Quantum Field Theory Course

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

Part VI, Chapter 1

Ultraviolet Divergences

When loop integrals blow up at high momentum

πŸ”—Course Connections

πŸ“šPrerequisites

QFT Part III
Feynman Diagrams
Understanding loop diagrams and momentum integration
β†’
QFT Part IV
Radiative Corrections
One-loop calculations in QED
β†’

1.1 The Problem: Infinite Quantum Corrections

In Part IV, we calculated radiative corrections in QED and encountered a disturbing phenomenon:loop diagrams produce infinite integrals. Consider the simplest one-loop correction to the photon propagator (vacuum polarization):

Vacuum Polarization Diagram:

~~~~~~~>~~~~~~~

| |

e⁻ e⁺

| |

<~~~~~~Ξ³~~~~~~~>

Virtual electron-positron loop contributes to photon self-energy

The amplitude involves an integral over the loop momentum k:

$$\Pi^{\mu\nu}(q) \sim \int \frac{d^4k}{(2\pi)^4} \frac{\text{Tr}[\gamma^\mu(\gamma \cdot k)\gamma^\nu(\gamma \cdot (k+q))]}{(k^2 - m_e^2)((k+q)^2 - m_e^2)}$$

At large momentum k β†’ ∞, the numerator behaves like kΒ² while the denominator goes like k⁴, giving:

$$\int \frac{d^4k}{(2\pi)^4} \frac{k^2}{k^4} = \int \frac{d^4k}{k^2} \sim \int_0^\Lambda \frac{k^3 dk}{k^2} = \int_0^\Lambda k\,dk \to \infty$$

The integral diverges logarithmically as we take the cutoff Ξ› β†’ ∞!

πŸ’‘Why do we integrate to infinity?

In quantum field theory, virtual particles can have any momentum k, no matter how large. The integral ∫d⁴k sums over all possible virtual particle configurations.

High momentum k corresponds to short distances Ξ”x ~ 1/k. Ultraviolet (UV) divergences signal our ignorance of physics at arbitrarily short distances - perhaps new physics appears at some scale!

1.2 Degree of Divergence: Power Counting

How badly does a Feynman diagram diverge? The superficial degree of divergence D tells us. For a diagram with:

  • L = number of loops
  • I_B = number of internal boson lines
  • I_F = number of internal fermion lines
  • E_B = number of external boson lines
  • E_F = number of external fermion lines
  • V = number of vertices

In d spacetime dimensions, the degree of divergence is:

$$D = dL - 2I_B - I_F$$

For d = 4 dimensions, using topological relations between loops, vertices, and propagators:

$$D = 4 - E_B - \frac{3}{2}E_F$$

Interpretation of D:

  • D > 0: Superficially divergent (power-law divergence: Ξ›^D)
  • D = 0: Logarithmically divergent (~ ln Ξ›)
  • D < 0: Superficially convergent

1.3 Examples in φ⁴ Theory

Consider scalar φ⁴ theory with interaction β„’_int = (Ξ»/4!)φ⁴. Let's analyze key diagrams:

(a) Vacuum Bubble

E_B = 0 external lines

$$D = 4 - 0 = 4 \quad \text{(quartic divergence!)}$$

(b) Tadpole (1-point function)

E_B = 1 external line

$$D = 4 - 1 = 3 \quad \text{(cubic divergence)}$$

(c) Self-Energy (2-point function)

E_B = 2 external lines

$$D = 4 - 2 = 2 \quad \text{(quadratic divergence)}$$

(d) Vertex Correction (4-point function)

E_B = 4 external lines

$$D = 4 - 4 = 0 \quad \text{(logarithmic divergence)}$$

(e) Higher-Point Functions (n β‰₯ 6)

E_B β‰₯ 6 external lines

$$D = 4 - E_B \leq -2 < 0 \quad \text{(convergent!)}$$

Key Observation:

In φ⁴ theory, only a finite number of diagram types are superficially divergent (vacuum, tadpole, 2-point, 4-point). This makes the theory renormalizable - we only need a finite number of counterterms!

1.4 Divergences in QED

In QED, we have both photons (spin-1) and electrons (spin-1/2). The three key divergent diagrams are:

1. Photon Self-Energy (Vacuum Polarization)

Virtual e⁺e⁻ pairs modify the photon propagator. Degree of divergence:

$$D = 4 - 2 = 2$$

Actually only logarithmically divergent due to gauge invariance (Ward identity constrains the form).

2. Electron Self-Energy

Virtual photon emission/absorption modifies the electron propagator:

$$D = 4 - 2 - \frac{3}{2}(2) = -1$$

Superficially convergent, but contains logarithmic subdivergences!

3. Vertex Correction

Virtual photon exchange corrects the Ξ³-e-e vertex:

$$D = 4 - 1 - \frac{3}{2}(2) = 0$$

Logarithmically divergent. This correction gives the anomalous magnetic moment of the electron!

1.5 Subdivergences and Overlapping Divergences

Power counting only identifies superficial divergences. A diagram with D < 0 can still diverge if it contains divergent subdiagrams!

Example: Two-Loop Self-Energy

Consider a two-loop scalar propagator correction. The overall diagram has D < 0, but the internal one-loop subdiagrams have D = 2 (quadratically divergent).

These subdivergences must be handled systematically using the BPHZ renormalization prescription (Bogoliubov-Parasiuk-Hepp-Zimmermann).

1.6 Renormalizability

A theory is renormalizable if only a finite number of coupling constants need counterterms (to all orders in perturbation theory).

Classification by Mass Dimension:

In d = 4 dimensions, operators have mass dimension [π’ͺ]. The coupling constant g in gπ’ͺ has dimension [g] = 4 - [π’ͺ]:

  • [g] > 0: Super-renormalizable (e.g., φ³ in d=6)
  • [g] = 0: Renormalizable (e.g., φ⁴, QED, QCD, Yang-Mills)
  • [g] < 0: Non-renormalizable (e.g., Fermi theory, General Relativity)

Standard Model is Renormalizable!

The entire Standard Model (QED + weak + QCD) is renormalizable. This is why it's so successful: a finite number of measurements (e, m_e, m_ΞΌ, Ξ±_s, etc.) determines infinitely many predictions.

1.7 Types of Divergences

UV (Ultraviolet)

  • β€’ From high momentum k β†’ ∞
  • β€’ Short distance Ξ”x β†’ 0
  • β€’ Fixed by renormalization
  • β€’ Examples: All loop diagrams

IR (Infrared)

  • β€’ From low momentum k β†’ 0
  • β€’ Long distance Ξ”x β†’ ∞
  • β€’ From massless particles (photons)
  • β€’ Resolved by Bloch-Nordsieck theorem

In this course, we focus on UV divergences and renormalization. IR divergences are handled separately (soft photon resummation, jets in QCD, etc.).

🎯 Key Takeaways

  • Loop integrals in QFT often diverge at high momentum (UV divergences)
  • Superficial degree of divergence: D = 4 - E_B - (3/2)E_F in d = 4
  • D > 0: power divergence; D = 0: log divergence; D < 0: convergent
  • Renormalizable theories: finite # of divergent diagram types
  • φ⁴, QED, QCD are renormalizable; Fermi theory is not
  • Subdivergences require careful treatment (BPHZ prescription)
  • UV divergences signal our ignorance of short-distance physics
  • Next: How to regulate these infinities and make sense of them!
← Part VI OverviewNext: Regularization β†’