Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part VII, Chapter 6

Introduction to Supersymmetry

The symmetry between bosons and fermions: a gateway to physics beyond the Standard Model

What is Supersymmetry?

Supersymmetry (SUSY) is a symmetry that relates bosons(integer spin) and fermions (half-integer spin). For every bosonic field, SUSY predicts a fermionic superpartner, and vice versa.

SUSY is the most studied extension of the Standard Model, with profound implications for hierarchy problem, dark matter, grand unification, and quantum gravity (string theory requires it!).

6.1 Why Supersymmetry?

1. The Hierarchy Problem

Why is the Higgs mass (125 GeV) so much lighter than the Planck scale (1019 GeV)? Quantum corrections should drive mH → MPl. SUSY cancels these corrections: boson loops cancel fermion loops!

2. Gauge Coupling Unification

In SUSY, the three gauge couplings (U(1), SU(2), SU(3)) unify beautifully at ~1016 GeV, suggesting a Grand Unified Theory (GUT).

3. Dark Matter Candidate

The lightest supersymmetric particle (LSP) is stable if R-parity is conserved, making it a natural dark matter candidate (e.g., neutralino).

4. String Theory

Consistent superstring theories require spacetime supersymmetry in 10 dimensions. SUSY is essential for quantum gravity candidates!

6.2 The Supersymmetry Algebra

The SUSY algebra extends the Poincaré algebra by adding supercharges Q (spinor generators):

\begin{align*} \{Q_\alpha, \bar{Q}_{\dot{\beta}}\} &= 2\sigma^\mu_{\alpha\dot{\beta}} P_\mu \\ \{Q_\alpha, Q_\beta\} &= 0 \\ \{\bar{Q}_{\dot{\alpha}}, \bar{Q}_{\dot{\beta}}\} &= 0 \\ [Q_\alpha, P_\mu] &= 0 \end{align*}

where Qα is a Weyl spinor (α = 1,2) and Pμ is the momentum operator.

Key Properties

  • Anticommutators: SUSY generators satisfy {Q, Q} not [Q, Q] (fermions!)
  • Square root of translation: {Q, Q̄} ~ P (SUSY charges are "square root" of momentum)
  • Spinor generators: Q changes spin by 1/2
  • Commutes with P: Superpartners have same mass and momentum

6.3 Supermultiplets

Particles organize into supermultiplets—irreducible representations of the SUSY algebra. Particles in the same multiplet are related by Q.

Chiral Multiplet (N=1 SUSY)

  • Bosonic component: 2 real scalars (or 1 complex scalar φ)
  • Fermionic component: 1 Weyl fermion ψ (2 components)
  • Degrees of freedom: 2 bosonic = 2 fermionic ✓
  • Example: Squark + quark, slepton + lepton

Vector Multiplet (N=1 SUSY)

  • Bosonic component: 1 gauge boson Aμ (2 polarizations)
  • Fermionic component: 1 Weyl fermion λ (gaugino, 2 components)
  • Degrees of freedom: 2 bosonic = 2 fermionic ✓
  • Example: Photon + photino, gluon + gluino

The key constraint: bosonic and fermionic degrees of freedom must matchin each multiplet!

6.4 Superspace and Superfields

Just as spacetime has coordinates xμ, superspace has additionalfermionic coordinates θα (Grassmann numbers):

$$\text{Superspace: } (x^\mu, \theta^\alpha, \bar{\theta}_{\dot{\alpha}})$$

Grassmann numbers anticommute:

$$\theta^\alpha \theta^\beta = -\theta^\beta \theta^\alpha, \quad (\theta^\alpha)^2 = 0$$

A superfield is a function on superspace. For example, a chiral superfield:

$$\Phi(x, \theta) = \phi(x) + \sqrt{2}\theta \psi(x) + \theta\theta F(x)$$

where:

  • φ(x): complex scalar field
  • ψ(x): Weyl fermion
  • F(x): auxiliary field (eliminates via equations of motion)

Why Superfields?

Superfields package entire supermultiplets into single objects. SUSY transformations are justtranslations in superspace! This makes manifest supersymmetry much easier to work with.

