Quantum Field Theory Course

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

Part VI, Chapter 5

The Renormalization Group

How physics changes with energy scale

The Big Idea

The renormalization group (RG) is one of the most profound concepts in physics. It describes how the effective description of a system changes when we look at it at different length/energy scales. This applies far beyond QFT - condensed matter, statistical mechanics, cosmology all use RG ideas.

5.1 The Renormalization Group Equation

Physical observables 𝒪 cannot depend on the arbitrary renormalization scale μ we introduced. This scale independence leads to the RG equation:

$$\mu\frac{d}{d\mu}\mathcal{O}(p_i, m(\mu), \lambda(\mu), \mu) = 0$$

Expanding the total derivative using the chain rule:

$$\left[\mu\frac{\partial}{\partial\mu} + \beta(\lambda)\frac{\partial}{\partial\lambda} + \gamma_m(\lambda) m \frac{\partial}{\partial m} - n\gamma(\lambda)\right]\mathcal{O} = 0$$

where we've defined:

Beta Function β(λ)

$$\beta(\lambda) = \mu\frac{d\lambda}{d\mu}$$

Describes how the coupling constant runs with scale

Anomalous Dimension γ(λ)

$$\gamma(\lambda) = \mu\frac{d\ln Z}{d\mu}$$

Describes how field normalization runs with scale

Mass Anomalous Dimension γ_m(λ)

$$\gamma_m(\lambda) = \mu\frac{1}{m}\frac{dm}{d\mu}$$

Describes how mass runs with scale

5.2 The Beta Function: Heart of the RG

The beta function determines whether a coupling increases or decreaseswith energy scale:

β(g) < 0

Asymptotic Freedom

  • • Coupling decreases at high energy
  • • Theory becomes weakly coupled (free) at E → ∞
  • • Example: QCD
  • • Can use perturbation theory at high E

β(g) > 0

Infrared Freedom

  • • Coupling increases at high energy
  • • Theory becomes strongly coupled at E → ∞
  • • Example: QED
  • • May hit Landau pole (g → ∞)

💡Physical picture: Screening vs. Anti-screening

In QED, virtual electron-positron pairs screen the electric charge, like a dielectric. At short distances, you penetrate the cloud and see a larger effective charge → β > 0.

In QCD, gluons carry color charge themselves. They anti-screen, making the effective charge smaller at short distances → β < 0. This is asymptotic freedom!

5.3 Calculating the Beta Function

To compute β(λ), we use the fact that bare coupling λ₀ is scale-independent:

$$\mu\frac{d\lambda_0}{d\mu} = 0$$

Since λ₀ = λ + δλ, and δλ depends on both λ and μ:

$$0 = \mu\frac{d\lambda}{d\mu} + \mu\frac{\partial\delta\lambda}{\partial\mu} + \mu\frac{\partial\delta\lambda}{\partial\lambda}\frac{d\lambda}{d\mu}$$

Solving for β = μ dλ/dμ:

$$\beta(\lambda) = -\frac{\mu\frac{\partial\delta\lambda}{\partial\mu}}{1 + \frac{\partial\delta\lambda}{\partial\lambda}}$$

Example: φ⁴ Theory Beta Function

For φ⁴ theory, the one-loop counterterm in MS-bar is:

$$\delta\lambda = \frac{3\lambda^2}{16\pi^2\epsilon} + \text{finite}$$

In MS-bar, μ enters through dimensional regularization. Working this out gives:

$$\beta(\lambda) = \frac{3\lambda^2}{16\pi^2} + O(\lambda^3)$$

Interpretation:

For φ⁴: β > 0, so the coupling increases at high energy. Eventually λ(μ) → ∞ (Landau pole). φ⁴ theory is likely only an effective field theory valid below some cutoff.

5.4 Beta Function in QED

In QED, vacuum polarization gives the one-loop beta function:

$$\beta(e) = \frac{e^3}{12\pi^2} + O(e^5)$$

In terms of the fine structure constant α = e²/(4π):

$$\beta(\alpha) = \mu\frac{d\alpha}{d\mu} = \frac{2\alpha^2}{3\pi} + O(\alpha^3)$$

Since β > 0, QED coupling increases at high energy.

Running of α in QED:

Solving the RG equation to one-loop:

$$\alpha(Q^2) = \frac{\alpha(\mu^2)}{1 - \frac{\alpha(\mu^2)}{3\pi}\ln\frac{Q^2}{\mu^2}}$$

At low energy: α(0) ≈ 1/137.036
At Z boson mass: α(M_Z²) ≈ 1/128
Measured effect of vacuum polarization!

5.5 Beta Function in QCD: Asymptotic Freedom

The breakthrough of Gross, Wilczek, and Politzer (2004 Nobel Prize): In non-abelian gauge theories like QCD, the beta function can be negative!

