Renormalization

Renormalization is the systematic absorption of UV divergences into redefinitions of the bare parameters in the Lagrangian. When physical observables are expressed in terms of renormalized (physically measured) parameters, all infinities cancel. Renormalizability — the property that only finitely many counterterms are needed — severely constrains the form of the Lagrangian and is satisfied by QED, QCD, and the Standard Model.

Key Concepts

Bare vs renormalized: φ₀ = Z^{½}φ, m₀² = m²+δm², λ₀ = μ^{2ε}(λ+δλ)
Counterterm Lagrangian: ℒ_CT = ½(Z−1)(∂φ)² − ½δm²φ² − δλ/4!φ⁴ added to cancel divergences
On-shell scheme: physical mass = pole of propagator; physical charge = coupling at q²=0
MS-bar scheme: subtract 1/ε + ln(4π) − γ_E only; coupling defined at scale μ
BPHZ theorem: all UV divergences of any renormalizable theory removable order by order
Renormalizability: all divergences proportional to operators already in the Lagrangian — no new counterterms needed

Key Equations

Field, mass, coupling renorm.

\phi_0=Z^{1/2}\phi,\quad m_0^2=m^2+\delta m^2,\quad\lambda_0=\mu^{2\varepsilon}(\lambda+\delta\lambda)

Renormalized propagator

\tilde{\Delta}(p^2)=\frac{i}{p^2-m^2-\Sigma_R(p^2)+i\varepsilon}

MS-bar subtraction

\delta\lambda=-\frac{3\lambda^2}{16\pi^2}\frac{1}{\varepsilon}+\text{finite}

Callan-Symanzik

\Bigl[\mu\partial_\mu+\beta(\lambda)\partial_\lambda-n\gamma\Bigr]G^{(n)}=0

Worked Example

Example Problem

Problem

In λφ⁴ theory at one loop, δm² = λm²/(16π²ε). Choosing on-shell renormalization, fix δm² to ensure the physical mass equals m.

Solution

On-shell: propagator pole at p²=m². Require Σ_R(m²) = 0. This fixes δm² = −λm²/(16π²)(1/ε + finite logs). The divergence is absorbed; the physical mass m is the measured pole mass.

Key Takeaways

UV divergences are absorbed into bare parameters; physical predictions expressed in renormalized parameters are finite
Different schemes (on-shell, MS, MS-bar) give different values for intermediate quantities but identical physical S-matrix elements
A theory is renormalizable if its divergences are proportional to operators already present — QED, QCD, and the Standard Model all satisfy this
BPHZ theorem guarantees all UV divergences cancel order by order in perturbation theory for renormalizable theories