Quantum Field Theory Course

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

Part III, Chapter 6

More on Perturbation Theory

Loop integrals, Feynman parametrization, and dimensional regularization

Page 2 of 3

6.4 Feynman Parametrization

Loop integrals involve products of propagators in the denominator. Feynman parametrization is a technique to combine multiple denominators into a single one, making the momentum integration tractable.

The Basic Identity

For two factors:

$$\boxed{\frac{1}{AB} = \int_0^1 dx\;\frac{1}{[xA + (1-x)B]^2}}$$

Proof: Let $D = xA + (1-x)B$. Then$\partial D/\partial x = A - B$ and:

$$\int_0^1 dx\;\frac{1}{D^2} = \left[-\frac{1}{(A-B)D}\right]_0^1 = -\frac{1}{(A-B)A} + \frac{1}{(A-B)B} = \frac{1}{AB}$$

General n-Factor Formula

For n propagator factors:

$$\frac{1}{A_1^{a_1}\cdots A_n^{a_n}} = \frac{\Gamma(a_1+\cdots+a_n)}{\Gamma(a_1)\cdots\Gamma(a_n)}\int_0^1 dx_1\cdots dx_n\;\delta\!\left(\sum x_i - 1\right)\frac{x_1^{a_1-1}\cdots x_n^{a_n-1}}{[x_1 A_1 + \cdots + x_n A_n]^{a_1+\cdots+a_n}}$$

After Feynman parametrization, we complete the square in the loop momentum to cast the integral into a standard form.

💡Strategy for Loop Integrals

The general procedure is: (1) Write the Feynman diagram as a momentum integral. (2) Use Feynman parameters to combine denominators. (3) Shift the loop momentum to complete the square. (4) Evaluate the resulting standard d-dimensional integral. (5) Perform the Feynman parameter integrals.

6.5 Dimensional Regularization

Loop integrals in 4 dimensions are often divergent. Dimensional regularizationis the most powerful and elegant regularization scheme: we analytically continue the spacetime dimension from d = 4 to $d = 4 - 2\varepsilon$.

Divergences manifest as poles in $\varepsilon$ (typically $1/\varepsilon$ for one-loop diagrams). Key advantages:

  • Preserves gauge invariance and Lorentz symmetry
  • No artificial mass scale or cutoff breaks symmetries
  • Power-counting is transparent
  • Algebraically systematic

Standard d-Dimensional Integrals

After Feynman parametrization and momentum shift, loop integrals reduce to standard forms. The fundamental Euclidean integral in d dimensions is:

$$\boxed{\int\frac{d^d\ell_E}{(2\pi)^d}\frac{1}{(\ell_E^2 + \Delta)^n} = \frac{1}{(4\pi)^{d/2}}\frac{\Gamma(n - d/2)}{\Gamma(n)}\frac{1}{\Delta^{n-d/2}}}$$

This is derived by going to d-dimensional spherical coordinates. The area of the unit (d-1)-sphere is $S_d = 2\pi^{d/2}/\Gamma(d/2)$, and the radial integral yields a Beta function that simplifies to Gamma functions.

Integrals with Numerator Momenta

For integrals with loop momenta in the numerator:

$$\int\frac{d^d\ell_E}{(2\pi)^d}\frac{\ell_E^\mu \ell_E^\nu}{(\ell_E^2 + \Delta)^n} = \frac{g^{\mu\nu}}{2}\frac{1}{(4\pi)^{d/2}}\frac{\Gamma(n-d/2-1)}{\Gamma(n)}\frac{1}{\Delta^{n-d/2-1}}$$

The factor of $g^{\mu\nu}/2$ arises from rotational symmetry in d dimensions:$\ell_E^\mu\ell_E^\nu \to g^{\mu\nu}\ell_E^2/d$ under angular averaging.

6.6 UV and IR Divergences

In $d = 4 - 2\varepsilon$, the Gamma function $\Gamma(n - d/2)$ develops poles when $n - d/2$ is a non-positive integer. For the most common case n = 2 (one-loop with two propagators):

$$\Gamma(2 - d/2) = \Gamma(\varepsilon) = \frac{1}{\varepsilon} - \gamma_E + O(\varepsilon)$$

where $\gamma_E \approx 0.5772$ is the Euler-Mascheroni constant. The pole$1/\varepsilon$ represents the ultraviolet divergence.

To maintain correct dimensions in $d \neq 4$, we introduce a mass scale$\mu$ via $\lambda \to \mu^{2\varepsilon}\lambda$:

$$\frac{\mu^{2\varepsilon}}{\Delta^\varepsilon} = 1 + \varepsilon\ln\frac{\mu^2}{\Delta} + O(\varepsilon^2)$$

The typical one-loop result takes the form:

$$\boxed{I = \frac{i}{16\pi^2}\left[\frac{1}{\varepsilon} - \gamma_E + \ln(4\pi) + \ln\frac{\mu^2}{\Delta} + \text{finite}\right]}$$

💡What the Pole Means

The $1/\varepsilon$ pole is not a physical infinity - it signals that the bare parameters of the Lagrangian need to be adjusted (renormalized) to absorb this divergence. The $\ln(\mu^2/\Delta)$ term gives physical, measurable logarithmic running of coupling constants with energy scale. The renormalization scale $\mu$ is arbitrary; physical observables are independent of $\mu$.

6.7 Power Counting and Renormalizability

The superficial degree of divergence D of a Feynman diagram determines whether it is UV divergent. For a diagram with L loops, I internal lines, and V vertices in d spacetime dimensions:

$$D = dL - 2I$$

Using the topological identity $L = I - V + 1$ and the relation between external legs E and vertices, for $\phi^4$ theory in d = 4:

$$\boxed{D = 4 - E}$$

This is independent of the number of loops! The divergent diagrams have:

E = 0 (vacuum)

D = 4, quartically div.

E = 2 (self-energy)

D = 2, quadratically div.

E = 4 (vertex)

D = 0, logarithmically div.

For E > 4, we have D < 0 and the diagram is superficially convergent. Since only a finite number of diagram types are divergent, $\phi^4$ theory in d = 4 is renormalizable: all divergences can be absorbed into a finite number of counterterms (mass, coupling, and field strength renormalization).

Criterion for Renormalizability

A theory with interaction $g\phi^n$ in d dimensions has coupling with mass dimension $[g] = d - n(d-2)/2$. The theory is:

$$[g] > 0: \;\text{super-renormalizable}, \qquad [g] = 0: \;\text{renormalizable}, \qquad [g] < 0: \;\text{non-renormalizable}$$

For $\phi^4$ in d = 4: $[\lambda] = 4 - 4(4-2)/2 = 0$, which is marginal (renormalizable). For $\phi^6$ in d = 4: $[g] = -2$, which is non-renormalizable.

Key Concepts (This Page)

  • Feynman parametrization combines propagator denominators: $1/AB = \int_0^1 dx\,[xA+(1-x)B]^{-2}$
  • Dimensional regularization: work in $d = 4 - 2\varepsilon$ dimensions
  • Standard d-dimensional integrals involve $\Gamma(n-d/2)$
  • UV divergences appear as $1/\varepsilon$ poles
  • Superficial degree of divergence: $D = 4 - E$ for $\phi^4$ in d = 4
  • Renormalizability requires $[g] \geq 0$ for coupling constants
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