Part VI, Chapter 2

Regularization Schemes

Making infinite integrals temporarily finite

2.1 The Need for Regularization

We've seen that loop integrals diverge. To extract physics, we need a systematic procedure:

Regularization Strategy:

Regularize: Introduce a parameter (cutoff Λ, dimension d, etc.) to make integrals finite
Calculate: Compute amplitudes as functions of the regulator
Renormalize: Add counterterms to cancel infinities
Remove regulator: Take Λ → ∞ (or d → 4), leaving finite physical results

The regularization scheme is the method we use to make integrals temporarily finite. Different schemes exist, each with advantages and disadvantages.

2.2 Momentum Cutoff Regularization

The simplest idea: just cut off the integral at some large momentum Λ:

$$I = \int_0^\infty \frac{d^4k}{(2\pi)^4} f(k) \quad \to \quad I_\Lambda = \int_0^\Lambda \frac{d^4k}{(2\pi)^4} f(k)$$

For example, the quadratically divergent integral:

$$\int \frac{d^4k}{(2\pi)^4} \frac{1}{k^2 - m^2} \sim \int_0^\Lambda \frac{k^3 dk}{k^2} = \frac{\Lambda^2}{32\pi^2}$$

⚠️ Problem: Lorentz Invariance

The cutoff |k| < Λ in Euclidean space is Lorentz invariant, but a sharp cutoff in Minkowski space breaks Lorentz invariance. The vector k² = E² - |k|² doesn't transform simply under boosts.

💡Physical Interpretation of the Cutoff

A momentum cutoff Λ corresponds to a shortest distance δx ~ ℏ/Λ that we can probe. Below this scale, our theory might break down (new physics appears).

For example, at the Planck scale Λ ~ M_Pl ~ 10¹⁹ GeV, quantum gravity becomes important. Our effective field theory of particles breaks down there.

2.3 Pauli-Villars Regularization

A clever trick: add fictitious heavy particles with masses M₁, M₂, ... that cancel the divergences. For a propagator:

$$\frac{1}{k^2 - m^2} \quad \to \quad \frac{1}{k^2 - m^2} - \sum_i \frac{c_i}{k^2 - M_i^2}$$

The coefficients c_i are chosen so that at large k:

$$\frac{1}{k^2} - \sum_i \frac{c_i}{k^2} = 0 \quad \Rightarrow \quad 1 - \sum_i c_i = 0$$

The divergence is regularized, and we take M_i → ∞ at the end.

Example: Logarithmic Divergence

For a log-divergent integral, use one Pauli-Villars field with c₁ = 1:

\begin{align*} I &= \int \frac{d^4k}{(2\pi)^4} \left(\frac{1}{k^2 - m^2} - \frac{1}{k^2 - M^2}\right) \\ &= \frac{1}{16\pi^2}\ln\frac{M^2}{m^2} + \text{finite} \end{align*}

The logarithm ln(M²/m²) → ∞ as M → ∞, but this infinity will be absorbed into counterterms.

Problem: Unphysical "Ghost" Fields

The Pauli-Villars fields have wrong-sign kinetic terms (c_i < 0 for some i), making them unphysical ghosts. They're purely mathematical artifacts and must decouple (M_i → ∞) at the end.

2.4 Dimensional Regularization (The Modern Standard)

The most elegant method: analytically continue the integral to d = 4 - ε dimensions, where it's finite for ε > 0, then take ε → 0.

Key Idea:

In d dimensions, the integral ∫d^d k has different UV behavior. What diverges in d = 4 becomes finite in d = 3.9, appearing as a pole 1/ε when we take d → 4.

How Dimensional Regularization Works

Step 1: Replace d⁴k → d^d k in all loop integrals. The measure is:

$$\int \frac{d^dk}{(2\pi)^d} = \mu^\epsilon \int \frac{d^dk}{(2\pi)^d}$$

where μ is a mass scale (dimensional regularization scale) introduced to keep coupling constants dimensionless, and ε = 4 - d.

Step 2: Use the master integral formula:

$$\int \frac{d^dk}{(2\pi)^d} \frac{1}{(k^2 - \Delta)^n} = \frac{i(-1)^n}{(4\pi)^{d/2}} \frac{\Gamma(n - d/2)}{\Gamma(n)} \frac{1}{(\Delta)^{n-d/2}}$$

Step 3: Expand Gamma functions near d = 4 (ε → 0):

$$\Gamma(n - 2 + \epsilon/2) = \Gamma(n-2)\left[1 + \frac{\epsilon}{2}\psi(n-2) + O(\epsilon^2)\right]$$

where ψ(z) = Γ'(z)/Γ(z) is the digamma function.

