Quantum Field Theory Course

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

Part VII, Chapter 2

Instantons & Tunneling

Non-perturbative quantum tunneling and the topology of the vacuum

What is an Instanton?

An instanton is a classical solution to the equations of motion in Euclidean spacetime (imaginary time) that describes quantum tunneling between different vacuum states. These are non-perturbative configurations that cannot be seen in any finite order of perturbation theory.

Instantons mediate quantum tunneling with amplitude proportional to e-SE, where SE is the Euclidean action. They reveal the rich topological structure of the vacuum in gauge theories.

2.1 Path Integral in Euclidean Time

The transition amplitude in quantum mechanics can be written as a path integral:

$$\langle q_f, t_f | q_i, t_i \rangle = \int \mathcal{D}q(t) \, e^{iS[q]}$$

To find classical solutions, we Wick rotate to imaginary time: t → -iτ (τ is Euclidean time). The action becomes:

$$S_E = -iS_M = \int d\tau \left[\frac{1}{2}\left(\frac{dq}{d\tau}\right)^2 + V(q)\right]$$

The path integral becomes:

$$\langle q_f | e^{-HT} | q_i \rangle = \int \mathcal{D}q(\tau) \, e^{-S_E[q]}$$

Now the action is real, and classical solutions minimize SE (not extremize S).

Euclidean vs Minkowski

  • Minkowski: eiS, oscillatory, saddle points
  • Euclidean: e-SE, damped, global minima
  • Euclidean solutions with finite action dominate tunneling amplitudes

2.2 Example: Double-Well Potential

Consider a particle in a double-well potential:

$$V(q) = \frac{\lambda}{4}(q^2 - a^2)^2$$

Two degenerate minima at q = ±a. In quantum mechanics, the particle can tunnelbetween the wells!

The instanton solution interpolates between the two vacua:

$$q_{\text{inst}}(\tau) = a \tanh\left(\frac{m\omega(\tau - \tau_0)}{\sqrt{2}}\right)$$

where ω = a√(λ) and τ₀ is the center of the instanton. As τ: -∞ → +∞, the field interpolates: -a → +a.

The Euclidean action of the instanton is:

$$S_E[\text{instanton}] = \frac{4ma^3\omega}{3\sqrt{2}}$$

The tunneling amplitude is suppressed by:

$$\mathcal{A}_{\text{tunnel}} \sim e^{-S_E} = e^{-4ma^3\omega/3\sqrt{2}}$$

This is non-perturbative: no power series in ℏ captures this behavior!

2.3 Instantons in Yang-Mills Theory

In 4D Yang-Mills theory, instantons are solutions to the Euclidean equations:

$$D_\mu F_{\mu\nu} = 0$$

The self-dual (or anti-self-dual) solutions minimize the action:

$$F_{\mu\nu} = \pm \tilde{F}_{\mu\nu}$$

where F̃μν = (1/2)εμνρσFρσ is the dual field strength.

Why Self-Dual?

The Yang-Mills action can be written as:

$$S_E = \frac{1}{4g^2}\int d^4x \, (F_{\mu\nu} \mp \tilde{F}_{\mu\nu})^2 \pm \frac{1}{g^2}\int d^4x \, F_{\mu\nu}\tilde{F}^{\mu\nu}$$

The first term is always ≥ 0. For fixed topology (second term), action is minimized when F = ±F̃!

2.4 The BPST Instanton

The BPST (Belavin-Polyakov-Schwartz-Tyupkin) instanton is the simplest instanton solution in SU(2) Yang-Mills theory:

$$A_\mu^a(x) = \frac{2\eta_{\mu\nu}^a (x-x_0)_\nu}{(x-x_0)^2 + \rho^2}$$

where ηaμν are 't Hooft symbols (related to Pauli matrices), ρ is theinstanton size, and x₀ is the instanton center.

