Quantum Field Theory Course

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

Part III, Chapter 6

More on Perturbation Theory

Worked examples: 1-loop scattering, the sunset diagram, and symmetry factors

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6.8 Worked Example: $\phi^4$ 4-Point Function at One Loop

We compute the one-loop correction to 2 $\to$ 2 scattering in $\phi^4$ theory. At tree level, the amplitude is simply $\mathcal{M}_0 = -\lambda$. At one loop, there are three channels (s, t, u) each contributing a "bubble" diagram.

Step 1: Write the s-channel loop integral

The s-channel bubble diagram with external momenta $p_1 + p_2$ entering gives:

$$iI_s = \frac{(-i\lambda)^2}{2}\int\frac{d^dk}{(2\pi)^d}\frac{i}{k^2-m^2+i\epsilon}\frac{i}{(k-p)^2-m^2+i\epsilon}$$

where $p = p_1 + p_2$ and $s = p^2$. The factor of 1/2 is the symmetry factor.

Step 2: Feynman parametrize

Using $1/(AB) = \int_0^1 dx\,[xA+(1-x)B]^{-2}$ with$A = k^2 - m^2$ and $B = (k-p)^2 - m^2$:

$$xA + (1-x)B = \ell^2 - \Delta, \qquad \ell = k - (1-x)p, \qquad \Delta = m^2 - x(1-x)s$$

Step 3: Wick-rotate and evaluate

After Wick rotation $\ell^0 \to i\ell_E^0$, the integral becomes:

$$I_s = \frac{\lambda^2}{2}\int_0^1 dx\int\frac{d^d\ell_E}{(2\pi)^d}\frac{1}{(\ell_E^2 + \Delta)^2}$$

Applying the standard integral formula:

$$I_s = \frac{\lambda^2}{2}\frac{1}{(4\pi)^{d/2}}\int_0^1 dx\;\frac{\Gamma(2-d/2)}{\Delta^{2-d/2}}$$

Step 4: Expand around d = 4

Setting $d = 4 - 2\varepsilon$ and using $\Gamma(\varepsilon) = 1/\varepsilon - \gamma_E + O(\varepsilon)$:

$$\boxed{I_s(s) = \frac{\lambda^2}{32\pi^2}\int_0^1 dx\left[\frac{1}{\varepsilon} - \gamma_E + \ln(4\pi) - \ln\frac{\Delta(x)}{\mu^2}\right] + O(\varepsilon)}$$

where $\Delta(x) = m^2 - x(1-x)s - i\epsilon$.

Step 5: Full one-loop amplitude

The complete one-loop result sums all three channels:

$$\boxed{\mathcal{M}_{\text{1-loop}} = -\lambda + I_s(s) + I_t(t) + I_u(u) + \text{counterterm}}$$

The counterterm $\delta_\lambda$ is chosen to absorb the $1/\varepsilon$ poles, leaving a finite, renormalized amplitude that depends on the renormalization scheme.

💡Physical Content of the Loop Correction

The finite part of the one-loop amplitude contains the logarithm$\ln(s/\mu^2)$, which means the effective coupling "runs" with energy. At high energies ($s \gg m^2$), the $\phi^4$ coupling grows logarithmically. This is the origin of the Landau pole and signals that $\phi^4$ theory is not asymptotically free.

6.9 The Sunset (Sunrise) Diagram

The sunset diagram (also called the sunrise diagram) is a two-loop self-energy diagram with three internal propagators forming a loop:

$$-i\Sigma_{\text{sunset}}(p^2) = \frac{(-i\lambda)^2}{6}\int\frac{d^dk}{(2\pi)^d}\frac{d^dq}{(2\pi)^d}\frac{i^3}{(k^2-m^2)(q^2-m^2)((p-k-q)^2-m^2)}$$

This involves two independent loop momenta (k and q). The symmetry factor of 1/6 comes from the 3! ways to assign the three propagators.