Concrete Example: One-Loop Scalar Integral

Consider the simplest divergent integral:

$$I = \int \frac{d^dk}{(2\pi)^d} \frac{1}{k^2 - m^2}$$

Using the master formula with n = 1, d = 4 - ε:

\begin{align*} I &= \frac{i}{(4\pi)^{2-\epsilon/2}} \frac{\Gamma(1 - 2 + \epsilon/2)}{\Gamma(1)} \frac{1}{(-m^2)^{-1+\epsilon/2}} \\ &= \frac{i}{(4\pi)^2} \frac{\Gamma(-1 + \epsilon/2)}{1} (-m^2)^{1-\epsilon/2} \\ &= \frac{i}{(4\pi)^2} m^2 \left[-\frac{2}{\epsilon} + \ln\frac{4\pi\mu^2}{m^2} - \gamma_E + O(\epsilon)\right] \end{align*}

The pole 1/ε represents the UV divergence! The finite part involves ln(μ²/m²).

2.5 Minimal Subtraction (MS) and MS-bar

In dimensional regularization, divergences appear as poles 1/ε. Two common subtraction schemes:

MS (Minimal Subtraction)

Subtract only the pole 1/ε:

$$I_{\text{ren}} = I - \text{(pole part)} = I - \frac{C}{\epsilon}$$

MS-bar (Modified Minimal Subtraction)

Subtract the pole plus the constant ln(4π) - γ_E:

$$I_{\text{ren}} = I - \frac{C}{\epsilon} - C(\ln 4\pi - \gamma_E)$$

MS-bar is the standard scheme used in modern calculations. It makes formulas cleaner and is used to define running couplings in the PDG.

2.6 Why Dimensional Regularization is Superior

Advantages of Dim Reg:

Preserves Lorentz invariance: No preferred frame
Respects gauge symmetry: Ward identities hold automatically
Clean calculations: Many integrals reduce to Gamma functions
Systematic: All divergences → 1/ε poles
Scheme-independent physics: Final results don't depend on regularization details

Caveat: γ₅ Problem

The completely antisymmetric tensor ε^μνρσ and the γ₅ matrix in d ≠ 4 dimensions require care. In chiral gauge theories (like the electroweak theory), dimensional regularization needs modifications to preserve chiral anomaly consistency.

2.7 Lattice Regularization (Brief Mention)

An alternative approach: discretize spacetime on a lattice with spacing a. Integrals become finite sums:

$$\int d^4x \to a^4 \sum_{\text{lattice sites}} \quad , \quad \int \frac{d^4k}{(2\pi)^4} \to \frac{1}{(2\pi)^4}\int_{-\pi/a}^{\pi/a} d^4k$$

The UV cutoff is Λ ~ π/a. Take a → 0 (continuum limit) to recover QFT.

Lattice QCD:

Lattice regularization allows non-perturbative calculations via Monte Carlo simulations. Lattice QCD has successfully computed hadron masses, the QCD phase diagram, etc. (Not covered in detail here - see dedicated lattice QFT courses.)

2.8 Comparison of Regularization Schemes

Regularization Methods: Pros and Cons

Different approaches to taming UV divergences

Aspect	Pros	Cons
Cutoff Regularization	Simple, physically intuitive	Breaks Lorentz invariance
Pauli-Villars	Preserves Lorentz invariance	Unphysical ghost fields required
Dimensional Regularization	Respects gauge symmetry, clean calculations	Abstract, γ₅ issues in chiral theories
Lattice Regularization	Non-perturbative, makes theory finite	Breaks Lorentz invariance, chiral fermions difficult

🎯 Key Takeaways

Regularization makes divergent integrals temporarily finite
Cutoff regularization: simple but breaks Lorentz invariance
Pauli-Villars: add heavy ghost fields to cancel divergences
Dimensional regularization: continue to d = 4 - ε dimensions (modern standard)
In dim reg, divergences appear as poles 1/ε
MS-bar scheme: subtract 1/ε + ln(4π) - γ_E (standard in PDG)
Physical results are regularization-scheme independent
Next: Adding counterterms to cancel the remaining infinities!

← UV Divergences Next: Counterterms →

Quantum Field Theory Course