For SU(N_c) gauge theory with n_f fermion flavors:

$$\beta(g) = -\mu\frac{dg}{d\mu} = -\frac{g^3}{16\pi^2}\left(\frac{11N_c - 2n_f}{3}\right) + O(g^5)$$

For QCD with N_c = 3 colors and n_f = 6 flavors:

$$\beta_0 = \frac{11 \cdot 3 - 2 \cdot 6}{3} = \frac{33 - 12}{3} = 7 > 0$$

🏆 Asymptotic Freedom (Nobel Prize 2004)

Since β₀ > 0, the QCD coupling α_s decreases at high energy:

$$\alpha_s(Q^2) = \frac{\alpha_s(\mu^2)}{1 + \frac{\alpha_s(\mu^2)}{2\pi}\beta_0\ln\frac{Q^2}{\mu^2}}$$
  • High energy (Q → ∞): α_s → 0 (asymptotically free, quarks behave almost freely)
  • Low energy: α_s → large (confinement, can't isolate quarks)
  • At M_Z: α_s(M_Z) ≈ 0.118
  • At 1 GeV: α_s ~ 0.5 (strong coupling!)

💡Why is QCD asymptotically free?

Contribution 1 (fermions): Quark loops screen the color charge, like in QED. This gives +2n_f term (screening).

Contribution 2 (gluons): Gluons carry color charge and self-interact. This creates anti-screening, dominating with -11N_c term.

For N_c = 3, the gluon contribution wins as long as n_f < 16.5. With n_f = 6 in nature, QCD is asymptotically free!

5.6 Callan-Symanzik Equation

The Callan-Symanzik equation is the RG equation applied to correlation functions:

$$\left[\mu\frac{\partial}{\partial\mu} + \beta(g)\frac{\partial}{\partial g} + n\gamma(g)\right]G^{(n)}(x_1,\ldots,x_n;m,g,\mu) = 0$$

This powerful equation relates correlation functions at different scales, allowing us to sum leading logarithms to all orders.

5.7 Fixed Points and Critical Behavior

A fixed point occurs when β(g*) = 0. At a fixed point, the coupling doesn't run:

UV Fixed Point

g* is reached at E → ∞ (asymptotic freedom). Theory becomes scale-invariant at high energies.

Example: QCD has UV fixed point at g* = 0 (free theory)

IR Fixed Point

g* is reached at E → 0 (infrared fixed point). Theory becomes scale-invariant at low energies.

Example: Critical points in statistical mechanics (Ising model)

Connection to Condensed Matter:

RG theory originated in condensed matter physics (Wilson, 1982 Nobel Prize). Critical phenomena at phase transitions (ferromagnets, liquid-gas) correspond to IR fixed points where the system becomes scale-invariant. The same mathematics governs both!

5.8 Wilsonian RG: Integrating Out Modes

Kenneth Wilson's approach: the RG describes coarse-graining. As we lower the energy scale, we integrate out high-energy degrees of freedom.

Wilsonian RG Procedure:

  1. Split modes: Separate high-energy (Λ/b < k < Λ) and low-energy (k < Λ/b) modes
  2. Integrate out: Perform path integral over high-energy modes
  3. Rescale: Rescale momenta k' = bk to restore original cutoff
  4. New effective theory: Get new Lagrangian with shifted couplings

The effective action at scale Λ/b has the form:

$$S_{\text{eff}}[\phi] = \int d^4x \left[\frac{Z(\Lambda)}{2}(\partial\phi)^2 - \frac{m^2(\Lambda)}{2}\phi^2 - \frac{\lambda(\Lambda)}{4!}\phi^4 - \sum_{n\geq 6} \frac{g_n(\Lambda)}{n!}\phi^n + \cdots\right]$$

As we lower Λ, an infinite number of operators are generated! But for renormalizable theories, only a few have dimensionless or relevant couplings that matter at low energies.

5.9 Operator Dimensions and Relevance

Operators are classified by their scaling dimension under RG flow:

Relevant (Δ < d)

Coupling grows at low energy. Dominates IR physics. Examples: mass term φ², φ⁴ coupling in d=4.

Marginal (Δ = d)

Dimensionless coupling. Need β function to determine running. Example: φ⁴ in d=4, gauge couplings.

Irrelevant (Δ > d)

Coupling decreases at low energy. Negligible in IR. Example: φ⁶, φ⁸, ... in d=4.

This classification explains why renormalizable theories work: at low energies, irrelevant operators (which we don't know/care about) become negligible!

5.10 Applications of RG

1. Running Couplings in Particle Physics

Predict how α, α_s, α_w evolve with energy. Crucial for precision tests and grand unification.

2. QCD Phenomenology

Asymptotic freedom explains why quarks appear free in deep inelastic scattering but are confined in hadrons. Parton distribution functions evolve via DGLAP equations (RG equations).

3. Effective Field Theory

RG justifies matching high-energy and low-energy theories. Heavy particles decouple, their effects encoded in renormalized couplings of light theory.

4. Critical Phenomena

RG explains universal behavior at phase transitions. Critical exponents are fixed-point properties, independent of microscopic details (Wilson's Nobel Prize work).

🎯 Key Takeaways

  • RG equation: μ d𝒪/dμ = 0 (physical observables scale-independent)
  • Beta function β(g) = μ dg/dμ governs coupling evolution
  • β < 0: asymptotic freedom (QCD); β > 0: IR freedom (QED, φ⁴)
  • QCD: β₀ = (11N_c - 2n_f)/3 > 0 → asymptotic freedom!
  • Callan-Symanzik equation for correlation functions
  • Fixed points: β(g*) = 0, scale invariance
  • Wilsonian RG: integrate out high-energy modes, generate effective theory
  • Operators classified: relevant, marginal, irrelevant
  • RG explains why renormalizable theories work at low energies
  • Next: Detailed study of running couplings in QED and QCD!
← Renormalization ConditionsNext: Running Couplings →