfies $\{Q,\bar Q\} \sim P$, 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 $M_X \approx 2\times10^{16}$ 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.[^gr-susy][^pk-susy]

[^gr-susy]: Griffiths, Ch. 12 (Afterword), introduces supersymmetry, the superpartner naming, and the role of the LSP.

[^pk-susy]: 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.

[^pdg-susy]: 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](https://pdg.lbl.gov).