Evaluation Strategy

The sunset diagram is considerably more complex than one-loop diagrams. The standard approach uses nested Feynman parametrization:

$$\frac{1}{A_1 A_2 A_3} = 2\int_0^1 dx\,dy\,dz\;\delta(x+y+z-1)\frac{1}{[xA_1+yA_2+zA_3]^3}$$

After parametrization and completing the square in both loop momenta sequentially, the momentum integrals yield:

$$\Sigma_{\text{sunset}}(p^2) = \frac{\lambda^2}{6}\frac{1}{(4\pi)^d}\frac{\Gamma(3-d)}{\Gamma(3)}\int_0^1 dx\,dy\,dz\;\delta(x+y+z-1)\;\frac{1}{[F(x,y,z)]^{3-d}}$$

where $F(x,y,z) = m^2 - p^2 xyz/(xy+yz+zx) \cdot (xy+yz+zx)$ (the exact form depends on the parametrization details). In $d = 4 - 2\varepsilon$:

$$\Gamma(3-d) = \Gamma(-1+2\varepsilon) \sim -\frac{1}{2\varepsilon} + \cdots$$

The sunset diagram has a sub-divergence (the inner one-loop bubble) in addition to the overall divergence. Proper renormalization requires first subtracting the sub-divergence (via counterterms from one-loop renormalization) before handling the overall pole.

6.10 Symmetry Factors for Complex Diagrams

The symmetry factor S of a Feynman diagram accounts for the number of ways the contractions in Wick's theorem produce the same topology. The diagram carries an overall factor of 1/S.

💡Counting Symmetry Factors

The symmetry factor equals the order of the automorphism group of the diagram (the number of permutations of internal lines and vertices that leave the diagram unchanged while keeping external legs fixed). For simple diagrams, count the interchangeable internal lines and vertices directly.

Rules for $\phi^4$ Theory

$$S = \prod_{\text{vertices}} (4!)^{n_v} \times \frac{1}{\text{(distinct Wick contractions giving this topology)}}$$

Equivalently, start from the Feynman rules with the factor $-i\lambda/4!$ per vertex and count combinatorics:

Common Symmetry Factors

  • 4-point tree vertex: S = 1 (4! from vertex / 4! ways to assign legs)
  • Self-energy bubble (1-loop): S = 2 (swap the two internal lines)
  • 4-point bubble (s-channel): S = 2
  • Sunset (2-loop self-energy): S = 6 (= 3! from permuting 3 lines)
  • Figure-eight vacuum diagram: S = 8

Systematic Method

  1. Draw the diagram with all lines distinct
  2. Count permutations of internal lines that map the diagram to itself
  3. Count permutations of vertices of the same type
  4. S = product of these factors

6.11 Systematic Calculation Procedure: Summary

We now have all the tools to perform perturbative calculations to any order. Here is the complete procedure:

Complete QFT Calculation Procedure

  1. Identify the process and order: External states and desired loop order
  2. Draw all topologically distinct diagrams: Connected and amputated, at the given order in $\lambda$
  3. Determine symmetry factors: For each diagram, compute 1/S
  4. Write the amplitude: Apply Feynman rules (propagators, vertices, momentum conservation)
  5. Feynman parametrize: Combine denominators using Feynman parameters
  6. Shift and Wick-rotate: Complete the square in loop momenta, rotate to Euclidean space
  7. Evaluate in $d = 4-2\varepsilon$: Use standard d-dimensional integral formulas
  8. Renormalize: Add counterterms to cancel $1/\varepsilon$ poles
  9. Take $\varepsilon \to 0$: Extract the finite, physical result
  10. Compute observables: Cross sections $\propto |\mathcal{M}|^2$, decay rates, etc.

Quick Reference: Key Formulas

Feynman parameters (2-denom): $1/(AB) = \int_0^1 dx\,[xA+(1-x)B]^{-2}$

d-dim integral: $\int d^d\ell_E/(2\pi)^d\;(\ell_E^2+\Delta)^{-n} = (4\pi)^{-d/2}\Gamma(n-d/2)/[\Gamma(n)\Delta^{n-d/2}]$

Pole structure: $\Gamma(\varepsilon) = 1/\varepsilon - \gamma_E + O(\varepsilon)$

Cross section: $d\sigma/d\Omega = |\mathcal{M}|^2/(64\pi^2 s)$

Key Concepts (This Page)

  • $\phi^4$ 4-point function at one loop: bubble diagrams in s, t, u channels
  • Result contains $1/\varepsilon$ pole plus $\ln(s/\mu^2)$ giving running coupling
  • Sunset diagram: two-loop self-energy with sub-divergences
  • Symmetry factors = order of automorphism group of the diagram
  • Complete 10-step procedure: diagrams $\to$ Feynman rules $\to$ dim reg $\to$ renormalize $\to$ observables
Prev: Loop Integrals & Dim Reg
End of Chapter 6