Properties of BPST Instanton

  • Localized in spacetime (falls off as 1/x²)
  • Euclidean action: SE = 8π²/g²
  • Topological charge: Q = 1 (one unit of winding number)
  • Collective coordinates: x₀ (center), ρ (size), gauge orientation

2.5 Topological Charge and Pontryagin Index

Instantons are classified by their topological charge (also called winding number):

$$Q = \frac{1}{32\pi^2}\int d^4x \, F_{\mu\nu}^a \tilde{F}^{a,\mu\nu}$$

This is the Pontryagin index, an integer that measures how the gauge field "wraps" around the compactified spacetime S⁴.

$$Q \in \mathbb{Z} \quad \text{(topological invariant)}$$

For a self-dual instanton (F = F̃):

$$S_E = \frac{8\pi^2}{g^2}|Q|$$

Physical Meaning

  • Q = 0: perturbative vacuum
  • Q = ±1: single instanton/anti-instanton
  • Q = ±n: multi-instanton configuration
  • Instantons connect topologically distinct vacua!

2.6 Theta Vacua and the Strong CP Problem

Because instantons connect different topological sectors, the true vacuum of QCD is asuperposition of all sectors:

$$|\theta\rangle = \sum_{n=-\infty}^{\infty} e^{in\theta}|n\rangle$$

where |n⟩ is the vacuum with topological charge n, and θ is the theta angle(0 ≤ θ < 2π).

This adds a term to the QCD Lagrangian:

$$\mathcal{L}_\theta = \frac{\theta g^2}{32\pi^2} F_{\mu\nu}^a \tilde{F}^{a,\mu\nu}$$

The Strong CP Problem

The θ term violates CP symmetry! It would give the neutron an electric dipole moment:

$$d_n \sim \theta \times 10^{-16} \, e \cdot \text{cm}$$

Experiments constrain: dn < 10-26 e·cm, implying θ < 10-10. Why is θ so incredibly small? This is the strong CP problem!

2.7 The Axion Solution

The leading solution to the strong CP problem is the Peccei-Quinn mechanism, which introduces a new particle: the axion.

How Axions Solve the Problem

  • Add new U(1) symmetry (Peccei-Quinn) that is spontaneously broken
  • Nambu-Goldstone boson = axion field a(x)
  • Axion couples to FF̃: a × FF̃
  • In QCD vacuum, a dynamically relaxes to cancel θ!
  • Result: θeff = 0 automatically

The axion is a dark matter candidate currently being searched for in experiments!

2.8 Physical Effects of Instantons

Despite being non-perturbative, instantons have measurable physical effects:

1. U(1) Problem in QCD

Explains why the η' meson is heavier than expected from naive SU(3) flavor symmetry. Instantons break the would-be U(1)A symmetry.

2. Baryon Number Violation

In the electroweak theory, instantons (sphalerons) can violate baryon and lepton number (though conserve B-L). Important for baryogenesis!

3. Confinement in QCD

Instanton-induced interactions contribute to the formation of quark condensates and may play a role in confinement.

4. Supersymmetry Breaking

In some SUSY theories, instantons generate non-perturbative superpotentials that can trigger SUSY breaking.

2.9 Semiclassical Approximation

To compute instanton contributions, we expand around the instanton solution:

$$A_\mu = A_{\mu,\text{inst}} + a_\mu$$

The path integral becomes:

$$\mathcal{Z} = e^{-S_E[\text{inst}]} \int \mathcal{D}a \, e^{-S_E[a]} \times (\text{collective coordinates})$$

This gives instanton effects of order:

$$\mathcal{O}(e^{-8\pi^2/g^2})$$

For QCD at low energies (g² ~ 1), this is ~10-34, extremely suppressed! But at asymptotic scales or in weak coupling theories, instantons can be important.

Key Takeaways

  • Instantons: Euclidean field configurations describing quantum tunneling
  • Non-perturbative: amplitude ~ e-SE
  • BPST instanton: fundamental solution in Yang-Mills theory
  • Topological charge Q = (1/32π²)∫FF̃ ∈ ℤ
  • Theta vacua: |θ⟩ = Σn einθ|n⟩
  • Strong CP problem: Why is θ < 10-10?
  • Axion solution: Peccei-Quinn mechanism
  • Physical effects: U(1) problem, baryon violation, confinement
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