6.5 The Wess-Zumino Model

The simplest SUSY theory is the Wess-Zumino model: a single chiral superfield Φ.

$$\mathcal{L} = \int d^4\theta \, \Phi^\dagger \Phi + \left(\int d^2\theta \, W(\Phi) + \text{h.c.}\right)$$

where W(Φ) is the superpotential. For example:

$$W(\Phi) = \frac{m}{2}\Phi^2 + \frac{g}{3}\Phi^3$$

In components, this becomes:

\begin{align*} \mathcal{L} &= \partial_\mu\phi^* \partial^\mu\phi + i\bar{\psi}\bar{\sigma}^\mu\partial_\mu\psi \\ &\quad - \frac{1}{2}|W'(\phi)|^2 - \frac{1}{2}(W''(\phi)\psi\psi + \text{h.c.}) \end{align*}

Key Features

  • Scalar potential: V(φ) = |W'(φ)|²
  • Yukawa coupling: determined by W''(φ)
  • All couplings related by SUSY—no free parameters!
  • Non-renormalization theorem: W receives no perturbative corrections

6.6 The Minimal Supersymmetric Standard Model (MSSM)

The MSSM is the minimal SUSY extension of the Standard Model. Every SM particle gets a superpartner:

SM ParticleSpinSuperpartnerSpin
Quarks (q)1/2Squarks (q̃)0
Leptons (ℓ, ν)1/2Sleptons (ℓ̃, ν̃)0
Gluon (g)1Gluino (g̃)1/2
W±, Z, γ1Wino, Zino, Photino1/2
Higgs (H)0Higgsino (H̃)1/2

The MSSM requires two Higgs doublets (Hu, Hd) instead of one, to give masses to both up-type and down-type quarks while preserving SUSY.

6.7 Supersymmetry Breaking

If SUSY were exact, superpartners would have identical masses. Since we don't see 125 GeV selectrons, SUSY must be broken!

The Problem

SUSY breaking must be soft (dimension-2 or 3 operators) to preserve the hierarchy solution. Hard breaking (dimension-4) reintroduces quadratic divergences!

Soft Breaking Terms

Allowed soft-breaking terms include:

  • Gaugino masses: M1,2,3λλ
  • Scalar masses: mφ²|φ|²
  • Trilinear couplings: A-terms
  • Bilinear terms: B-term for Higgses

Mediation Mechanisms

How is SUSY breaking communicated to the MSSM?

  • Gravity mediation: via Planck-suppressed operators
  • Gauge mediation: via messenger fields in gauge multiplets
  • Anomaly mediation: via super-Weyl anomaly

6.8 R-Parity and Dark Matter

R-parity is a discrete symmetry that prevents dangerous processes like rapid proton decay:

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

where B is baryon number, L is lepton number, and s is spin.

  • Standard Model particles: R = +1
  • Superpartners: R = -1

Consequences

  • Sparticles produced in pairs
  • Lightest Supersymmetric Particle (LSP) is stable
  • LSP is neutral, weakly interacting → dark matter candidate!
  • Common LSP: neutralino (mixture of photino, Zino, Higgsinos)

6.9 Experimental Searches

The LHC has searched extensively for supersymmetry:

Current Limits (as of 2024)

  • Gluinos: mg̃ > 2.3 TeV (no signal)
  • First/second generation squarks: mq̃ > 1.8 TeV
  • Charginos/neutralinos: constraints depend on decay modes
  • Stop quarks: mt̃ > 1.2 TeV (varies with decay)

No evidence yet, but parameter space remains. Future: High-Luminosity LHC, future colliders?

6.10 Beyond the MSSM

Next-to-MSSM (NMSSM)

Adds singlet superfield to solve μ problem. Richer Higgs sector, additional neutralinos.

Split SUSY

Scalars at very high mass (>>TeV), gauginos/higgsinos at TeV scale. Preserves gauge unification, gives dark matter.

Gauge Mediation

SUSY breaking mediated by gauge interactions. Predicts gravitino LSP, different phenomenology.

RPV SUSY

R-parity violation: LSP unstable, different collider signatures. Can explain neutrino masses!

Key Takeaways

  • SUSY: symmetry relating bosons and fermions via spinor generators Q
  • SUSY algebra: {Q, Q̄} ~ P (square root of translations)
  • Supermultiplets: bosonic d.o.f. = fermionic d.o.f.
  • Superspace: (xμ, θα), superfields package multiplets
  • Wess-Zumino model: simplest SUSY theory, superpotential W(Φ)
  • MSSM: every SM particle has superpartner
  • SUSY breaking: must be soft to preserve hierarchy solution
  • R-parity: LSP stable → dark matter candidate
  • LHC: no signal yet, limits mSUSY > ~1-2 TeV
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